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I remember an instructor explaining during the math olympiad training program: "A proof doesn't have to be in any particular format. A proof is an argument that

by lacker 8y ago

I remember an instructor explaining during the math olympiad training program: "A proof doesn't have to be in any particular format. A proof is an argument that can convince other mathematicians." At this point, it seems clear that we do not have a proof of the ABC conjecture. Perhaps someone will be able to improve this to make a real proof - after all, there were some errors in the first Wiles proof of Fermat's Last Theorem, but they ended up being minor errors that were fixed up once they were found. But what we have is not a proof.

westoncb 8y ago
It's interesting that this article doesn't touch on this aspect of the situation at all, but when reading about Mochizuki's proof in the past it was presented as clear that proving the ABC conjecture was one application of a new general theory—not that all of Mochizuki's innovations are tied specifically to that problem. I was given to understand that his work was more like Grothendieck's, where he's developing this entirely new, super general system. But you can imagine if Grothendieck for example had done all of his work for years and years in isolation and then presented Schemes, Topoi, and Motives to everyone all at once, using them in combination to prove something familiar, and expecting other mathematicians to just learn and understand them all in order to verify a single proof. I think in that case, folks would not have been immediately so interested in the new constructions Grothendieck had come up, viewing it initially as mountains of unnecessarily alien concepts built up just to give other mathematicians a hard time ;) (Not to identify the two mathematicians overly—I'm not sure how appropriate that really is—but I remember that being my impression when first digging into Mochizuki's work, and I haven't seen it mentioned here yet.)
- Sniffnoy 8y ago
  This is where Terry Tao's comment from December seems relevant: https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/#comment-4563 https://galoisrepresentations.wordpress.com/2017/12/17/the-a...
- emileokada 8y ago
  I believe the article explicitly steers away from this because nearly all the discourse on this proof has been precisely about this (c.f. remark Scholze that 'The whole discussion surrounding the proof had gotten “too sociological,”').
kstrauser 8y ago
I think you're exactly right. Mochizuki's work may or may not turn out to be sound, but in the mean time he's failed to communicate it to other experts, even when they've met with him in person for the specific goal of understanding his reasoning. It's not sufficient to have a great idea, or even to be right about it. You have to be able to explain it thoroughly enough that other qualified peers can understand your logic enough to weigh in on it. That seems not to have happened yet hear.
- fjsolwmv 8y ago
  It's sufficient for Mochizuki. Everyone else can decide for themselves what is sufficient for them.
  - busyant 8y ago
    > It's sufficient for Mochizuki. Everyone else can decide for themselves This sounds like a variant of the Turing test. Mochizuki and your pet dog both claim to have proven the ABC conjecture. The ABC conjecture has been proven when you can understand one proof but not the other.
    - falcor84 8y ago
      I love your idea in principle, but it would be very problematic, and very sensitive to the tester's prior experience. I think that many people would (un)naturally be more trustful of mathematically-looking squiggles than a dog's barks, while others, especially if they got burned by damn lies before, would inherently distrust mathematics. In any case, people are very prone to trying to complete gaps in their understanding of a speaker based on prior conceptions.
    - busyant 8y ago
      I can't edit my post, but I should have phrased it more pithily: Mochizuki and your pet dog both claim to have proven the ABC conjecture. The ABC conjecture has been proven when you can tell the difference between the two proofs.
  - 8y ago
konschubert 8y ago
And it's unlikely that there will be anything following from it if the article is right that it's basically "Trivialities to confuse the reader + an esoteric leap of faith"
ken47 8y ago
FYI, the proof has convinced other mathematicians.

by lacker 8y ago

westoncb 8y ago

It's interesting that this article doesn't touch on this aspect of the situation at all, but when reading about Mochizuki's proof in the past it was presented as clear that proving the ABC conjecture was one application of a new general theory—not that all of Mochizuki's innovations are tied specifically to that problem. I was given to understand that his work was more like Grothendieck's, where he's developing this entirely new, super general system. But you can imagine if Grothendieck for example had done all of his work for years and years in isolation and then presented Schemes, Topoi, and Motives to everyone all at once, using them in combination to prove something familiar, and expecting other mathematicians to just learn and understand them all in order to verify a single proof. I think in that case, folks would not have been immediately so interested in the new constructions Grothendieck had come up, viewing it initially as mountains of unnecessarily alien concepts built up just to give other mathematicians a hard time ;) (Not to identify the two mathematicians overly—I'm not sure how appropriate that really is—but I remember that being my impression when first digging into Mochizuki's work, and I haven't seen it mentioned here yet.)

Sniffnoy 8y ago
This is where Terry Tao's comment from December seems relevant: https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/#comment-4563 https://galoisrepresentations.wordpress.com/2017/12/17/the-a...
emileokada 8y ago
I believe the article explicitly steers away from this because nearly all the discourse on this proof has been precisely about this (c.f. remark Scholze that 'The whole discussion surrounding the proof had gotten “too sociological,”').

kstrauser 8y ago

I think you're exactly right. Mochizuki's work may or may not turn out to be sound, but in the mean time he's failed to communicate it to other experts, even when they've met with him in person for the specific goal of understanding his reasoning. It's not sufficient to have a great idea, or even to be right about it. You have to be able to explain it thoroughly enough that other qualified peers can understand your logic enough to weigh in on it. That seems not to have happened yet hear.

fjsolwmv 8y ago
It's sufficient for Mochizuki. Everyone else can decide for themselves what is sufficient for them.
- busyant 8y ago
  > It's sufficient for Mochizuki. Everyone else can decide for themselves This sounds like a variant of the Turing test. Mochizuki and your pet dog both claim to have proven the ABC conjecture. The ABC conjecture has been proven when you can understand one proof but not the other.
  - falcor84 8y ago
    I love your idea in principle, but it would be very problematic, and very sensitive to the tester's prior experience. I think that many people would (un)naturally be more trustful of mathematically-looking squiggles than a dog's barks, while others, especially if they got burned by damn lies before, would inherently distrust mathematics. In any case, people are very prone to trying to complete gaps in their understanding of a speaker based on prior conceptions.
  - busyant 8y ago
    I can't edit my post, but I should have phrased it more pithily: Mochizuki and your pet dog both claim to have proven the ABC conjecture. The ABC conjecture has been proven when you can tell the difference between the two proofs.
- 8y ago

konschubert 8y ago

And it's unlikely that there will be anything following from it if the article is right that it's basically "Trivialities to confuse the reader + an esoteric leap of faith"

ken47 8y ago

FYI, the proof has convinced other mathematicians.