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?
Fermat's Last Theorem Earns Andrew Wiles the Abel Prize
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Re: Fermat's Last Theorem Earns Andrew Wiles the Abel Prize
#22I 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.
Re: Fermat's Last Theorem Earns Andrew Wiles the Abel Prize
#23I 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.
> "travellers" I sincerely do not understand what was meant by this.
A travel writer or vacation journalist. An Idiot Abroad, etc.
Re: Fermat's Last Theorem Earns Andrew Wiles the Abel Prize
#24Well, 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.
Re: Fermat's Last Theorem Earns Andrew Wiles the Abel Prize
#25I 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 dunno, the prize might be a little overblown. All-in it seems like a pretty marginal improvement if you ask me.
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.
Re: Fermat's Last Theorem Earns Andrew Wiles the Abel Prize
#26Earlier quoted context omitted.
> "travellers" I sincerely do not understand what was meant by this.
A travel writer or vacation journalist. An Idiot Abroad, etc.
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")
Re: Fermat's Last Theorem Earns Andrew Wiles the Abel Prize
#27There's a really nice documentary on Andrew Wiles and Fermat's Last Theorem https://vimeo.com/18216532 It even gives a high level description of the methods used for those who are interested.
"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…
David Hilbert agreed:
"Good, he did not have enough imagination to become a mathematician".
— Hilbert's response upon hearing that one of his students had dropped out to study poetry.[11]
Re: Fermat's Last Theorem Earns Andrew Wiles the Abel Prize
#28Earlier quoted context omitted.
I dunno, the prize might be a little overblown. All-in it seems like a pretty marginal improvement if you ask me.
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.
"marginal improvement" was the key phrase
Re: Fermat's Last Theorem Earns Andrew Wiles the Abel Prize
#29I 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.
> in an age of professional lifestyle bloggers, "travellers" and other celebrities
Many of whom work equally hard at what they do, and bring a lot of joy to a lot of people.
Re: Fermat's Last Theorem Earns Andrew Wiles the Abel Prize
#30There's a really nice documentary on Andrew Wiles and Fermat's Last Theorem https://vimeo.com/18216532 It even gives a high level description of the methods used for those who are interested.
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.