I wonder if language is a factor in this at all? I know mathematics is written entirely in English but most likely discussions still happen in local languages. Could part of the proof be lost in translation?
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Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
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Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
#12I have a MSc in mathematics so I am by no means an expert in mathematics proofs but I get the gist of it. To me a proof is literally a logical argument that you can follow to "believe" that a theorem is true. I do worry how many people actually understand or verify mathematics proofs. How many people have actually read and verified Perelman's or Wiles' proofs (I'm only singling them out because they're famous, not because they're particularly sketchy). We seem to think that once a proof is published in a reputable journal then it is definitely true, but really we should only think "a small number of qualified people have read it and think it is fine". The foundations actually seem a bit shakier to me than people seem to appreciate. Perhaps machine-verified proofs will be the answer one day.
Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
#13So what would this imply over the larger body of Mochizukis work? As far as I have understood the situation, the abc proof is more of a sideproduct of him developing this completely new system(?) for maths rather than the main thrust of the works. Is it all falling down like house of cards, or is there still workable and useful bits there?
Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
#14https://galoisrepresentations.wordpress.com/2017/12/17/the-a... Terence Tao and Peter Scholze (and others) join the discussion in the comments section.
Tao's commentary is especially interesting for me as a non-mathematician:
>It seems bizarre to me that there would be an entire self-contained theory whose only external application is to prove the abc conjecture after 300+ pages of set up, with no smaller fragment of this setup having any non-trivial external consequence whatsoever.
Likely such is the understatement-ish way of a mathematician to say "This is clearly wrong, but I have not the time to dig in and find where exactly"
Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
#15I have a MSc in mathematics so I am by no means an expert in mathematics proofs but I get the gist of it. To me a proof is literally a logical argument that you can follow to "believe" that a theorem is true. I do worry how many people actually understand or verify mathematics proofs. How many people have actually read and verified Perelman's or Wiles' proofs (I'm only singling them out because they're famous, not be…
> We seem to think that once a proof is published in a reputable journal then it is definitely true, but really we should only think "a small number of qualified people have read it and think it is fine". The foundations actually seem a bit shakier to me than people seem to appreciate.
I think you are understating the certainty of fundamental proofs like this.
Once such a fundamental proof is considered "true", it almost always has applications beyond what the original proof was used for. It is often used to try to prove things that we already know are "true" in other ways.
So, any "true" proof generally gets tested from multiple directions.
Wile's proof is a good example of this. It doesn't just prove Fermat's Last Theorem. Quoting Wikipedia: "Wiles' path to proving Fermat's Last Theorem, by way of proving the modularity theorem for the special case of semistable elliptic curves, established powerful modularity lifting techniques and opened up entire new approaches to numerous other problems."
Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
#16I think this would count as esoteric if you ignore the drama portion of it. Just to get to the point of understanding what they're talking about would probably take months of dedicated uninterrupted work. Neat. I have no idea where to even start. https://en.wikipedia.org/wiki/Anabelian_geometry
Years even. And really, that's kind of the issue at hand. This stuff is so difficult that very few mathematicians really understand it, and Mochizuki's work is so much more esoteric than that, that basically no one understands it, even recent Fields Medallists.
Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
#17https://galoisrepresentations.wordpress.com/2017/12/17/the-a... Terence Tao and Peter Scholze (and others) join the discussion in the comments section.
Tao's commentary is especially interesting for me as a non-mathematician: >It seems bizarre to me that there would be an entire self-contained theory whose only external application is to prove the abc conjecture after 300+ pages of set up, with no smaller fragment of this setup having any non-trivial external consequence whatsoever. Likely such is the understatement-ish way of a mathematician to say "This is clearly…
I read it more as "This is probably wrong, because it differs in this important way from every other true groundbreaking proof."
Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
#18https://galoisrepresentations.wordpress.com/2017/12/17/the-a... Terence Tao and Peter Scholze (and others) join the discussion in the comments section.
Tao's commentary is especially interesting for me as a non-mathematician: >It seems bizarre to me that there would be an entire self-contained theory whose only external application is to prove the abc conjecture after 300+ pages of set up, with no smaller fragment of this setup having any non-trivial external consequence whatsoever. Likely such is the understatement-ish way of a mathematician to say "This is clearly…
I think it might be more akin to a programmer complaining about a monolithic/black-box chunk of code that can't easily be split into understandable and/or unit-testable functions.
Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
#19I wonder if language is a factor in this at all? I know mathematics is written entirely in English but most likely discussions still happen in local languages. Could part of the proof be lost in translation?
It sounds like what is lost in translation is somebody's face and/or reputation, purely judging the presented situation socially. That is all we can do as non-mathematicians.
Re: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
#20I have a MSc in mathematics so I am by no means an expert in mathematics proofs but I get the gist of it. To me a proof is literally a logical argument that you can follow to "believe" that a theorem is true. I do worry how many people actually understand or verify mathematics proofs. How many people have actually read and verified Perelman's or Wiles' proofs (I'm only singling them out because they're famous, not be…
> We seem to think that once a proof is published in a reputable journal then it is definitely true, but really we should only think "a small number of qualified people have read it and think it is fine". The foundations actually seem a bit shakier to me than people seem to appreciate. I think you are understating the certainty of fundamental proofs like this. Once such a fundamental proof is considered "true", it al…
True, but even then, how large do you think the field is? And how many people verify these proofs?
I have a PhD in an esoteric field myself. There's literally <50 people worldwide who write/verify proofs in it. I've spotted errors in my own work or other's work that got past peer review. And I don't think I'm particularly special. There's just not enough eyeballs, time and incentives sometimes.