Fermat's Last Theorem (22 page)

Decades later, in
Portraits from Memory
, Bertrand Russell reflected on his reaction to Gödel's discovery:

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest,
I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.

Although Gödel's second statement said that it was impossible to prove that the axioms were consistent, this did not necessarily mean that they were inconsistent. In their hearts many mathematicians still believed that their mathematics would remain consistent, but in their minds they could not prove it. Many years later the great number theorist Andre Weil would say: ‘God exists since mathematics is consistent, and the Devil exists since we cannot prove it.'

The proof of Gödel's theorems of undecidability is immensely complicated, and in fact a more rigorous statement of the First Theorem should be:

To every ω-consistent recursive class
k
of
formulae
there correspond recursive
class-signs r
, such that neither ν Gen
r
nor Neg(ν Gen
r
) belongs to Flg(
k
) (where ν is the
free variable
of
r
).

Fortunately, as with Russell's paradox and the tale of the librarian, Gödel's first theorem can be illustrated with another logical analogy due to Epimenides and known as the
Cretan paradox
, or
liar's paradox.
Epimenides was a Cretan who exclaimed:

‘I am a liar!'

The paradox arises when we try and determine whether this statement is true or false. First let us see what happens if we assume that the statement is true. A true statement implies that Epimenides is a liar, but we initially assumed that he made a true statement and therefore Epimenides is not a liar – we have an inconsistency. On
the other hand let us see what happens if we assume that the statement is false. A false statement implies that Epimenides is not a liar, but we initially assumed that he made a false statement and therefore Epimenides is a liar – we have another inconsistency. Whether we assume that the statement is true or false we end up with an inconsistency, and therefore the statement is neither true nor false.

Gödel reinterpreted the liar's paradox and introduced the concept of proof. The result was a statement along the following lines:

This statement does not have any proof.

If the statement were false then the statement would be provable, but this would contradict the statement. Therefore the statement must be true in order to avoid the contradiction. However, although the statement is true it cannot be proven, because this statement (which we now know to be true) says so.

Because Gödel could translate the above statement into mathematical notation, he was able to demonstrate that there existed statements in mathematics which are true but which could never be proven to be true, so-called undecidable statements. This was the death-blow for the Hilbert programme.

In many ways Gödel's work paralleled similar discoveries being made in quantum physics. Just four years before Gödel published his work on undecidability, the German physicist Werner Heisenberg uncovered the uncertainty principle. Just as there was a fundamental limit to what theorems mathematicians could prove, Heisenberg showed that there was a fundamental limit to what properties physicists could measure. For example, if they wanted to measure the exact position of an object, then they could measure the object's velocity with only relatively poor accuracy. This is because in order to measure the position of the object it
would be necessary to illuminate it with photons of light, but to pinpoint its exact locality the photons of light would have to have enormous energy. However, if the object is being bombarded by high-energy photons its own velocity will be affected and becomes inherently uncertain. Hence, by demanding knowledge of an object's position, physicists would have to give up some knowledge of its velocity.

Heisenberg's uncertainty principle only reveals itself at atomic scales, when high-precision measurements become critical. Therefore much of physics could carry on regardless while quantum physicists concerned themselves with profound questions about the limits of knowledge. The same was happening in the world of mathematics. While the logicians concerned themselves with a highly esoteric debate about undecidability, the rest of the mathematical community carried on regardless. Although Gödel had proved that there were some statements which could not be proven, there were plenty of statements which could be proven and his discovery did not invalidate anything proven in the past. Furthermore, many mathematicians believed that Gödel's undecidable statements would only be found in the most obscure and extreme regions of mathematics and might therefore never be encountered. After all Gödel had only said that these statements existed; he could not actually point to one. Then in 1963 Gödel's theoretical nightmare became a full-blooded reality.

Paul Cohen, a twenty-nine-year-old mathematician at Stanford University, developed a technique for testing whether or not a particular question is undecidable. The technique only works in a few very special cases, but he was nevertheless the first person to discover specific questions which were indeed undecidable. Having made his discovery Cohen immediately flew to Princeton, proof in hand, to have it verified by Gödel himself. Gödel, who by now was
entering a paranoid phase of his life, opened the door slightly, snatched the papers and slammed the door shut. Two days later Cohen received an invitation to tea at Gödel's house, a sign that the master had given the proof his stamp of authority. What was particularly dramatic was that some of these undecidable questions were central to mathematics. Ironically Cohen proved that one of the questions which David Hilbert declared to be among the twenty-three most important problems in mathematics, the continuum hypothesis, was undecidable.

Gödel's work, compounded by Cohen's undecidable statements, sent a disturbing message to all those mathematicians, professional and amateur, who were persisting in their attempts to prove Fermat's Last Theorem – perhaps Fermat's Last Theorem was undecidable! What if Pierre de Fermat had made a mistake when he claimed to have found a proof? If so, then there was the possibility that the Last Theorem was undecidable. Proving Fermat's Last Theorem might be more than just difficult, it might be impossible. If Fermat's Last Theorem were undecidable, then mathematicians had spent centuries in search of a proof that did not exist.

Curiously if Fermat's Last Theorem turned out to be undecidable, then this would imply that it must be true. The reason is as follows. The Last Theorem says that there are no whole number solutions to the equation

If the Last Theorem were in fact false, then it would be possible to prove this by identifying a solution (a counter-example). Therefore the Last Theorem would be decidable. Being false would be inconsistent with being undecidable. However, if the Last Theorem were true, there would not necessarily be such an unequivocal way
of proving it so, i.e. it could be undecidable. In conclusion, Fermat's Last Theorem might be true, but there may be no way of proving it.

Pierre de Fermat's casual jotting in the margin of Diophantus'
Arithmetica
had led to the most infuriating riddle in history. Despite three centuries of glorious failure and Gödel's suggestion that they might be hunting for a non-existent proof, some mathematicians continued to be attracted to the problem. The Last Theorem was a mathematical siren, luring geniuses towards it, only to dash their hopes. Any mathematician who got involved with Fermat's Last Theorem risked wasting their career, and yet whoever could make the crucial breakthrough would go down in history as having solved the world's most difficult problem.

Generations of mathematicians were obsessed with Fermat's Last Theorem for two reasons. First, there was the ruthless sense of one-upmanship. The Last Theorem was the ultimate test and whoever could prove it would succeed where Cauchy, Euler, Kummer, and countless others had failed. Just as Fermat himself took great pleasure in solving problems which baffled his contemporaries, whoever could prove the Last Theorem could enjoy the fact that they had solved a problem which had confounded the entire community of mathematicians for hundreds of years. Second, whoever could meet Fermat's challenge could enjoy the innocent satisfaction of solving a riddle. The delight derived from solving esoteric questions in number theory is not so different from the simple joy of tackling the trivial riddles of Sam Loyd. A mathematician once said to me that the pleasure he derived from solving mathematical
problems is similar to that gained by crossword addicts. Filling in the last clue of a particularly tough crossword is always a satisfying experience, but imagine the sense of achievement after spending years on a puzzle, which nobody else in the world has been able to solve, and then figuring out the solution.

These are the same reasons why Andrew Wiles became fascinated by Fermat: ‘Pure mathematicians just love a challenge. They love unsolved problems. When doing maths there's this great feeling. You start with a problem that just mystifies you. You can't understand it, it's so complicated, you just can't make head nor tail of it. But then when you finally resolve it, you have this incredible feeling of how beautiful it is, how it all fits together so elegantly. Most deceptive are the problems which look easy, and yet they turn out to be extremely intricate. Fermat is the most beautiful example of this. It just looked as though it had to have a solution and, of course, it's very special because Fermat said that he had a solution.'

Mathematics has its applications in science and technology, but that is not what drives mathematicians. They are inspired by the joy of discovery. G.H. Hardy tried to explain and justify his own career in a book entitled
A Mathematician's Apology
:

I will only say that if a chess problem is, in the crude sense, ‘useless', then that is equally true of most of the best mathematics … I have never done anything ‘useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about its value.

The desire for a solution to any mathematical problem is largely fired by curiosity, and the reward is the simple but enormous satisfaction derived from solving any riddle. The mathematician E.C. Titchmarsh once said: ‘It can be of no practical use to know that π is irrational, but if we can know, it surely would be intolerable not to know.'

In the case of Fermat's Last Theorem there was no shortage of curiosity. Gödel's work on undecidability had introduced an element of doubt as to whether the problem was soluble, but this was not enough to discourage the true Fermat fanatic. What was more dispiriting was the fact that by the 1930s mathematicians had exhausted all their techniques and had little else at their disposal. What was needed was a new tool, something that would raise mathematical morale. The Second World War was to provide just what was required – the greatest leap in calculating power since the invention of the slide-rule.

When in 1940 G.H. Hardy declared that the best mathematics is largely useless, he was quick to add that this was not necessarily a bad thing: ‘Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers.' Hardy was soon to be proved wrong.

In 1944 John von Neumann co-wrote the book
The Theory of Games and Economic Behavior
, in which he coined the term
game theory.
Game theory was von Neumann's attempt to use mathematics to describe the structure of games and how humans play them. He began by studying chess and poker, and then went on to try and model more sophisticated games such as economics. After the
Second World War the RAND corporation realised the potential of von Neumann's ideas and hired him to work on developing Cold War strategies. From that point on, mathematical game theory has become a basic tool for generals to test their military-strategies by treating battles as complex games of chess. A simple illustration of the application of game theory in battles is the story of the
truel.