Fermat's Last Theorem

Mathematicians, Romantic Heroes of the Past, Present and Future


Last October I found myself in Chicago taking part in an hour long radio interview about my book, “Fermat’s Last Theorem,” the story of an ancient mathematical riddle. Before the interview began I commented to the host that I was surprised that he was brave enough to schedule a programme about mathematics, which was not exactly guaranteed to be a ratings hit. He shrugged his shoulders, and replied, “Well tonight is Halloween, and we reckon that nothing is scarier than maths!”

The mere mention of Pythagoras’ theorem or Euclidean geometry is enough to send a shudder down the spine of the vast majority of the population, who wear their innumeracy with pride and wonder why anybody would ever choose to devote their life to the study numbers – surely, mathematicians are nothing more than human calculators, soulless creatures capable of nothing more than adding up long lists of tedious numbers. As a result of this persecution, mathematicians have tended to withdraw into their own communities, believing that there is safety in numbers. However, a spate of books over the last twelve months has sought to dispel the popular stereotype, encouraging mathematicians to come out of the closet and take pride in their calculations. These books reveal that mathematicians can be as passionate as any poet and as obsessive as any artist, their lives being a rich mixture of ecstasy and tragedy, madness and romance.

My own interest in mathematics began with the story of Fermat’s Last Theorem, a problem invented in the 17th century by the French scholar and judge Pierre de Fermat. Fermat wrote in the margin of his book that he had a proof that could solve the problem, but, annoyingly, he explained that there was insufficient space to write down his proof. Following his death and the discovery of his marginal note, generations of mathematicians attempted to rediscover Fermat’s proof, which resulted in rivalries, rich prizes, tragedy, suicide, duelling at dawn, and three centuries of failure. Then, in 1963, a ten year old boy read about Fermat’s Last Theorem and promised himself that he would devote his life to finding Fermat’s proof. Andrew Wiles’s childhood dream dominated his life, and, eventually, in 1986, he realised a potential strategy for attacking the problem. He spent the next seven years working in secrecy, abandoning everything except mathematics and his family – his wife only learnt about his obsession during their honeymoon. In 1993, with his proof apparently complete, Wiles announced his success to the rest of the world, but then the discovery of an error during the refereeing process meant that his entire logical framework collapsed, leading to professional embarrassment and public humiliation. He was forced to return to his study, where he spent a year struggling to correct the mistake. Just when he was on the point of admitting defeat, a brilliant insight provided him with the fix he needed and his proof was complete. At last, he had achieved his childhood dream.

For me, Wiles’s story includes the essence of a romantic tale: a lost treasure, a childhood dream, ruthless ambition, hope in the face of adversity, failure, and triumph. Furthermore, Wiles was not searching for riches, but for a solution to a purely intellectual problem. His desire was not fuelled by greed, but by curiosity. Pure mathematics has few applications in the real world, rather it consists of a series of conundrums which are challenges to the mathematician. Wiles’s success will not lead to patents, rather it is a tribute to the human spirit. In 1996 I and a colleague, John Lynch, made a BBC Horizon documentary on the subject, which begins with Wiles recalling the moment his odyssey was complete, at which point he is overcome with emotion and turns away from the camera. Mathematicians are not soulless.

Wiles’s proof was achieved at the age of 41, which makes him a rather elderly addition to the pantheon of mathematical heroes. It is remarkable that the majority of brilliant mathematicians do their greatest work when they are in their twenties, which is convenient for those geniuses who die young. For example, Niels Henrik Abel, the nineteenth century Norwegian, made his greatest contribution to mathematics at the age of nineteen, and then died in poverty just eight years later, struck down by tuberculosis. Another victim of tuberculosis was the Indian prodigy Srinivasa Ramanujan, a largely untutored villager from southern India, who discovered some of the most beautiful proofs of the early twentieth century. He was invited to Cambridge to work with the most eminent professors in the Empire, but the harsh English winters took their toll, and by the age of 33 he was dead.

Arguably the most romantic example of the “live fast, die young” mathematician is Evariste Galois. Born near Paris in 1881, Galois was already exploring new mathematical territory at the age of 18, but his ideas were so advanced that his tutors could not comprehend their true significance. Disillusioned by their failure to recognise his talent, Galois left mathematics and joined the “Friends of the People,” an outlawed republican organisation which fought against the increasing power of the monarchy and the church. He was soon arrested, but shortly afterwards a cholera outbreak forced the authorities to empty the prisons. Upon his release, he fell in love with Stéphanie-Félicie Poterine du Motel, which was unfortunate because she was already engaged to be married. To make matters worse, her fiancé happened to be one of the finest marksmen in Paris, and he challenged Galois to a duel at dawn. The following day Galois was shot in the stomach and killed. Some have speculated that the duel was not the result of a simple love affair, rather that his seduction was planned by his political enemies who realised that it would inevitably lead to his death.

Galois’s life was full of political intrigue and passion, and yet, for him, his proudest achievement was his mathematics. The night before the duel, when he realised that death was unavoidable, he spent his final hours committing his mathematical ideas to paper, in the hope that they would not be forgotten by future generations. For those who hate mathematics, Galois provides a paradox. How can anybody who would fight for political freedom and die for love have his emotions stirred by numbers? How can something as logical as mathematics, something so divorced from sex and violence, attract somebody as romantic as Galois?

First, although mathematics is logical, that does not exclude it from being creative, intuitive and inspirational. In the case of Fermat’s Last Theorem, mathematicians suspected that it was true, but the challenge was to prove it, i.e., develop a logical sequence of arguments which demonstrate the validity of the Last Theorem. Unfortunately, there is no recipe for building such a proof, and so mathematicians have to use their intuition, following trails which might take them round in circles, often pursuing leads which may be nothing more than dead ends. The proof itself might be purely logical, but finding it is not.

According to Sir Arthur Eddington, a great English scientist from the early part of this century, “Proof is an idol before which the mathematician tortures himself.” More recently an anonymous mathematician posted the following note on the internet: “Sex and drugs? They’re nothing compared with a good proof!” Mathematicians worship proof because it is the ultimate arbiter of truth, and, because it relies on logic rather than measurement or experiment, it is untainted by uncertainty and inaccuracy. Hence, once a proof is established it remains true forever. In “A Mathematician’s Apology,” written in 1940 by G.H. Hardy as a justification of his life, the author claims that, “Immortality may be a silly word, but probably a mathematician has the best chance of whatever it may mean.”

Building proofs is a passionate pursuit, and furthermore, the proofs themselves can exhibit great beauty. The way individual arguments are woven together, the manner in which a particular method is brought to bear when the proof looks on the point of collapse, the involvement of surprising techniques, and sudden twists in the reasoning all contribute to an elegant intellectual construction. G.H. Hardy’s own work was guided by the ethos that mathematical truth was equivalent to beauty:

“Beauty is the first test: there is no permanent place in the world for ugly mathematics… A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs it is becausethey are made with ideas. A painter makes patternswith shapes and colours, a poet with words … A mathematician, on the other hand, has no material to work with but ideas, and so his patterns are likely to last longer.” The proofs of Pythagoras are as valid today as they were when he invented them over two thousand years ago.

The search for mathematical proofs has captured the imagination of many men and women, and in my opinion many of these folk qualify as romantic heroes, encompassing all aspects of romance and heroism. For example, in the 19th century Sophie German defied prejudice and discrimination in order to establish her reputation during an era when women were excluded from studying mathematics. For long periods, she was forced to take on the identity of a man, signing herself as Monsieur Leblanc in order to be taken seriously. Another anonymous hero, who was also discriminated against, was Alan Turing. His mathematics was pivotal in helping break the German codes during the Second World War, but the secrecy surrounding British Intelligence meant that his work and that of his colleagues could not be publicly acknowledged. After the war, the government was concerned that his homosexuality was a security risk, which meant that Turing’s movements were being constantly monitored. In 1952 he was arrested for violating British homosexuality statutes, and thereafter his life became intolerable, leading to his suicide at the age of 42. An equally tragic story is that of John Nash, whose work in 1949 (at the age of just 21) would eventually lead to overdue recognition and a Nobel Prize in 1994. In the intervening decades, Nash suffered from severe paranoid schizophrenia, which led to the loss of his family and freedom. Remarkably, he survived the horrific treatments of the 1960s, such as electroshock therapy, and made a spontaneous recovery in the late 1980s. He is now coming terms with his new life, living with Alicia Nash, his former wife, who divorced him, but who never abandoned him.

However, if I had to choose one mathematician to be the epitome of romantic heroism, it would be Paul Erdös. He was the most prolific mathematician in history, working 19 hour days, operating largely on coffee and amphetamines, a lifestyle he continued right up to his death in 1996, at the age of 83. Paul Hoffman’s biography of Erdös is entitled “The Man Who Loved Only Numbers,” which, if anything, is an understatement. Here was a man who worshipped numbers, he was “a mathematical monk, who renounced physical pleasure and material possessions for an ascetic, contemplative life.” He gave away all his money, he never had a relationship, he never learnt to drive, never had a house, and travelled around the world with all his worldly belongings in two suitcases, each only one third full.

Erdös devoted his life to the search for mathematical truth, and he sacrificed everything in order to achieve his goal. He had an immense curiosity and an intense belief in the honesty of numbers. Indeed, he felt sorry for those people who were blind to the beauty of numbers. When asked why he thought numbers are beautiful, he replied, “It’s like asking why Beethoven’s Ninth Symphony is beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.”