"He was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction. I tried to imitate him. But I've realized that it's very difficult to make good mistakes."
- Goro Shimura's profound description of how Yutaka Taniyama approached mathematics.
I never understood the amount of creativity and ingenuity that goes into good math until I learned some number theory and abstract algebra back in my college days, and read about the many math greats throughout history.
Thanks for the link to the documentary. This was well worth the time.
There's a striking moment at about 41:00 where Wiles tearfully describes his ultimate breakthrough. Mathematicians and others in the hard sciences are often perceived as cold and joyless, but this is a remarkable counterexample.
Exactly. Proving the theorem was his dream and kick started his passion for maths. This is a man who has achieved his childhood dream. That's no different from a singer who's achieved their childhood dream.
For many, I imagine maths is cold and joyless so they associate that with those who enjoy it. Nobody can see the internal experience, which probably is just as warm and joyful as the singer or writer who achieved their dreams.
I had the great fortune of being shown this documentary during highschool calculus class. Even though I didn't understand the details at all, it was my first glimpse into what "real math" (so to speak) was about.
I know he has other sources of income but the fact that this extremely smart and dedicated person, in an age of professional lifestyle bloggers, "travellers" and other celebrities, is rewarded for his efforts and can just go out and buy himself a brand new Ferrari with the prize money makes me smile.
I don't want to take anything away from anyone, but I did not mean writers/journalists, I meant people who travel the world on their parents' money and post photos of themselves on their voyages on instagram (labelling themselves as "travellers")
I don't know if you're being sarcastic, but Wiles had to first prove the Modularity theorem for a certain subset of elliptic curves, which lay the ground work for proving the Modularity theorem in general. This is one of the major achievements of modern mathematics.
Indeed. The curvature of my humor might have been too subtle. Easily mistakable for a troll. Luckily I'm often too late to threads--as you can see--to be subject to getting mobbed with down votes ;-)
Ha. I've read it as "Fermat's Last Theorem" - guess that's the UK name.
For anyone who hasn't read them Simon Singh's books are great introductions to subjects by tracking their history. In this case mathematics, but he also has one about cryptography [1] and another on the space and the universe [2].
Seconded, the subject matter is fascinating and the human story of Wiles' difficult journey and ultimate accomplishment is gripping. One of the books I've most enjoyed ever on any topic, let alone in popular science / maths. It's Singh's best work.
I have a question about these hundreds-of-pages long proofs. Who can check such long proofs and be confident about it? To study a document of this size and advanced content rigorously, you would need years - and even then, it is not certain that you didn't overlook something. How can I ever be confident in such proofs until they are checked in, say, Coq or something?
This is one of the reasons it takes a lot of training and talent to become a mathematician. If you wonder how mathematicians can do this when programmers write such buggy programs and the two processes seem so similar, I would note that I've found that mathematicians are remarkably careful, if slow, programmers and they make mistakes at a much lower rate than most programmers. If you consider moreover that mathematics is entirely free of many of the things that make programming hard – such as any kind of mutable state, input/output or concurrency – then you can see why proof may be a more reliable process than programming.
Another part of is that one spends a huge amount of time questioning and testing every single step of a proof. There's a slow back-and-forth process to proving things:
1. I think this is true – let me try to prove it.
2. I get stuck on something in the proof – let me try to construct a counterexample to the specific thing I'm currently trying to prove.
3. If I can't construct a counterexample, why? That reason itself is a proof of the point I was stuck on.
And so on, back-and-forth until the whole proof is done or you've managed to find some insane, devious counterexample. So by the time you've proved something, you've looked at it from every angle and considered every possibility. Mathematicians reading a proof do something very similar, but a bit more passively – it's still a very active process where you try to poke holes in every single claim and convince yourself that you can't before moving on. But a lot of the steps will be completely obvious to a trained mathematician – it's the sequence of them that's non-obvious, but once presented in order, the arc of reasoning becomes crystal clear and irrefutable.
In this case, each chapter of the proof was sent off to different people so they could spend time studying a smaller section of the proof. Famously a significant mistake was discovered during this review that took Wiles a year to fix.
I have the same questions when reading something like this. Expert verification seems more like an appeal to authority and intuition instead of a definite proof with formal rigorosity. Locally, there is always structure, but in pure mathematics even these steps require such an in depth understanding that trusting experts as an outsider seems to be impossible, it could as well be magic. As a programmer, I have a quite mechanistic understanding of the world, projects like http://us.metamath.org/index.html seem to make it much more understandable.
I'm wondering, if the longest path to the theorem "2 + 2 = 4" is 150 layers deep in this system, how deep would this proof be?
Huh, I thought I had read somewhere that a machine checked proof of Fermat's Last Theorem had already been created. I must have mixed it up with something else.
Well, the margin of Arithmetica is certainly too narrow to contain it. Can any mathematicians confirm that it is truly marvellous? Or is there another proof out there?
It's obviously impossible to know what Fermat actually had, but consensus has always been that Fermat may have had a proof in mind, but almost certainly not a correct one.
A few hundred years of brilliant minds throwing themselves at it didn't find anything simpler than Wiles' approach, and now that the first proof has been found it seems fairly unlikely that any serious science will get expended specifically towards finding another proof of the Theorem.
Lots of fresh crackpot non-proofs available, though, if you're into those! There are plenty of simply-stated problems in number theory that do not yield to straightforward proof, but Fermat hit marketing gold by saying that he had such a proof already. Who wouldn't want to think that a few years of the common core gives you at least the mathematical skills of a 17th century fancylad?
A common theory about Fermat's claimed "proof" is that he mistakenly assumed that the technique he used to prove the cases for n=3 and n=4 (he called it "infinite descent", basically proof by induction + contradiction) would generalise to higher n. You might broadly think of it like saying "I have a technique for finding prime numbers! You take any integer then double it and add one". Well, you get lucky for a few cases but the machinery isn't there at the back end. Long and short of it is that until some magnificent new areas of the science open up, Wiles' proof is probably the simplest we're likely to get; indeed it doesn't get much simpler than just citing it [Wiles 1995].
BTW there's a pretty great blog at http://fermatslasttheorem.blogspot.co.uk/ which tries to present Wiles' theorem in bite-size chunks, if you're interested in exploring whether you'd regard it as truly marvellous.
https://vimeo.com/18216532
It even gives a high level description of the methods used for those who are interested.