As discussed here a couple months ago, Peter Scholze and Jakob Stix believe they have found a serious problem with Mochizuki’s claimed proof of the abc conjecture, and traveled to Kyoto in March to discuss it with him. Their write-up is now available here. Mochizuki has made public his response to this, creating a web-page available here. There’s also an updated version of Ivan Fesenko’s take on the story, as well as a possibly relevant FAQ on IUTeich from Go Yamashita.

Erica Klarreich has an excellent long and detailed article about this story at Quanta.

**Update**: Looking through these Scholze/Stix/Mochizuki documents, my non-expert opinion is that Mochizuki does not seem to effectively address the Scholze-Stix objections, which are aimed at a very specific piece of his argument. Unfortunately, he also does his own credibility a huge amount of damage by including over-the-top attacks on the competence of Scholze and Stix, in typefaces that make him look unserious. For instance, there’s

I can only say that it is avery challenging task to document the depth of my astonishmentwhen I first read this Remark! This Remark may be described as abreath-takingly (melo?)dramatic self-declaration,on the part of SS, of theirprofound ignoranceof theelementary theory of heights,at the advanced undergraduate/beginning graduate level.

or the last couple pages of his report.

**Update**: More of the same about IUT from Fesenko available here. His argument is that the overwhelming majority of leading experts in arithmetic geometry who are skeptical of the purported abc proof should be ignored, since they haven’t put in the two years of continuous study of IUT necessary. I don’t think this collection of ad hominem arguments will do anything to change anyone’s mind. I also don’t see why he doesn’t instead produce what could change minds: a clear and convincing technical refutation of the Scholze-Stix argument.

This is very interesting, thank you. It is discouraging, on one hand, that agreement cannot be reached even when things are boiled down like this to (seemingly) one small issue. On the other hand, I feel more optimism than before that this will get finally resolved fairly soon. At the least, Scholze and Stix are doing a great service by highlighting and publicly discussing the problem area. Hopefully more mathematicians will now get involved.

Thanks for posting the update, it is the same as my impression of Mochizuki’s comments, which I didn’t want to write out of politeness.

S,

Thanks. My impression is that for most people these documents will conclusively resolve the issue, in favor of the Scholze/Stix side that this proof has a serious flaw. Mochizuki does not do an effective job of answering their objections. His only hope is that some others interested in this will now be able to focus on the very specific problem that Scholze/Stix point out, and if they find that Scholze/Stix are mistaken, do a better job than Mochizuki of explaining why.

This has been a very strange and unusual story since the beginning, fascinating because it illuminates well how mathematics normally works, because of the failures that occurred here. Here again, the situation is remarkable and illuminating. I’ve never heard of something like this happening before (experts writing long competing documents arguing about whether a specific part of a technical proof is correct). In other subjects you have experts publishing competing papers claiming the other person is wrong, with no resolution coming out of this. This doesn’t happen in mathematics: experts discussing a technical question about a proof are supposed to converge on an understanding of what is correct and what isn’t. In particular, I don’t think any journal would even consider publishing both of these, the editor would just say “one or both of them is wrong, if we publish both, we’d be publishing an incorrect paper”.

That’s probably right, Peter; I lack competence to judge, and clearly there is a very subtle issue (either real or imagined) about whether and when to identify non-canonically-identified things, so it’s hard for me to say for sure.

One thing for certain is that your update is right. I hadn’t read Mochizuki’s reply carefully last time I posted. When I did, I was simply aghast at the remark you quote. How can anybody dare say such things about… well, honestly, ANY competent fellow-mathematician? But much less a Fields medalist widely regarded as the star of his generation? It really does feel like Mochizuki is working through some issues in public, but that of course is a sidelight to the mathematical questions here. (Except that, as you say, that impression does not underwrite confidence in his mathematical judgments here.)

What I find most telling is that Scholze–Stix provide explanations of some of Mochizuki’s objects in terms that humans can understand, explain where their own simplifications may have been too strong, and where the simplifications are perfectly safe, whereas Mochizuki supplies yet more analogies based on manipulations of inequalities based on unspecified quantities, and then pages and pages of explanations about the importance of distinct labels. While Mochizuki is welcoming actual technical and precise discussion (while dismissing it at misunderstanding his work), I find his report

not technically precise enough. As a category theorist I feel I’m capable of recognising the distinctions one makes between objects considered in different categories linked by various forgetful functors, and his treatment is less than clear. It’s entirely possible there was a simplification made that accidentally identifies two isomorphic objects by an incorrect isomorphism, and Scholze–Stix are aware of this, but aren’t convinced by M’s protestations on their specific choices. At this point, M could make a very short and concrete rebuttal of their rebuttal, but pointing to justonesuch incorrect choice, but no, it needs a 41-page report…Also this quote:

!

Voevodsky’s proof of the Milnor conjecture was likewise discussed around the world in seminars for a six years (and was much better received and understood) before Deligne discovered an incorrect lemma that had to be replaced. The number of people who claim to understand IUTT is surely that much smaller… (Not to mention the Kapranov–Voevodsky paper on the homotopy hypothesis that V admitted he took over a decade to accept had a mistake.)

As a casual non-at-all-expert observer, I find myself oddly reassured to see that mathematics is performed by flesh-and-blood human beings, faults and all. 🙂

Wow. I thought that Fesenko’s criticisms of people who don’t understand Mochizuki were harsh, but now I see that also when it comes to hurling insults, he is just an apprentice of the great IUT master.

Your quote does not seem to appear in the current version of the document on Mochizuki’s website.

Also, the document seems full of highly technical content, I think you are doing a disservice to the discussion by highlighting the small bit of “sensationalistic” remarks (assuming they appeared in a previous version of the same document). One wonders whether you formed what you call your “non-expert opinion” by focusing on the mathematical content, or, as seems to be the norm in this saga (and the source of rightful protests from Mochizuki), the non-mathematical aspects of the discussion.

MK,

The quote is not from the “Report”, but from his second response to Scholze-Stix, see

http://www.kurims.kyoto-u.ac.jp/~motizuki/Cmt2018-08.pdf

and it is still there.

I wonder if Scholze and Stix expect their paper to be published when it contains comments like:

“When it comes to the more drastic simplifications indicated below […], these are inessential to the point we are making, and Mochizuki was not able to convince us during the week why such a simplification was not allowed.”(SS2018-08 2.1 (3))However

“drastic simplifications”should be proven, otherwise they are just conjectures. Claiming that“Mochizuki was not able to convince us”doesn’t prove these conjectures. On the other hand, if the“drastic simplifications”are not needed to refute the proof, then they prevent the claimed refutation from being published, which also doesn’t make sense, if the aim of the authors really is to have the paper published.Indeed, rather than aiming at publishing their paper, might it be the case, that Scholze and Stix aim “lower”, i.e. at preventing Mochizuki’s proof from being published?Mochizuki states on his web page (http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html), that

“it does not necessarily appear realistic to expect that further substantial efforts of the sort just described will be made by the authors of these files [SS2018-05], [SS2018-08]”Why otherwise would they write their paper and then walk away from the discussion?

What Mochizuki claims and what Scholze and Stix themselves admit being possible, is that Scholze and Stix with their simplifications have constructed the mathematical equivalent of a Straw man argument.

Here’s what wikipedia has to say about the Straw man:

“The typical straw man argument creates the illusion of having completely refuted or defeated an opponent’s proposition through the covert replacement of it with a different proposition (i.e., “stand up a straw man”) and the subsequent refutation of that false argument (“knock down a straw man”) instead of the opponent’s proposition.[2][3]This technique has been used throughout history in polemical debate, particularly in arguments about highly charged emotional issues where a fiery “battle” and the defeat of an “enemy” may be more valued than critical thinking or an understanding of both sides of the issue.”https://en.wikipedia.org/wiki/Straw_man

I wrote a short Science fiction script, somewhat inspired by this remarkable discussion. You can read it here: https://twitter.com/zbornikp/status/1033370752756719619

zbornikp,

I don’t think Scholze and Stix wrote that for publication. See the comments from David Roberts about the “simplification” issue. Yes, Scholze and Stix might be wrong about this, but if so Mochizuki should be able to point to a specific error and convincingly show it is an error. He doesn’t appear to be able to do so.

It is true that one motivating reason for Scholze and Stix to get involved was the possibility that PRIMS might accept and publish the Mochizuki proof, even though Scholze (and many others) believed the proof was flawed. I think these documents are best thought of as standard sorts of referee reports and an author’s response, which for unusual reasons are being made public. Up to the editor of PRIMS or another journal, but I don’t see how an editor, faced with these reports, could possibly accept the proof and publish it. As in such cases, it’s not the responsibility of the referees to keep working on this, once they feel they have found an error, explained what it is, and the author has no convincing response.

I think it makes an interesting case that the Scholze-Stix paper is made public on the website of Mochizuki, together with a parallel emission of technically inconclusive yet strongly opinionated pieces by himself. This feels very paradoxal. It seems to me that all previous chapters of the story, including Mochizuki’s refusal to travel abroad as well as his fabulous text “On the verification of inter-universal Teichmüller theory: a progress report (as of December 2014)” [ http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202014-12.pdf%5D, which – if stripped out of the context – could be as well an artwork of a mathematically informed yet satirical western author, indicate some sort of difference of cultural values and mindset, rather than a personal oddity. I wonder, from the perspective of cultural anthropology (albeit I’m not an expert), to what extent such activity should be read in terms of the acting within the bounds of taboo of ‘losing the face’ (which is essential in Chineese and Japaneese cultures) or, more specifically and more speculatively, within the bounds of some sort of modernisation of samurai ethics. This would make it very different situation from e.g. the case of error(s) in Voevodsky’s work(s). So, while the question how much of it can become useful in the long term run for the ‘canonical’ international mathematical community remains open, this story seems to carry also a meta-theoretical level that maybe could feed some PhD theses in (mathematically informed) social sciences.

As for science-fication of the inter-universal breath-taking (melo?)drama in the Hodge theatre [sic], here is a little fictional piece about the possible response to Scholze-Stix critique arriving from Mochizuki’s camp, which follows the above anthropological speculation: http://www.fuw.edu.pl/~kostecki/inter-universal_samurai.html

While I am by no means an expert on the subject (and agree that the over the top statements are not helpful), I have otherwise a somewhat different impression from reading the reports.

Scholze and Stix say at the beginning explicitly, that they use certain radical simplifications of IUT to get to the core of the proof and then show that this simplified procedure leads to nothing meaningful.

It seems to me that Mochizuki thinks, that if one makes these simplifying assumptions then one indeed arrives at a meaningless conclusion (specifically §12 (AD),(IUAD) in his report), but he disagrees that this would still be the case, if one does not make these simplifying assumptions (SSADFs).

I think this addresses the assertion that the procedure leads to a “meaningless result” by saying: Yes, the result of this procedure *under these simplifying assumptions* leads to meaningless result.

Mochizuki also attempts to explain why his procedure (without these simplifying assumptions) is supposed to be substantially different from a simplified procedure.

Mochizuki seems to argue about a (more or less) specific simplified version of the theta link (§10 (SSid) in his report). He claims that while this specific simplified version (SSid) of the theta link is able (and well adapted) to satisfy a certain “switching condition” (SW), it is not sufficiently compatible with “large parts” of the ring structures on both sides of the theta link (i.e. he claims (SSidFs) it cannot satisfy condition ‘Theta’CR from his §9).

I cannot judge the merits of this, but he seems to make specific claims, which properties fail in a certain simplified model (SSid). (I don’t know how close/far (SSid) is from the simplified model Scholze and Stix consider in their reports).

Maybe relatedly, a rather stark contrast between the positions seems to be whether and to what degree anabelian geometry enters Mochizuki’s argument: For Scholze and Stix it does not really seem to enter at all (their Remark 9), while Mochizuki seems to insist ((C5) in his July comments), that it is used in an essential way to guarantee certain symmetries of etale-like structures in the log-theta lattice.

These statements (and Q1,Q2 thereafter) seem to be purely about (non)existence of various versions of “theta links” with various properties (and about which compatibility properties of “theta links” are necessary to deduce Corollary 3.12). Thus they come logically before the disputed Corollary 3.12 (whose proof is based on the existence of a theta link with various properties). Corollary 3.12 seems to be the place, where the main (potential) problems seem to occur, since the reasoning in Corollary 3.12 leads (in the report of Scholze and Stix) to the meaningless inequality (1.6).

This could move the discussion a bit away from Corollary 3.12, whose proof may look quite initimidating, as it looks like it consists just of a summary/combination of nearly everything, which was achieved before.

These are clearly ad hominem attacks leveled at people who are politely and rightly asking for clarification; the part about the group laughing out loud is particularly troubling as it has a sort of staged feel to it “a remarkably unanimous response of utter astonishment and even disbelief (at times accompanied by bouts of laughter!).” In addition, there is a the material that implies a lower academic level and “profound ignorance.” These ad hominen attacks are now being extended to a wider circle. In this case, I think it is actually a good thing to point out what is obvious: the personal stuff weakens the argument. People should just focus on the math.

UF (and others whose comments I’ve deleted…),

I don’t think it’s helpful for non-experts to try and argue here about the competing technical claims in these documents. Scholze and Stix are experts, have worked hard at this, interacted extensively and directly with Mochizuki, and are well aware of his response to their claims. They explicitly say that Mochizuki does not have a proof. Whatever Mochizuki says, the problem is not that they’re ignorant people who aren’t able to appreciate his arguments. It is of course possible that they are wrong and he is right. To show this, someone will have to identify a serious flaw in their argument, in a convincing way. Doing this will require a high level of expertise and some serious work. We’re not going to see it in comments from non-experts on blogs a day after the documents came out.

I am surprised that one of Mochizuki’s friends or the university pr department has not gently suggested he revise his home page http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html . The presence on the site of nonsensical legal proscriptions (“The author of this web site prohibits the use of the contents, such as images, of this site, as well as all linked sites, by the mass media”) does not give a professional impression. The “Safety Confirmation Information for Shinichi Mochizuki” graphic on the “What’s New” section is also rather strange (when clicked on, the graphic confirms that Mochizuki was safe on September 4, 2018, at 11:00 PM).

Some people, no matter how brilliant, are poor judges of how graphic and personal communications are received by others. Not sure why his supporters are not assisting him a bit here in an area – web page design – that is not his strength.

Is the optimism voiced by e.g. S above really warranted? True, actual mathematics has entered the discussion, but at the end we’re left with Scholze&Stix saying that

The inequality derived in our simplified setup is trivialand Mochizuki:Indeed it is! And this shows that your simplification is wrong!and he then dumps another 41 pages of analogies that doesn’t seem to convince anyone (besides the people already sold on IUTeich). It’s worth noting that S&S say thattheyare convinced the same problem they identify remains even without their simplifications.So, is this situation any different from what has been going on all along? With the reaction of most experts being

We don’t understand what Mochizuki is talking aboutand his response:You need to stop everything else you’re doing and reflect on IUTeich in solemn contemplation for a few months and then you will see the Truth!Peter. Will you post something on Atiyah “proof” of the RH?

I’m still quite surprised that Scholze and Stix traveled to Japan at all… the only conclusion to draw from that is that they must have thought the gap could be fixed but the severely negative tone they take when explaining the gap in their write-up suggests otherwise, so I’m still rather mystified by all this…

Simple,

No, and I don’t think others should be publicizing this story, for reasons that I won’t discuss here, but shouldn’t be hard to figure out.

sdf,

I think the reason Scholze and Stix went to Japan is pretty clear. They had decided it was important to resolve the issue of whether this was a proof, and the only way they were going to get to the bottom of things and find out if Mochizuki had an answer to the problem they saw was by talking to him in depth about it. He wasn’t going to leave Kyoto, so they had to go there.

Is not the whole point of a proof that it is an argument that convinces someone else?!

If you can’t prove a thing to at least one other person, have you not by definition failed to prove anything?

Failure to be understood is the fault of the speaker!

Dear Peter (and others),

At this point (just a few days in), what is the state of expert opinion concerning the likelihood that detailed study of the two new papers will lead reasonably smoothly to a consensus on who is right? Focusing only on the substantive content, is there a sense that SM’s response has a level of explanatory clarity comparable with the remark attributed to Scholze in Quanta that other number theorists “would have totally been able to follow the discussions that we had had this week with Mochizuki”? Or does SM’s document seem, ro experts, to be just as impenetrable as the original papers? Or are things somewhere in between those two extremes?

As a number theorist whose has previously put some effort into attempting to read the IUT papers, here are some miscellaneous thoughts after spending ~6 hours with the Scholze-Stix document and Mochizuki’s report(s).

-First and foremost, Mochizuki does not explicitly address the main issue raised by Scholze-Stix, namely the necessity of differentiating between what they call “concrete” and “abstract” pilot objects. Nowhere in his report does one find any use of the words “concrete” or “abstract” in this context: he only refers to “*the* Theta-pilot object” and “*the* q-pilot object”.

-The Remark which filled Mochizuki with undocumentable astonishment IS in fact a bit silly. However, it is completely immaterial w/r/t the substance of Scholze-Stix’s objection, and could have been dropped from their document without affecting anything else.

-I find the discussion of “histories of operations performed on mathematical objects” in Mochizuki’s report to be largely meaningless. A mathematical object is an object in some category, and/or a set with some extra structures; it does not have a “history”. (Of course there are mathematical objects where one speaks of their histories, e.g. solutions of PDEs or trajectories of dynamical systems, but this is a precise technical use of the word.)

Aubrey de Grey,

As far as I can tell, there already is a consensus. In a case like this, the burden is on the mathematician who believes they have a proof to convince other experts, and Mochizuki has conclusively failed to meet this burden. One can argue about the wide variety of claims he makes in response to the Scholze/Stix document (which is short, focused and precise), but the bottom line is that they found them unconvincing. They clearly put quite a bit of time into this, traveling to Kyoto and giving him a week to try and convince them. Mochizuki tries to explain away the problem of not convincing them by arguing that they are incompetent and need to spend more months studying his work. The first of these arguments is both ludicrous and offensive. The second is both not very plausible and likely to discourage anyone else from paying attention. If Mochizuki is saying you need to devote multiple Scholze-months of intellectual effort to understand why he is right, most mathematicians will figure that Scholze-months are equivalent to their years, so no point in even trying.

At this point the only way I see things changing is if some of those who supposedly understand the proof do a better job than Mochizuki of answering Scholze/Stix. The “IUTeich FAQ” of Yamashita and Fesenko’s recent article seem to be attempts along this line, but they’re even less convincing than Mochizuki himself.

DH,

SM addresses the issue of concrete/abstract pilot objects (to some degree) in (C16) in his comments to [SS2018-05].

The mathematical community should ignore what is coming out of Kyoto. A number of very fine mathematicians have put in an inordinate amount of time and hard work in trying to understand Mochizuki’s work and it looks like they have wasted their time. It is time for the community to move on!

Wouldn’t it be the right way for Scholze/Stix (or others) to respond to the mathematical content on the latest documents by Mochizuki (which is a response to the SS document) to move forward? Of course Mochizuki shouldn’t try to make them look incompetent and this makes things suspicous in my view. But why is the mathematical content of the latest Mochizuki document ‘not convincing’? This should be motivated mathematically.

Marcus,

No, Scholze and Stix have no obligation to keep arguing with Mochizuki if they feel he doesn’t have an answer to the problem with his proof they have pointed out. They have other, more important things to spend their time on. Scholze has been revolutionizing his subfield of mathematics and that’s what he should be spending his time on, not arguing with an author who insults him and won’t admit there’s a problem with his manuscript.

One aspect of this whole story that hasn’t been mentioned is that Mochizuki has made no substantive changes to his manuscript since his March meetings with Scholze and Stix. This is extremely odd. Normally if an expert tells you he thinks there’s a gap or mistake in your manuscript giving a proof, even if you think this expert is wrong, you take this as an indication that you did not explain things well enough. Mochizuki is not just taking the attitude that there’s nothing wrong with the logic of his proof, he’s also taking the attitude that he does not need to even try to improve its exposition.

I just find the comment of Mizan R Khan is bit offensive and political. It sounds like Univeristy of Kyoto is not a good at mathematics which is not true. I wonder why this comment is not sensored while other comments were taken out.

“The mathematical community should ignore what is coming out of Kyoto. ….” You can take this comment out if you wish.

HK,

I’ve deleted lots of comments, pro and anti-Mochizuki, on the grounds that they add nothing, just seem intent on carrying on a pointless argument. I interpreted Mizan R. Khan’s “coming out of Kyoto” to just be a shorthand reference to Mochizuki and those around him who claim this is a proof, but have never been able to explain it satisfactorily to others. His “time to move on” take on this is a widely held one.

There is one other sense in which a complaint about “Kyoto” is justified. One of the main sources of the problem here seems to me to be the PRIMS refereeing of the paper. In the end, it appears they have decided not to accept the paper for publication, despite early reports they were going to do so, but that seems to have required the intervention of Scholze and other outsiders. The current state of affairs should have been reached much earlier, by expert referees in contact with Scholze and others who were pointing to the questions about Corollary 3.12.

HK: I am very sorry. I should have put more thought into the phrasing of my statement about Kyoto. I did not mean any disrespect to the mathematical school in Kyoto. Clearly it is very strong mathematically.

When I said Kyoto I was solely referencing Mochizuki’s work on abc. Mochizuki is

undoubtedly an extraordinarily talented mathematician. However, the enormous demands and expectations he has made of mathematicians of the stature of Scholze, Stix, et al is utterly ridiculous and self-centered! No matter how brilliant one is, one needs to show some consideration for other people’s time and effort.

Peter, you wrote, “I’ve never heard of something like this happening before (experts writing long competing documents arguing about whether a specific part of a technical proof is correct).” Perhaps the closest analogue was the controversy over Hsiang’s claim to have proved the Kepler conjecture. (Googling “hsiang hales rejoinder” will give you an entry point to the literature.) As far as I am aware, Hsiang still maintains that his proof is correct, but I am not aware of any other professional mathematician who publicly defends Hsiang’s proof. One notable difference in this case is that quite a few mathematicians say that they understand Mochizuki’s proof and believe that it is correct. If the current situation persists, with no argument materializing that persuades Stix and Scholze but with a sizable group of researchers continuing to study IUT in its current form, then I think that really would be unprecedented (at such a high level of mathematics, at least).

Timothy Chow,

Thanks, that is an interesting historical analog, one that I was unaware of.

As for the claim “quite a few mathematicians say that they understand Mochizuki’s proof and believe that it is correct”, I’ve mainly heard that coming from Fesenko and Mochizuki, not from mathematicians themselves, especially not from well-known experts in the field. The Scholze-Stix paper may also change the minds of some who in the past thought the Mochizuki proof was probably right. At this point, saying you understand the proof and believe it is correct implies you know why Scholze-Stix are wrong, and can explain that to others.

This is off-topic but the ICM plenary videos are now up (https://tinyurl.com/y96a9cn6). I’ve only watched Geordie Williamson’s so far but IMO it is brilliant and of course there’s also Scholze’s!

I wrote some notes while trying understand the situation. They may be of interest to some people.

From Fesenko’s new article:

Well if this isn’t an anonymous pointer at myself and Urs Schreiber, then I don’t know what is. Let it be put on the record that I do not claim Scholze and Stix’s notes are incorrect, just that it is not clear how far they have moved from Mochizuki’s work by their simplifications, because I cannot say for sure what Mochizuki’s work is really doing, shorn of its verbiage, whereas I generally understand what Scholze and Stix have done.