When to Stop Counting

Aheroic story: Andrew Wiles, a Cambridge mathematician living in the United States, emerges after seven years of self-incarceration and paranoid secrecy from his Princeton attic, clutching two hundred pages of hieroglyphics. He is triumphant. He has cracked the most famous problem in number theory: Fermat’s Last Theorem, which has eluded some of the finest efforts of mathematicians for over three hundred years. He is beside himself with anticipated glory, but holds off, maintaining secrecy until the right moment. Serendipitously, a conference on exactly the branch of number theory covering his work is to take place in Cambridge. Choosing an unrevealing title, he lectures on three consecutive days. In the final minute of the final hour, he is able to utter the magic claim that the puzzle of Fermat’s celebrated theorem is now solved. Shocked silence and then a standing ovation from his astonished international colleagues. In the ruthlessly competitive world of research mathematics, Wiles has pulled off one of the most dramatic successes anyone could hope for.

As a sixth-former, I tried to prove Fermat’s Last Theorem. A little later, at university, I tried to prove Goldbach’s conjecture, disprove the twin-prime conjecture, generalise Riemann’s zeta function, settle the four-colour problem and solve various other enigmas, not least that of uncountable infinity, locked inside Cantor’s Continuum Hypothesis. I failed in all of these, but not being able to crack such puzzles only confirmed their attraction. Tackling them was self-confirmation not combat, making sure they were real, that I was kindred to all the other great names who’d tried and failed, that mathematics was as illustrious and difficult as it was said to be. Over the next ten years, working on my PhD and then teaching and researching, I spent hundreds of hours in silent rapport with the symbols that live in the depths. Sometimes, a whole day would be consumed as I paced and sat and scribbled and stared at my symbols, head filled with pictures, shapes, movements, patterns, connections on the edge of perception and language that I’d be trying to coax or bully into some stable, and ultimately logical/communicable, form. In those days, I found the meditations and imaginary journeys intensely satisfying.

This is presumably the kind of experience Andrew Wiles was having up there in his attic. What exactly was he doing? Anybody who has ever been fascinated, however briefly, by the extraordinary patterns numbers make has some idea. Number theory has been called the most ‘beautiful and treacherous’ of all the branches of mathematics: it is as easy to ask a profoundly difficult question as it is to demonstrate an elegant regularity. For example, one might observe that each even number is a sum of two primes (6 = 3 + 3, 8 = 3 + 5, 10 = 3 +7, 20 = 7 +13 and so on) and guess that all even numbers have this property. Nobody has found an even number which doesn’t have it, and nobody has proved that all numbers must have it; the hypothesis that they do, Goldbach’s conjecture, remains unresolved more than two centuries after its first appearance. Simple to state and understand, and seemingly impossible to settle, the example could be multiplied many times over.

Basically, number theory is theoretical arithmetic: it studies such things as the behaviour of prime numbers or integer solutions to equations. The pursuit is as old as mathematics: the Babylonians listed several sets of integer solutions to the famous equation we know as Pythagoras’ theorem, namely x² + y² = z² (putting 3, 4, 5, say, for x, y, z); the Greeks proved that the prime numbers were as unlimited as the numbers themselves. The subject revived in the 16th century, with the discovery and translation from Greek of Diophantus’ Arithmetica. One of the simpler problems Diophantus records asks for a method that would generate the triples of integers satisfying Pythagoras’ equation. Pierre de Fermat, a professional jurist, passionate part-time mathematician and author of many splendid number theorems, thought about the obvious extension of Diophantus’ question: could a cube, for example, 27 (= 3³) or 1000 (= 10³), be split into a sum of two cubes and, more generally, could any higher power be split into a sum of two whole numbers each of that power? In other words (or rather symbols), Fermat was interested in whether the equation xⁿ + yⁿ = zⁿ could have any integer solutions when the exponent n was 3 or more. He decided that it couldn’t, that no whole numbers could be found to replace x, y and z, which would make the equation true for any value of n other than 2. Notoriously, he didn’t give a proof. Instead, he apparently scribbled in the margin of his copy of Arithmetica that he had a proof but that it was too long to be written in the margin. That, at least, is the story: Fermat’s annotated copy of Diophantus has never been found.

Thus began the history of Fermat’s Last Theorem. Ever since, mathematicians have wondered whether Fermat really did have a proof. He certainly could not have had one of the kind Wiles constructed, and contemporary opinion is that he was mistaken in thinking he had one at all. Since the 17th century, many have attacked the problem, prizes have been offered for its solution, and thousands of false proofs have found their way onto the desks of eminent mathematicians. Why this particular problem should have attracted so much attention is not obvious. Doubtless, Fermat’s tantalising claim and its superficial resemblance to Pythagoras’ theorem played their part. In any event, the supposed theorem became the site of new theories invented to solve it which have had consequences far outside its orbit. To say more, one would need to explain the role of transformations of the complex plane as well as deeply technical (even for most mathematicians) connections between the two great rivers of arithmetic and geometry that run through mathematics. Until the 17th century these were independent of each other: a mutual autonomy that ended when Descartes’s co-ordinate geometry created a translation between curves in the plane and equations. After that, other joinings of the two – algebraic topology, geometry of numbers – ensured that problems about numbers, particularly ones like Fermat’s which involve integers satisfying equations, could never be separate from the spatio-visual insights of geometry/topology. Thus, Mordell’s famous conjecture from the Twenties, whose recent settlement was a stepping-stone to Wiles’s result, hypothesises a connection between the number of holes on a certain surface naturally connected to an equation and the question whether the equation has an infinity of solutions.

These are highly technical considerations whose explication for a non-specialist audience presents a serious challenge. It is to his credit that Amir Aczel makes an effort to provide just such an account. In his delightfully simple and brief book, part detective chase and part mathematical popularisation, he manages to convey not the substance of these connections (which is impossible) but the illusion of it – an outline of the path leading to Wiles’s achievement – for the mathematically illiterate. They will not, however, make much of Euler Systems, Horizontal Iwasawa Theory and the Shimura-Taniyama conjecture and, excluded from such arcana, may be left wondering about the whole business of proving theorems, and about the mysteriously powerful attraction these problems have. Surely, something more than symbolic game-playing is involved.

Certainly, mathematicians see themselves as frying much grander fish. As discoverers of an objective mathematical reality, they speak of truth, the kind of irrefutable truth that only rigorous logical justification can deliver. Once proved, a mathematical result, from the most trivial proposition of Euclid’s geometry to Fermat’s theorem, is true (must always have been true) and stays true for ever. From outside, mathematicians could seem like high priests of abstract thought, an impression that is confirmed when they are asked to explain the nature of the objects they study. What or where is a number? The answer that comes back is metaphysical. Numbers are not anything real: instances of them can be manifested but they themselves are outside time, space, the vicissitudes of material presence and any kind of historical change. They are ideal forms. Mathematicians, in other words, are latter-day Platonists. Nobody created numbers, nobody needs to ensure they exist or go on existing, they extend and always have extended, to infinity.

Platonism allows number theorists (perhaps its most ardent adherents) to believe that Fermat’s theorem, for example, is a transcendental fact. Take away the Platonism and the theorem is a less glamorous, more complicatedly earthbound, social/cultural event in history, part of a practice of thinking and writing which allows imaginary objects to be created by manipulating signs that invoke and control them. Both descriptions can be used of the same piece of mathematics; but they have different consequences. Because it sees mathematics as extra-human – pan-galactic no less – Platonism gives credibility to the idea of using it to communicate with aliens; an idea that, on the culturist reading, is ludicrous. Without Platonism’s imprimatur, we have no transcendental carrot that will make up for the years in the attic; little encouragement to Wiles’s long and devoted effort: non-believers don’t give their lives to build cathedrals.

The point here, as I can testify, is a question of belief. Like everyone, I was once told that numbers go on for ever, that they are ‘infinite’. After a further period of enculturation, learning how to manipulate their signs and so imagine all manner of infinite mathematical objects inside my head, I spent a dozen years proving theorems about them. It was enthralling while it lasted. At some point, however, I ceased to believe. Not so much in the whole business of proving theorems, though that, too, was beginning to pall, but in the idea of infinity. The reality and obviousness of it lost its transparency: belief in the infinite progression of numbers became opaque and unavailable. I shed the Platonic God’s-eye view, gave up playing enthralling games with infinity, and tried to imagine how numbers might look from inside this universe. I switched from theorem-proving believer to apostate – a renegade blowing the whistle on the metaphysical interpretation of numbers.

At the time this switch happened, computers were beginning their colonisation of the nervous system. One consequence of this is to make us reconceive what it means to iterate, reckon and count. Computers (outside the Platonic fantasies of mathematical logic’s Turing machines, with their infinitely long input tapes) are material devices: they require energy, are subject to error and to entropy, operate in real time and space. All of which means they are hostile to unrealisable idealisations, not least counting endlessly to ‘infinity’.

Meanwhile, for mathematicians in their offices and attics, it’s very much metaphysical business as usual: God is in his Platonic heaven, the integers are as classically ‘out there’ and endless as they ever were, and number theorists will no doubt continue to devote years of their lives to trying to reveal the transcendental mysteries of their presence. How long it can last, I don’t know, but if the transformative impact of computers and their material logic continues, I’d say the days of classical number theory and its picture of numbers as ideal entities are (as it were) numbered. But this is to advance into a speculative future in which the computer – the brainchild of mathematical logic – becomes the dog that wags the tail of pure mathematics.