Fermat's Last Theorem

The Joy of X

The Observer
28th September 1997

Forget life on Mars and ignore the hype of Dolly the cloned sheep, because according to many science journalists the hippest research is currently going on in the arcane world of mathematics. This mathematical mania began in 1995 when Professor Andrew Wiles, a spectacled, softly spoken Englishman, solved Fermat’s Last Theorem, the most notorious problem in mathematics. After making front-page headlines around the world and being interviewed on CNN, he appeared in a special issue of People magazine, which listed him among “The 25 Most Intriguing People of the Year’, alongside such luminaries as Oprah Winfrey. The ultimate accolade came from an international clothing chain who asked the shy genius to endorse their new range of menswear – Wiles turned down the offer.

Although he is not in the same league as Madonna, Wiles has become a celebrity – the first mathematical celebrity since the nineteenth century. His achievements have been the focus of a BAFTA award-winning documentary made for the BBC series Horizon, and there is even discussion of a feature film. My own book about Wiles’ exploits, Fermat’s Last Theorem, became a No.1 best-seller. But why has Wiles caught the public imagination, and why is maths suddenly sexy?

First of all, it is important to appreciate that mathematical maths is different from scientific maths. Even Stephen Hawking’s most complex astronomical calculations would appear trivial to the most mediocre of mathematicians. Real maths is not about exploiting numbers for the benefit of science, but rather understanding numbers at the most fundamental level, to the extent of even discovering (or perhaps inventing) new types of numbers – as well as so-called natural numbers, there now exist imaginary, surreal and transcendental numbers.

Second, mathematics has a unique purity and innocence, exemplified by the problem that obsessed Professor Wiles. Fermat’s Last Theorem is nothing more than a conundrum, a challenge to the intellect, a riddle to test the human spirit. Wiles discovered the problem as a ten year old schoolboy browsing in his local library, and to him it was nothing more than a wonderful puzzle. As Wiles grew up he was well aware that solving this puzzle would not lead to a cure for cancer and neither would it ever result in a weapon of mass destruction, but it would generate an immense sense of achievement. Like mountaineers, mathematicians solve problems simply because they are there.

This is not to say that mathematics does not have applications in the real word – mathematics is the language of science and engineering. The last time maths hit the headlines was when chaos theory showed how apparently simple equations can lead to unpredictable results. Applied to the real world, this allowed meteorologists and others to justify their failed forecasts. However, pure mathematicians, such as Wiles, do not look for applications in their work, instead they develop theories for the sheer intellectual thrill of it. In an increasingly materialistic world, the mathematician remains ethereal.

Wiles is also an attractive character because he is very much the lone genius battling single-handed against a mighty riddle. In recent decades, much of the rest of science has become a collaborative exercise, in which the practitioners lose their individuality and leaders have to be elected to speak on behalf of a giant collaboration. In comparison, mathematics still has plenty of room for heroic boffins scribbling away in their garrets. For seven years Wiles worked in complete secrecy and isolation, confiding only in his wife. Mathematicians strike a romantic chord, because they seem to have more in common with the tortured artist than the lab-coat brigade.

The practice of doing mathematics has changed little through the centuries. In many ways Wiles’ lifestyle mirrors that of Pierre de Fermat, who in seventeenth century created the problem that would torment generations of mathematicians. Fermat was a judge by profession, and in order to remain impartial he was discouraged from socialising. As a result he spent his evenings alone in his study, and took up mathematics as a hobby. Known as the Prince of Amateurs, he went on to make some of the most important discoveries in the history of numbers.

The Last Theorem, his most famous legacy, was casually scribbled in the margin of a book he happened to be reading. Unfortunately, he did not bother to write down the proof behind the theorem, the vital justification without which no theorem can be accepted. After Fermat’s death, his note describing the Last Theorem was discovered, and ever since mathematicians have sought to rediscover the proof behind it. This challenge obsessed minds for over three centuries, resulting in rivalries, rich prizes, duelling at dawn, suicide, and even transvestism – in the eighteenth century Sophie Germain was forced to take on the identity of man in order to conduct her research in an era of discrimination toward women academics. Where all others had failed, Wiles succeeded.

Part of the attraction for Wiles, Germain and the others was that mathematics, more than any other subject, is timeless. The Cambridge professor Godfrey Harold Hardy wrote in 1940: “Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. Immortality may be a silly word, but probably a mathematician has the best chance of whatever it may mean.” Hardy was referring to the fact that mathematics relies solely on absolute, undeniable, logical proof, and therefore theorems, once proved, remain true forever.

On the other hand, scientific theories rely on fallible experiments, and as such are only probably true at best. For centuries the scientific establishment accepted Isaac Newton’s theory of gravity, but in the twentieth century the work of Albert Einstein showed that it was only an approximation to the truth – General Relativity has since become the dominant theory. Although Einstein’s theories appear to be true today, there is no guarantee that they will be considered true a century from now. However, Wiles’ proof of Fermat’s Last Theorem is valid today, and it will remain so until the end of time. Pythagoras’ theories about medicine and astronomy are no longer taken seriously, whereas Pythagoras’ Theorem regarding right-angled triangles is still considered a basic mathematical truth.

If nothing else, perhaps this current interest in mathematics will go some way toward destroying the fear and prejudice so many people feel toward such a noble pursuit. In particular, this terror of numbers exists amongst English youngsters, who still struggle compared to the rest of the world, coming 17th out of 26 countries in recent mathematical tests. The top four places were taken by countries in the Far East, but then there are a whole host of other Western countries before the Scots appear, one place above the even less numerate English.

People fear maths because it dares us to examine an abstract world full of bizarre geometries, unimaginable numbers, and, most terrifying of all, infinity. These entities only exist in the the mind of the mathematician. Even the most mundane of all numbers, “1”, is an abstract concept – you can hold a piece of paper with a symbol representing “1” written upon it, you can hold “1” pebble, but you cannot hold “1” itself. Mathematics is a deeply philosophical pursuit, and as such it, of all the sciences, is closest to the arts. The abstraction of mathematics should not be an excuse to shun it, but rather a reason to embrace it. Perhaps in the years to come, the gawky sixth former who studies A’ level maths will no longer be perceived as the class wimp, but rather as the heroic explorer venturing into an unknown world.