The Mind of the Mathematician

Introduction

ix 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 mathematician 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 develops a sensitivity to the notion of simplicity that opens up access to new, wholly unsuspected regions of the mathematical landscape.’’ 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, psychology , 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 mathematician August Ferdinand Möbius), sought evidence of diagnostic categories 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 wellknown 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 period , 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 distinctive culture. To counter the popular...