Fermat's Last Theorem (37 page)

For Nick Katz, too, this was a tense period: ‘By October the only people who knew about the error were myself, Illusie, the other referees of other chapters and Andrew – in principle that was all. My attitude was that as a referee I was supposed to act in confidentiality. I certainly didn't feel that it was my business to discuss this matter with anyone except Andrew, so I just didn't say a word about it. I think externally he appeared normal but at this point he was keeping a secret from the world, and I think he must have been pretty uncomfortable about it. Andrew's attitude was that with just
another day he would solve it, but as the fall went on, and no manuscript was available, rumours started circulating that there was a problem.'

In particular, Ken Ribet, another of the referees, began to feel the pressure of keeping the secret: ‘For some completely accidental reason I became known as the “Fermat Information Service”. There was an initial article in the
New York Times
, where Andrew asked me to speak to the reporter in his place, and the article said, ‘Ribet who is acting as a spokesperson for Andrew Wiles …', or something to that effect. After that I became a magnet for all kinds of interest in Fermat's Last Theorem, both from inside and outside the mathematics community. People were calling from the press, from all around the world in fact, and also I gave a very large number of lectures over a period of two or three months. In these lectures I stressed what a magnificent achievement this was and I outlined the proof and I talked about the parts that I knew best, but after a while people started getting impatient and began asking awkward questions.

‘You see Wiles had made this very public announcement, but no one outside of the very small group of referees had seen a copy of the manuscript. So mathematicians were waiting for this manuscript that Andrew had promised a few weeks after the initial announcement in June. People said, “Okay, well this theorem has been announced – we'd like to see what's going on. What's he doing? Why don't we hear anything?” People were a little upset that they were being held in ignorance and they simply wanted to know what was going on. Then things got even worse because slowly this cloud gathered over the proof and people kept telling me about these rumours, which claimed there was a gap in
chapter 3
. They'd ask me what I knew about it, and I just didn't know what to say.'

With Wiles and the referees denying any knowledge of a gap, or at the very least refusing to comment, speculation began to run wild. In desperation mathematicians began sending e-mails to each other in the hope of getting to the bottom of the mystery.

In every tea-room of every mathematics department the gossip surrounding Wiles's proof escalated every day. In response to the rumours and the speculative e-mails some mathematicians tried to return a sense of calm to the community.

Despite the calls for calm, the e-mails continued unabated. As well as discussing the putative error, mathematicians were now arguing over the ethics of pre-empting the referees' announcement.

While the furore over his elusive proof was increasing, Wiles did his best to ignore the controversy and speculation. ‘I really shut myself off because I didn't want to know what people were saying about me. I just went into seclusion but periodically my colleague Peter Sarnak would say to me, “You know that there's a storm out there.” I'd listen, but, for myself, I really just wanted to cut myself off completely, just to focus completely on the problem.'

Peter Sarnak had joined the Princeton Mathematics Department at the same time as Wiles, and over the years they had become close friends. During this intense period of turmoil Sarnak was one of the few people in whom Wiles would confide. ‘Well, I never knew the exact details, but it was clear that he was trying to overcome this one serious issue. But every time he would fix this one part of the calculation, it would cause some other difficulty in another part of the proof. It was like he was trying to put a carpet in a room where the carpet might be bigger than the room. So Andrew could fit the carpet in any one corner, only to find that it would pop up in another corner. Whether you could or could not
fit the carpet in the room was not something he was able to decide. Mind you, even with the error, Andrew had made a giant step. Before him there was no one who had any approach to the Taniyama–Shimura conjecture, but now everybody got really excited because he showed us so many new ideas. They were fundamental, new things that nobody had considered before. So even if it couldn't be fixed this was a very major advance – but of course Fermat would still be unsolved.'

Eventually Wiles realised that he could not maintain his silence forever. The solution to the mistake was not just round the corner, and it was time to put an end to the speculation. After an autumn of dismal failure he sent the following e-mail to the mathematical bulletin board:

Few were convinced by Wiles's optimism. Almost six months had passed without the error being corrected, and there was no reason to think anything would change in the next six months. In any case, if he really could ‘finish this in the near future', then why bother issuing the e-mail? Why not just maintain the silence for a few more weeks and then release the finished manuscript? The February lecture course which he mentioned in his e-mail failed to give any of the promised detail, and the mathematical community suspected that Wiles was just trying to buy himself extra time.

The newspapers leapt on the story once again and mathematicians were reminded of Miyaoka's failed proof in 1988. History was repeating itself. Number theorists were now waiting for the next e-mail which would explain why the proof was irretrievably flawed. A handful of mathematicians had expressed doubts over the proof back in the summer, and now their pessimism seemed to have been justified. One story claims that Professor Alan Baker at the University of Cambridge offered to bet one hundred bottles of wine against a single bottle that the proof would be shown to be invalid within a year. Baker denies the anecdote, but proudly admits to having expressed a ‘healthy scepticism'.

Less than six months after his lecture at the Newton Institute
Wiles's proof was in tatters. The pleasure, passion and hope that carried him through the years of secret calculations were replaced with embarrassment and despair. He recalls how his childhood dream had become a nightmare: ‘The first seven years that I worked on this problem I enjoyed the private combat. No matter how hard it had been, no matter how insurmountable things seemed, I was engaged in my favourite problem. It was my childhood passion, I just couldn't put it down, I didn't want to leave it for a moment. Then I'd spoken about it publicly, and in speaking about it there was actually a certain sense of loss. It was a very mixed emotion. It was wonderful to see other people reacting to the proof, to see how the arguments could completely change the whole direction of mathematics, but at the same time I'd lost that personal quest. It was now open to the world and I no longer had this private dream which I was fulfilling. And then, after there was a problem with it, there were dozens, hundreds, thousands of people who wanted to distract me. Doing maths in that kind of rather overexposed way is certainly not my style and I didn't at all enjoy this very public way of doing it.'

Number theorists around the world empathised with Wiles's position. Ken Ribet had himself been through the same nightmare eight years earlier when he tried to prove the link between the Taniyama–Shimura conjecture and Fermat's Last Theorem. ‘I was giving a lecture about the proof at the Mathematical Sciences Research Institute in Berkeley and someone from the audience said, “Well, wait a minute, how do you know that such and such is true?” I responded immediately giving my reason and they said, “Well that doesn't apply in this situation.” I had an immediate terror. I kind of broke out into a sweat and I was very upset about it. Then I realised that there was only one possibility for justifying this, which was to go back to the fundamental work on the subject
and see exactly how it was done in a similar situation. I looked in the relevant paper and I saw that the method did indeed apply in my case, and within a day or two I had the thing all fixed up. In my next lecture I was able to give the justification. But you always live with this fear that if you announce something important, a fundamental mistake can be discovered.

‘When you find an error in a manuscript it can go two ways. Sometimes there's an immediate confidence and the proof can be resurrected with little difficulty. And sometimes there's the opposite. It's very disquieting, there's a sinking feeling when you realise that you've made a fundamental error and there's no way to repair it. It's possible that when a hole develops the theorem really just falls apart completely, because the more you try to patch it the more trouble you get into. But in Wiles's case each chapter of the proof was a significant article in its own right. The manuscript was seven years' work, it was basically several important papers pieced together and each one of the papers has a great deal of interest. The error occurred in one of the papers, in
chapter 3
, but even if you take out
chapter 3
what remained was absolutely wonderful.'

But without
chapter 3
there was no proof of the Taniyama–Shimura conjecture and therefore no proof of Fermat's Last Theorem. There was a sense of frustration in the mathematical community that the proof behind two great problems was in jeopardy. Moreover, after six months of waiting still nobody, beyond Wiles and the referees, had access to the manuscript. There was a growing clamour for more openness, so everyone could see for themselves the details of the error. The hope was that somebody somewhere might see something that Wiles had missed, and conjure up a calculation to fix the gap in the proof. Some mathematicians claimed that the proof was too valuable to be left in the hands of just one man. Number theorists had become the butt of jibes
from other mathematicians, who sarcastically questioned whether or not they understood the concept of proof. What should have been the proudest moment in the history of mathematics was turning into a joke.