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Fermat's Last Theorem – how it’s going

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Re: Fermat's Last Theorem – how it’s going

#41

Post by bell-cot »

boothby wrote: post #27

This reminds me of a fun experience I had in grad school. I was working on writing some fast code to compute something I can no longer explain, to help my advisor in his computational approach to the Birch and Swinnerton-Dyer conjecture. I gave a talk at a number theory seminar a few towns over, and was asked if I was doing this in hopes of reinforcing the evidence behind the conjecture. I said with a grin, "well, no…

> The crowd went wild; I've never made a group of experts so angry as...

Also not a number theorist...but I'd bet those so-called experts had invested far, far too many of their man-years in that unproven conjecture. All of which effort and edifice would collapse into the dumpster if some snot-nosed little upstart like you, using crude computation, achieved overnight fame by finding a counter-example.

(If I could give my many-decades-ago younger self some advice for math grad school, one bit of that would be: For any non-trivial "Prove X" assignment, start by spending at least 1/4 of my time budget looking for counter-examples. For academic assignments, that's 99% likely to fail. But the insight you'll get into the problem by trying will be more worth it. And the other 1% of the time you'll look like a genius. And - as soon as you attempt real math research, those odds shift enormously, in favor of the counterexample-first approach.)

Re: Fermat's Last Theorem – how it’s going

#42

Post by TechDebtDevin »

Etheryte wrote: post #13
bee_rider wrote: post #7

This hurts my engineer-brain. Just a reminder than the gap between us and mathematicians is about the same as the gap between us and everybody else (edit: in terms of math skills), just in the other direction, haha. Oh well, hopefully when they get the machines to solve math, they’ll still want them to run a couple percents faster every year.

[flagged]

I don't even understand how you interpreted that to have anything to do with being "smarter or making more money". Get some more coffee dude.

Re: Fermat's Last Theorem – how it’s going

#43

Post by moomin »

I remember when I was a student a friend of mine telling me this guy was giving a seminar and he'd just completed day one and everyone was really excited he was going to prove FLT. Of course, the guy in question was Andrew Wiles. He then spent months patching up problems they found prior to publication and finally the whole thing got published. It was a hugely exciting thing when you were studying mathematics.

All of which is a long way of saying the line "the old-fashioned 1990s proof" makes me feel _really_ old.

Re: Fermat's Last Theorem – how it’s going

#44

Post by hyperthesis »

vouaobrasil wrote: post #24

> The experts are in agreement that the important ideas are robust enough to withstand knocks like this, but the details of what is actually going on might not actually be where you expect them to be. Past researcher in pure math here. The big problem is that mathematicians are notorious for not providing self-contained proofs of anything because there is no incentive to do so and authors sometimes even seem proud to…

There'a a story that when someone wrote up a famous mathematician's work (Euler?), he found many errors, some quite serious. But all the theorems were true anyway.

Sounds like Tao's third stage, of informed intuition.

Re: Fermat's Last Theorem – how it’s going

#45

Post by isotypic »

bell-cot wrote: post #41
boothby wrote: post #27

This reminds me of a fun experience I had in grad school. I was working on writing some fast code to compute something I can no longer explain, to help my advisor in his computational approach to the Birch and Swinnerton-Dyer conjecture. I gave a talk at a number theory seminar a few towns over, and was asked if I was doing this in hopes of reinforcing the evidence behind the conjecture. I said with a grin, "well, no…

> The crowd went wild; I've never made a group of experts so angry as... Also not a number theorist...but I'd bet those so-called experts had invested far, far too many of their man-years in that unproven conjecture. All of which effort and edifice would collapse into the dumpster if some snot-nosed little upstart like you, using crude computation , achieved overnight fame by finding a counter-example. (If I could gi…

> All of which effort and edifice would collapse into the dumpster

Except it wouldn't, because the work towards the BSD would still be right and applicable to other problems. If someone proved the Riemann hypothesis false, all of our math (and there is a lot of it) surrounding the problem isn't immediately made worthless. The same is true for any mathematical conjecture.

I don't doubt the rest of your comment might have played a role, however.

Re: Fermat's Last Theorem – how it’s going

#46

Post by tux3 »

bell-cot wrote: post #41
boothby wrote: post #27

This reminds me of a fun experience I had in grad school. I was working on writing some fast code to compute something I can no longer explain, to help my advisor in his computational approach to the Birch and Swinnerton-Dyer conjecture. I gave a talk at a number theory seminar a few towns over, and was asked if I was doing this in hopes of reinforcing the evidence behind the conjecture. I said with a grin, "well, no…

> The crowd went wild; I've never made a group of experts so angry as... Also not a number theorist...but I'd bet those so-called experts had invested far, far too many of their man-years in that unproven conjecture. All of which effort and edifice would collapse into the dumpster if some snot-nosed little upstart like you, using crude computation , achieved overnight fame by finding a counter-example. (If I could gi…

Then again it might just be a misunderstanding. And the so-called experts, in aggregate, turn out to be right more often than not.

Say I show up to a physics conference and proclaim that I hope my computational effort will disprove some major physical law. Well, you better have a good delivery, or the joke might not land well!

Sometimes people take things a little too literally after attending hours of very dry talks at serious seminars. I wouldn't read too much into it.

Re: Fermat's Last Theorem – how it’s going

#47

Post by moomin »

brobdingnagians wrote: post #38
uffjedn wrote: post #36

Earlier quoted context omitted.

Studied math a long time ago and one of my profs was proud about not going into the details. He said "Once you did something 100 times, you can go and say -as easily observable- and move on."

I loved that some of the proof solutions in the back of mathematical logic book said, "Observe that ..." as the start of the proof. Our little study group definitely did not see how what followed was an observation we would have made.

Baker's little book of number theory made me work to get through every page...

Re: Fermat's Last Theorem – how it’s going

#48

Post by impendia »

vouaobrasil wrote: post #24

> The experts are in agreement that the important ideas are robust enough to withstand knocks like this, but the details of what is actually going on might not actually be where you expect them to be. Past researcher in pure math here. The big problem is that mathematicians are notorious for not providing self-contained proofs of anything because there is no incentive to do so and authors sometimes even seem proud to…

Speaking as a current researcher in pure math -- you're right, but I don't think this is easily resolved.

Math research papers are written for other specialists in the field. Sometimes too few details are provided; indeed I commonly gripe about this when asked to do peer review; but to truly provide all the details would make papers far, far longer.

Here is an elementary example, which could be worked out by anyone with a good background in high school math. Prove the following statement: there exist constants C, X > 0 such that for every real number x > X, we have

log(x^2 + 1) + sqrt(x) + x/exp(sqrt(4x + 3)) i.e. the left-side is big-O of the right.

This flavor of statement comes up all the time in analytic number theory (my subject), would be regarded as obvious to any specialist, and in a paper would be invariably stated without proof.

To come up with a complete, rigorous proof would be long, and not interesting. No professional would want to read one.

I agree this attitude comes with some tradeoffs, but they do seem manageable.

Re: Fermat's Last Theorem – how it’s going

#49

Post by mostly_a_lurker »

bell-cot wrote: post #41
boothby wrote: post #27

This reminds me of a fun experience I had in grad school. I was working on writing some fast code to compute something I can no longer explain, to help my advisor in his computational approach to the Birch and Swinnerton-Dyer conjecture. I gave a talk at a number theory seminar a few towns over, and was asked if I was doing this in hopes of reinforcing the evidence behind the conjecture. I said with a grin, "well, no…

> The crowd went wild; I've never made a group of experts so angry as... Also not a number theorist...but I'd bet those so-called experts had invested far, far too many of their man-years in that unproven conjecture. All of which effort and edifice would collapse into the dumpster if some snot-nosed little upstart like you, using crude computation , achieved overnight fame by finding a counter-example. (If I could gi…

> Also not a number theorist...but I'd bet those so-called experts had invested far, far too many of their man-years in that unproven conjecture. All of which effort and edifice would collapse into the dumpster if some snot-nosed little upstart like you, using crude computation, achieved overnight fame by finding a counter-example.

Are you maybe confusing math academia for psychology or social sciences? There is no replication crisis in math, no house of cards of self-proclaimed experts riding on bullshit. Mathematicians are _actually experts_ at a deep and extremely rigorous technical field -- many of them are even experts at computational approaches to problems! -- and when outsiders and upstarts resolve old conjectures, mathematicians generally react by celebrating them and showering them with fame, job offers and gushing articles in Quanta.

Re: Fermat's Last Theorem – how it’s going

#50

Post by jmpetroske »

ur-whale wrote: post #26
jmpetroske wrote: post #22

Earlier quoted context omitted.

The reason Hilbert failed is because it’s mathematically impossible (Gödels incompleteness theorem).

> because it’s mathematically impossible I guess the Lean devs can all pack and go home then. Also: how could they possibly miss Gödel's theorem, how careless of them.

Look, you’re taking my response out of context. I’m just stating that the reason Hilbert failed is not that he didn’t have a computer. Lean devs aren’t trying to do what Hilbert was.
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