10 ms·

Mochizuki's proof of the ABC conjecture accepted for publication

116 points by fovc 7y ago

avip 7y ago
The first comment in https://news.ycombinator.com/item?id=15971802 https://news.ycombinator.com/item?id=15971802 (one of many HN posts discussing this ongoing situation) is intriguing.
- westoncb 7y ago
  From Terry Tao's comment: > 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. This seems like the crux of the controversy: what is the true value in the 300+ "pages of set up"? Clearly he's providing a new framing for the problem. If that new framing ends up being applicable to other problems (which was presumably the author's intent), then the "pages of set up" are the true value here, not the proof of the abc conjecture itself. When Tao says, "no smaller fragment of this setup having any non-trivial external consequence whatsoever"—this is a comment about the state of things so far—but whether that will change is unknown. Maybe all of that "set up" ends up having no more general utility whatsoever, or maybe we shouldn't even expect to have found external consequences yet: very fundamental re-framings may be very disconnected from applications. Maybe Mochizuki has found one path from the new framing back into an area of contemporary mathematical interest, and maybe further exploration will yield an abundance of new paths—maybe some highways. Anyway just playing devil's advocate since the prevailing stance on this seems to be against Mochizuki in a way that feels odd to me.
gizmondo 7y ago
> Acceptance of the work in Publications of the Research Institute for Mathematical Sciences (RIMS) — a journal of which Mochizuki is chief editor, published by the institute where he works at Kyoto University That's exceptionally bad optics.
- timkam 7y ago
  I agree, not necessarily because publishing in a journal one is involved in is uncommon, but rather because putting one's work there although this work is disputed in the community clearly signals that one has not managed to build consensus and does not want/is not able to fight the good fight. If it was an incremental addition to well-established work that hardly anyone in the community can dispute, it would be a different story.
  - cwzwarich 7y ago
    It would also be extremely strange to publish a result like a proof of the ABC conjecture in a third-tier journal where you are an editor (actually, the editor-in-chief) rather than a top-tier journal.
- battery_cowboy 7y ago
  From TFA: > Mathematicians often publish papers in journals where they are editors. As long as the authors recuse themselves from the peer-review process “such a case is not a violation of any rule, and is common”, says Hiraku Nakajima, a mathematician at the Kavli Institute for the Physics and Mathematics of the Universe in Tokyo formerly part of Publications of RIMS’s editorial board. Mehrmann confirms that this would not violate EMS guidelines. > Kashiwara said that Mochizuki had recused himself from the review process, and had not attended any of the editorial board meetings about the paper. The journal has previously published papers from other members of the journals’ editorial board, he said.
  - lisper 7y ago
    Speaking as someone who used to make a living by publishing (though not in mathematics): the problem with this is that even though you may have recused yourself, everyone who is still in the room knows that you will be passing judgement on their papers at some point in the future. So recusal does not entirely remove the potential for conflict of interest.
    - 7y ago
mellosouls 7y ago
TL;DR: obscure, long "proof" of major conjecture by respected mathematician to be published in a journal he is closely associated with despite several years of scepticism by the wider maths community his attempt is successful. Complicated somewhat by possible language and cultural barriers, and his perceived reluctance to fully engage with his critics or the maths world outside his home country. It's an interesting and odd story that has been rumbling on for the last few years.
- throwlaplace 7y ago
  There's no language barrier Mochizuki is completely fluent in English.
  - mellosouls 7y ago
    He seems to insist on only communicating his ideas in Japanese though.
    - throwlaplace 7y ago
      The paper is written in English and scholz for example traveled to Japan and communicated with him in English
      - mellosouls 7y ago
        Although if you look at his rather colourful webpage, the strong, almost defensive, warning that Japanese will be the language used (at a Japanese institution of course) doesn't detract from the anecdotal suggestions that have dogged the saga of a reluctance to engage more fully with the world beyond his local region. Scholze and Stix had to go to him. http://www.kurims.kyoto-u.ac.jp/~motizuki/students-english.html http://www.kurims.kyoto-u.ac.jp/~motizuki/students-english.h...
        throwlaplace 7y ago
        I'm not argusing that he's not cagey. Simply that there is no issues of a language barrier (instruction in Japanese at Japanese university is a complete red herring when talking about what language he engages with the international mathematical community in).
gautamcgoel 7y ago
Math student here. This whole process reflects very poorly on Mochizuki, IMO. The proper thing to do when claiming to have solved a major open problem is to make yourself available to the mathematical community, for example by giving lectures on your work. Releasing a preprint on your website and expecting other mathematicians to drop what they're doing and devote years of their time to understand your obscure paper just screams of arrogance. Also, publishing your work in a journal where you are the editor isn't a good look; it gives the impression of a conflict of interest. The critique by Scholze et al is the final nail in the coffin as far as I'm concerned.
- antonzabirko 7y ago
  Not even all that - it's that he hasn't addressed the issues in his paper.
- wolfgke 7y ago
  The whole situation is in my opinion much more complicated. Here are some texts from the internet concerning this: David Michael Roberts - A Crisis of Identification: https://inference-review.com/article/a-crisis-of-identification https://inference-review.com/article/a-crisis-of-identificat... Two Quora posts: https://www.quora.com/Did-Peter-Scholze-and-Jakob-Stix-really-find-a-serious-flaw-in-Shinichi-Mochizukis-proof-of-ABC-conjecture/answer/Xiaowei-Xu-1 https://www.quora.com/Did-Peter-Scholze-and-Jakob-Stix-reall... https://www.quora.com/What-do-you-think-about-Stix-and-Scholze-coming-out-and-saying-Mochizukis-proof-of-the-abc-conjecture-is-wrong/answer/Xiaowei-Xu-1 https://www.quora.com/What-do-you-think-about-Stix-and-Schol... Also relevant: https://thehighergeometer.wordpress.com/2019/01/18/taylor-dupuy-on-mochizukis-iutt-infamous-corollary-3-12/ https://thehighergeometer.wordpress.com/2019/01/18/taylor-du...
  - hyperpape 7y ago
    Thanks for this. Tao’s comment made quite an impression on me, but important for outsiders like me to remember that the facts on the ground can change as time goes on.
    - 7y ago
rkagerer 7y ago
Maybe try publishing it first, then see if it's newsworthy. This isn't something you hype like a iPhone.
- abnry 7y ago
  Add to it that it has been known for a long time that this paper would be published in this journal.
hn_throwaway_99 7y ago
> But one mathematician who prefers to be quoted anonymously says that editors and referees handling these papers might have been in a nearly impossible situation. “If the best mathematicians spend time trying to work out what’s going on and fail, how can one referee on his own have any chance?” I totally disagree. When someone is making extraordinary claims, the burden is on them to go the extra mile to explain their reasoning, none of which Mochizuki has done. The easy decision in this case should be to leave things at the current "default" state unless a higher burden of proof is met. I think it reflects especially poorly that when confronted with criticisms that Mochizuki just waved the criticisms away with what was basically a "you mere mortals misunderstood my greatness", without taking the effort to engage and explain himself. I'm not in the math community so could be misunderstanding, but that's certainly the sense I got reading this article.
- pixiemaster 7y ago
  > the burden is on them to go the extra mile to explain their reasoning that’s not how extraordinary math worked in the past
  - QuesnayJr 7y ago
    They go the extra mile to explain it to other experts. When Perelman produced his proof of the Poincare conjecture, he traveled all over the place explaining his result.
deleted 7y ago
[deleted]
confuseshrink 7y ago
I'm not a mathematician but coming from the software world if one guy wrote a massive program (I'm assuming 600 pages is massive) in "an impenetrable, idiosyncratic style" you could virtually guarantee it would not be correct.
- wolfgke 7y ago
  Surely not the best example, but Terry A. Davis' (RIP) TempleOS was written in a very idiosyncratic style (he even invented his own programming language (HolyC) for this).
  - voldacar 7y ago
    TempleOS is awesome Being able to mix text, images, 3d models, etc in a terminal session is something that modern OSes still haven't gotten around to doing. Terry may have been crazy but he was right about a lot of things and unlike many, he had the skills to implement his ideas
- AzzieElbab 7y ago
  Massive hard to read code makes up 99% of software in the real world
- lostmsu 7y ago
  Most programs are not correct. You'd have to refer to number of bugs.
- Barrin92 7y ago
  It definitely poses a challenge to the question of what a mathematical proof actually is. One of the things about mathematics is in a way its 'obviousness', there's a way in which once something is proven its intelligble to mathematicians in an immediate, direct sense. A 600 page proof that requires essentially a new branch of idiosyncratic mathematics which as an end result is barely understandable even by peers in the field almost moves it from mathematics into the realm of empirical science, where people are often for years occupied with interpretation of data and discussions about how significant a finding is. As mathematics moves on to tackle more and more complicated questions I think it's interesting to ask if there will be a push back against complicated solutions, focus on simplicity as integral to solving a mathematical problem, and so on.
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.
dang 7y ago
Previous threads on this: https://hn.algolia.com/?dateRange=all&page=0&prefix=true&query=comments%3E1%20abc%20conjecture&sort=byDate&type=story https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que... https://hn.algolia.com/?dateRange=all&page=0&prefix=true&query=comments%3E1%20mochizuki&sort=byDate&type=story https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...
- vosper 7y ago
  If anyone on iOS Safari gets a blank white page following these links: you may have to turn off your content blocker(s) (in my case Firefox Focus)
ur-whale 7y ago
Once again, this is a rather perfect example of why there is a great need in Math for a universal formal proof language that can be verified by computers. If this existed, the burden of proof would be on Mochizuki to present his proof in a language that can actually be understood by others and by machines.
- QuesnayJr 7y ago
  The problem is that producing a formal proof that can be verified by a computer is incredibly boring. Since it's incredibly boring, nobody is going to do it. It's not a matter of a universal formal proof language existing (there are lots of frameworks that are powerful enough), it's matter of one being easy enough to use that people will willingly use it.
  - ur-whale 7y ago
    > The problem is that producing a formal proof that can be verified by a computer is incredibly boring One can't help but wonder: as boring as reading through inter-universal teichmuller theory?
    - Grue3 7y ago
      Well, suppose you want to prove another result similar to abc conjecture. What would you rather read, a formal proof (basically low-level machine code), or an actual human description of a theory used to produce the result?
  - gravypod 7y ago
    What would need to exist to make such a proof specification utility less boring? I know many engineers who never thought they'd use unit tests until I was able to demonstrate immediate workflow benefit that made their job far easier than before. If you have experience in higher math, what would need to exist to make such specification useful?
    - QuesnayJr 7y ago
jesuslop 7y ago
Had we a slick method of formalizing proofs for computer verification, no polemics would be possible.
- Koshkin 7y ago
  Good point; was it not possible in this case?
cronocr 7y ago
It seems mathematicians are not aware of code obfuscation. Mochizuki objective is to keep this knowledge in Japan and protect this technological advantage.
- papeda 7y ago
  If his objective is to "keep this knowledge in Japan", circulating the preprint and then publishing it in the journal he edits is a weird strategy.
  - cronocr 7y ago
    Weird, why? Since he can't keep this information from leaking around the world, the best strategy is to obfuscate it and show his trusted partners the concepts required to understand it. By gaining notoriety around the world, he is also attracting more partners, and he can filter them out (ie. he knows english, but will only do lectures in japanese, inside the country). If at some point the information gets fully decoded, he will be recognized as having solved the problem anyway. In the meanwhile there is a technological advantage to keep.
    - knolax 7y ago
      What is the technological advantage of proving the ABC conjecture? Seems like any application for it would comparatively take much more time to develop than whatever time advantage can be gained through obsfucation.
haecceity 7y ago
He should have written the proof in Lean or something.
- JulianWasTaken 7y ago
  As far (though little) as I know, that's way beyond feasible still -- lean has a ton I think, but is nowhere near covering the "seam" of mathematics -- many fundamental objects are not even defined themselves yet, so you'd have to do lots of extra work there, not to mention all the work to define your own new things. Lean is amazing (and I played around with it a bit, should do so more) but I don't suspect it's realistic to expect new theorems of this magnitude to be written in it at this point yet, doing so is a huge huge effort on top of the life-altering effort that the theorem itself requires.
akvadrako 7y ago
For a decent take in support of the publication, see this recent philosophy of mathematics paper: https://www.maths.nottingham.ac.uk/plp/pmzibf/rpp.pdf https://www.maths.nottingham.ac.uk/plp/pmzibf/rpp.pdf
doall 7y ago
This is Mochizuki's informal blog post in Japanese about the current situation. https://plaza.rakuten.co.jp/shinichi0329/diary/202001050000/ https://plaza.rakuten.co.jp/shinichi0329/diary/202001050000/ Although he had lived in the US for more than a decade and has no problem with the English language, he seem to have a kind of "western culture allergy" that is written in detail in the post below: https://plaza.rakuten.co.jp/shinichi0329/diary/201711210000/ https://plaza.rakuten.co.jp/shinichi0329/diary/201711210000/ I think the "allergy thing" is the reason he doesn't want to follow the ordinary "western approved way" and do a tour in the US. Also I have read somewhere that he is open to mathematical discussions via online or if you visit him in Japan.
peter_d_sherman 7y ago
Excerpt: "The saga began when Mochizuki, a respected number theorist quietly posted his preprints on 30 August 2012 — not on arXiv.org, mathematicians’ preferred repository, but on his own webpage at RIMS. Written in an impenetrable, idiosyncratic style, the papers seemed to entirely consist of mathematical concepts that were completely unfamiliar to the rest of the community — “like you might be reading a paper from the future, or from outer space”, wrote Jordan Ellenberg, a number theorist at the University of Wisconsin–Madison, on his blog soon after the papers appeared." ...Which makes it all the more a subject of curiousity -- and worth looking at...
credit_guy 7y ago
The top 2 math journals are, in many people's minds, Annals and Inventiones (i.e. Annals of Mathematics and Inventiones Mathematicae). The fact that this work, which is supposed to be of utmost importance, was not published in one of these two, is not confidence inspiring. That it was published in a journal where the author is chief editor is downright scandalous.

116 points by fovc 7y ago

avip 7y ago

The first comment in https://news.ycombinator.com/item?id=15971802 https://news.ycombinator.com/item?id=15971802 (one of many HN posts discussing this ongoing situation) is intriguing.

westoncb 7y ago
From Terry Tao's comment: > 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. This seems like the crux of the controversy: what is the true value in the 300+ "pages of set up"? Clearly he's providing a new framing for the problem. If that new framing ends up being applicable to other problems (which was presumably the author's intent), then the "pages of set up" are the true value here, not the proof of the abc conjecture itself. When Tao says, "no smaller fragment of this setup having any non-trivial external consequence whatsoever"—this is a comment about the state of things so far—but whether that will change is unknown. Maybe all of that "set up" ends up having no more general utility whatsoever, or maybe we shouldn't even expect to have found external consequences yet: very fundamental re-framings may be very disconnected from applications. Maybe Mochizuki has found one path from the new framing back into an area of contemporary mathematical interest, and maybe further exploration will yield an abundance of new paths—maybe some highways. Anyway just playing devil's advocate since the prevailing stance on this seems to be against Mochizuki in a way that feels odd to me.

gizmondo 7y ago

> Acceptance of the work in Publications of the Research Institute for Mathematical Sciences (RIMS) — a journal of which Mochizuki is chief editor, published by the institute where he works at Kyoto University That's exceptionally bad optics.

timkam 7y ago
I agree, not necessarily because publishing in a journal one is involved in is uncommon, but rather because putting one's work there although this work is disputed in the community clearly signals that one has not managed to build consensus and does not want/is not able to fight the good fight. If it was an incremental addition to well-established work that hardly anyone in the community can dispute, it would be a different story.
- cwzwarich 7y ago
  It would also be extremely strange to publish a result like a proof of the ABC conjecture in a third-tier journal where you are an editor (actually, the editor-in-chief) rather than a top-tier journal.
battery_cowboy 7y ago
From TFA: > Mathematicians often publish papers in journals where they are editors. As long as the authors recuse themselves from the peer-review process “such a case is not a violation of any rule, and is common”, says Hiraku Nakajima, a mathematician at the Kavli Institute for the Physics and Mathematics of the Universe in Tokyo formerly part of Publications of RIMS’s editorial board. Mehrmann confirms that this would not violate EMS guidelines. > Kashiwara said that Mochizuki had recused himself from the review process, and had not attended any of the editorial board meetings about the paper. The journal has previously published papers from other members of the journals’ editorial board, he said.
- lisper 7y ago
  Speaking as someone who used to make a living by publishing (though not in mathematics): the problem with this is that even though you may have recused yourself, everyone who is still in the room knows that you will be passing judgement on their papers at some point in the future. So recusal does not entirely remove the potential for conflict of interest.
  - 7y ago

mellosouls 7y ago

TL;DR: obscure, long "proof" of major conjecture by respected mathematician to be published in a journal he is closely associated with despite several years of scepticism by the wider maths community his attempt is successful. Complicated somewhat by possible language and cultural barriers, and his perceived reluctance to fully engage with his critics or the maths world outside his home country. It's an interesting and odd story that has been rumbling on for the last few years.

throwlaplace 7y ago
There's no language barrier Mochizuki is completely fluent in English.
- mellosouls 7y ago
  He seems to insist on only communicating his ideas in Japanese though.
  - throwlaplace 7y ago
    The paper is written in English and scholz for example traveled to Japan and communicated with him in English
    - mellosouls 7y ago
      Although if you look at his rather colourful webpage, the strong, almost defensive, warning that Japanese will be the language used (at a Japanese institution of course) doesn't detract from the anecdotal suggestions that have dogged the saga of a reluctance to engage more fully with the world beyond his local region. Scholze and Stix had to go to him. http://www.kurims.kyoto-u.ac.jp/~motizuki/students-english.html http://www.kurims.kyoto-u.ac.jp/~motizuki/students-english.h...
      - throwlaplace 7y ago
        I'm not argusing that he's not cagey. Simply that there is no issues of a language barrier (instruction in Japanese at Japanese university is a complete red herring when talking about what language he engages with the international mathematical community in).

gautamcgoel 7y ago

Math student here. This whole process reflects very poorly on Mochizuki, IMO. The proper thing to do when claiming to have solved a major open problem is to make yourself available to the mathematical community, for example by giving lectures on your work. Releasing a preprint on your website and expecting other mathematicians to drop what they're doing and devote years of their time to understand your obscure paper just screams of arrogance. Also, publishing your work in a journal where you are the editor isn't a good look; it gives the impression of a conflict of interest. The critique by Scholze et al is the final nail in the coffin as far as I'm concerned.

antonzabirko 7y ago
Not even all that - it's that he hasn't addressed the issues in his paper.
wolfgke 7y ago
The whole situation is in my opinion much more complicated. Here are some texts from the internet concerning this: David Michael Roberts - A Crisis of Identification: https://inference-review.com/article/a-crisis-of-identification https://inference-review.com/article/a-crisis-of-identificat... Two Quora posts: https://www.quora.com/Did-Peter-Scholze-and-Jakob-Stix-really-find-a-serious-flaw-in-Shinichi-Mochizukis-proof-of-ABC-conjecture/answer/Xiaowei-Xu-1 https://www.quora.com/Did-Peter-Scholze-and-Jakob-Stix-reall... https://www.quora.com/What-do-you-think-about-Stix-and-Scholze-coming-out-and-saying-Mochizukis-proof-of-the-abc-conjecture-is-wrong/answer/Xiaowei-Xu-1 https://www.quora.com/What-do-you-think-about-Stix-and-Schol... Also relevant: https://thehighergeometer.wordpress.com/2019/01/18/taylor-dupuy-on-mochizukis-iutt-infamous-corollary-3-12/ https://thehighergeometer.wordpress.com/2019/01/18/taylor-du...
- hyperpape 7y ago
  Thanks for this. Tao’s comment made quite an impression on me, but important for outsiders like me to remember that the facts on the ground can change as time goes on.
  - 7y ago

rkagerer 7y ago

Maybe try publishing it first, then see if it's newsworthy. This isn't something you hype like a iPhone.

abnry 7y ago
Add to it that it has been known for a long time that this paper would be published in this journal.

hn_throwaway_99 7y ago

> But one mathematician who prefers to be quoted anonymously says that editors and referees handling these papers might have been in a nearly impossible situation. “If the best mathematicians spend time trying to work out what’s going on and fail, how can one referee on his own have any chance?” I totally disagree. When someone is making extraordinary claims, the burden is on them to go the extra mile to explain their reasoning, none of which Mochizuki has done. The easy decision in this case should be to leave things at the current "default" state unless a higher burden of proof is met. I think it reflects especially poorly that when confronted with criticisms that Mochizuki just waved the criticisms away with what was basically a "you mere mortals misunderstood my greatness", without taking the effort to engage and explain himself. I'm not in the math community so could be misunderstanding, but that's certainly the sense I got reading this article.

pixiemaster 7y ago
> the burden is on them to go the extra mile to explain their reasoning that’s not how extraordinary math worked in the past
- QuesnayJr 7y ago
  They go the extra mile to explain it to other experts. When Perelman produced his proof of the Poincare conjecture, he traveled all over the place explaining his result.

deleted 7y ago

[deleted]

confuseshrink 7y ago

I'm not a mathematician but coming from the software world if one guy wrote a massive program (I'm assuming 600 pages is massive) in "an impenetrable, idiosyncratic style" you could virtually guarantee it would not be correct.

wolfgke 7y ago
Surely not the best example, but Terry A. Davis' (RIP) TempleOS was written in a very idiosyncratic style (he even invented his own programming language (HolyC) for this).
- voldacar 7y ago
  TempleOS is awesome Being able to mix text, images, 3d models, etc in a terminal session is something that modern OSes still haven't gotten around to doing. Terry may have been crazy but he was right about a lot of things and unlike many, he had the skills to implement his ideas
AzzieElbab 7y ago
Massive hard to read code makes up 99% of software in the real world
lostmsu 7y ago
Most programs are not correct. You'd have to refer to number of bugs.
Barrin92 7y ago
It definitely poses a challenge to the question of what a mathematical proof actually is. One of the things about mathematics is in a way its 'obviousness', there's a way in which once something is proven its intelligble to mathematicians in an immediate, direct sense. A 600 page proof that requires essentially a new branch of idiosyncratic mathematics which as an end result is barely understandable even by peers in the field almost moves it from mathematics into the realm of empirical science, where people are often for years occupied with interpretation of data and discussions about how significant a finding is. As mathematics moves on to tackle more and more complicated questions I think it's interesting to ask if there will be a push back against complicated solutions, focus on simplicity as integral to solving a mathematical problem, and so on.

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.

dang 7y ago

Previous threads on this: https://hn.algolia.com/?dateRange=all&page=0&prefix=true&query=comments%3E1%20abc%20conjecture&sort=byDate&type=story https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que... https://hn.algolia.com/?dateRange=all&page=0&prefix=true&query=comments%3E1%20mochizuki&sort=byDate&type=story https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...

vosper 7y ago
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ur-whale 7y ago

Once again, this is a rather perfect example of why there is a great need in Math for a universal formal proof language that can be verified by computers. If this existed, the burden of proof would be on Mochizuki to present his proof in a language that can actually be understood by others and by machines.

QuesnayJr 7y ago
The problem is that producing a formal proof that can be verified by a computer is incredibly boring. Since it's incredibly boring, nobody is going to do it. It's not a matter of a universal formal proof language existing (there are lots of frameworks that are powerful enough), it's matter of one being easy enough to use that people will willingly use it.
- ur-whale 7y ago
  > The problem is that producing a formal proof that can be verified by a computer is incredibly boring One can't help but wonder: as boring as reading through inter-universal teichmuller theory?
  - Grue3 7y ago
    Well, suppose you want to prove another result similar to abc conjecture. What would you rather read, a formal proof (basically low-level machine code), or an actual human description of a theory used to produce the result?
- gravypod 7y ago
  What would need to exist to make such a proof specification utility less boring? I know many engineers who never thought they'd use unit tests until I was able to demonstrate immediate workflow benefit that made their job far easier than before. If you have experience in higher math, what would need to exist to make such specification useful?
  - QuesnayJr 7y ago

jesuslop 7y ago

Had we a slick method of formalizing proofs for computer verification, no polemics would be possible.

Koshkin 7y ago
Good point; was it not possible in this case?

cronocr 7y ago

It seems mathematicians are not aware of code obfuscation. Mochizuki objective is to keep this knowledge in Japan and protect this technological advantage.

papeda 7y ago
If his objective is to "keep this knowledge in Japan", circulating the preprint and then publishing it in the journal he edits is a weird strategy.
- cronocr 7y ago
  Weird, why? Since he can't keep this information from leaking around the world, the best strategy is to obfuscate it and show his trusted partners the concepts required to understand it. By gaining notoriety around the world, he is also attracting more partners, and he can filter them out (ie. he knows english, but will only do lectures in japanese, inside the country). If at some point the information gets fully decoded, he will be recognized as having solved the problem anyway. In the meanwhile there is a technological advantage to keep.
  - knolax 7y ago
    What is the technological advantage of proving the ABC conjecture? Seems like any application for it would comparatively take much more time to develop than whatever time advantage can be gained through obsfucation.

haecceity 7y ago

He should have written the proof in Lean or something.

JulianWasTaken 7y ago
As far (though little) as I know, that's way beyond feasible still -- lean has a ton I think, but is nowhere near covering the "seam" of mathematics -- many fundamental objects are not even defined themselves yet, so you'd have to do lots of extra work there, not to mention all the work to define your own new things. Lean is amazing (and I played around with it a bit, should do so more) but I don't suspect it's realistic to expect new theorems of this magnitude to be written in it at this point yet, doing so is a huge huge effort on top of the life-altering effort that the theorem itself requires.

akvadrako 7y ago

For a decent take in support of the publication, see this recent philosophy of mathematics paper: https://www.maths.nottingham.ac.uk/plp/pmzibf/rpp.pdf https://www.maths.nottingham.ac.uk/plp/pmzibf/rpp.pdf

doall 7y ago

This is Mochizuki's informal blog post in Japanese about the current situation. https://plaza.rakuten.co.jp/shinichi0329/diary/202001050000/ https://plaza.rakuten.co.jp/shinichi0329/diary/202001050000/ Although he had lived in the US for more than a decade and has no problem with the English language, he seem to have a kind of "western culture allergy" that is written in detail in the post below: https://plaza.rakuten.co.jp/shinichi0329/diary/201711210000/ https://plaza.rakuten.co.jp/shinichi0329/diary/201711210000/ I think the "allergy thing" is the reason he doesn't want to follow the ordinary "western approved way" and do a tour in the US. Also I have read somewhere that he is open to mathematical discussions via online or if you visit him in Japan.

peter_d_sherman 7y ago

Excerpt: "The saga began when Mochizuki, a respected number theorist quietly posted his preprints on 30 August 2012 — not on arXiv.org, mathematicians’ preferred repository, but on his own webpage at RIMS. Written in an impenetrable, idiosyncratic style, the papers seemed to entirely consist of mathematical concepts that were completely unfamiliar to the rest of the community — “like you might be reading a paper from the future, or from outer space”, wrote Jordan Ellenberg, a number theorist at the University of Wisconsin–Madison, on his blog soon after the papers appeared." ...Which makes it all the more a subject of curiousity -- and worth looking at...

credit_guy 7y ago

The top 2 math journals are, in many people's minds, Annals and Inventiones (i.e. Annals of Mathematics and Inventiones Mathematicae). The fact that this work, which is supposed to be of utmost importance, was not published in one of these two, is not confidence inspiring. That it was published in a journal where the author is chief editor is downright scandalous.