The mind of a mathematician

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 ofCongress Cataloging-in-Publication Data Fitzgerald, Michael, 1946The 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
3

Chapter 1. Mathematicians and Their World Chapter 2. Mathematical Ability
24

Chapter 3. The Dynamicsof Mathematical Creation

42

PART II }

TWENTY MATHEMATICAL PERSONALITI ES
67 88
105

Chapter 4. Lagrange, Gauss, Cauchy, and Dirichlet Chapter 5. Hamilton, Galois, Byron, and Riemann

Chapter 6. Cantor, Kovalevskaya, Poincare, and Hilbert Chapter 7. Hadamard, Hardy, Noether, and Ramanujan Chapter 8. Fisher, Wiener, Dirac, and G6del
149

131

References Index
175

163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 psychology of mathematicians and related matters, such as the source of mathematical creativity. By limiting both mathematical and psychological technicalities, or explaining them when necessary, we seek to make our review of research in thisfield easily readable by both mathematicians and psychologists. 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 haveconsulted 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

SonyaKovalevskaya.

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, observedthat 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 BertrandRussell (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...