The Mind of the Mathematician THE JOHNS HOPKINS UNIVERSITY PRESS } BALTIMORE Michael Fitzgerald and Ioan James T H E M I N D O F T H E M A N AT H E M AT I C I ∫ 2007 Michael Fitzgerald and Ioan James All rights reserved. Published 2007 Printed in the United States of America on acid-free paper 987654321 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 Ioan James. p. cm. Includes bibliographical references and index. isbn-13: 978-0-8018-8587-7 (hardcover : acid-free paper) isbn-10: 0-8018-8587-6 (hardcover : acid-free paper) 1. Mathematicians—Psychology. 2. Mathematics—Psychological aspects. 3. Mathematical ability. 4. Mathematical ability—Sex di√erences. I. James, I. M. (Ioan Mackenzie), 1928– II. Title. bf456.n7f58 2007 510.1%9—dc22 2006025988 A catalog record for this book is available from the British Library. Contents Preface vii 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 PERSONALITIES Chapter 4. Lagrange, Gauss, Cauchy, and Dirichlet 67 Chapter 5. Hamilton, Galois, Byron, and Riemann 88 Chapter 6. Cantor, Kovalevskaya, Poincaré, and Hilbert 105 Chapter 7. Hadamard, Hardy, Noether, and Ramanujan 131 Chapter 8. Fisher, Wiener, Dirac, and Gödel 149 References 163 Index 175 This page intentionally left blank 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 This page intentionally left blank 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 Poincaré, in his famous 1908 lecture to the Société 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 di√erent traditions may emphasize di√erent 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 x Introduction Nearly seventy years, ago the Scottish-American mathematician Eric Temple Bell set out to write about mathematicians in a way that would grip the imagination. In the introduction to his immensely readable Men of Mathematics, first published in 1937, he begins by emphasizing, ‘‘The lives of mathematicians presented here are addressed to the general reader and to others who might wish to see what sort of human beings the men were who created modern mathematics.’’ Unfortunately, it leaves the impression that many of the more notable mathematicians of the past were self-serving and quarrelsome. This is partly because Bell selected his subjects accordingly, but also because he was not above distorting the facts to make a good story. He was a man of strong opinions, not simply reflecting the prejudices of a bygone age. While the history of mathematics goes back thousands of years, psychol- ogy, in the modern sense, only originated in the nineteenth century. A special interest in mathematicians was present from the early years of the subject. The Leipzig neurologist Paul Julius Möbius (grandson of the math- ematician August Ferdinand Möbius), sought evidence of diagnostic cate- gories that might be related to the creative behavior of mathematicians in his Uber die Anlage zur Mathematik (‘‘on the gift for mathematics’’) of 1900. A little later the Swiss psychologists Edouarde Claparède and Théodore Flournoy organized an inquiry into mathematicians’ working methods, while in 1906 Poincaré gave the seminal lecture on mathematical invention mentioned earlier. Psychoanalysts have displayed relatively little interest in mathematicians, although they have a lot to say about creativity in general, with illustrations mainly from the arts. As Storr points out in his well- known book The Dynamics of Creation (1972), psychologists are primarily concerned with the causes that lie behind creativity, rather than the reasons that drive those who enjoy creative gifts to make full and e√ective use of them. The word genius often occurs in the literature, usually meaning much the same as ‘‘exceptionally able.’’ Some Reflections on Genius, by Lord Russell Brain (1960), provides a good introduction; other works are listed in the bibliography. The term has gradually changed in meaning over a long pe- riod, stretching back to classical times. Today the meaning seems rather uncertain, and we avoid using it ourselves. The material in the tour of the literature that forms the first part of this book is of necessity somewhat miscellaneous in character, but we have arranged the di√erent topics under three broad chapter headings. In the Introduction xi first chapter we describe the special attraction of mathematics and its dis- tinctive culture. To counter the popular impression that mathematicians are not interested in anything else, we give some examples of mathemati- cians who contributed to music and the other arts. For the second part of this book, we are dependent on reliable biographical information. For- tunately some excellent biographies of mathematicians have been written, and we mention some of these. Another popular misconception is that mathematicians spend their time making calculations. While this is not the case in general, there are some exceptional individuals, including a few mathematicians, who have an ex- traordinary ability to do so. These ‘‘lightning calculators,’’ who are known as savants, have intrigued psychologists for many years. Alfred Binet’s classic account of his investigations of two lightning calculators and others with savant skills, Psychologie des grands calculateurs et joueurs d’échecs (‘‘psy- chology of the great calculators and chess-players’’) of 1894, is still worth reading. We summarize what is known about them and then go on to discuss various other phenomena that have led some investigators to believe that there may be modules in the brain that specialize in calculation. In chapter two we present a summary of the vast literature on mathe- matical education, and we also discuss the vexed question of gender di√er- ence. Child prodigies sometimes emerge in mathematics, as they do in music and languages, although they do not always develop into full-fledged mathematicians. We give some striking examples. Finally we review the literature on the decline of productivity with age. This is well established across a wide range of activities; we present evidence to support the general belief that mathematics is no exception. In the last chapter of the first part of this book, we turn to the fascinating problem of mathematical creativity and the role of intuition: intuition is perception via the unconscious, according to Carl Jung. There is no better place to start than the well-known memoir, The Psychology of Invention in the Mathematical Field (1945), by Poincaré’s disciple Jacques Hadamard, which takes up many of the interesting questions raised by Poincaré in his famous lecture. At the same time that this book appeared, the Viennese psychiatrist Hans Asperger published a thesis in which he gave a clinical description of the mild form of autism to which his name has been at- tached, and this has turned out to have an important bearing on the study of creativity, especially in mathematics. xii Introduction Asperger syndrome, as the disorder is called, is much more common than classical autism.