Chapter 2 } Mathematical Ability

196 Pages·2009·12.83 MB·English

The Mind of the Mathematician
THE JOHNS HOPKINS UNIVERSITY PRESS BAlTIMORE
Michael FitzgeraLd and loan James
© 2007 Michael Fitzgerald and loan James
All rights reserved. Published 2007
Printed in the United States of America on acid-free paper
9 87 6 5 432
The Johns Hopkins University Press
2715 North Charles Street
Baltimore, Maryland 21218-4363
www.press.jhu.edu
Library of Congress Cataloging-in-Publication Data
Fitzgerald, Michael, 1946-
The mind of the mathematician / Michael Fitzgerald and loan
James.
p.cm.
Includes bibliographical references and index.
ISBN-13: 978-0-8018-8587-7 (hardcover: acid-free paper)
ISBN-lO: 0-8018-8587-6 (hardcover: acid-free paper)
1. Mathematicians-Psychology. 2. Mathematics-Psychological
aspects. 3. Mathematical ability. 4. Mathematical ability-Sex
differences. I. James, I. M. (loan Mackenzie), 1928-II. Title.
BF456.N7F582007
510.1' 9-dc22 2006025988
A catalog record for this book is available from the British Library.
Contents
Preface Vll
Introduction ix
PART I } TOUR OF THE LITERATURE
Chapter 1. Mathematicians and Their World 3
Chapter 2. Mathematical Ability 24
Chapter 3. The Dynamics of Mathematical Creation 42
PART II } TWENTY MATHEMATICAL PERSONALITI ES
Chapter 4. Lagrange, Gauss, Cauchy, and Dirichlet 67
Chapter 5. Hamilton, Galois, Byron, and Riemann 88
Chapter 6. Cantor, Kovalevskaya, Poincare, and Hilbert 105
Chapter 7. Hadamard, Hardy, Noether, and Ramanujan 131
Chapter 8. Fisher, Wiener, Dirac, and G6del 149
References 163
Index 175

Preface
Psychologists have long been fascinated by mathematicians and their world.
In this book we start with a tour of the extensive literature on the psychol
ogy of mathematicians and related matters, such as the source of mathe
matical creativity. By limiting both mathematical and psychological techni
calities, or explaining them when necessary, we seek to make our review of
research in this field easily readable by both mathematicians and psycholo
gists. In the belief that they might also wish to learn about some of the
human beings who helped to create modern mathematics, we go on to
profile twenty well-known mathematicians of the past whose personalities
we find particularly interesting. These profiles serve to illustrate our tour of
the literature.
Among the many people we have consulted in the course of writing this
book we would particularly like to thank Ann Dowker, Jean Mawhin, Allan
Muir, Daniel Nettle, Brendan O'Brien, Susan Lantz, and Mikhail Treisman.
We also wish to thank Ohio University Press, Athens, Ohio (www.ohio
.edu/oupress), for granting permission to reprint an excerpt from Don H.
Kennedy's biography of Sonya Kovalevskaya, Little Sparrow: A Portrait of
Sonya Kovalevskaya.
vii

Introduction
Mathematics, according to the Marquis de Condorcet, is the science that
yields the most opportunity to observe the workings of the mind. Its study,
he wrote, is the best training for our abilities, as it develops both the power
and the precision of our thinking. Henri Poincare, in his famous 1908
lecture to the Societe de Psychologie in Paris, observed that mathematics is
the activity in which the human mind seems to take least from the outside
world, in which it seems to act only of itself and on itself. He went on to
describe the feeling of the mathematical beauty of the harmony of numbers
and forms, of geometric elegance-the true aesthetic feeling that all real
mathematicians know. According to the British mathematical philosopher
Bertrand Russell (1910), mathematics possesses "not only truth but supreme
beauty-a beauty cold and austere ... yet sublimely pure, and capable of a
stern perfection such as only the greatest art can show." In the words of
Courant and Robbins (1941): "Mathematics as an expression of the human
mind reflects the active will, the contemplative reason, and the desire for
aesthetic perfection. Its basic elements are logic and intuition, analysis and
construction, generality and individuality. Though different traditions may
emphasize different aspects, it is only the interplay of these aesthetic forces
and the struggle for their syntheses that constitute the life, usefulness, and
supreme value of mathematical science." The modern French mathemati
cian Alain Connes, in Changeux and Connes (1995), tells us that "exploring
the geography of mathematics, little by little the mathematician perceives
the contours and structures of an incredibly rich world. Gradually he de
velops a sensitivity to the notion of simplicity that opens up access to new,
wholly unsuspected regions of the mathematical landscape."
ix