The Mathematician's Brain: A Personal Tour Through the Essentials of Mathematics and Some of the Great Minds Behind Them

168 Pages·2007·1.31 MB·English

by  David Ruelle

The
Mathematician's
Brain

DAVID RUELLE
The Mathematician's Brain

DAVID RUELLE

P R I N C E T O N U N IV E R S I T Y P R E S S

P R IN C E T O N A N D O X F O R D
Copyright © 2007 by Princeton University Press

Published by Princeton University Press, 41 William Street,

Princeton, New Jersey 08540

In the United Kingdom: Princeton University Press, 3 Market Place,

Woodstock, Oxfordshire OX20 ISY

All Rights Reserved

Library of Congress Cataloging-in-Publication Data

Ruelle, David.

The mathematician's brain

p.

/ David Ruelle.

cm.

Includes bibliographical references and index.

ISBN-13: 978-0-691-12982-2 (cl : acid-free paper)

ISBN-I0: 0-691-12982-7 (cl : acid-free paper)

1. Mathematics-Philosophy. 2. Mathematicians-Psychology. 1.

QA8.4.R84

51D--dc22

2007

2006049700

British Library Cataloging-in-Publication Data is available

This book has been composed in Palatino

Printed on acid-free paper. =

press.princeton.edu

Printed in the United States of America

1 3 5 7 9 10 8 6 4 2

Title.
Preface

1. Scientific Thinking
2. What Is Mathematics?
3. The Erlangen Program
4. Mathematics and Ideologies
5. The Unity of Mathematics
6. A Glimpse into Algebraic Geometry and Arithmetic
7. A Trip to Nancy with Alexander Grothendieck
8. Structures
9. The Computer and the Brain
10. Mathematical Texts
11. Honors
12. Infinity: The Smoke Screen of the Gods
13. Foundations
14. Structures and Concept Creation
15. Turing's Apple
16. Mathematical Invention: Psychology and Aesthetics
17. The Circle Theorem and an InfiniteDimensional Labyrinth
18. Mistake!
19. The Smile of Mona Lisa
20. Tinkering and the Construction of Mathematical
Theories
21. The Strategy of Mathematical Invention

v

vii

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5
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85

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108

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C O N T E N T S

22. Mathematical Physics and Emergent Behavior
23. The Beauty of Mathematics

Notes

Index

vi

119
127

131
157
.:. Preface

.:.

ArEOMETPHTOL MHLlEIL EILITO

ACCORDING TO TRADITION, Plato put a sign at the entrance
of the Academy in Athens: "Let none enter who is ignorant of
mathematics./I Today mathematics still is, in more ways Th-an
one, an essential preparation for those who want to understand
the nature of things. But can one enter the world of mathematics
without long and arid studies? Yes, one can to some extent, because what interests the curious and cultivated person (in older
days called a philosopher) is not an extensive technical knowledge. Rather, the old-style philosopher (i.e., you and me) would
like to see how the human mind, or we may say the mathematician's brain, comes to grips with mathematical reality.
My ambition is to present here a view of mathematics and
mathematicians that will interest those without training in mathematics, as well as the many who are mathematically literate. I
shall not attempt to follow majority views systematically. Rather,
I shall try to present a coherent set of facts and opinions, each
of which would be acceptable to a fair proportion of my mathematically active colleagues. In no way can I hope to make a complete presentation, but I shall exhibit a variety of aspects of the
relation between mathematics and mathematicians. Some of
these aspects will turn out to be less than admirable, and perhaps I should have omitted them, but I felt it more important to
be truthful than politically correct. I may also be faulted for my
emphasis on the formal and structural aspects of mathematics;
these aspects, however, are likely to be of most interest to the
reader of the present book.
Human communication is based on language. This method of
communication is acquired and maintained by each of us
through contact with other language users, against a background of human experiences. Human language is a vehicle of
truth but also of error, deception, and nonsense. Its use, as in
the present discussion, thus requires great prudence. One can
improve the precision of language by explicit definition of the

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terms used. But this approach has its limitations: the definition
of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating
some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in
principle rely on a human language. It can use, instead, a formal
presentation in which the validity of a deduction can be checked
mechanically and without risk of error or deception.
Human language carries some concepts like meaning or beauty.
These concepts are important to us but difficult to define in
general. Perhaps one can hope that mathematical meaning
and mathematical beauty will be more accessible to analysis
than the general concepts. I shall spend a little bit of time on
such questions.
The contrast is striking between the fallibility of the human
mind and the infallibility of mathematical deduction, the deceptiveness of human language and the total precision of formal
mathematics. Certainly this makes the study of mathematics a
necessity for the philosopher, as was stressed by Plato. But while
learning mathematics was, in Plato's view, an essential intellectual exercise, it was not the final aim. Many of us will concur:
there are more things of interest to the philosopher (i.e., you and
me) than the mathematical experience, however valuable that
experience is.
This book was written for readers with all kinds of mathematical expertise (including minimal). Most of it is a nontechnical
discussion of mathematics and mathematicians, but I have also
inserted some pieces of real mathematics, easy and less easy. I
urge the reader, whatever his or her mathematical background,
to make an effort to understand the mathematical paragraphs or
at least to read through them rather than jumping straight ahead
to the other chapters.
.
Mathematics has many aspects, and those involving logic, algebra, and arithmetic are among the most difficult and technical.
But some of the results obtained in those directions are very
striking, are relatively easy to present, and have probably the
greatest philosophical interest to the reader. I have thus largely

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emphasized these aspects. I should, however, say that my own
fields of expertise lie in different areas: smooth dynamics and
mathematical physics. The reader should thus not be astonished
to find a chapter on mathematical physics, showing how mathematics opens to something else. This something else is what
Galileo called the /I great book of nature," which he spent his life
studying. Most important, the great book of nature, Galileo said,
is written in mathematical language.

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Scientific Thinking

My DAILY WORK consists mostly of research in mathematical
physics, and I have often wondered about the intellectual processes that constitute this activity. How does a problem arise?
How does it get solved? What is the nature of scientific thinking?
Many people have asked these sorts of questions. Their answers
fill many books and come under many labels: epistemology, cognitive science, neurophysiology, history of science, and so on. I
have read a number of these books and have been in part gratified, in part disappointed. Clearly the questions I was asking are
very difficult, and it appears that they cannot be fully answered
at this time. I have, however, come to the notion that my insight
into the nature of scientific thinking could be usefully complemented by analyzing my own way of working and that of my
professional colleagues.
The idea is that scientific thinking is best understood by
studying the good practice of science and in fact by being a scientist immersed in research work. This does not mean that popular beliefs of the research community should be accepted uncritically. I have, for example, serious reservations with regard to
the mathematical Platonism professed by many mathematicians.
But asking professionals how they work seems a better starting
point than ideological views of how they should function.
Of course, asking yourself how you function is introspection,
and introspection is notoriously unreliable. This is a very serious
issue, and it will require that we be constantly alert: what are
good and what are bad questions you may ask yourself? A physicist knows that trying to learn about the nature of time by introspection is pointless. But the same physicist will be willing to
explain how he or she tries to solve certain kinds of problems
(and this is also introspection). The distinction between acceptable and unacceptable questions is in many cases obvious to a
working scientist and is really at the heart of the so-called scientific method, which has required centuries to develop. I would
thus refrain from saying that the distinction between good and

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bad questions is always obvious, but I maintain that scientific
training helps in making the distinction.
Enough for the moment about introspection. Let me state
again that I have been led by curiosity about the intellectual processes of the scientist and in particular about my own work. As
a result of my quest I have come to a certain number of views
or ideas that I have first, naturally, discussed with colleagues. 1
Now I am putting these views and ideas in writing for a more
general audience. Let me say right away that I have no final theory to propose. Rather, my main ambition is to give a detailed
description of scientific thinking: it is a somewhat subtle and
complex matter, and absolutely fascinating. To repeat: I shall discuss my views and ideas but avoid dogmatic assertions. Such
assertions might give nonprofessionals the false impression that
the relations between human intelligence and what we call reality have been clearly and finally elucidated. Also, a dogmatic
attitude might encourage some professional colleagues to state
as firm and final conclusions their own somewhat uncertain beliefs. We are in a domain where discussion is necessary and
under way. But we have at this time informed opinions rather
than certain knowledge.
After all these verbal precautions, let me state a conclusion
that I find hard to escape: the structure of human science is largely
dependent on the special nature and organization of the human brain.
I am not at all suggesting here that an alien intelligent species
might develop science with conclusions opposite to ours. Rather,
I shall later argue that what our supposed alien intelligent species would understand (and be interested in) might be hard to
translate into something that we would understand (and be interested in).
Here is another conclusion: what we call the scientific method is
a different thing in different disciplines. This will hardly surprise
those who have worked both in mathematics and in physics or
in physics and in biology. The subject matter defines to some
extent the rules of the game, which are different in different
areas of science. Even different areas of mathematics (say, algebra and smooth dynamics) have a very different feeL I shall
in what follows try to understand the mathematician's brain.
This is not at all because I find mathematics more interesting
than physics and biology. The point is that mathematics may be

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