Chapter Four
Mathematics Challenges
Philosophy: Galileo, Kepler,
and the Surveyors
I Natural philosophy - the only game in town?
Bacon's notion of an operational natural philosophy took its lead from
the kinds of natural philosophy taught in the schools. Bacon attempted a
radical reformation of natural philosophy, but it was still a reformation
rather than a completely different enterprise. This fact might suggest that
the available scope for rethinking the study of nature was severely
restricted - as indeed it was. But natural philosophy was not the only model
provided by learned culture for the study of nature. There were other
relevant areas of inquiry too, areas that could be turned to account by
people dissatisfied by (or uninterested in) the enterprise of the physicists.
Recall that Aristotelian physics aimed at understanding qualitative
processes. Quantities were at best peripheral to it, because they failed to
speak of the essences of things - of what kinds of things they were. Measurements,
whether of dimensions or of numbers, were purely descriptive,
while the natural philosopher's job was defined by its attempt to explain,
not merely describe.
During the sixteenth century, certain Aristotelian philosophers had
denigrated the mathematical enterprise on precisely these grounds.
Scholars like Alessandro Piccolomini, and prominent natural philosophers
like Benito Pereira, published critiques of mathematics that contrasted it
unfavourably with physics. Mathematics, they said, did not demonstrate
its conclusions through causes. This disqualified mathematical proofs from
Iwing scientific in Aristotle's sense, because Aristotle had specified that true
scientific demonstration always proceeded through the identification of
a rl'lcvant explanatory cause for its conclusion. Such causes, falling under
OIW of Aristotle's fOUf categories of formal, final, efficient, and material,
Wl'fl' what made a proof into n piece of science.' None of these kinds of
l'illISl' was utilill('d in mnllwmntics, its critics claimed, and hence math-
1'l1lutks was not 11 sdl'l1l1lk dlloldplhw. Indl'('d, the most damning 8hort-
66 Revolutionizing the Sciences
coming of all was mathematics' failure to speak of fonnal causes, that is,
explanatory causes that relied on specifying the kind of thing that was
involved. In other words, mathematics did not get at the true natures of its
objects, and was restricted to discussing only superficial quantitative properties
(in Aristotelian terminology, quantitative accidents unrevealing of a
thing's nature, or essence).
Needless to say, there were contemporary mathematicians who resented
such assertions. They wished to portray their own discipline as a "science"
because that was the highest grade of knowledge; they did not want
second-class status behind the physicists. Accordingly, several mathematical
writers in the later sixteenth century and the early seventeenth century
produced counter-arguments to establish, against the natural philosophers,
that mathematical proofs were indeed causal and properly scientific. Foremost
among them were mathematicians belonging to the Catholic religious
order called the Society of Jesus - the Jesuits.
During the second half of the sixteenth century the Jesuits (founded by
Ignatius Loyola in 1540) became the foremost teaching order in the Catholic
world. Their colleges quickly sprang up all over Europe, with a reputation
for excellence that was second to none. The education that the Jesuit colleges
offered was comparable to the arts education available at universities.
Apart from the explicitly religious aspects, which underlay the whole,
Jesuit education thus consisted of a great deal of humanist training in
ancient languages and literature, as well as education in the traditional
scholastic subjects based on the texts of Aristotle - physics, metaphysics,
and ethics, together with the subjects of the quadrivium, that is, math-
ematics.2
The Jesuit mathematicians were frequently different people from
those who taught natural philosophy, and some of them objected to the
belittling characterizations of their specialty found even in the writings of
their own philosophical brothers, such as Pereira. The earliest concerted
defence came from the leading Jesuit mathematician of the late sixteenth
century, Christoph Clavius, professor of mathematics at the Jesuits' flagship
college in Rome, the Collegio Romano. Clavius explicitly rejected the
claims of the philosophers concerning mathematics, and pointed out the
pedagogical harm that could be caused by their teachings on the subject.
There were those, he complained in the 1580s, who told their pupils that
"mathematical sciences are not sciences, do not have demonstrations,
abstract from being and the good, etc.".3 Clavius wanted the teachers of
mathematics to be accorded as much respect as the teachers of natural philosophy
and metaphysics, and the scurrilous charges against mathematical
knowledge hindered this goal. As regards substantive responses to the
hated arguments, Clavius himself was less effective, although he established
a position in support of mathematics that was subsequently widely
imitatt'd by otht'r JI'suit mntlwmnticians. He reliL'd \'spl'l'inlly on Aristotle's
own disl'wlsions, pointinj.\ mIt Ihnt Aristotle had indlldl'd mathemntics as
.m inh'jI,TilI pilrt 01 phlloNllf1hy nlol1j.\sidl' niltllTill phlloi'llll'hy, Iht'rl'by imply-
Mathematics Challenges Philosophy: Galileo, Kepler, and the Surveyors 67
ing that it had an equivalent cognitive status, and that Aristotle had
described the mixed mathematical disciplines (astronomy, music, and so
on) as being "subordinate sciences"; that is, sciences that relied on results
borrowed from other higher sciences - meaning arithmetic and geometry.
There could thus be no doubt that Aristotle regarded mathematics as truly
scientific.
Later Jesuit mathematical writers supplemented Clavius's appeals for
fair play with philosophically-based refutations of the anti-mathematical
arguments. A former student of Clavius, Giuseppe Biancani, in a work of
1615, wrote at some length on the question, denying the view that mathematical
demonstrations did not employ causal proofs and that mathematical
objects (geometrical figures or numbers) lacked true essences - in
effect, that they were not real things. Biancani says that, on the contrary,
geometry defines its objects in such a way as to express their essences. He
means that a triangle, for example, is a figure composed of three right lines
in the same plane that intersect one another to yield three internal angles
- that is what a triangle is. Similarly, geometrical figures have their own
matter (the subject of material-cause explanations), in this case quantity.
Using such arguments, Biancani attempted to refute the philosophical
critics of mathematics, while also following Clavius in claiming a certain
superiority for mathematical demonstrations over those of natural philosophy.
This superiority flowed from the generally accepted certainty of mathematical
proofs, which by common consent exceeded that of other kinds of
philosophical argument.
Thanks initially to Clavius, these sorts of arguments were well known,
especially among Jesuit mathematicians, in the early seventeenth century.
They served as a means of increasing the confidence of mathematicians
that their sciences were not only on a par with natural philosophy but were
perhaps in some ways even better at making reliable knowledge of nature.
One such mathematician was an Italian friend of Clavius, Galileo GalileL
II Galileo the mathematical philosopher
Galileo was born at Pisa, the second city of the Grand Duchy of Tuscany
in northern Italy, in 1564. He was the son of a musician, Vincenzo Galilei,
who was from Tuscany's capital city, Florence, and Giulia Ammannati, and
the family held minor noble status derived from its Florentine forebears.
Galileo attended the University of Pisa to study medicine, but his lack of
vocation conspired with his aptitude for mathematics to cause him to
leave in 1585; he subsequently returned to the university in 1589 to take
up a chair in mathematics. The chair had been secured on the strength of
personal recommendations from established mathematicians, especially
Cuidobaldo dal Monte (Gnlilco had also met Clavius by this time, on a visit
to Roml' in 1.'iH7):'
MUl'h llr (;nlilt'o'H HIl!lHl'qlll'n( t'ilft'('r musl bt' ('xplnil1t'd by n·f(·nonce to
68 Revolutionizing the Sciences
his aggressive and ambitious personality. His approach, however, and the
values that he expressed, were not idiosyncratic, but can be understood
as part of the outlook of a university mathematician of his time and place.
Although other people in similar positions failed to acquire Galileo's fame,
Galileo did what many of them would no doubt have liked to achieve he
stood up to the higher-paid, more prestigious natural philosophers and
refused to concede to their expertise.
The earliest example of this dates from around 1590, during Galileo's
professorship at Pisa. An early manuscript treatise surviving from that
period, usually known as De motu ("On Motion," composed in Latinl,
signals by its very title that Galileo intends to take on the despised Aristotelian
physicists. Motion, as an example of change, was a central topic of
Aristotelian physics. The natural philosopher spoke of motion so as to
explain why things moved, and one of the typical kinds of such explanations
invoked an appropriate final cause. In particular, to explain the free
fall of a heavy body, Aristotle had described it as a natural motion, since it
is in the nature of heavy bodies to fall when unimpeded. But why do they
fall? Aristotle decided that they fell because they were seeking their proper
place at the centre of the universe. Fall thus appeared as a process of travel,
wherein the moving body set off from its starting place in an endeavour to
reach its goal. That goal, the centre of the universe, coincided in Aristotle's
cosmos with the centre of the earth - because the earth is simply the accretion
of all heavy bodies bunched together around their natural place,
towards which they strive.
One of the Aristotelian rules governing fall that emerged from this
conceptualization was that the heavier a body, the faster it falls. Weight
expressed the motive tendency of the body, so if weight increased, so too
should the speed of descent. A body that weighs twice as much as another
ought therefore to descend twice as quickly as the lighter body. Galileo,
in De motu, argues that this familiar Aristotelian claim is false, and he
provides a number of arguments intended to show it. One, for example,
imagines two independently falling bodies becoming linked together by
a piece of cord as they fall. Becoming connected, they should now constitute
a single aggregate body. Such a body, being heavier than either of
its original components, would, according to Aristotelian doctrine, fall
more rapidly than either one. And yet, Galileo urges, it is not conceivable
that the two pieces would suddenly speed up as soon as the cord linked
them.
Galileo's strategy becomes clearer when he calls on the precedent of
the ancient mathematician Archimedes to aid him.s In On Floating Bodies,
Archimedes coi,~idered the relationship between the specific gravity (or
density) of a body and tlmt of Uw mt'dium in which it was immenil'd. Ill'
used this relationship to ddt·rmilll' whether the body should float or sink:
if the body was denser thnll 1111' llwdium, it sank; if less dense, it Jloatl'd.
Galill'o takl's llw Sl\lnL' ApproAdl til hiM own discUSMioll of fnlllllH blldlt'j04 ...
Mathematics Challenges Philosophy: Gali/eo, Kepler, and the Surveyors 69
in effect, he treats falling bodies as if they were all sinking in a common
medium, the air, and compares their rates of fall by comparing their specific
gravities in relation to air's.
Galileo, notably, does not ask the question "why do heavy bodies fall?"
That would have been a natural philosopher's question. Galileo, the mathematician,
asks only how fast they fall, and what the relationship is
between their densities and that of the medium; like Archimedes, Galileo
does not ask what weight is. Against Aristotle, he concludes, first of all,
that two bodies of differing weights - say, differently sized iron balls - will
nonetheless fall at identical speeds. The speeds are a function of the balls'
specific gravities in a common medium, air; since both balls are made of
the same material, solid iron, their speeds too are the same.
In 1591 Galileo left the university at Pisa to take up a similar, although
rather more illustrious, professorship at the great thirteenth-century
university of Padua. The city of Padua, in north-eastern Italy, was at this
time a part of the independent republic of Venice, and Galileo's academic
position fell under the control of the Venetian senate. For nearly two
decades Galileo remained at Padua, lecturing on mathematical subjects
and engaging in occasional controversies with Aristotelian philosophers
there. He supplemented his income by making and selling mathematical
instruments designed for surveying work, an activity that was a common
feature of practical mathematical pursuits at the time.6
By 1609 he had
developed to a high degree his work on the motion of heavy bodies, including
the famous doctrines of the uniform acceleration of freely falling bodies
and the parabolic paths of projectiles. This work, however, was not to
be published until 1638, in his Discorsi ("Discourses and Demonstrations
Concerning Two New Sciences," often referred to in English as the Two
New Sciences).7 His aversion, as a mathematician, to the natural philosophy
of his Aristotelian colleagues continued to motivate him, and probably
contributed to his readiness, from the 1590s onwards, to entertain the
unorthodox doctrines of another mathematician, Nicolaus Copernicus.
Galileo's interest in Copernicanism existed from at least 1597, when he
mentions Copernicus in two letters. One of these letters was sent to the
weat astronomer Johannes Kepler, acknowledging receipt of the latter's
Copernican book Mysterium cosmographicum ("Cosmographical Mystery")
of 1596; Galileo, famously, claims to Kepler that he too was a Copernican,
and had been "for many years:,8 It was not until the first decade of the
seventeenth century, however, that Galileo took up astronomical and cosIllulogical
issues in a serious way, especially from 1609 onwards when he
I'l'gan to use a telescope to make astronomical observations.9
CopernicaniSIll
St'ems to have appealed to Galileo above all because it was a useful
1001 for nttflckin~ the Aristotelian physicists. First, it advocated the accepl.nln'
of i1 sun-centred universe, which would tear to shreds the physical
world-pidurl' on which tlw t~nlire Aristotelian system was based. If the
"llrlh wen' no \onll;l'r at till' l'l'ntn' of the univenll', for l~xnmpll" thl' fnll of
70 Revolutionizing the Sciences
heavy bodies (and the rise of light bodies) could no longer be explained by
their desire to reach a destination defined in terms of the centre of the universe,
because the latter would no longer coincide with the earth's centre.10
Secondly, the chief arguments in favour of Copernicanism were astronomical
rather than cosmological: that is, they were the arguments of a mathematician,
concerned with reducing the apparent motions of the heavens
to order, rather than those of the physicist, concerned with the nature of
the heavens and the explanation of their movements. At the same time,
Copernicus and a few followers of his doctrine, such as Kepler, had
embraced the cosmological inferences that they nonetheless dared to draw
from the new astronomical system.l1
Galileo therefore attempted to use Copernican astronomy as a mathematician's
means of subverting Aristotelian cosmology. He trampled on
the usual demarcation between physics and mathematics by stressing that
the natural philosopher had to take into account the discoveries of the
mathematical astronomer, since the latter concretely affected the content of
the natural philosopher's theorizing - the astronomer told the physicist
what the phenomena were that required explanation. In his Letters on
Sunspots (1613), Galileo made this point strongly in arguing for the presence
of variable blemishes on the sun's surface. The Aristotelian heavens
were held to be perfect and substantively unchangingi all they did was to
wheel around eternally, exhibiting no generation of new things or passing
away of old. The marks first seen on the face of the sun by Galileo and
others in 1611 did not appear to show the permanence and cyclicity charflcteristic
of celestial bodies, and Galileo took the opportunity to argue that
they were, in fact, dark blemishes that appeared, changed, and disappeared
irregularly on the surface of the sun. It was important to the argument
that the spots be located precisely on the sun's surface itself. The Jesuit
Christoph Scheiner, Galilee's main rival for the glory of their discovery, at
first thought that the spots were actually composed of small bodies akin to
moons, which orbited around the sun in swarms so numerous as to elude,
thus far, reduction to proper order. Accordingly, Galileo presented careful,
~eometrically couched observational reasoning to show, first of all, that
there was an apparent shrinkage of the spots' width as they moved across
the face of the sun from its centre towards the limb (and corresponding
widening as they appeared from the other limb and approached the centre)i
and secondly, that this effect, interpreted as foreshortening when the spots
were seen near the edges of the sun's disc, was consistent with their having
i1 location on the very surface of the sun itself. The precise appearances, he
ilrglwd, would be noticeably different if these necessarily flat patches were
ilny distance abo...·e the sun."
Calileo's argument Icads to till' followin~ point: if it is established that
tlw SUI1'S surfflCl' is blt'mislwd by dark patches that mflnifestly appear
from nothin~ flnd ultimnll'ly vnnillh, I1W11 it bc('ol11(';; undl'niflbk' thnt
1I11'I"(' is, conll'l1ry 10 ArlMtllh·ltnn dod rllll', 1I.('lwration and corruptiol1 in tlw
1ll'llV('nM, (;111111'0 hnM 1l1llVI'd f!'1I1ll II "l11lllhl'l11alil'i1I" ('xl'licillion or 11ll'
Mathematics Challenges Philosophy: CaWeo, Kepler, and the Surveyors 71
---1--------.--
---- -------------
Figure 4.1 Calileo's reasoning concerning the foreshortening ofsunspots as they approach
the sun's limb, to show that they are on the sun's surface.
external properties of things (here, the apparent size, shape, and motion
of the sunspots) to a properly physical conclusion about the matter of the
heavens.
As he explained elsewhere in his published contributions to the debate
with Scheiner, the true essences of things as distant as the celestial bodies
cannot be determined by the senses, and indeed the same should be understood
also of bodies near at hand: "I know no more about the true essences
of earth or fire than about those of the moon or sun, for that knowledge is
withheld from us, and is not to be understood until we reach the state of
blessedness."13 Hence all that remains to us is knowledge of those manifest
properties which are accessible to the senses.
Hence I should infer that although it may be in vain to seek to determine
the true substance of the sunspots, still it does not follow that we cannot
know some properties of them, such as their location, motion, shape, size,
opacity, mutability, generation, and dissolution. These in turn may
become the means by which we shall be able to philosophize better about
other and more controversial qualities of natural substances.14
Not only l'uuld lIll' manifl'st (and mmsurable) properties of bodies be
known, but Huch knowll'dHI' would l'nnbll'Ill'lll'r philoHophizin~.TIn> work
or till' nlnllll'l11nlll'lulI, IhuIIH, \'oll1d ~lIilll' Ihllt or 1111' I'hyHklMI.
72 Revolutionizing the Sciences
III The rising status and cognitive ambitions of the mathematical
sciences: Galileo and Kepler
Galileo sometimes used the self-descriptive label "philosophical astronomer"15
to represent the kind of work that he purported to be achieving in his
work on sunspots and on the Copernican world-system. There is a hint of
continuing deference to the category ofnatural philosopher, ifnot to natural
philosophers themselves, in the way he liked to characterize himself.
While negotiating with the Tuscan court in 1610 over the terms of his new
service to the Medici (see Chapter 6, section 11, below), Galileo insisted that
his official title be that of court "philosopher and mathematician." It was
common for a princely court to retain a mathematician (Tycho Brahe and
Kepler both played that role), but this was dearly insufficient for Galileo. He
wanted to be recognized also, and perhaps first, as a philosopher, someone
who had things to say about the nature, not just the disposition, of the
universe.
The Jesuit Biancani's arguments for the full causal character of mathematical
demonstration expressed very much the same sentiment. In
Biancani's case, however, there was no real attempt (Clavius's paean to the
peculiar certainty of mathematics notwithstanding) to set up the techniques
of mathematicians as potentally superior alternatives to those of the physicists.
The Jesuit mathematicians' goal seems to have been one of achieving
parity with their natural-philosophical colleagues; Galileo's goal was to
reform natural philosophy itself, so that it would be recognized as a discipline
for mathematicians. Either way, such promotion of mathematical
sciences as exemplary ways of learning about the natural world typifies
a widespread movement in the first half of the seventeenth century. It was
a movement that began to be recognizable through its gradual adoption
of an identifying label: "physico-mathematics."
The value of this label sprang from its imprecision. It served to unite the
notion of the physical with that of the mathematical, but the nature of the
juxtaposition was ambiguous. It apparently designated a kind of mathematics
(in the broad contemporary understanding of that word) that was
in some way of physical relevance. There were older, pre-existent terms for
what looks like the same thing, as we have seen in Chapter 1, section II:
"mixed mathematics" was perhaps the most common. And yet there seems
to have been a felt need for the new term. Why?
This is where Galileo is such a useful figure. His endeavours help us to
understand what the spread of "physico-mathematics" meant to those who
eagerly adopted the term. Galileo's polemics and propaganda set into
relief, perhaps'h"l exaggerated form, those issues the debating of which form
the core of what we can call the Scil~ntj(jc Revolution. These issues concerned
the question of thl' prnpt'r t'harac[er of natural philosophy: what
!l!wu!d it be olxlUt, how MhllUld II lw pursued, and why? Chapter 3 conHldl'fl'd
till' nttl'mplM of lwoph·lIk,' Jlmnd" I3i1CO!1 to reform notions of whi1t
Mathematics Challenges Philosophy: Cali/eo, Kepler, and the Surveyors 73
the purpose of natural philosophy should be. In arguing that it ought to be
directed towards practical utility, Bacon at the same time effectively altered
the ways in which it should be conducted, as well as how its knowledgeclaims
should be constituted and presented (his new definition of "forms").
The endeavours of the mathematicians, while different in focus and scope,
acted in concert with this new stress on knowledge for practical use to
promote a view of natural philosophy that emphasized the operational. In
doing so, they came close to rejecting natural philosophy in its old sense in
favour of an entirely different enterprise, simply applying to it an old name
borrowed from the rejected discipline.
The case of Galileo illustrates how this complete break in fact failed to
take place. He, in common with users of the contemporary term "physicomathematics,"
retained a claim to the label of natural philosopher. The
properties that he and other mathematicians wished to attribute to mathematical
knowledge, properties that they resented the physicists for
denying to it, were lifted from natural philosophy itself. Mathematicians
did not simply declare the virtues of the mathematical sciences in isolation
from those of physics; the relative status of the two disciplinary areas meant
that mathematicians would still have been left - however certain their
demonstrations - in command of what most others saw as an inferior kind
of knowledge. In this regard the mathematicians resembled the craftsmen.
The change in values expressed by Bacon involved the investing of practical,
artisanal knowledge with a higher social status. It had been (and to a
considerable extent continued to be) associated with low-status work manual
labour. Bacon in particular argued for a higher evaluation of utility
by claiming its importance for the state, as well as through moral and
religious arguments that associated it with Christian charity. And yet he
wanted this newly-upgraded practical knowledge to receive the prestige
already possessed by natural philosophy. His solution was to argue as if
"natural philosophy" were a category much broader in scope than usually
admitted by academics, one that included practical knowledge; he then
chased out purely contemplative knowledge by criticizing the goals of the
latter, thus leaving the field to his own proposed endeavour.
Similarly, Galileo and other mathematicians rejected the disciplinary
boundary between natural philosophy and mathematics by arguing that
mathematics was crucially important in drawing legitimate physical conclusions.
In effect, the label "physico-mathematics" served to signal that
the mathematicians' own expertise would not thereby be subsumed to
that of the natural philosophers. Instead, the cuckoo's egg of physicomathematics
would (if Galileo had his way) serve to expel most of the
original occupants of the natural-philosophical nest, so as to leave the
mathematicians in the position formerly occupied by the physicists. In both
this and the previous case, the established category "natural philosophy"
was a valuable resource for those who wanted to raise the status of their
own fnvourl,d kind of knllwll'd~l"
74 Revolutionizing the Sciences
Another important advocate of the central place of mathematics in
natural philosophy was the Copernican astronomer Johannes Kepler.
Kepler's approach to astronomy was, like any astronomer of the time, fundamentally
mathematical. But he went much further in his promotion of
mathematics than most of his colleagues: for Kepler, the mathematics that
structured astronomical theory was the very mathematics that underlay
the structure of the universe itself. Thus, in his work as a mathematical
astronomer, Kepler at the same time endeavoured to create a mathematical
physics. For Kepler, the universe is properly intelligible in mathematical
terms; it is mathematics, especially geometry, which allows insight into the
mind of God, the Creator, and hence into the deepest realms of natural
philosophy. In one of his last publications, a work of 1618 called Epitome
astronomiae Copernicanae ("Epitome of Copernican Astronomy"), Kepler
describes his own special field as a part of physics:
What is the relation between this science [astronomy] and others? 1. It is
a part of physics, because it seeks the causes of things and natural occurrences,
because the motion of the heavenly bodies is amongst its subjects,
and because one of its purposes is to inquire into the form of the structure
of the universe and its parts.... To this end, [the astronomer] directs
all his opinions, both by geometrical and by physical arguments, so that
truly he places before the eyes an authentic form and disposition or furnishing
of the whole universe.16
Kepler put these principles into effect in his restructuring of Copernican
astronomy. As a student at the Lutheran university in the German town of
Tiibingen, he had become convinced of the truth of the new Copernican
cosmology from his teacher in astronomy, Michael Mastlin. Belief in the
literal truth of the Copernican system, as opposed to a recognition of the
value of Copernicus's De revolutionibus in the practical computational work
of mathematical astronomy, was not widespread among astronomers at
this time, and Kepler's early guidance by one of the exceptions to this rule
is therefore noteworthy. Kepler's metaphysical and theological predilections
expressed themselves in relation to Copernican astronomy in his first
publication, the Mysterium cosmographicum ("Cosmographical Mystery")
of 1596, when Kepler was working as a school teacher in Austria. The
most noteworthy feature of the work is its presentation of Kepler's proud
discovery of a relationship between the dimensions of the planetary orbits
(calculated according to the Copernican system) and certain interrelationships
among the so-called "perfect" or "Platonic" or "regular" solids.
The latter were solid figures that had been demonstrated by Euclid to
be restricted to precisely five in number. They were solids that are conlilil1l'd
by identical {Clcets which Clre themselves regular polygons, such as
l'l)uililtl'ral triangles, squnrl's, or p('ntagons. The five solids, as Euclid had
.~hIlWIl, w('r(' lhl' tdralwdl'Oll, 1111' rube, the octahedron, the dodecahedron,
Mathematics Challenges Philosophy: Cali/eo, Kepler, and the Surveyors 75
Figure 4.2 The nested perfect solids structuring the universe, from Kepler's Mysterium
cosmographicum.
and the icosahedron, of four, six, eight, twelve, and twenty faces respectively.
The fact that these five solids were unique of their kind implied to
Kepler that t1wy n'pn'sented something profound about the nature of space
and of t1w W'ol111'II'il'il1 principles on the basis of which God had created
the univl'rl4l', III till' M,II~I/',.il(/I/ cos/twgraphicltm, he shows that (imaginary)
Iip!wn'l4 wlI'd III 1"'prl'Nl'nl 11\1' n,lillivI' sin's of tlw various Copl'rnienn plnn-
Revolutionizing the Sciences
,'IIII'Y orbits around the sun are separated by various distances that closely
nl'nllnmodate the perfect solids as spacers between the spheres. Using
IIvaiiable data, Kepler was able to show that the sizes of the planetary orbits
dosely fit the sizes allowed by the intercalated solids, to within an error of
lu'ound five per cent. In 1600 he joined Tycho Brahe in Prague so as
10 gain access to Tycho's famed data on planetary motions, which Kepler
hoped would enable him to reduce the error still further, Furthermore,
Kepler's model accounted for there being, in a Copernican universe,
precisely six planets - the number that could be adequately spaced by five
intervening solids.
Kepler was enormously proud of this result, which he believed brought
him nearer to an intimate understanding of the structure of God's Creation.
The role of geometry in his argumentation was fundamental: geometry was
not simply a tool for calculating dimensions and motions in astronomy; it
was capable of providing explanations of why things in the world are as
they are. The geometry of the five perfect solids serves not only to describe
the number of the planets and their distances from the sun, but to make
sense of those facts. Kepler believed in a fundamentally mathematical constitution
to the universe, in the sense that mathematical intelligibility of the
kind provided by the perfect solids accounted for why certain things are as
they are. The nature of such an explanation is not, in the present case, one
that provides mathematical, demonstrative necessity to the things that it
explains (as with showing, as Euclid does, why the base angles of an isosceles
triangle are equal to one another); but it does show, Kepler believed,
what was in God's mind when He chose to create things in the way that
He did. In many respects, in fact, Kepler's entire astronomical career was
one directed towards gaining an understanding of God's mind, of corning
closer to God through the medium of astronomical study. This was natural
philosophy in its starkest, most theocentric form.
Kepler's major work was the Astronomia nova of 1609. It was the published
result of a project that he had originally undertaken at the behest
of Tycho, to determine a satisfactory astronomical model for the motion
of Mars. Mars had always been a planet whose motion was particularly
troublesome to model with exactness, and since Tycho's great observational
project had been designed as the foundation for much more accurate planetary
models, the continuing recalcitrance of Mars was a source of especial
concern to him. Tycho was particularly interested in having Kepler solve
the difficulties in terms of Tycho's own favoured cosmological system, a
kind of compromise between Ptolemy and Copernicus that he had first
published in a book of 1588. This scheme had the moon and sun in orbit
around a centra~ stationary earth, but with the planets orbiting that moving
S\II1, The resultant relative motions thus remained the same as in Copernil'US'S
system (djsrcgClrdjn~ the issue of the fixed stars), with Copernicus's
nnl1l1l11 orbit of the earth nruund tlw sun being exactly mirrored in the
nlll\\u\!lll"billlf til\' !'lun IIro\llld tilt' \'I\rlh. K\'p!l'r n'spondl'd to the chnlll'ngl'
MatherlUltics Challenges Philosophy: Galileo, Kepler, and the Surveyors 77
by producing models that could be expressed in PtolemaIc, Copernican, or
Tychonic terms (simply by shifting reference-frames). But, for Kepler, the
Copernican remained the true account.
Several years of intensive work by Kepler resulted in an achievement
that was remarkable in several ways. First, Kepler produced a model for
the motion of Mars of unparalleled accuracy, both as determined by comparison
with Tycho's observations and as confirmed over time by its predictions.
Second, in doing so, he had come to abandon the classical Greek
astronomical requirement, followed proudly by Copernicus as well as by
Tycho himself, that the component motions used in creating astronomical
models each be a uniform motion around a circle. Third, Kepler developed
his new laws governing planetary motion on a basis that involved speculation
about the physical causes that brought about that motion.
His new planetary orbits around the sun took the form of ellipses, with
one focus of each ellipse located on the sun itself. He knew the geometry
of the ellipse, one of the conic sections, from the treatise on conic sections
written by the Greek astronomer and mathematician Apollonius of Perga,
and Kepler's desire to find mathematics written in the fabric of the universe
was thoroughly satisfied by this result, even though it meant abandoning
circles. Furthermore, his elliptical orbits were traversed by the
planets (including the earth) in such a way that the space swept out by the
line joining the planet to the sun was uniform - equal areas swept out in
equal times.
Equally importantly for Kepler, however, he had achieved these results
in continual dialogue with ideas on the causes of planetary motions. These
included the idea of a motive force emanating from the sun that drove the
planets around in their orbits, together with an idea about a kind of magnetic
attraction and repulsion between the sun and the two poles of each
planet that served to explain why planetary orbits were not perfectly circular.
Making explicit reference to William Gilbert, Kepler used his notion
of the earth as a giant magnet to explain why planets successively approach
and depart from the sun in the course of their elliptical orbits. The celestial
spheres were gone (Tycho had already rejected them); Kepler's planets
moved independently through space.
Kepler's views on the place of mathematics in understanding the
physical world were thus more directly related to a purely philosophical,
as opposed to practical, conception of natural knowledge than were
GalileD's. The very nature of the mixed mathematical sciences, however,
was such as to encourage, even in Kepler, a concern with some operational
criteria of knowledge. The instrumental function of optics in assisting
astronomical investigations was a major part of his justification for publishing
Ad Vitellionem paralipomena quibus astronomiae pars optica traditur
("Additions to Witelo, in which the Optical Part of Astronomy is Treated"),
in lfj04.17
Kepk~r considers the imperfection of sciences such as astronomy
nnd optics, MIl'Ompared til till' demonstralive ideal of gt'ometry, but "rgues
78 Revolutionizing the Sciences
Figure 4.3 Kepler's elliptical orbit for the planets, and his area-law. The planet, P, pursues
its elliptical path with the sun, 5, at one focus. The line joining the planet to the sun sweeps
out equal areas in equal times, 50 that the distance traversed In) the planet when nearer to
the sun (P,-P4) is greater than that traversed when farther from the sun (P,-P,). From
Marie Boas, The Scientific Renaissance 1450-1630 (New York: Harper and Brothers,
1962), © 1962 by Marie Boas.
that optical theorems should be sufficient to satisfy an astronomer's
nceds.'R
IV Knowing, doing, and mathematics
Aathematics was itself traditionally related to practical endeavours such
ns land-surveying or the building of fortifications. Both fell under the
Iwnding of "mixed mathematics," along with such others as astronomy and
l11l'chanics. The latter too were of great practical importance. Astronomy
had been valu~~d in Latin Europe since the Middle Ages for its use in marine
l1avigntion nnd in astrology, a practical art much used in learned medieval
nll'didill'. Ml'chanks ('ol1l,t'l'lwd machines themselves (such as wind or
wall'r mills), but morl' l'Hpl'l'iilIly disl'llSSl'd the classical domain of the socalil'd
simpil' Illilchilll'tl, whkh l'ollsidl'J'l'd cl'rlain dl'vin's nnd Iechniqlll~s
Mathematics Challenges Philosophy: Galileo, Kepler, and the Surveyors 79
(such as levers or pulleys) that made work easier. The practical and artisanal
associations of many of the mathematical sciences were thus very
hard to miss.
During the second half of the sixteenth century, mathematicians, especially
in England, had begun to make strong claims for their discipline that
revolved around its practical dimensions rather than focusing on the more
philosophical justifications preferred by increasing numbers of bookish
mathematicians. In 1570 there appeared a new translation into English of
Euclid's Elements, bearing a preface written by John Dee of Mortlake. He
used this opportunity to praise the branches of mathematics for their
usefulness "in the Common lyfe and trade of men," as witnessed by the
practices of many and diverse occupations.19
Dee had himself already had
dealings with one such endeavour, navigation; the interrelated concerns
of navigation (including in this period increasing interest in the magnetic
compass and terrestrial magnetism) and of cartography were important,
and unassailably mathematical, subjects of books by a number of English
authors in the decades around 1600, such as Robert Recorde, Thomas
Digges, and Edward Wright. Most such authors wrote in English rather
than Latin, and presented themselves as men of practical rather than contemplative
bent. Typical examples of the genre include works on surveying
techniques, the demand for which seems to have grown during the
second half of the sixteenth century, in concert with the increasing enclosure
of formerly common land and the surveying of church lands now
seized by the Crown following the English Reformation.
Mathematics thus had, besides its association with learned classical treatises
and the niceties of formal demonstration, a practical, computational
image somewhat at odds with the academic, philosophical discipline promoted
by scholars such as Clavius. At the same time, its leaning towards
practicality enabled it to appeal to the same sensibilities that Bacon's propaganda
exploited. The kind of knowledge that mathematical practices
tended to promote was not simply utilitarian, however: its elevation to
philosophical importance by such as Galileo implied a revaluing of mathematical
characteristics as being peculiarly important to true understanding
of nature.