Mochizuki's proof of the ABC conjecture accepted for publication
nature.com
nature.com
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...
The problem is: Scholze and Stix formulated their objections to a "simplified" version of IUTT (i.e. they did quite some identifications). Their claim is that these simplifcations do not matter for whether IUTT holds or not, but Mochizuki claims that the details that Scholze and Stix "wiped away" do matter. No side could convince the other side of their position.
If you want to dive into the gory details:
http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf (Scholze & Stix)
http://www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf (Mochizuki)
If you love flamewars, read: Ivan Fesenko: Remarks on Aspects of Modern Pioneering Mathematical Resarch; https://www.maths.nottingham.ac.uk/plp/pmzibf/rapm.pdf
We'll probably never know if he actually solved it. Few have the knowledge, fewer had the patience and nobody has had rebuttal. Scholze and Stix conclusions hold exactly as much weight as Mochizuki's. They are unwilling to change their mind and recant, just as much as Mochizuki is.
This whole fiasco reflects very poorly on the mathematical community in general.
http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-...
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.
Taylor Dupuy together with Anton Hilado is attempting this:
https://thehighergeometer.wordpress.com/2019/01/18/taylor-du...
That's exceptionally bad optics.
> 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.
Contrast with Wiles, where they _did_ understand it, they _did_ find a gap in his proof and he fixed it to everyone’s satisfaction.
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.
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.
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.
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.
On the other hand, there was a similar case with a German mathematician beforehand, and he turned out to be correct.
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.
And those are papers that an expert can skim to get an idea about the flow of the proofs. This one, if you jump in, says you things like “every fooable bazz is a bar”, where foo, bar and baz are new terms no mathematician has any intuition for that, likely, got defined in terms of other hitherto unknown terms qux, quux and quuz, making it impossible to judge whether that statement has merit, or how it leads to proving the abc conjecture.
And yes, an automated proof checker would be nice to have, but we aren’t there yet, by a wide margin. Getting there would make this proof a lot longer.
(For a loose analogue: imagine that the published solution to a “mate in 2” problem in chess wouldn’t just be “Queen g1, check, King h8, rook h2, mate”, but “whites queen is on field d1, by rule r queen can move horizontally across empty fields, there are no empty fields between d1 and g1, that doesn’t bring white’s king in check (blacks pawn on e4 can’t take it because…, blacks knight on c6 can only move to… because of rules…, so white can move their queen to g1. That brings black in check because of rules…, etc)
Who is that?
> 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-identificat...
[2] http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20...
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
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.
that’s not how extraordinary math worked in the past
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
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.
> 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.
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.
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.
Scholze and Stix had to go to him.
http://www.kurims.kyoto-u.ac.jp/~motizuki/students-english.h...
However, one could view it as the only common language between you and someone else and use it for that reason. In my mind the one that know more languages are able to more easily communicate with more people more accurately, and as such have a leg up on teaching and learning from others. These ideas might be able to travel faster since they can travel by English and Japanese. Though maybe that is just wishful thinking from my side.
Of course, it could be coincidental, and he could also have been unfairly represented in his willingness to engage.
I was actually careful to exclude that particular job title from my OP to avoid the unfairness you are implying in my initial summary.
However, your point is correct in ordinary situations. This is no ordinary situation.
"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...
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.
This isn't something you hype like a iPhone.
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.
One can't help but wonder: as boring as reading through inter-universal teichmuller theory?
So, I believe that this would make it highly complicated to even formulate some very new math at the borderlands of our knowledge.
One way to solve this problem is to replace ZFC by its non-conservative extension Tarski–Grothendieck set theory
https://en.wikipedia.org/w/index.php?title=Tarski%E2%80%93Gr...
to ensure that a Grothendieck universe
https://en.wikipedia.org/w/index.php?title=Grothendieck_univ...
always exists.
Grothendieck needed an extra axiom because he wanted to consider large Grothendieck topologies. The consensus seems to be that universes are avoidable: https://mathoverflow.net/questions/35746/inaccessible-cardin...