Simon Singh and John Lynch: Fermat’s last theorem. Horizon (26 January 1996), BBC2
The Proof. NOVA (1997), PBS
This award-winning documentary tells us the story of Fermat’s last theorem, and how it was proved by Andrew Wiles.
It starts with a metaphor: Doing research in mathematics is like stumbling around in a dark mansion, bumping into all kinds of things that are in there. Step by step, one learns some of the things and where they are, until finally one finds the light switch, which illuminates everything. One can now see the exit of the room, leading to the next, still dark room, and the whole thing starts again.
We learn about the history of Fermat’s last theorem, and how even young students in school can understand it quite easily, starting from Pythagoras’ theorem. This theorem states that in any right triangle, the area of the square of the longest side is the sum of the areas of the squares of the two other sides, or, put in a formula, that a^2+b^2=c^2. There are infinitely many integer solution to this equation, and what Fermat’s last theorem claims is that the generalized equation a^n+b^n=c^n cannot be satisfied by any integers for n greater than two.
John Conway gives us a look at Fermat’s famous marginal note in which he claims that he had a proof for this theorem, but that the margin was too small to contain it. Generations of mathematicians were trying to find that proof, with no success. We see that a computer cannot solve this problem—it can only ever check finitely many solutions, but we would have to check infinitely many possible solutions to find out whether the equation really has no solutions. Instead, we need a mathematical proof of the theorem, for “knowing it with absolute certainty […] That’s what mathematics is about,” says Nick Katz.
We learn about the young Andrew Wiles, both from himself and from his supervisor. Proving Fermat’s Last Theorem was one of Wiles childhood dreams, but by the time Wiles was starting to do research work with his supervisor John Coates, nobody had any idea about how to tackle the problem. After many years of failed attempts to prove it, not only from professionals, but also from laymen, it was deemed as too inaccessible, and mathematicians were very skeptical about any attempted solution. So Coates told Wiles to better work on more mainstream maths, elliptic curves, mathematical structures that look like donuts.
Next, we learn from Barry Mazur about the seemingly different, very strange world of modular forms, and how the Tainyama–Shimura conjecture connects this world to the world of elliptic curves. Goro Shimura explains to us how he found this conjecture together with a friend, Yutaka Taniyama, who was very talented in speculating into the right direction. (Taniyama later committed suicide for no apparent reason, saying in a suicide note that it was a spontaneous idea that confused even himself.) The conjecture claims that every elliptic curve is modular. So if it is true, then we can translate all the ideas, intuitions and concepts from the elliptic curve world into the world of modular forms.
Now we see how much luck Wiles had: By studying elliptic curves, something the Tainyama–Shimura conjecture talks about, he had actually been working all the time on a topic that was related to Fermat’s Last Theorem in the most intimate manner. But that had not occurred to anyone before Gerhard Frey, a mathematician from Saarland University in Germany, conjectured that Fermat’s Last Theorem was actually a consequence of Tainyama–Shimura: If Fermat’s Last Theorem is false, and there is a solution to the equation, then we can construct an elliptic curve with some very strange properties—so strange that is seems not to be modular. Frey did not have a proof for this ‘epsilon conjecture’ himself, but Ken Ribet worked it out soon.
When Wiles heard in passing that this had been achieved, he immediately realized that attempting a proof was now realistic, by proving Taniyama–Shimura. We are told how he stopped working on everything but this problem. We see his working place at home, and how he was working on the problem there without a computer. An obvious way is to count elliptic curves and modular forms, and show that there is the same number of both. But that does not work; and Wiles’ crucial idea was to use Galois representations instead of elliptic curves. We see a lot of intricate details of his work on the proof (although still at a very high level of abstraction), and some of the difficulties he faced. After six years, he let Nick Katz and a little bit later Peter Sarnak know about what he was working on, and started a lecture on it, without letting attendants know that it was really about Fermat’s Last Theorem. The lecture was so hard to understand that after a few weeks, none of the students were left.
Soon, he was quite confident to have finished the proof and gave a series of lectures on a conference. Immediately, the press started making him a star.
But during the rigorous review of the proof for journal publication, a fundamental error turned up, having been missed in the earlier lectures. We see how complicated and difficult the problem was and how emotionally depressing it was for Wiles to fight it with all the public pressure.
Of course, he finally found the solution: By examining very carefully exactly where the problem was, and by trying to prove that the proof could not work, he finally had the idea of how it could be fixed. We see the deep emotions involved and how satisfying it was for Wiles to finally have things done, the ultimate triumph.
The documentary closes by noting that Fermat could not possibly have known this proof, because it depends on so many modern results from 20th century mathematics that it was impossible for him to figure them out all by himself. This is made very clear with an impressive list of all the people that contributed results that the proof depended on.
The documentary certainly has earned its awards, without question. It is a great achievement to make such a difficult topic available to the layman, while at the same time still being sufficiently accurate, bringing the emotions across and yet be entertaining. It is also a good portrait of the major figures of the Princeton maths department.
There are some minor issue thoughs: The documentary is a little bit crowded at the end, starting from the transition when it becomes clear that the first version of the proof is incorrect. Further, that transition itself is a little bit waggly and could have been done with better tension.
I dislike how they talk about Wiles doing a calculation. This is really a bit misleading and invokes the old misconception of mathematics being essentially about calculating things. It is even more puzzling given how they explain nicely that just letting some computer calculate is not an approach that works.
Wiles’ views seem a bit contradictory: His dark mansion metaphor presents mathematics as a static world covered by the fog of war. Step by step, we uncover this world, learn about some room, first some small part of it merely by stumbling around, then more and more, and finally we are enlightened, seeing everything, when we have found the light switch. On the other hand, he admits that “there’d been setbacks often, there’d been things that had seemed insurmountable.” This implies that not everything was just uncovered, but that there were failed attempts to uncover at least some of the things. It also implies that his work involved the goal-oriented attempt to solve certain (very hard) problems, not merely to stumble around rather randomly and crash into the solutions. All this might seem a little bit pedandic, but it really is significant from an epistemological point of view. A better version of the metaphor—not restricted to mathematics—is given by Karl Popper in his video interview with the open universiteit. He speaks of “a black man, who, in a black room, looks for a black hat—which may not be there”.
Katz’s philosophy that proofs give you absolute certainty seems to take the opposite view. It may be a classic view in the philosophy of mathematics, but just as any other claims of certainty, it contradicts the simple fact of logic that every deductive reasoning starts from assumptions—axioms (which includes definitions)—that are unproven. In fact, they may even be self-contradictory, and they also may not be adequate with respect to the background of the problems against which they were put forward. Sure, logic may be valid, and so may be a proof, and axioms may be true, consistent and adequate, and so may be a theorem. But for none of these attributes, there is any source of certainty.
On the other hand, Taniyama’s approach, as given by Shimura’s tongue-in-cheek description, seems entirely valid: “Taniyama was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction. So eventually, he got right answers. And I tried to imitate him. But I found out that it is very difficult to make good mistakes.”