Fermat’s Enigma: The Quest to Solve the World’s Greatest Mathematical Problem

The books for this month are a holiday gift list: books to broaden the library and the mind, to provide pleasure and enjoyment, to give to oneself and others.

The capacity for sustained attention to an abstract train of thought is most highly developed in mathematicians. This pleasingly compact book allows the general reader to understand, in simplified terms, the astonishing solution, over an 8-year period of mostly self-imposed solitude, of a problem that seventeenth-century French mathematician Pierre de Fermat claimed he had proved, namely, that there are no solutions of Xⁿ+Yⁿ=Zⁿ for n greater than 2. Fermat, in his characteristically superior and annoying way, had said he had “a truly marvelous demonstration of this proposition which this margin is too narrow to contain” (p. 62). Indeed, Wiles’s prize-winning proof took more than 100 pages. It also employed a remarkable breadth of the developments in mathematics since Fermat’s day and could not have been Fermat’s solution if he had one.

This provides the pretext for Singh to trace all the roots of Wiles’s solution through much of the history of mathematics, a human enterprise marked by black-and-white judgment, fierce competition, and no small amount of bloodshed. A member of the Pythagorean Brotherhood was drowned for breaking his secrecy oath by publishing his discovery of the dodecahedron. The severity afflicted even the occasional female geniuses who stumbled into mathematics, such as Hypatia, whose flesh was scraped from her bones with oyster shells. And then there was Galois, distracted from the promise of his genius by his tragic allegiances to the nineteenth-century French Republican cause. After his father was shamed into suicide, Galois found himself facing an unwinnable duel over a mysterious woman with a superior duelist who may have been a government agent. The night before the duel in which he died, Galois scribbled out solutions to the problems of quintic equations, a precursor in group theory to the solution of Fermat’s enigma. More recently, the suicide in 1958 of Taniyama of the Taniyama-Shimura conjecture, unexplained, incomprehensible to Taniyama himself in his suicide note, and surely psychopathological, deprived the world of a Japanese genius. Taniyama linked elliptical equations and modular forms, directly presaging Wiles’s solution and making progress toward the grand unification of all fields of mathematics envisioned in 1960 by another Prince­ton mathematician, Robert Langlands.

Wiles’s approach to his own solution speaks volumes about the fierce competition of the mathematical world. Having prepared all his life to prove Fermat’s last theorem, Wiles found it necessary to throw his competitors off his track. He prepared a series of papers based on his research in the 1980s on a type of elliptical equation, and, instead of publishing them together, he eked them out bit by bit to give the appearance that he was slowly working on this rather than his true obsession, as Singh describes it. Wiles also worked in total secrecy, telling no one but his wife, and withdrawing from all but a minimum of his teaching duties. After 6 years in isolation, during which he fathered two children, he let another professor, Nick Katz, in on it in order to check his extensive use of the Kolyvagin-Flach method. Wiles and Katz decided on a series of lectures for graduate students to test out Wiles’s extensive, secret work. It was as if Wiles had to speak to someone to finish his proof.

When he finally had the proof completed and announced it to the world, Wiles became an instant—and mathematics’ only living—celebrity. But one of the six referees required to check the enormous proof found a gap in it. Had Wiles’s 7 years of effort been wasted? He stalled on releasing the flawed proof, fearing another mathematician would find the answer and get the final credit, humiliating him. Again he turned to another mathematician, Richard Taylor, for help. But it was Wiles himself who finally saw that his earlier, discarded method based on Iwasawa theory, taken together with the Kolyvagin-Flach method, each inadequate on its own, together gave the answer.

This intellectual saga with its triumphant conclusion has great charm. It is fun to read in fairly recent books about the recalcitrance of the problem (1). If one wished merely to know the gross structure of the proof, the description of it by Singh and Ribet in Scientific American (2) would do. But the human factors in the story, so well drawn by Singh, made me think of our own most difficult enigma in psychiatry, the problem of how psychotherapy works. Reading Wiles’s story, I began to glimpse a solution to that ineluctable problem. Alas, there is insufficient space in this review to record it now.