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https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ https://galoisrepresentations.wordpress.com/2017/12/17/the-

by sajid 8y ago

https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ https://galoisrepresentations.wordpress.com/2017/12/17/the-a... Terence Tao and Peter Scholze (and others) join the discussion in the comments section.

avip 8y ago
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"
- bsder 8y ago
  > 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" This is a statement of "If this thing has no other application, it isn't worth my time to dig through what is likely an impenetrable pile of garbage." Basically, proof of the abc conjecture should have a bunch of follow on consequences. If those aren't materializing, that makes taking the time to understand the proof quite a bit less enticing (ie. the Bayesian prior on "true" goes down a lot).
- AnimalMuppet 8y ago
  I read it more as "This is probably wrong, because it differs in this important way from every other true groundbreaking proof."
- neuronexmachina 8y ago
  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.
  - sorokod 8y ago
    For the analogy, I'd replace 'unit-testable' with reusable
    - shoo 8y ago
      I'd argue for something stronger: both reusable and useful (outside of the initial application) and novel. It's possible to write structurally reusable components that are useless, i.e. they do a thing that no one wants or needs.
- currymj 8y ago
  I thought he was more suggesting that the proponents could bolster their case by using their weird mathematical machinery to prove some other, less daunting, results or otherwise show that it has more general uses. (Not asserting that this is impossible, just complaining that it hasn’t been done.)
- nabla9 8y ago
  I think Tao describes important metaheuristic. Important conjecture like ABC expressess some fundamental feature in the field. It's not just a arbitrary brainteaser. How to solve this one problem should give insights to many related problems. Thought experiment: Imagine a modern mathematician/time traveller who want's to impress Carl Friedrich Gauss by solving problems CFG can't solve, but at the same time the not wanting to give away any clues to modern mathematics. Traveler would need to devise complex and convoluted ways to proof things. Proofs would probably piss off GFG more than impress him.
- strikelaserclaw 8y ago
  Tao has always been a proponent of the fact that proving a theorem establishes powerful tools that might help mathematicians work on other problems, i think his comment is that this does not do that
- zakk 8y ago
  To me it sounds more like: I don't like the idea of this theory working, but I don't have a refutation. Honestly I am surprised of hearing such an argument from one of the world's best mathematicians.
  - Retra 8y ago
    Well, that's not what he said, and he still is one of the world's best mathematicians. So one wonders why you would take that surprise as indicative of a mistake on his part rather than on yours?
    - zakk 8y ago
      Well, in science there’s a simple rule: your substantiate your claims. A plausibility argument like Tao’s one is simply not an argument. To show that Mochizuchi is wrong you need to point out which equation of his work is wrong, which by the way is Peter Scholze’s plan. Tao by his own admission does not have the background to evaluate Mochizuchi’s theory and still gives this worthless plausibility argument, appealing to his knowledge of other unrelated proofs. I’m astonished.
      - proto-n 8y ago
        Except that he doesn't say it is incorrect, he says that "it seems bizarre" to him that seemingly Michozuki's paper doesn't have any uses beyond proving the ABC conjuncture. The rest of this comment is just explaining how it's usually not like this.
        zakk 8y ago
        When the world's most brilliant mathematician (or at least one of the most brilliant) says "it would be rather bizzarre if your argument worked" the implications are absolutely clear. As a scientist I would never make such a statement, moreso for a theory I admittedly don't understand. Tangentially, it's not Tao's case, but I have seen a lot of bullying in the academia along these lines... "I am not saying it's wrong, just very bizzarre", "I am not saying it's wrong, but my Ph.D. student worked on it for two years and couldn't solve it", "I am not saying it's wrong, actually I don't even understand the details, but please recheck everything..." etc...
guelo 8y ago
That discussion is from before Scholze & Stix wrote their criticism.
- p1esk 8y ago
  Scholze wrote it right there: https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/#comment-4619 https://galoisrepresentations.wordpress.com/2017/12/17/the-a...
  - deleted 8y ago
    [deleted]

by sajid 8y ago

avip 8y ago

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"

bsder 8y ago
> 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" This is a statement of "If this thing has no other application, it isn't worth my time to dig through what is likely an impenetrable pile of garbage." Basically, proof of the abc conjecture should have a bunch of follow on consequences. If those aren't materializing, that makes taking the time to understand the proof quite a bit less enticing (ie. the Bayesian prior on "true" goes down a lot).
AnimalMuppet 8y ago
I read it more as "This is probably wrong, because it differs in this important way from every other true groundbreaking proof."
neuronexmachina 8y ago
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.
- sorokod 8y ago
  For the analogy, I'd replace 'unit-testable' with reusable
  - shoo 8y ago
    I'd argue for something stronger: both reusable and useful (outside of the initial application) and novel. It's possible to write structurally reusable components that are useless, i.e. they do a thing that no one wants or needs.
currymj 8y ago
I thought he was more suggesting that the proponents could bolster their case by using their weird mathematical machinery to prove some other, less daunting, results or otherwise show that it has more general uses. (Not asserting that this is impossible, just complaining that it hasn’t been done.)
nabla9 8y ago
I think Tao describes important metaheuristic. Important conjecture like ABC expressess some fundamental feature in the field. It's not just a arbitrary brainteaser. How to solve this one problem should give insights to many related problems. Thought experiment: Imagine a modern mathematician/time traveller who want's to impress Carl Friedrich Gauss by solving problems CFG can't solve, but at the same time the not wanting to give away any clues to modern mathematics. Traveler would need to devise complex and convoluted ways to proof things. Proofs would probably piss off GFG more than impress him.
strikelaserclaw 8y ago
Tao has always been a proponent of the fact that proving a theorem establishes powerful tools that might help mathematicians work on other problems, i think his comment is that this does not do that
zakk 8y ago
To me it sounds more like: I don't like the idea of this theory working, but I don't have a refutation. Honestly I am surprised of hearing such an argument from one of the world's best mathematicians.
- Retra 8y ago
  Well, that's not what he said, and he still is one of the world's best mathematicians. So one wonders why you would take that surprise as indicative of a mistake on his part rather than on yours?
  - zakk 8y ago
    Well, in science there’s a simple rule: your substantiate your claims. A plausibility argument like Tao’s one is simply not an argument. To show that Mochizuchi is wrong you need to point out which equation of his work is wrong, which by the way is Peter Scholze’s plan. Tao by his own admission does not have the background to evaluate Mochizuchi’s theory and still gives this worthless plausibility argument, appealing to his knowledge of other unrelated proofs. I’m astonished.
    - proto-n 8y ago
      Except that he doesn't say it is incorrect, he says that "it seems bizarre" to him that seemingly Michozuki's paper doesn't have any uses beyond proving the ABC conjuncture. The rest of this comment is just explaining how it's usually not like this.
      - zakk 8y ago
        When the world's most brilliant mathematician (or at least one of the most brilliant) says "it would be rather bizzarre if your argument worked" the implications are absolutely clear. As a scientist I would never make such a statement, moreso for a theory I admittedly don't understand. Tangentially, it's not Tao's case, but I have seen a lot of bullying in the academia along these lines... "I am not saying it's wrong, just very bizzarre", "I am not saying it's wrong, but my Ph.D. student worked on it for two years and couldn't solve it", "I am not saying it's wrong, actually I don't even understand the details, but please recheck everything..." etc...

guelo 8y ago

That discussion is from before Scholze & Stix wrote their criticism.

p1esk 8y ago
Scholze wrote it right there: https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/#comment-4619 https://galoisrepresentations.wordpress.com/2017/12/17/the-a...
- deleted 8y ago
  [deleted]