There has been a remarkable discussion going on for the past couple weeks in the comment section of this blog posting, which gives a very clear picture of the problems with Mochizuki’s claimed proof of the Szpiro conjecture. These problems were first explained in the 2018 Scholze-Stix document Why *abc* is still a conjecture.

In order to make this discussion more legible, and provide a form for it that can be consulted and distributed outside my blog software, I’ve put together an edited version of the discussion. I’ll update this document if the discussion continues, but it seemed to me to now be winding down.

Depending on one’s background, one will be able to get less or more out of trying to follow this discussion, but it seems to me that it makes an overwhelmingly convincing case that Mochizuki’s articles do not contain a proof of the conjecture and should not be published by PRIMS. No one involved in the discussion claims that there is an understandable and convincing proof in the articles. The discussion is rather about Scholze’s argument that there is no way that the kind of thing Mochizuki is doing can possibly work. While Scholze may not have a fully rigorous, loophole-free argument (and given the ambiguous nature of many of Mochizuki’s claims, this may not be possible) the burden is not on him to do this.

To justify the PRIMS decision to publish the proof, one needs to assume that the referees have some understood and convincing counterargument to that of Scholze, one that nobody has made publicly anywhere. If this really is the case, the editors of PRIMS need to make public these counterarguments, and those mathematicians who find them convincing need to be able to explain them.

*A note on comments*: if someone has further technical comments on the mathematical issues being discussed at the earlier posting, they should be submitted there. For discussion of issues surrounding publication of the claimed Mochizuki proof, this would be the right place (and I’ve moved a couple recent ones to here). For comments about Szpiro and his conjecture, the posting about him would be an appropriate one.

**Update**: I hear that the editors of PRIMS are aware of the recent discussion of the problems with the Mochizuki proof, but have decided to go ahead with the publication of the proof anyway. They do not seem to intend to release any information about their editorial process, in particular what counter-arguments to Scholze’s they considered. In effect, they are taking the stand that they have convincing evidence that Scholze is wrong about the mathematics here, but cannot make it public for confidentiality reasons.

Note that the discussion in the comment thread itself has some later entries after the ones gathered in the pdf document I created.

This is not in response to any particular comments but to the overall discussion.

It seems to me that there is a psychological issue at play here which might have not been made explicit. This is that for anyone who has devoted enough time to understand the basic setup and language of Mochizuki’s papers, it is very hard to accept that in the end there is no proof and one has wasted a huge amount of time. I myself devoted most of three months way back in 2013 to going through the papers and it was a shock, on reaching Corollary 3.12, to realise that nothing is really proved. (One might say that this shows my lack of understanding, but the point is that Mochizuki himself says that this is the key part of his proof and if you see how the proof is written it is clear that there is something seriously wrong here.)

Given the time that has passed since Mochizuki’s manuscripts appeared and the objections that have been raised by many experts, the obvious interpretation of the lack of any serious revision by Mochizuki (going by Occam’s Razor) is that there is actually nothing that readers have missed, it’s simply that there is no proof. However, because of Mochizuki’s past achievements, this is hard to accept (it was for me), perhaps especially for those who know him personally. This seems to have led to some people assuming that it is because of their lack of understanding of what Mochizuki “really means” and so devoting a lot more time to IUTT. Once one has invested too much time, I think it becomes psychologically almost impossible to give up hope, and the lack of precision in the papers sadly ensures that there is no simple statement in them which can be shown to be wrong and make it impossible to maintain faith.

Now that Scholze has explained what is going on with Mochizuki’s manuscripts, it’s time for other arithmetic geometry experts to revisit the papers and weigh in. Those of us in the field know that almost all the experts definitively side with Scholze-Stix, so it’s frustrating that the popular articles (which most people base their opinion on!) depict the situation as if it were much more ambiguous, because experts are afraid of taking a stand. And when mathematicians do get themselves quoted, it’s with painfully noncommittal lines like “I will withhold my judgement on the publication of this work until it actually happens, as new information might emerge” and “In spite of all the difficulties over the years, I still think it would be great if Mochizuki’s ideas turned out to be correct”. The purpose of granting tenure was so that academics could speak honestly, rather than hide behind these sorts of vague equivocations. With increasing resources being allocated to IUT, it’s clear that the situation will only become more and more damaging to mathematics at large.

One reason for the situation in the press that Ibj is frustrated with is simply that the publication hasn’t publicly appeared. Experts are rightly hesitant to say something definitive about a paper that they haven’t seen. If it turns out that PRIMS publishes a paper that, as is widely expected, just leaves the existing gap in the proof of Cor 3.12 essentially unchanged, then we’d see some more forceful statements from experts. Of course, by then the popular press will have moved on.

Of course this means holding a public press conference without releasing the manuscript is pretty shocking and irresponsible in this context. The article I’d like to read is what on earth explains the willingness of these other great mathematicians at RIMS to go along with this charade.

The comment of @naf hits the nail on the head. But I would go a step further: when confronted with a truly massive edifice of highly technical mathematics, nearly all experts need some kind of motivation to persevere beyond the final goal at the end of the tunnel. For example, a powerful heuristic to give confidence in the strategy, or some kind of intuitive guide to grab onto along the way to have a feeling of making progress (or at least interesting achievements along the way in the absence of a global guide). But here there is nothing of the sort, not even a compelling mathematical reason to believe at the outset that investing a huge amount of time is going to reap satisfying mathematical understanding. There is only patience to keep oneself going, and it can be very hard to rely on that alone after a lot of time.

This practical (albeit psychological) concern came to mind almost immediately after I was asked early on to be on the referee team for the IUT papers. I have great respect for Mochizuki’s mathematical talent, and no doubt in the sincerity of his belief that he has a proof of the main result. But I could see that the referees would not only have to check the details of an extremely long work written in a very obscure style (which didn’t provide insightful reasons for confidence in the approach being used). They would also have to engage in a herculean effort to get the writing substantially changed. It was too much, so I declined and communicated my concerns to the editorial board. (I recommended immediate rejection with a demand that the work be completely rewritten before it could be reconsidered.)

I am very sorry to see all these years later that neither the referees who were eventually obtained nor the editorial board obtained any real improvement on the clarity of the way the material is presented, not even at least an Introduction presenting key new insights in some conventional manner (to compensate for the way the technical material is presented). I hope the editors of PRIMS and the senior faculty at RIMS will reflect on their responsibility to the field of mathematics, and reconsider what they are doing.

Professor Woit: minor typo – 2108 should be 2018

“These problems were first explained in the 2108 Scholze-Stix document Why abc is still a conjecture. “

derek,

Thanks! Fixed.

Thanks, OP. I wish the other referees would be so forthcoming, if it were indeed possible. Or if not, the journal could release their reports.

I note that Nick Katz was public about his refereeing of Wiles’ first FLT paper, and his finding of the mistake, and it’s public knowledge Gábor Fejes Tóth was one of the referees for Hales’ Kepler conjecture paper that appeared in the Annals with a disclaimer. This is an extraordinary case, and warrants extraordinary actions, IHMO.

@Peter W – I notice you link, in your pdf, to my Inference essay, but not the technical note I pointed you to. I’d prefer the technical note is used, not the essay, which was written for the general public and much below the level of the comments here. In any case, I should have given you this link: https://doi.org/10.25909/5c5ce1fda4b7c instead of the one I originally sent.

David Roberts,

Thanks for pointing this out! Fixed.

@David Roberts:

Just to clarify: I wasn’t a referee on the IUT papers, but rather was invited to serve as one, and I declined (giving the editorial board my recommendation for how I thought would be best to proceed). As for Katz’ work as a referee on the initial FLT paper, my impression is that this only became public knowledge sometime after the fix was found and the corrected version had gone through the review process (e.g., maybe via the BBC video that was made about it).

@OP sure. By ‘other referees’ I mean the ones that accepted the job.

For more sociological understanding , please let me put about some curious social situation around this issue in Japan.

On the official announcement of acceptance of the papers in Kyoto, two mathematicians Tamagawa and Kashiwara attended there, and Tamagawa is actually well-known expert in the anabelian geometry.But It seems likely that he can not explain about the mathematical matter of IUT in public so far.

In addition , Rigid geometer Fumiharu Kato already published the book “in 2019” about IUT “treated as correct theory” for general audience amazingly, but actually it wouldn’t be sufficient to offer any insight for professional mathematicians.Furthermore , as Tamagawa is , he also has never wrote any rigorous mathematical papers about IUT as far as I know at present.

These facts seems to be indeed mysterious that is , ” whoever did understand and can defense the theory sufficiently?” .

Anyway needless to say , since IUTT use too many and overlevel terminology ,so that if it cannot be expressed with crucial idea for proof , other mathematicians wouldn’t accept it at all.

Peter, I wonder if you could provide a little historical context for the non-mathematician. Are situations like this one common in the modern history of mathematics? Occasional? Rare? Unprecedented? Is there another situation you can compare this to?

About the situation in Japan: do Japanese mathematicians in general accept the “proof” as it is? Or is it just (at least most) mathematicians at RIMS?

Jeff Berkowitz,

This is unprecedented. I know of no other example in the history of the field of a reputable journal publishing a proof of a major result over the objections of experts in the field who have publicly argued that the proof is flawed, even that it cannot possibly work.

Very few Japanese experts in this field have expressed publicly an opinion that the proof is correct. I hear (second hand) that many of them privately say they are embarrassed by this situation. It would be helpful if they would say this publicly.

I have been following this drama for quite a while now and there’re a couple of things I just don’t get:

1. Why do people assume that the referees (if there were any) really understood the papers and should thus come out and explain it? Most of you have been refereeing papers yourself. Do you read and check every line? That’s impossible, and I’m open about this in every report I write. Even if you do think you got it all, does your judgment make the paper correct? We’re all just humans and prone to make errors. As an amusing reminder, here’s an (incomplete) list of “incomplete proofs”: https://en.wikipedia.org/wiki/List_of_incomplete_proofs.

2. What’s the matter with statements like the following?

* “well-known to every one at RIMS” (K. Joshi)

* “these German mathematicians” + many more (I. Fesenko)

* “a deep sense of discomfort, or unfamiliarity, with new ways of thinking about familiar mathematical objects” + many more (S. Mochizuki)

Did you all lose your minds? What is this?

3. The papers by Mochizuki consist of building up tons of highly ambiguous language, and when it comes to a proof, it’s all obvious from the definitions. Why does anyone actually take this serious? What’s comical is that Mochizuki himself complains in his pamphlet that “negative positions concerning IUTch were always discussed in highly non-mathematical terms, i.e., by focusing on various aspects of the situation that were quite far removed from any sort of detailed, well-defined, mathematically substantive content.” Yeah, sorry, but how are we supposed to talk about this thing then?

4. Scholze and Stix tried to make some sense out of this and pointed out a specific part they don’t understand; see also the comments above. There has been no reply except comments along the lines of point 2 above. So, what? Do you want to force them with a baseball bat to understand it? What are they supposed to do? What do you want?

Mochizuki: If you can’t explain your stuff, no one will care, period.

Jay Watt,

For many of the things you are not getting, I think most mathematicians feel the same way. About point 1 though (referees):

First of all, the standard of refereeing expected for a claimed proof of this kind is much higher than normal. But the reason for demanding an explanation from the editors and referees is that two years ago they were presented by Scholze and Stix with a strong argument about a specific flaw in this proof, as well as the information that Scholze and Stix spent a week discussing this flaw with the author and left convinced the author had no answer for how to fix the proof. In response to the problem identified by Scholze/Stix, the author seems to have made no substantive changes to the argument to fix it (Scholze says the problem is still there).

What’s incomprehensible here is that the editors decided to publish the proof anyway, and even held a press conference to promote this. The editors and referees involved in this have a very specific responsibility to the math community that they are not meeting: they need to produce a convincing explanation for why Scholze/Stix are wrong (presumably they have this from the referees, otherwise they would not have decided to publish). Absent that, what we have here is a strong argument that the proof is wrong, met by a press conference saying the editors disagree, but will not say why.

Regarding whether this an unprecedented situation, I agree that there are many ways in which it is unprecedented. However, I would say that it is far from unprecedented for a flawed paper (even one solving a major open problem) to be published even when experts have seen a preprint and believe it to be at best incomprehensible. If an author has a strong reputation then mathematicians are typically very leery of saying that a paper is wrong unless they can point to a provably false statement. They have been socialized all their lives that it is taboo to say something definitive unless they can back it up with a rigorous mathematical proof. So if a paper is not even wrong then mathematicians will dance around the issue and generally refrain from delivering a verdict that it should not be published, unless they have formally accepted the task of refereeing the paper. Note, for instance, that Taylor Dupuy has said that he does not think that the Scholze–Stix manuscript is a sufficient reason for rejecting Mochizuki’s proof, and he wants to see the final published version before making a final judgment. In general (i.e., setting aside the specifics of the IUTT papers), this is a typical attitude, and leads to quite a number of dubious papers by famous people getting published even though experts know that there are serious and perhaps fatal problems with them.

I think that what’s most unprecedented about the current situation is that two mathematicians, who are not referees, have gone public with a major criticism prior to the publication of the article. This is the really unusual part of the story, and it might not have played out this way had there not been a false rumor a couple of years ago about the imminent publication of the paper. Had this rumor not been circulated, and had Mochizuki’s papers simply appeared in PRIMS without a prior public announcement, then the situation would not be that different from, say, Hsiang’s claimed proof of the Kepler conjecture and the ensuing controversy.

Jay Watt:

Not sure this is accurate. The M remark you quote was mentioned in a 2018 Quanta piece—perhaps that is where you are getting it from? Its original source is (apparently) the 2018 report by M on the discussions he had by Scholze & Stix. This report has comments way beyond what you quote in terms of mathematical content.

Also, reading the comments from people who actually interacted with him, it seems that M does engage into detailed mathematical discussions with people who contact him about his work. This way of representing his responses as no-math-content, dismissive commentary seems inaccurate.

As for point-by-point factual summaries of the state of the problem, this description by D. Roberts seems to have a more neutral/objective wording than some of the summaries given during the current comment thread:

Timothy Chow,

I think you’re making an interesting point, but the question then becomes: when is a proof “published”, i.e. “made public”?

You write “two mathematicians, who are not referees, have gone public with a major criticism prior to the publication of the article.” But Peter Scholze is always careful to point out that he’s analyzing what is public now, and not the publication in PRIMS. And what is public now, is entirely the responsibility of Mochizuki. This unprecedented situation is created by him, not by these two mathematicians.

Timothy Chow,

The whole story here is very unusual. Yes, in an alternate universe where Ivan Fesenko hadn’t distributed an email claiming the IUT papers had been accepted for publication, which caused me and others to blog about the publication news , which caused Peter Scholze to speak up publicly about the problem, much of this would have taken place only privately before publication, and the public mess would have been, like in the Hsiang case, post-publication.

But that’s not what happened, and one reason that it’s not what happened is that, unlike Hsiang’s proof, the Mochizuki proof received a huge amount of attention from experts from 2012 on, leading most of them to conclude that the proof was incomprehensible. As it became increasingly clear that experts in the subject could not get a comprehensible explanation from Mochizuki (or those around him who claimed to understand the proof), many of them became worried that the journal would publish the proof despite its incomprehensiblity and were debating what to do about this.

It should be emphasized again that at the time of the abortive announcement, what experts were upset about was the unusual prospect of publication of an incomprehensible claimed proof, but what has happened now is much worse than that: publication of a proof the community is convinced is flawed, by editors who are well aware of this.

I don’t know the story of the refereeing of the Hsiang paper, but find it hard to believe that editors of the paper were aware of any privately expressed opinion by experts that that the proof wasn’t complete and decided to publish anyway. In any case, I don’t think comparison to the Hsiang/Kepler conjecture story is very apt. For one thing, the Hsiang proof was not based on claims of a completely new (as in Mochizuki’s IUT) approach to the problem. He was following a well-known line of attack. Note that in the skeptical Mathscinet review of the Hsiang paper, you can read

“I think there is hope that Hsiang’s strategy will work: at least the main inequalities seem to hold. As far as details are concerned, my opinion is that many of the key statements have no acceptable proofs. Typically, we are given arguments such as “the most critical case is…” followed by a statement that “the same method will imply the general case”. The problem with arguments of this kind is not only that they require the reader to redo some pages of calculations, but, notoriously, that they occur at places where we expect difficulties and most frequently it is impossible to see how the same method works in the general case.”

In the Mochizuki case, Scholze’s claim is not that important details haven’t been worked out, but that the whole approach is fundamentally flawed. By the way, if you read the exchange with Scholze you will see that Dupuy does not at all claim there is a proof in the Mochizuki papers, he is just trying to argue the Scholze has not yet rigorously shown that Mochizuki’s approach can’t possibly work.

MZ,

I don’t think the Roberts quote gives an accurate picture of the argument between Scholze and Mochizuki. I strongly recommend reading Scholze’s summary of the problem here

https://www.math.columbia.edu/~woit/wordpress/?p=11709&cpage=1#comment-235940

and if you don’t find it convincing, read the entire discussion, linked to in this posting.

What Scholze is claiming is not that “if you simplify the proof it won’t work”, but that there is a fundamental mathematical issue of principle for why nothing like this kind of proof can work. He explains this issue clearly in the second and third paragraphs of his comment. The later discussion is all about whether these paragraphs give a rigorous argument, or whether maybe Mochizuki has some way (which no one else has been able to understand) around the argument.

Peter,

you are quite right about what you said about my point 1 (referees). Let me say I’m more surprised by the fact, it seems, that people indeed expect this may be helpful and would lead to anything. If a referee could actually clarify all/some of the issues (or just the specific one raised by Scholze&Stix), then why not do it here, anonymously? I think they can’t, and any attempt would likely go along the lines “Well, Mochizuki probably meant (some vague statement)”, as all the other discussions went so far. All this is not getting anywhere. It’s the nature of the papers and the “theory” that makes it impossible to deal with on a rigorous and verifiable basis. It’s an eel.

@MZ: The “deep sense of discomfort” quote is by M, at the end of http://www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf. But this whole thing: why all these philosophical meta phrases? Why not focus on the subject? What is this all about?

Peter Woit: I agree that there are disanalogies with the Hsiang case. But I do want to point out that it is actually disturbingly common for mathematical papers by well-known people, claiming to solve important problems, to be published even when the editors have received private indications from experts that the papers have serious gaps, or even that the entire approach is fundamentally flawed. It’s usually not talked about openly because unless the author admits that there is an irreparable gap, you’ll make enemies by making such a claim. I also don’t know what happened behind the scenes with the publication of Hsiang’s paper, but if the editors privately knew that other experts had serious reservations and decided to publish anyway, it wouldn’t have been the first time that that sort of thing happened.

If Mochizuki’s papers had been published in, say, 2016, after “only” four years, I don’t think the situation would be quite so unprecedented. But yes, the current circumstances are unprecedented.

By the way, just now I looked again at Hales’s 1994 Intelligencer article about Hsiang’s proof. He makes it clear that he presented Hsiang with serious objections several times in the interim between 1990, when Hsiang announced his proof, and 1993, when Hsiang’s proof was published. So it seems likely that the editor of the journal was aware, prior to publication, that other experts in the field found the proof unacceptable.

Jay Watt,

I also find it highly unlikely that any of the referees or anyone the editors consulted post Scholze/Stix had a convincing argument for how the Mochizuki proof overcomes their objections. I do think it’s important that they be confronted with a demand to produce such an argument. If it exists, they can vindicate themselves by making it publicly available. If it doesn’t, they have two choices: change their decision about publication or double down on unethical behavior, claiming “there is an argument but we have to keep it secret”.

I agree with Timothy Chow that incorrect proofs being published, even when concerns have been privately discussed with the editors, isn’t _that_ unusual. Also, in mathematics retraction of (well-known to be) faulty papers seems even more difficult than in other disciplines

Of course, the IUT saga has been very unusual or even unique in mathematics because of the huge amount of publicity it has received, and the behaviour of some people surrounding the case. It’s definitely strange that besides the author there are several people who vouch (even very, very, fervently) for the proof, yet are completely unable to explain it in any meaningful way.

math_jin@ retweeted a report that the papers to be published in P(RIMS) are essentially identical to the versions on M’s website, with the exception of a single footnote on the first page.

Hi Peter Woit,

Yesterday, David Roberts on his blog “theHigherGeometer” reported that papers relevant to Mochizuki’s Corollary 3.12 are finally beginning to appear on Taylor Dupuy’s homepage. See “Dupuy and Hilado’s work on unravelling Mochizuki”:

https://thehighergeometer.wordpress.com/2020/04/27/dupuy-and-hilados-work-on-unravelling-mochizuki/

The same twitter account mentioned above by mgflax (which may or may not be a source of reliable information) retweeted another message a few days ago identifying one specific mathematician at RIMS who has apparently gone through Mochizuki’s papers and is reported as not being convinced of the validity of the proof.

Although this may only count as anecdotal evidence of internal dissent (were it true), it would suggest that not everyone at RIMS fully endorses the belief that the Szpiro conjecture will become a theorem when the papers get published by PRIMS.

JE,

At this point it’s a factual matter that no one other than Mochizuki has come forward publicly with an argument explaining why the proof is correct, despite Scholze’s arguments. I assume that those mathematicians at RIMS with any interest in the subject are well aware of this situation, and many if not most are privately drawing the obvious conclusions.

@JE is this ‘specific mathematician’ Teruhisa Koshikawa?

David Roberts,

I assume JE is referring to this tweet

https://twitter.com/MugaShohou/status/1253341200054054912

Besides Mochizuki, the others listed there as understanding the proof are Hoshi, Yamashita and Saidi. As far as I know, the documents written by Hoshi (in Japanese) and Yamashita (in English) do not address the problems that Scholze raises, and I am not aware of anything on the topic written by Saidi.

@Peter, that was my guess, too. (I seriously wish that Hoshi’s notes were in English, as his other work seems to be very reasonable, as far as I can tell.)

@Peter and David,

Yes, that’s the tweet. Peter, you’re probably right about the uneasy situation created for some mathematicians at RIMS, especially after Kashiwara’s and Tamagawa’s appearance at the press conference held in early April to announce the publication, which could have been interpreted as a clear sign of support for Mochizuki’s proof from the institution as a whole (Kashiwara was presented as head of the team that examined professor Mochizuki’s theory, see e.g. https://www.japantimes.co.jp/news/2020/04/04/national/japanese-mathematician-shinichi-mochizuki/#.XqvfxGgzaUk). As things currently stand there, silence can be eloquent. We already knew that Hoshi, Yamashita and Saidi endorsed the proof, but the fact that one mathematician at RIMS has been publicly reported as not being convinced of it (at this point of the discussion) may be even more eloquent.

Hi Peter, I think it worth including *in the pdf document* a mention that the discussion continued, and give a link back. You mentioned that there was content before the first comment you included, so why not also say the discussion continued? Not everyone reading that document will come to it by a link from your blog post where you mention this. I say this merely from the point of view of having a coherent scholarly record (whatever one’s view of the various positions).

David Roberts,

I was planning on updating the pdf document to a final form, didn’t get around to it today until just now. It’s a beautiful day here in New York, and earlier a long bike ride seemed more pressing than that particular task.

Peter Woit, I noticed that in the PDF document you say that the PRIMS editors have decided that the Mochizuki proof should be deemed correct and complete. Are those the editors’ precise words, or is that an inference you’re making from their decision to go ahead and publish the proof?

Timothy Chow,

That’s my inference from their decision to publish (reiterated after consideration of the latest discussion with Scholze). I suppose it’s possible that the PRIMS editors have an unusual understanding of the role of a mathematics journal and have decided to publish a proof they know to be possibly incorrect or incomplete. If so, perhaps at publication they will include some sort of text explaining such an unusual choice.