Chapter 3
Fractals, Graphics, and
Mathematics Education1
Benoit B. Mandelbrot
1 Introduction
The fundamental importance of education has always been
very clear to me and it has been very frustrating, and certainly
not a good thing in itself, that the bulk of my working
life went without the pleasures and the agonies of teaching.
On the other hand, there is every evidence, in my case, that
being sheltered from academic life has often been a necessary
condition for the success of my research. An incidental
consequence is that some of the external circumstances that
dominated my life may matter to the story to be told here, and
it will be good to mention them, in due time.
But past frustrations are the last thing to dwell upon in this
book. Watching some ideas of mine straddle the chasm between
the research frontier and the schools overwhelms me
with a feeling of deep accomplishment. Clearly, for better or
worse, I have ceased to be alone in an observation, a belief,
and a hope, that keep being reinforced over the years.
The observation is that fractals--together with chaos, easy
graphics, and the computer--enchant many young people and
make them excited about learning mathematics and physics.
In part, this is because an element of instant gratification
happens to be strongly present in this piece of mathematics
called fractal geometry. The belief is that this excitement can
help make these subjects easier to teach to teenagers and to
beginning college students. This is true even of those students
who do not feel they will need mathematics and physics
in their professions. This belief leads to a hope--perhaps
megalomaniac--concerning the abyss which has lately separated
the scientific and liberal cultures. It is a clich´e, but
1Adapted from a closing invited address delivered at the Seventh International
Congress of Mathematics Education (ICME-7), held in 1992 at Laval
University of Quebec City. The text remains self-contained and preserves
some of the original flavor; it repeats some points that were already used
elsewhere in this book but bear emphasis.
one confirmed by my experience, that scientists tend to know
more of music, art, history, and literature, than humanists
know of any science. A related fact is that far more scientists
take courses in the humanities than the other way around.
So let me give voice to a strongly held feeling. An element
of instant gratification happens to be strongly present in this
piece of mathematics called fractal geometry. Would it be extravagant
to hope that it could help broaden the small band
of those who see mathematics as essential to every educated
citizen, and therefore as having its place among the liberal
arts?
The lost unity of liberal knowledge is not just something
that old folks gather to complain about; it has very real social
consequences. The fact that science is understood by few
people other than the scientists themselves has created a terrible
situation. One aspect is a tension between conflict of
interest and stark ignorance: that vital decisions about science
and technology policy are all too often taken either by
people so closely concerned that they have strong vested interests,
or by people who went through the schools with no
math or science. Thus, every country would be far better off
if understanding and appreciation for some significant aspect
of science could become more widespread among its citizens.
This demands a liberal education that includes substantial instruction
in math.
Fractals prove to have many uses in technical areas of
mathematics and science. However, this will not matter in
this chapter. Besides, if fractals' usefulness in teaching is
confirmed and proves lasting, this is likely to dwarf all their
other uses.
This chapter shall assume all of you to have a rudimentary
awareness or knowledge of fractals, or will one day become
motivated to acquire this knowledge elsewhere. My offering
is my book, Mandelbrot (1982), but there are many more
21
M.L. Frame & B.B. Mandelbrot
Washington D.C. Mathematical Association of America, 2002
Fractals, Graphics, & Mathematics Education
22 Chapter 3. Fractals, Graphics, and Mathematics Education
sources at this point. For example, the website
http://classes.yale.edu/math190a/
Fractals/Welcome.html
is a self-contained short course on basic fractal geometry.
I shall take up diverse aspects of a basic and very concrete
question about mathematics education: what should be the
relations--if any--between (a) the overall development of
mathematics in history, (b) the present status of the best and
brightest in mathematics research, and (c) the most effective
ways of teaching the basics of the field?
2 Three mutually antagonistic
approaches to education
By simplifying (strongly but not destructively), one can distinguish
three mutually antagonistic approaches to mathematical
education. The first two are built on a priori doctrine: the
old math, dominated by (a) above, and the new math, dominated
by (b). (I shall also mention a transitional approach
between old and new math.) To the contrary, the approach I
welcome would be resolutely pragmatic. It would encourage
educational philosophy to seek points of easiest entry. In this
quest, the questions of how mathematics research began and
of its present state, are totally irrelevant.
To elaborate by a simile loaded in my favor, think of the
task of luring convinced nomads into hard shelter. One could
tempt them into the kinds of shelters that have been built
long ago, in countries that happened to provide a convenient
starting point in the form of caves. One could also try to
tempt them into the best possible shelters, those being built
far away, in highly advanced countries where architecture is
dominated by structurally pure skyscrapers. But both strategies
would be most ill-inspired. It is clearly far better to tempt
our nomads by something that interests them spontaneously.
But such happens precisely to be the case with fractals, chaos,
easy graphics, and the computer. Hence, if their effectiveness
becomes confirmed, a working pragmatic approach to mathematics
education may actually be at hand. We may no longer
be limited to the old and new math. Let me dwell on them for
a moment.
2.1 The old math approach
to mathematics education
The old math approach to mathematics education saw the
teacher's task as that of following history. The goal was to
guide the child or young person of today along a simplified
sequence of landmarks in the progress of science throughout
the history of humanity. An extreme form of this approach
prevailed until mid-nineteenth century in Great Britain, the
sole acceptable textbook of geometry being a translation of
Euclid's Elements.
The folk-psychology behind this approach asserted with a
straight face that the mental evolution of mankind was the
product of historical necessity and that the evolution of an
individual must follow the same sequence. In particular, the
acquisition of concepts by the small child must follow the
same sequence as the acquisition of concepts by humankind.
Piaget taught me that such is indeed the case for concepts
that children must have acquired before they start studying
mathematics.
All this sounds like a version of "ontogeny recapitulates
phylogeny," but it is safe to say that people had started developing
mathematics well before Euclid. As a matter of
fact, those who edited the Elements were somewhat casual
and left a number of propositions in the form of an archaeological
site where the latest strata do not completely hide
some tantalizing early ones. To be brief, what we know of
the origin of mathematics is too thin and uncertain to help the
teacher.
Be that as it may, an acknowledged failing of old math
was that the teacher could not conceivably move fast enough
to reach modern topics. For example, the school mathematics
and science taken up between ages 10 and 20 used to be
largely restricted to topics humanity discovered in antiquity.
As might be expected, teachers of old lit heard the same criticism.
A curriculum once reserved to Masters of Antiquity
was gradually changed to leave room for the likes of Shakespeare
and of increasingly modern authors; in the USA, it
had to yield room to American Masters, then to multicultural
programs.
2.2 Transitional approaches
to mathematics education
Concerns about old math are an old story. Consider two examples.
In Great Britain, unhappiness with Euclid's Elements
as a textbook fueled the reforms movement that led in 1871 to
the foundation of the Association for the Improvement of Geometrical
Teaching (in 1897 it was renamed the Mathematical
Association). As a student in France around 1940, I heard
about a reform movement that had flourished before 1900. It
motivated Jacques Hadamard (1865­1963), a truly great man,
to help high school instruction by writing Hadamard (1898),
a modern textbook of geometry that stressed the notion of
transformation. I was given a copy and greatly enjoyed it, but
the consensus was that it was far above the heads of those it
hoped to please. In Germany, there was the book by Hilbert
and Cohn­Vossen (1952).
But the 20th century witnessed a gradual collapse of geometry.
Favored topics became arithmetic and number theory;
they have ancient roots, are one of the top fields in
today's mathematical research, and include large portions that
are independent of the messy rest of mathematics. Therefore,
3. The purely utilitarian argument for widespread literacy in mathematics and science 23
they are central to many charismatic teachers' efforts to fire
youngsters' imaginations towards mathematics.
2.3 The 1960s and the new math approach to
mathematics education
Far bolder than those half-hearted attempts to enrich the
highly endowed students with properly modern topics was the
second broad approach to mathematics education exemplified
by the new math of the 1960s.
Militantly anti-historical, I viewed the state of mathematics
in the 1960s, and the direction in which it was evolving at
that particular juncture in history, as an intrinsic product of
historical necessity. This is what made it a model at every
level of mathematics education. If the research frontier of the
1960s had not been historically necessary, new math would
have lost much of its gloss or even legitimacy. The evidence,
however, is that the notion of historical necessity as applied to
mathematics (as well as other areas!) is merely an ideological
invention. This issue is important and tackled at length in
Chapter 4 of this volume.
In any event, new math died a while ago, victim of its obvious
failure as an educational theory. The Romans used to say
that "of the dead, one should speak nothing but good." But
the new math's unmitigated disaster ought at least teach us
how to avoid a repetition. However, it is well known that failure
is an orphan (while success has many would-be parents),
that is, no responsibility for this historical episode is claimed
by anyone, as of today.
Take for example the French formalists who once flourished
under the pen-name of Bourbaki (I shall have much
more to say about them). They nurtured an environment in
which new math became all but inevitable, yet today they
join everyone else in making fun of the outcome, especially
when it hurts their own children or grandchildren. This denial
of responsibility is strikingly explicit in a one-hour story
a French radio network devoted to the Bourbaki a few years
ago. (Audio-cassettes may be available from the Soci´et´e
Math´ematique de France.) One hears in it that the Bourbaki
bear no more responsibility than the French man in the street
(failure is indeed an orphan), and that they have never made
a statement in favor of new math. On the other hand, having
paid attention while suffering through the episode as the
father of two sons, I do not recall their making a statement
against new math, and I certainly recall the mood of that time.
Be that as it may, it is not useful to wax indignant, but important
to draw a lesson for the future. The lesson is that no
frontier mathematics research must again be allowed to dominate
mathematics education. At the other unacceptable extreme,
needless to say, I see even less merit in the notion that
one can become expert at teaching mathematics or at writing
textbooks, yet know nothing at all about the subject. Quite to
the contrary, the teachers and the writers must know a great
deal about at least some aspects of mathematics.
Fortunately, mathematics is not the conservatives' ivory
tower. As will be seen in Chapter 4, I see it as a very big
house that offers teachers a rich choice of topics to study and
transmit to students. The serious problem is how to choose
among those topics. My point is that this choice must not be
left to people who have never entered the big house of mathematics,
nor to the leaders of frontier mathematics research,
nor to those who claim authority to interpret the leaders' preferences.
Of course, you all know already which wing of the
big house I think deserves special consideration. But let me
not rush to talk of fractals, and stop to ask why the big house
deserves to be visited.
3 The purely utilitarian argument for
widespread literacy in mathematics
and science
My own experiences suggest, and all anecdotal reports confirm,
that traditional mathematics (of the kind described in
the section before last) does marvels when a very charismatic
teacher meets ambitious and mathematically gifted children.
Helping the very gifted and ambitious is an extraordinarily
important task, both for the sake of those individuals and of
the future development of math and science. But (as already
stated) I also believe that math and science literacy must extend
beyond the very gifted pupils.
Unfortunately, as we all know, this belief is not shared
by everyone. How can we help it become more widely accepted?
All too often, I see the need for math and science
literacy referred to exclusively in terms of the needs (already
mentioned) of future math and science teaching and research,
and those of an increasingly technological society. To my
mind, however, this direct utilitarian argument fails on two
accounts: it is not politically effective; and it is not sufficiently
ambitious.
First of all, if scientific literacy is valuable and remains
scarce, it has always been hard to explain why the scientifically
literate fail (overall) to reap the financial rewards of
valuable scarcity. In fact, scientific migrant workers, like
agricultural ones, keep pouring in from poorer countries.
Recent years were especially unkind to the utilitarian argument
since many engineers and scientists are becoming unemployed
and had to move on to fields that do not require
their specialized training.
Even though this is an international issue, allow me to center
the following comments on the conditions in the USA. In
its crudest form, very widespread only a few years ago, the
utilitarian argument led many people to compare the United
States unfavorably to countries, including Russia, France, or
Japan, with far more students in math or science. Similarly
unfavorable comparisons concerned foreign language instruction
in the USA to that in other countries. The explanation
24 Chapter 3. Fractals, Graphics, and Mathematics Education
in the case of the languages of Hungary or Holland is obvious:
the Hungarians are not genetically or socially superior to
the Austrians, but the Austrians speak German, a useful language,
while Hungarian is of no use elsewhere; hence, multilingual
Hungarians receive unquestioned real-life rewards.
Similarly, school programs heavy in compulsory math are
tolerated in France and Japan because they provide unquestioned
great real-life rewards to those who do well in math.
For example, many jobs in France that require little academic
knowledge to be performed are reserved (by law) for
those who pass a qualifying examination. The exam seeks objectivity,
and ends up being heavy on math. There are many
applicants, the exams are difficult, and the students are motivated
to be serious about preparing for them.
Some of these jobs are among the best possible. For example,
in many French businesses one cannot approach the
top unless one started at the Ecole Polytechnique, the school
I attended. (I first entered Ecole Normale Sup´erieure, but left
immediately). For a time after Polytechnique was founded
(in 1794), it first selected and judged its students on the broad
and subjective grounds ideally used in today's America, but
later the criteria for entrance and ranking became increasingly
objective--that is, mathematical. One reason was the justified
fear of nepotism and political pressure, another the skill of
Augustin Cauchy (1789­1857), a very great mathematician
and also a master at exerting self-serving political pressure.
The result was clear at the forty-fifth and fiftieth reunions
of my class at Polytechnique. For a few freshly retired classmates
a knowledge of science had been essential. But most
had held very powerful positions to general acclaim, yet
hardly remembered what a complex number is--because it
has not much mattered to them. They gave no evidence of an
exceptionally strong love of science. (I do not know what to
make of the number of articles our Alumni Monthly devotes to
the paranormal.) But my classmates could never have reached
those powerful positions without joining the Polytechnique
"club"; to be a wizard at math, at least up to age twenty, was
part of the initiation and a desired source of homogenity.
The United States of America also singles out an activity
that brings monetary rewards and prestige that continue
through a person's life--independent of the person's profession.
This activity is sports. In France it is math. For example,
one of my classmates (Val´ery Giscard d'Estaing) became
President of France, his goal since childhood; to help himself
along, he chose to go to a college even more demanding than
MIT.
For a long time France recognized a second path to the top:
a mastery of Greek or Latin writers and philosophers. But
by now this path has been replaced by an obstacle course in
public administration. A competition continues between the
two ways of training for the top, but no one claims that either
mathematics or the obstacle courses is important per se. You
see how little bearing this French model has on the situation
in the USA.
Needless to say, many French people have always complained
that their school system demands more math than is
sensible; other French people complain that the teaching of
math is poor. And I heard the same complaints on a trip to
Japan. So my feeling is that the real problem may not involve
embarrassing national comparisons.
4 In praise of widespread literacy
in mathematics and science
Lacking the purely utilitarian argument, what could one conceivably
propose to justify more and better math and physics?
When I was young some of my friends were delighted to reserve
real math to a small elite. But other friends and I envied
the historians, the painters, and the musicians. Their fields
also involved elite training, yet their goals seemed blessed by
the additional virtue of striking raw nerves in other human beings.
They were well understood and appreciated by a wide
number of people with comparatively minimal and unprofessional
artistic education. To the contrary, the goals of my
community of mathematicians were becoming increasingly
opaque beyond a circle of specialists. Tongue in cheek, my
youthful friends and I dreamt of some extraordinary change
of heart that would induce ordinary people to come closer to
us of their own free will. They should not have to be bribed
by promises of jobs and money, as was the case for the French
adolescents. Who can tell, a popular wish to come closer to us
might even induce them to buy tickets to our performances!
When our demanding dream was challenged as ridiculous
and contrary to history and common sense, we could only
produce one historical period when something like our hopes
had been realized. Our example is best described in the following
words of Sir Isaiah Berlin (Berlin 1979):
"Galileo's method ... and his naturalism, played a crucial
role in the development of seventeenth-century thought, and
extended far beyond technical philosophy. The impact of
Newton's ideas was immense: whether they were correctly
understood or not, the entire program of the Enlightenment,
especially in France, was consciously founded on Newton's
principles and methods, and derived its confidence and its
vast influence from his spectacular achievements. And this,
in due course, transformed--indeed, largely created--some
of the central concepts and directions of modern culture in
the West, moral, political, technological, historical, social--
no sphere of thought or life escaped the consequences of this
cultural mutation. This is true to a lesser extent of Darwin ....
Modern theoretical physics cannot, has not, even in its most
general outlines, thus far been successfully rendered in popular
language as Newton's central doctrines were, for example,
by Voltaire."
Voltaire was, of course, the most celebrated French writer
of the eighteenth century and mention of his name brings to
mind a fact that is instructive but little known, especially out-
6. A new kind of science: Chaos and fractals 25
side France: it concerns the first translation of Newton into
French, which appeared in Voltaire's time. Feminists, listen:
the translator was Gabrielle ´Emilie le Tonnelier de Breteuil,
marquise du Ch^atelet-Lomont (1706­49). Madame du
Ch^atelet was a pillar of High Society: her salon was among
the most brilliant in Paris.
In addition, the XVIIIth century left us the letters that
the great Leonhard Euler (1707­83) wrote to "a German
princess" on topics of mathematics. Thus a significantly
broad scientific literacy was welcomed and conspicuously
present in a century when it hardly seemed to matter.
5 Contrasts between two patterns
for hard scientific knowledge:
Astronomy and history
Why is there such an outrageous difference between activities
that appeal to many (like serious history), and those which
only appeal to specialists? To try and explain this contrast, let
me sketch yet another bit of history, comparing knowledge
patterned after astronomy and history.
The Ancient Greeks and the medieval scholastics saw a
perfect contrast between two extremes: the purity and perfection
of the Heaven, and the hopeless imperfection of the
Earth. Pure meant subject to rational laws which involve
simple rules yet allow excellent predictions of the motion
of planets and stars. Many civilizations and individuals believe
that their lives are written up in full detail in a book
and hence can in theory be predicted and cannot be changed.
But many others (including Ancient Greeks) thought otherwise.
They expected almost everything on Earth to be
a thorough mess. This allowed events that were in themselves
insignificant to have unpredictable and overwhelming
consequences--a rationalization for magic and spells. This
sensitive dependence became a favorite theme of many writers;
Benjamin Franklin's Poor Richard's Almanac (published
in 1757), retells an ancient ditty as follows:
"A little neglect may breed mischief.
For lack of a nail, the shoe was lost;
for lack of a shoe, the horse was lost;
for lack of a horse, the rider was lost;
for lack of a rider, the message was lost;
for lack of a message, the battle was lost;
for lack of a battle, the war was lost;
for lack of a war, the kingdom was lost;
and all because of one horseshoe nail."
From this perspective, it seems to me that belief in astrology,
and the hopes that continue to be invested today in diverse
would-be sciences, all express a natural desire to escape
the terrestrial confusion of human events and emotions by
putting them into correspondence with the pure predictability
of the stars.
The beautiful separation between pure and impure (confused)
lasted until Galileo. He destroyed it by creating a
terrestrial mechanics that obeyed the same laws as celestial
mechanics; he also discovered that the surface of the Sun is
covered with spots and hence is imperfect. His extension
of the domain of order opened the route to Newton and to
science. His extension of the domain of disorder made our
vision of the universe more realistic, but for a long time it
removed the Sun's surface from the reach of quantitative sci-
ence.
After Galileo, knowledge was free from the Greeks' distinction
between Heaven and Earth, but it continued to distinguish
between several levels of knowledge. At one end was
hard knowledge, a science of order patterned after astronomy.
At the other end, is soft knowledge patterned after history, i.e.,
the study of human and social behavior. (In German, the word
Wissenschaft stands for both knowledge and science; this may
be one of several bad reasons why the English and the French
often use science as a substitute for knowledge.)
Let me at this point confess to you the envy I experienced
as a young man, when watching the hold on minds that is the
privilege of psychology and sociology, and of my youthful
dreams of seeing some part of hard science somehow succeed
in achieving a similar hold. Until a few decades ago, the
nature of science made this an idle dream. Human beings (not
all, to be sure, but enough of them) view history, psychology,
and sociology as alive (unless they had been smothered by
mathematical modeling). Astronomy is not viewed as alive;
the Sun and the Moon are superhuman because of their regularity,
therefore gods. In the same spirit, many students view
math as cold and dry, something wholly separate from any
spontaneous concern, not worth thinking about unless they
are compelled. Scientists and engineers must know the rules
that govern the motions of planets. But these rules have limited
appeal to ordinary humans because they have nothing to
do with history or the messy, everyday life, in which, let me
repeat, the lack of a nail can lose a horse (a battle, a war, and
even a kingdom) or a bride.
6 A new kind of science:
Chaos and fractals
Now we are ready for my main point. In recent years the
sharp contrast between astronomy and history has collapsed.
We witness the coming together, not of a new species of science;
nor even (to continue in taxonomic terms) a new genus
or family, but a much more profound change. Towards the
end of the 19th century, a seed was sowed by Poincar´e and
Hadamard; but practically no one paid attention, and the seed
failed to develop until recently. It is only since the 1960s
that the study of true disorder and complexity has come onto
26 Chapter 3. Fractals, Graphics, and Mathematics Education
Figure 1: Two close-up views of parts of the Mandelbrot set.
the scene. Two key words are chaos and fractals, but I shall
keep to fractals. Again and again my work has revealed cases
where simplicity breeds a complication that seems incredibly
lifelike.
The crux of the matter is a geometric object that I first saw
in 1979, took very seriously, and worked hard to describe in
1980. It has been named the Mandelbrot set. It starts with a
formula so simple that no one could possibly have expected
so much from it. You program this silly little formula into
your trusty personal computer or workstation, and suddenly
everything breaks loose. Astronomy described simple rules
and simple effects, while history described complicated rules
and complicated effects. Fractal geometry has revealed simple
rules and complicated effects. The complication one sees
is not only most extraordinary but is also spontaneously attractive,
and often breathtakingly beautiful. See Figure 1.
Besides, you may change the formula by what seems a tiny
amount, and the complication is replaced by something altogether
different, but equally beautiful.
The effect is absolutely like an uncanny form of white
magic. I shall never forget the first time I experienced it. I
ran the program over and over again and just could not let
it go. I was a visiting professor at Harvard at the time and
interest in my pictures immediately proved contagious. As
the bug spread, I began to be stopped in the halls by people
who wanted to hear the latest news. In due time, the Scientific
American of April 1985 published a story that spread the
news beyond Harvard.
The bug spread to tens, hundreds, and thousands of people.
I started getting calls from people who said they loved those
pictures so much that they simply had to understand them;
where could they find out about the multiplication of complex
numbers? Other people wrote to tell me that they found
my pictures frightening. Soon the bug spread from adults to
children, and then (how often does this happen?) from the
kids to teachers and to parents.
Lovable! Frightening! One expects these words to be applied
to live, warm bodies, not to mere geometric shapes.
Would you have expected kids to go to you, their teachers,
and ask you to explain a mathematical picture? And be eager
enough to volunteer to learn more and better algebra? Would
you expect strangers to stop me in a store downtown, because
they just have to find out what a complex number is?
Next, let me remind you that the new math fiasco started
when a committee of my elders, including some of my
friends, all very distinguished and full of goodwill, figured
out among themselves that it was best to start by teaching
small kids the notions that famous professors living in the
1950s viewed as being fundamental, and therefore simple.
They wanted grade schoolers to be taught the abstract idea
of a set. For example, a box containing five nails was given
a new name: it became a set of five nails. As it happened,
hardly anyone was dying to know about five-nail sets.
On the other hand, the initial spread of fractals among students
and ordinary people was neither planned nor supported
by any committee or corporation, least of all by IBM, which
supported my scientific work but had no interest in its graphic
or popular aspects. This spread was one of the most truly
spontaneous events I ever heard of or witnessed. People could
not wait to understand and master the white magic and find
out about those crazy Mandelbrot sets. The five-nail set was
rejected as cold and dry. The Mandelbrot set was welcomed
almost as if it were alive. Everything suggests that its study
can become a part of liberal knowledge!
Chaotic dynamics meets the same response. There is no
fun in watching a classical pendulum beat away relentlessly,
but the motion of a pendulum made of two hinged sticks is
endlessly fascinating. I believe that this contrast reveals a basic
truth that every scientist knows or suspects, but few would
concede. The only trace of historical necessity in the evolution
of science may be that its grand strategy is to begin with
questions that are not necessarily the most exciting, but are
simple enough to be tackled at a given time.
The lesson for the educator is obvious. Motivate the students
by that which is fascinating, and hope that the resulting
enthusiasm will create sufficient momentum to move
them through material that must be studied but is less widely
viewed as fun.
8. The computer is the teacher's best friend in communicating the meaning of rigor 27
7 Just beyond the easy fractals lurk
overwhelming challenges
This last word, "fun," deserves amplification. The widely perceived
difficulty of mathematics is a reason for criticism by
the outsider. But for the insider it is a source of pride, and
mathematics is not viewed as real unless it is difficult. In that
sense, fractal geometry is as real as can be, but with a few
uncommon wrinkles.
The first uncommon wrinkle has already been mentioned:
hardly any other chapter of mathematics can boast that even
to the outsider its first steps are fun.
Pushing beyond the first steps, a few additional ones led
me (and soon led others beyond counting) to stunning observations
that the eye tells us must be true, but the mind tells us
must be proven.
A second uncommon wrinkle of fractal geometry is that
those observations are often both simple and new; at least,
they are very new within recent memory. Hardly any other
chapter of mathematics can boast of simple and new observations
worth making. Therefore, fractal geometry has provided
multitudes with the awareness that the field of mathematics is
alive.
A third uncommon wrinkle of fractal geometry is that, next
to simple and new observations that were easy to prove, several
revealed themselves beyond the power of the exceptionally
skilled mathematicians who tackled them. Thus, some
of my earliest observations about the Mandelbrot set remain
open. Furthermore, no one knows the dimensions of selfaffine
sets beyond the simplest. In physics, turbulence and
fractal aggregates remain mysterious. The thrills of frontier
life can be enjoyed right next to the thriving settlements.
Hardly any other chapter of mathematics can boast of so
many simple but intractable conjectures.
A fourth wrinkle concerns the easy beginnings of fractal
geometry. Thanks to intense exposure, it is quite true that
much about fractals appears obvious today. But yesterday the
opposite view was held by everyone. My writings have--
perhaps with excessive verve--blamed mathematicians for
having boxed themselves and everyone else in an intellectual
environment where constructions now viewed as protofractal
were once viewed as pathological and anything but
obvious. This intellectual environment was proud of having
broken the connections between mathematics and physics.
Today there is a growing consensus that the continuity of the
links between mathematics and physics is obvious, but the
statements ring false in the mouths of those who denied and
destroyed this continuity; they sound better in the mouths of
those who rebuilt it.
To conclude this section, fractals may be unrepresentative.
This is not a drawback but rather a very great strength from
the viewpoint of education. If it is true that "math was never
like that," it is also true that "this is more lifelike than any
other branch of math."
8 The computer is the teacher's best
friend in communicating the
meaning of rigor
One passionate objection to the computer as the point of entry
into real mathematics is the following: if the young replace
solving traditional problems by computer games, they
will never be able to understand the fundamental notion of
mathematical rigor. This fear is based on an obvious chain of
associations: the computer started as a tool of applied mathematics,
applied mathematicians spurn rigor, the friend of my
enemies is my enemy, therefore, the computer is the enemy
of rigor.
With equal passion I think that the precise contrary is true:
rain or shine, the computer is rigor's only true friend. True, a
child can play forever with a ready-made program that draws
Mandelbrot sets and never understand rigor, nor learn much
of any value. But neither does the child who always does
his mathematical homework with access to the teacher's answer
book. On the contrary, the notion of rigor is of the
essence for anyone who has been motivated to write a computer
program--even a short one--from scratch.
When I was a student a non-rigorous proof did not scream
look out at me and I soon realized that even my excellent
teachers occasionally failed to notice clearcut errors in my
papers. In the case of a computer program, on the contrary,
being rigorous is not simply an esthetic requirement;
in most cases, a non-rigorous program fails completely, and
the slightest departure from absolute rigor makes it scream
"Error!" at the programmer. No wonder that the birth of the
computer was assisted by logicians and not mainstream mathematicians.
(This topic is discussed in Mandelbrot 1993a.) It
is true that, on occasion, a nonrigorous program generates
meaningless typography or graphics, or--worse--sensiblelooking
output that happens to be wrong. But those rare
examples only prove that programming requires no less care
than does traditional proof.
Moreover, the computer programmer soon learns that a
program that works on one computer, with its operating
system, will not work on another. He will swear at the discrepancies,
but I cannot imagine a better illustration of the
changeability and arbitrariness of axiomatic systems.
Many other concepts used to be subtle and controversial
before the computer made them become clear. Thus, computer
graphics refreshes a distinction between fact and proof,
one that many mathematicians prefer not to acknowledge but
that Archimedes described wonderfully in these words: "Certain
things first became clear to me by a mechanical method,
although they had to be demonstrated by geometry afterwards
because their investigation by the said mechanical method did
not furnish an actual demonstration. But it is of course easier,
when the method has previously given us some knowledge of
the questions, to supply the proof than it is to find it without
28 Chapter 3. Fractals, Graphics, and Mathematics Education
any previous knowledge. This is a reason why, in the case of
the theorems that the volumes of a cone and a pyramid are
one-third of the volumes of the cylinder and prism (respectively)
having the same base and equal height, the proofs of
which Eudoxus was the first to discover, no small share of
the credit should be given to Democritus who was the first to
state the fact, though without proof."
The first two sentences might easily have been written
in our time by someone describing renascent experimental
mathematics, but Archimedes lived from 287 to 212 BC,
Democritus from 460 to 370 BC and Eudoxus from 408 to
355 BC. (Don't let your eyes glaze over at the names of these
Ancient heroes. This chapter is almost over.)
When a child (and why not an adult?) becomes tired of
seeing chaos and fractal games as white magic and draws up
a list of observations he wants to really understand, he goes
beyond playing the role of Democritus and on to playing the
role of Eudoxus. Moreover, anyone's list of observations is
bound to include several that are obviously mutually contradictory,
stressing the need for a referee. Is there a better way
of communicating another role for rigor and a role for further
experimentation?
9 Conclusion
As was obvious all along, I am a working scientist fascinated
by history and education, but totally ignorant of the literature
of educational philosophy. I hope that some of my thoughts
will be useful, but many must be commonplace or otherwise
deserve to be credited to someone. One area where I claim no
perverse originality is the historical assertions: they are documented
facts, not anecdotes made up to justify a conclusion.
Now to conclude. The best is to quote myself and to ask
once again: Is it extravagant to hope that, starting with this
piece of mathematics called fractal geometry, we could help
broaden the small band of those who see mathematics as essential?
That band ought to include every educated citizen
and therefore to have mathematics take its place among the
liberal arts. A statement of hope is the best place to close.