4 ms·

I mean, I’m enjoying the show, but this changes nothing. Very few people claim to understand the proof and some of the few that do think it’s wrong. Contrast w

by moomin 7y ago

I mean, I’m enjoying the show, but this changes nothing. Very few people claim to understand the proof and some of the few that do think it’s wrong. Contrast with Wiles, where they _did_ understand it, they _did_ find a gap in his proof and he fixed it to everyone’s satisfaction.

yodsanklai 7y ago
> Very few people claim to understand the proof I don't have any idea what this proof looks like and I can assume it's very complex. But, could it be modular enough that you can check the proof without understanding it? For instance, imagine the proof is made of 50 lemmas. One could check the main theorem derives from the 50 lemmas. And checking each individual lemma could be left to other mathematicians.
- the8472 7y ago
  > His series of papers, which total more than 500 pages, are written in an impenetrable style, and refer back to a further 500 pages or so of previous work by Mochizuki IANAM, but formalizing mathematical proofs that they can be machine-checked is hard, tedious and I guess doesn't get you much fame either. Doing this for such a monstrosity of proof must be an epic task. And as far as non-machine-checked analysis goes, other mathematicians have claimed that specific parts contain flaws (TFA mentions this) but the author disagrees.
- Someone 7y ago
  https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/ https://www.quantamagazine.org/titans-of-mathematics-clash-o...: Despite multiple conferences dedicated to explicating Mochizuki’s proof, number theorists have struggled to come to grips with its underlying ideas. His series of papers, which total more than 500 pages, are written in an impenetrable style, and refer back to a further 500 pages or so of previous work by Mochizuki, creating what one mathematician, Brian Conrad of Stanford University, has called “a sense of infinite regress.” It’s not possible to “check the main theorem derives from the 50 lemmas” if each of the lemmas uses terminology one doesn’t understand or even has never heard of.
  - throwaway_pdp09 7y ago
    I would expect the base axioms to be specified by the author of the paper. That is, by the guy who claims to understand it. After that the steps of the proof should be just turning the handle, and perhaps the distance between steps to be bridged automatically. Maybe you wouldn't understand what was going on and likely not understand the end product any better but at least by symbol manipulation alone you should be able to make a path and verify that start via steps arrives at conclusion Of course I am not a mathematician and haven't a clue but in principle it seems viable and if it isn't I'd really love to know why.
    - ttz 7y ago
      The problem is that the base "axioms" (really, definitions) he's working off of are basically alien language, and almost nobody understands why they are relevant. Add to that he is not saying "turn the crank like this", but is effectively saying "trust me when I say the cranks turn and out pops the answer, but if you dig into the machinery yourselves you'll see what I mean". On the other hand, there was a similar case with a German mathematician beforehand, and he turned out to be correct.
      - throwaway_pdp09 7y ago
        I don't buy that the base axioms are alien and it doesn't matter anyway. I can give you an alien language of logical axioms, and a series of steps, none of which a random person wouldn't be able to understand, but could still lead them through a series of steps, showing that the outcome could be derived from the input via the axioms. Now, if nobody could understand the axioms and steps of this but they still got a consistent outcome, mathematicians would still be interested if for no other reason than it was self-consistent. That alone would be a result. But if they had that, I suspect that would also give them a framework for understanding it. > "trust me when I say the cranks turn and out pops the answer, but if you dig into the machinery yourselves you'll see what I mean". automate that and there'll be no need to trust him.
        7y ago
- papeda 7y ago
  Not a mathematician, but I've kept up with this story as a non-mathematician. > For instance, imagine the proof is made of 50 lemmas. One could check the main theorem derives from the 50 lemmas. And checking each individual lemma could be left to other mathematicians. To my understanding, some mathematicians have gone ever further and checked the work themselves. Peter Scholze (a 32-year-old who won the Fields medal a couple of years ago and seems to be in the sweet spot of expertise and stamina to really audit this thing) has identified a specific result ("Corollary 3.12") that he says does not follow from the results leading up to it [1]. This is pretty weird because "Corollary" is usually reserved for things that are immediate consequences of previous results. Based on the featured article, Scholze's position hasn't changed. There is a specific objection and the two sides just...disagree. For those interested, you can look at the papers yourself if you want to. Here's the one containing the controversial result [2]. Theorem 3.11 is on p153, and the statement alone is 5 pages long. The proof is 3 lines and asserts that it's obvious from definitions and previous statements. Corollary 3.12 is on p173. As an outsider, I am amazed at Scholze's ability to work through all of this and identify a specific criticism. [1] https://inference-review.com/article/a-crisis-of-identification https://inference-review.com/article/a-crisis-of-identificat... [2] http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20...
- QuesnayJr 7y ago
  There are theorems like that. (The classification of finite simple groups, whose proof is famously thousands of pages long, is like that.) But Mochizuki's proof is not like that. He creates a lot of definitions, and it's hard to figure out what the content of the definitions are.
- geofft 7y ago
  The comment from "PS" and the reply from "BCnrd" on this blog post https://www.galoisrepresentations.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/#comment-638 https://www.galoisrepresentations.com/2017/12/17/the-abc-con... (which was linked elsewhere in this comment section) claim, more or less, that there are 49 lemmas which are presented in sufficient detail to be computer-checkable, and one lemma which is not presented in enough detail for either computers or (most) humans to check, but is stated to follow straightforwardly if you understand the overall argument made by the other 49 lemmas. There are a couple of people besides Mochizuki who say "Yes, it does follow immediately, it's pretty clear," and they haven't been able to communicate why it's pretty clear to anyone else. That was in 2017: in 2018, Peter Scholze and Jakob Stix spent a week with Mochizuki trying to understand the proof and published their claim that the proof is flawed, which focuses on that very same unclear portion. In their understanding, that portion of the proof claims something much weaker and uninteresting, and they haven't been convinced of the intended claim. http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf

by moomin 7y ago

yodsanklai 7y ago

> Very few people claim to understand the proof I don't have any idea what this proof looks like and I can assume it's very complex. But, could it be modular enough that you can check the proof without understanding it? For instance, imagine the proof is made of 50 lemmas. One could check the main theorem derives from the 50 lemmas. And checking each individual lemma could be left to other mathematicians.

the8472 7y ago
> His series of papers, which total more than 500 pages, are written in an impenetrable style, and refer back to a further 500 pages or so of previous work by Mochizuki IANAM, but formalizing mathematical proofs that they can be machine-checked is hard, tedious and I guess doesn't get you much fame either. Doing this for such a monstrosity of proof must be an epic task. And as far as non-machine-checked analysis goes, other mathematicians have claimed that specific parts contain flaws (TFA mentions this) but the author disagrees.
Someone 7y ago
https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/ https://www.quantamagazine.org/titans-of-mathematics-clash-o...: Despite multiple conferences dedicated to explicating Mochizuki’s proof, number theorists have struggled to come to grips with its underlying ideas. His series of papers, which total more than 500 pages, are written in an impenetrable style, and refer back to a further 500 pages or so of previous work by Mochizuki, creating what one mathematician, Brian Conrad of Stanford University, has called “a sense of infinite regress.” It’s not possible to “check the main theorem derives from the 50 lemmas” if each of the lemmas uses terminology one doesn’t understand or even has never heard of.
- throwaway_pdp09 7y ago
  I would expect the base axioms to be specified by the author of the paper. That is, by the guy who claims to understand it. After that the steps of the proof should be just turning the handle, and perhaps the distance between steps to be bridged automatically. Maybe you wouldn't understand what was going on and likely not understand the end product any better but at least by symbol manipulation alone you should be able to make a path and verify that start via steps arrives at conclusion Of course I am not a mathematician and haven't a clue but in principle it seems viable and if it isn't I'd really love to know why.
  - ttz 7y ago
    The problem is that the base "axioms" (really, definitions) he's working off of are basically alien language, and almost nobody understands why they are relevant. Add to that he is not saying "turn the crank like this", but is effectively saying "trust me when I say the cranks turn and out pops the answer, but if you dig into the machinery yourselves you'll see what I mean". On the other hand, there was a similar case with a German mathematician beforehand, and he turned out to be correct.
    - throwaway_pdp09 7y ago
      I don't buy that the base axioms are alien and it doesn't matter anyway. I can give you an alien language of logical axioms, and a series of steps, none of which a random person wouldn't be able to understand, but could still lead them through a series of steps, showing that the outcome could be derived from the input via the axioms. Now, if nobody could understand the axioms and steps of this but they still got a consistent outcome, mathematicians would still be interested if for no other reason than it was self-consistent. That alone would be a result. But if they had that, I suspect that would also give them a framework for understanding it. > "trust me when I say the cranks turn and out pops the answer, but if you dig into the machinery yourselves you'll see what I mean". automate that and there'll be no need to trust him.
      - 7y ago
papeda 7y ago
Not a mathematician, but I've kept up with this story as a non-mathematician. > For instance, imagine the proof is made of 50 lemmas. One could check the main theorem derives from the 50 lemmas. And checking each individual lemma could be left to other mathematicians. To my understanding, some mathematicians have gone ever further and checked the work themselves. Peter Scholze (a 32-year-old who won the Fields medal a couple of years ago and seems to be in the sweet spot of expertise and stamina to really audit this thing) has identified a specific result ("Corollary 3.12") that he says does not follow from the results leading up to it [1]. This is pretty weird because "Corollary" is usually reserved for things that are immediate consequences of previous results. Based on the featured article, Scholze's position hasn't changed. There is a specific objection and the two sides just...disagree. For those interested, you can look at the papers yourself if you want to. Here's the one containing the controversial result [2]. Theorem 3.11 is on p153, and the statement alone is 5 pages long. The proof is 3 lines and asserts that it's obvious from definitions and previous statements. Corollary 3.12 is on p173. As an outsider, I am amazed at Scholze's ability to work through all of this and identify a specific criticism. [1] https://inference-review.com/article/a-crisis-of-identification https://inference-review.com/article/a-crisis-of-identificat... [2] http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20...
QuesnayJr 7y ago
There are theorems like that. (The classification of finite simple groups, whose proof is famously thousands of pages long, is like that.) But Mochizuki's proof is not like that. He creates a lot of definitions, and it's hard to figure out what the content of the definitions are.
geofft 7y ago
The comment from "PS" and the reply from "BCnrd" on this blog post https://www.galoisrepresentations.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/#comment-638 https://www.galoisrepresentations.com/2017/12/17/the-abc-con... (which was linked elsewhere in this comment section) claim, more or less, that there are 49 lemmas which are presented in sufficient detail to be computer-checkable, and one lemma which is not presented in enough detail for either computers or (most) humans to check, but is stated to follow straightforwardly if you understand the overall argument made by the other 49 lemmas. There are a couple of people besides Mochizuki who say "Yes, it does follow immediately, it's pretty clear," and they haven't been able to communicate why it's pretty clear to anyone else. That was in 2017: in 2018, Peter Scholze and Jakob Stix spent a week with Mochizuki trying to understand the proof and published their claim that the proof is flawed, which focuses on that very same unclear portion. In their understanding, that portion of the proof claims something much weaker and uninteresting, and they haven't been convinced of the intended claim. http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf