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.
> 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
> 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.
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.
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.
> 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.
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.
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.
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).
> 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.
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:
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.
If the TLDR is long enough to include that it's in a journal the author was associated with then it's long enough to also mention that this isn't an abnormal thing in mathematics. Otherwise both details should be excluded.
"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...
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.
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.
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.
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.
It seems mathematicians are not aware of code obfuscation. Mochizuki objective is to keep this knowledge in Japan and protect this technological advantage.
[+] [-] gautamcgoel|6 years ago|reply
[+] [-] wolfgke|6 years ago|reply
Here are some texts from the internet concerning this:
David Michael Roberts - 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-reall...
https://www.quora.com/What-do-you-think-about-Stix-and-Schol...
Also relevant:
https://thehighergeometer.wordpress.com/2019/01/18/taylor-du...
[+] [-] antonzabirko|6 years ago|reply
[+] [-] bsder|6 years ago|reply
It's worse than that. At this point, the person who bridges that gap will almost certainly get his name added to the proof--that's a big incentive.
The fact that nobody seems to be able to bridge that gap is a gigantic glaring flag that something is wrong.
[+] [-] akvadrako|6 years ago|reply
[+] [-] gizmondo|6 years ago|reply
That's exceptionally bad optics.
[+] [-] battery_cowboy|6 years ago|reply
> 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.
[+] [-] timkam|6 years ago|reply
[+] [-] moomin|6 years ago|reply
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|6 years ago|reply
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.
[+] [-] hn_throwaway_99|6 years ago|reply
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|6 years ago|reply
that’s not how extraordinary math worked in the past
[+] [-] confuseshrink|6 years ago|reply
[+] [-] Barrin92|6 years ago|reply
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.
[+] [-] AzzieElbab|6 years ago|reply
[+] [-] unknown|6 years ago|reply
[deleted]
[+] [-] lostmsu|6 years ago|reply
[+] [-] wolfgke|6 years ago|reply
[+] [-] avip|6 years ago|reply
[+] [-] westoncb|6 years ago|reply
> 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.
[+] [-] dang|6 years ago|reply
https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...
https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...
[+] [-] vosper|6 years ago|reply
[+] [-] doall|6 years ago|reply
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/
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.
[+] [-] akvadrako|6 years ago|reply
https://www.maths.nottingham.ac.uk/plp/pmzibf/rpp.pdf
[+] [-] mellosouls|6 years ago|reply
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|6 years ago|reply
[+] [-] sudurbeh|6 years ago|reply
[+] [-] peter_d_sherman|6 years ago|reply
"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...
[+] [-] unknown|6 years ago|reply
[deleted]
[+] [-] credit_guy|6 years ago|reply
[+] [-] haecceity|6 years ago|reply
[+] [-] JulianWasTaken|6 years ago|reply
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.
[+] [-] jesuslop|6 years ago|reply
[+] [-] Koshkin|6 years ago|reply
[+] [-] rkagerer|6 years ago|reply
This isn't something you hype like a iPhone.
[+] [-] abnry|6 years ago|reply
[+] [-] ur-whale|6 years ago|reply
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|6 years ago|reply
[+] [-] wolfgke|6 years ago|reply
So, I believe that this would make it highly complicated to even formulate some very new math at the borderlands of our knowledge.
[+] [-] cronocr|6 years ago|reply
[+] [-] papeda|6 years ago|reply