ABC is Still a Conjecture

Just a reminder that the abc conjecture is still a conjecture, there is no known valid proof (don’t believe what you might read in an EMS journal). For more about why one attempted proof doesn’t work, see here and here. For extensive background on this, you could start at this blog posting and work backwards, to the first announcement of a claimed proof back in 2012. By 2018 Scholze and Stix had shown that the claimed argument was flawed, and since then the math community has lost interest and moved on. Devotion to the idea that the proof is valid seems now restricted to a small circle of die-hards based in Kyoto and Nottingham who are doing what they can to try and pretend the hole pointed out in the proof does not exist. There will be an IUT Summit in Kyoto in September, but the organizers don’t seem to have found anyone from outside Kyoto or Nottingham willing to participate.

Update: Mochizuki today on his website has put out a 65 page manuscript dealing with criticisms of his proof, it’s entitled:
ON THE ESSENTIAL LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMULLER THEORY IN TERMS OF LOGICAL AND “∧”/LOGICAL OR “∨” RELATIONS: REPORT ON THE OCCASION OF THE PUBLICATION OF THE FOUR MAIN PAPERS ON INTER-UNIVERSAL TEICHMULLER THEORY

I’ve taken a quick look at this document, and I don’t think it will convince anyone Scholze is wrong about the flaw in Mochizuki’s proof. There’s a long third and final technical section, but the first two sections do a great deal of damage to Mochizuki’s credibility. Nowhere in the document do the names Scholze or Stix appear (they are referred to as “RCS: the redundant copies school”), but it starts off with statements such as

the response of all of the mathematicians with whom I have had technically meaningful discussions concerning the assertions of the RCS was completely uniform and unanimous, i.e., to the effect that these assertions of the RCS were obviously completely mathematically inaccurate/absurd, and that they had no idea why adherents of the RCS continued to make such manifestly absurd assertions.

and

the assertions of the RCS are nothing more than meaningless, superficial misunderstandings of inter-universal Teichmuller theory on the part of people who are clearly not operating on the basis of a solid, technically accurate understanding of the mathematical content and essential logical structure of inter-universal Teichmuller theory.

Before going on to the more technical third part, the second part is an extensive discussion of elementary mathematical errors, as some sort of “explanation” of what’s wrong with Scholze and Stix.

Essentially the claim Mochizuki is making in these first two sections is that the most accomplished and talented young mathematician in his field is an ignorant incompetent, and that everyone Mochizuki has consulted about this agrees with him. It’s hard to imagine a more effective way to destroy one’s own credibility and to convince people not to bother to try and make sense of the third section.

There’s no direct reference to the Scholze-Stix document, just a reference to Mochizuki’s own web-page about March 2018. Mochizuki has even gone to some trouble to stop anyone from accessing the Scholze-Stix document without first reading his own web-page.

As for the long discussion by Scholze and others of the problems with the proof that was hosted here and gathered here, the only apparent reference to this is

More recently, one mathematician with whom I have been in contact has made a quite intensive study of the mathematical content of recent blog posts by adherents of the RCS.

followed by

Despite all of these efforts, the only justification for th logical cornerstone RCS-identification of (RC-Θ) that we [i.e., I myself, together with the many mathematicians that I have discussed these issues with] could find either in oral explanations during the discussions of March 2018 or in subsequent written records produced by adherents of the RCS [i.e., such as the 10pp. manuscripts referred to above or various blog posts] were statements of the form

“I don’t see why not”.


Update
: To take a look at the preface, see here.

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78 Responses to ABC is Still a Conjecture

  1. lroberth says:

    Funny, I was just thinking about the conjecture and visited your page for the first time in months to see if you’d weighed in… sure enough! If I understand in layman terms, Mochizuki wants to imbue his objects with a history and this was flawed; regardless, many of us marvel how the most advanced mathematicians simply cannot agree on what should be the rules of logical argument itself. Is math fuzzy or just too hard to understand beyond this level except for just a few?

  2. Peter Woit says:

    lroberth,
    I think you’re misunderstanding the situation. The experts in this field are in close to unanimous agreement there is no proof. The math is not fuzzy and the rules of logical argument are clear. What experts in the field are finding hard to understand is why anyone is still claiming there’s a proof and why any journal would publish such a claim.

  3. lroberth says:

    @Peter, no I get it, I just don’t get it why Mochizuki doesn’t! He is an accomplished and brilliant mathematician after all, so why doesn’t a widely-agreed logical disproof work with him and his supporters? This is the fascinating meta to a lot of us.

  4. Peter Woit says:

    lroberth,
    Mathematicians are human beings, not calculating machines. Over the years, I’ve become all too familiar with how hard it is to get someone who has invested their soul in an idea to realize that it doesn’t work. The abc story is nothing compared to some others I could think of…

  5. Simon L. says:

    Oh, I wasn’t aware that this is going to be published by the EMS! Why on earth did they do this? They should have been aware of the context. Ugh.

    I cannot read the preface due to paywall. Is there any context provided at least?

  6. anon says:

    This might be nitpicking, but I wouldn’t say that PRIMS is an EMS journal. It’s published by the EMS Publishing House, but I don’t think that makes it an EMS journal (unlike, for example, JEMS).

  7. Peter Woit says:

    Simon L./anon,

    The EMS name dominates the web-page for this publication. I understand that contractually they may not be able to affect what PRIMS publishes, but I don’t understand why they haven’t issued a statement explaining the situation, but instead recently put out this
    https://ems.press/updates/2020-11-16-prims-special-issues-2021

    I still haven’t accessed the journal through the paywall, but somewhere I can’t now find did see the first page of the preface. It appears to be the only introductory material and is just a statement that Mochizuki did not participate in review of the papers, and I believe states that there were five referees. Nothing about Scholze-Stix or any indication that there’s any controversy over the proof.

  8. This journal is one of those old-fashioned ones that there used to be lots of, run by an academic institution and published by a commercial publisher who didn’t interfere. We have become too used to the modern situation in which the commercial publisher interferes far too much in the running of a journal. To add to the confusion, this commercial publisher is linked to a mathematical society, which gives the false impression that the EMS endorses what is published in this journal. Nevertheless, I would consider it to be inappropriate for either the EMS or the EMS Press (which are separate organisations) to make any comment of the kind you suggest: it would amount to exactly the kind of interference from a publisher that you would normally suggest should be strongly condemned.

  9. Florian Stümpfl says:

    Just skimmed through the first pages of the Mochizuki manuscript mentioned in the update. Nothing of substance really, all he does is slander Scholze and Stix. So nothing new.

  10. JE says:

    Just a brief quote on the EMS reaction to the announcement of the upcoming publication of Mochizuki’s papers on ABC, taken from an article published in Nature on April 3, 2020: “If the editors of the journal “waved away these criticisms” and published the paper without major revisions, it would reflect badly on them and on Mochizuki himself, says Volker Mehrmann, the president of the European Mathematical Society (EMS), which publishes the journal on behalf of the RIMS. (The EMS has no editorial control over the journal’s content, Mehrmann says, and he was unaware that an announcement was imminent until contacted by Nature.)”

    What strikes me as mystifying in this whole Mochizuki affair is not the fact that a highly reputed mathematician who has invested his soul in an idea, as Peter rightly put it, is finding big trouble to realize that it doesn’t work, even to the point of engaging in ad hominem attacks on two equally reputed mathematicians in a seemingly desperate attempt to whisk away criticism on his work, but rather that he has been able to convince a small but important group of die-hard mathematicians in Kyoto and Nottingham to ignore the criticism and side unequivocally with him, and have two other highly reputed mathematicians, like Kashiwara and Tamagawa, making the announcement.

  11. Peter Woit says:

    Robert A. Wilson,
    The EMS Press is owned by the EMS, and its president is the president of the EMS (this is Volker Mehrmann, mentioned by JE).

    Yes, editorial independence is an important principle, but no matter what the context, publishers have some basic level of responsibility for what they publish. I’m not surprised that the EMS Press felt it could not intervene in a Kyoto editorial decision, was surprised that they thought it a good idea to advertise this issue and put it in a positive light.

  12. xyz says:

    The EMS publishes Portugaliae Mathematica and Rendiconti del Seminario Matematico della Università di Padova too. Some people have focused rather on the fact that the editor-in-chief of PRIMS is Mochizuki himself.

  13. Peter Woit says:

    xyz,
    If you read Mochizuki’s new document, you’ll find his explanation of why his journal and his colleagues at RIMS were the only people “technically qualified” to referee his papers.

  14. ….and even tell people how they can buy their own special copy of the papers:

    Individuals interested in purchasing a print copy of the PRIMS special issue on Inter-universal Teichmüller Theory may contact us at preorders@ems.press.

    as if it’s a souvenir!

  15. random reader says:

    The next step seems obvious: set up an open online bulletin board, independent of the involved parties, for discussing and disputing the papers.

    If Mochizuki is confident that he is right, he can not only vindicate his credit for solving ABC, but publicly “own” (pwn!) the entire mathematics community simply by answering questions in a public, archived forum. It would be the biggest flex in mathematical history, maybe all of modern academic history.

    But if he were not entirely confident of his position, he would need to find a justification for not taking any further questions, and somehow de-legitimize any further scrutiny lest things get out of hand. The anonymity and obscurity of the refereeing process creates a circular, perpetual excuse —- if the referees approved the paper, then it must have been sufficiently well explained, and thus Mochizuki does not have to explain it to anyone else if he doesn’t want to.

    If referee comments (not identities) were published this illusion would be harder to pull off, even as a pretense.

  16. Peter Woit says:

    random reader,
    Mochizuki and the PRIMS editors have here taken unjustifiable advantage of the traditional confidentiality of the refereeing process. In their statement publishing the papers, they do not acknowledge the existence of the Scholze-Stix objection or the consensus of experts in the field that the proof is flawed. They explain that a committee was formed to evaluate the proof and imply that it considered the objections, but have refused to make public anything from this committee justifying its decision. All we have to go on is Mochizuki’s new document, and I don’t think that’s going to convince anyone.

    As for an “open online bulletin board” debate, that’s essentially already happened, hosted here April 6-19 of last year. I’m sure Mochizuki and those around him in Kyoto were aware it was going on, and they would have been welcome to participate, either anonymously or identifying themselves, but they chose not to. One could argue that the moderator (me) had a particular point of view, but I stand by the decisions I made about moderating comments, which emphasized allowing anyone who actually understood the mathematics involved to have their say. At this point, anyone who wants to evaluate this for themselves can read the earlier Scholze-Stix document and Mochizuki’s response, + the discussion here, and now Mochizuki’s new contribution. I think if one does this, it’s clear there is currently no proof of abc, and that Scholze and others have much better ways to spend their time than engage with Mochizuki’s claims that they are ignorant and just don’t understand.

  17. W says:

    Another response to Mochizuki’s earlier work that I was disappointed he didn’t engage with was David Roberts’ notes discussed
    here https://thehighergeometer.wordpress.com/2018/09/28/on-mochiukis-report-on-discussions/ which elegantly explains how the confusions/contradictions/difficulties he describes in various simplified examples in his response to Scholze and Stix can be easily resolved using the standard mathematical language of categories and functors.

    The new document contains many other simple examples trying to get the underlying point across, and if there really is an underlying point being communicated, the first order of business would be to explain why the usual mathematical language is insufficient.

    One part of the document I found interesting is when he says

    ” In fact, however, the RCS-identifications of (RC-log), (RC-Θ) do not resolve
    such issues [i.e., of relating corresponding objects in (Θ±ellNF-)Hodge theaters at
    distinct coordinates “(n, m)” of the log-theta-lattice] at all [cf. the discussion of
    symmetries in Example 2.3.1, (iii)!], but rather merely have the effect of
    translating/reformulating such issues of relating corresponding objects in
    (Θ±ellNF-)Hodge theaters at distinct coordinates “(n, m)” of the log-theta lattice into issues of tracking the effect on objects in (Θ±ellNF-)Hodge theaters as one moves along the paths constituted by various composites
    of Θ- and log-links.”

    This is striking to me, because to my understanding “tracking the effect on objects as one moves along the paths” is precisely what Scholze, Stix, etc. propose to do! This seems like conceding that their approach could be made to work, if one tracks these effects appropriately. (Which, of course, makes me think I am reading it wrong.)

    The fundamental issue here is not that working with a single concrete object and tracking how it transforms along paths (i.e. compositions of various automorphisms) is better in some abstract, logical, philosophical sense, but rather that it is much less likely to conceal hidden errors, than Mochizuki’s approach of defining an infinite number of distinct objects and identifying some of them by definition, even ones written by quite different notation, meaning that objects with quite similar notation mean quite different things.

    Another quote I found striking was

    “it should be emphasized that it is entirely unrealistic
    to attempt to obtain the inequality of the final numerical estimate as the result of
    concatenating some chain of intermediate inequalities since this is simply not the
    way in which the logical structure of inter-universal Teichmuller theory is organized.
    That is to say, in a word, the logical structure of inter-universal Teichmuller theory
    does not proceed by concatenating some sort of chain of intermediate inequalities.
    Rather, […] the logical structure of inter-universal Teichmuller theory proceeds by
    observing a chain of AND relations “∧””

    Similarly, the key point here is that (1) any argument involving combining a bunch of relations between points of different torsors for the real number can be logically translated into an argument involve combining a bunch of equalities and inequalities of real numbers by fixing an identification of each torsor with R, (2) any symmetries of the situation that existed prior to doing this are preserved, at the cost of introducing terms in the formulas describing those symmetries to account for isomorphisms between torsors that are incompatible with the identifications, (3) an argument written this way is less likely to conceal errors.

  18. Charlie G says:

    Not a mathematician here. All of this is far over my level, but while ABC is still a conjecture, I wonder if Mochizuki’s attempted proof has revealed any techniques that are of more general value for other purposes – in the way Wiles’ proof has – I gather – opened up useful avenues for other research.

  19. anon says:

    To be fair to Mochizuki, his manuscript has a section two for non-specialists like myself which shows he’s sincere in trying to get people to understand his ideas.

    Initially it didn’t make sense to me and I was tempted to arrogantly dismiss it as gibberish; but after reading it a few more times its main points are becoming profoundly clearer: how apparent contradictions disappear in mathematics once the overall structure is restructured in some part, despite the total information content remaining the same.

  20. Peter Woit says:

    anon,
    Mochizuki likely is sincere in his belief that those who think his proof is flawed are making elementary mistakes in their reasoning. The problem is that this is not even slightly credible. If you know anything about how the best mathematicians work, his trying to explain why Scholze disagrees with him by listing elementary mistakes a student might make comes off as just bizarre.

    One thing that strikes me about this document is that Mochizuki sadly now appears to lack colleagues willing or able to tell him not to do things that destroy his own credibility.

  21. André says:

    Peter, Charlie G,

    the question whether Mochizuki has produced valuable insights or tools is the central one. The question will decide whether, in the future, people will treat him like Wiles or like a crackpot. Is there any information on this? As an outsider, I have a second question: the term “inter-universal” makes no sense at all. Why is it called that way?

  22. Pedro says:

    From the 65-page document:

    “There is a fundamental difference between criticism of a mathematical theory that is based on a solid, technically accurate understanding of the content and logical structure of the theory and criticism of a mathematical theory that is based on a fundamental ignorance of the content and logical structure of the theory.”

    That is totally right. But pretending that the time employed by Stix-Scholze (who have more things to do) just led them to “fundamental ignorance” says a lot about his thinking process. And about the clarity of his writing.

    Also all the praise of PRISM and its history and the editors in the preface to the publication, and dismissing other journals (Annals, Inventiones…) as unsuitable for publication only point the reader to doubting the refereeing process.

  23. The Vlad says:

    What’s the deal with the odd stylistic structure of the Mochizuki document (i.e. littered with bold text, italics, randomly dispersed quotation marks, no numbering of equations)? I’ve never seen anything like it in the math literature. Is this just some quirk of his or is it standard in some math fields, or something else? I found it very distracting.

  24. Jon1 says:

    Apparently there are also some people interested by Mochizuki’s work in Lille, he will answer questions in a seminar in a few weeks and other folks will talk there later on, see this page https://math.univ-lille1.fr/d7/sarit

  25. UF says:

    Let me draw attention to p.58/ step 3 of the descent process in Mochizukis manuscript, which is (very superficially) perhaps the most counterintuitive of the descent process described by Mochizuki: Here one passes from (0,\circ)^\perp to (0,0)^\perp. How can this be “passing to weaker information”, if we go from unspecific vertical \circ position to the specific vertical position 0?

    As far as I understand, the point is the following: we can in the reconstruction algorithm reconstruct all data \pi_1, monoids and the action on monoids etc. up to some indeterminacies. However, keeping in mind the algorithm these data were reconstructed from, these reconstructed data (are claimed to) contain more information than the information contained just in the output data (forgetting about the algorithm constructing this output). One may thus decide to forget some of this information, and replace some (not all) of the reconstructed objects by abstract, unrelated ones. In this step 3 one replaces reconstructed monoids by abstract monoids and builds with the algorithm some “shells” in these abstract monoids (instead of in the reconstructed monoids). The claim is that the reconstruction algorithm still works (up to further indeterminacies Ind2) even if the monoids are unrelated, and not reconstructed like other data showing up in the algorithm.

    If one accepts this, one can take advantage of this by taking as the unrelated monoid the specific one which is shared via the theta-link from the 1-column. Whereas before (in step 2) the input data into the algorithm was vertically invariant, it is no more so with this specific choice of abstract monoid. On the other hand the (via step 3) reconstructed data has now some relation to the q-pilot in the 1-column: the monoid in which the shells are constructed comes from the q-pilot via the theta-link. (Breaking this vertical invariance of the input data is essential to prove anything meaningful related to the q-pilot, which is located at a specific vertical position).

    This may also indicate a reason to pass from X to \pi_1(X): In IUT, one decides to share across the theta-link a “part” of $\pi_1(X)$ (or X), which is “easily” describeable in terms of \pi_1(X) (but not so much in terms of X), namely for instance the monoid on which \pi_1(X) acts (this monoid can be viewed as “part” of \pi_1(X), since a copy of it can be reconstructed by $\pi_1(X)$).

  26. Peter Woit says:

    The Vlad,
    It’s a quirk of Mochizuki’s. Again, I think an increasingly large part of his problem is that he has no one around him willing or able to advise him to not do things that don’t help with his credibility.

    Jon1,
    This is part of a program organized by Collas out of Kyoto, see
    http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/IUT-schedule.html
    In all the materials linked to there, again, no mention even of the existence of Scholze-Stix, much less any discussion of the problem with the proof they have pointed out. The Kyoto standpoint is pretty uniformly to try and pretend that Scholze and Stix don’t exist.

    Charlie G/Andre,

    One big problem with publishing the IUT papers without acknowledging that the consensus of the field is that they contain serious error is that people will now have trouble using anything in those papers to build on and do something else with. If you use results from those papers in a proof, you’re going to need to show that they’re independent of the problematic parts of the papers if you want to publish your results outside of Kyoto. Essentially this may create a fork in the math literature.

  27. xyz says:

    Charlie G/Andre,

    You’re raising a very good question, the answer to which (so far) strengthens the sceptics’ case: Months after Wiles’ first announcement, many original, universally accepted results were produced on the basis of his methods. (That was already before the full proof of the part of modularity conjecture required for Fermat’s Last Theorem). In the claimed proof of the ABC-conj. nothing remotely like this has happened during all those years after its announcement. This may indicate something about the applicability/generality/strength of the tools and insights employed for the claimed proof.

  28. A. Rukhin says:

    “ Both Professors Kashiwara and Tamagawa have an outstandingly high international reputation, built up over distinguished careers that span several decades.”

    The mention of these two stands out as somewhat interesting: what have these two said publicly about the latest developments? Apparently, they supported the decision to publish…

  29. Peter Woit says:

    A. Rukhin,
    Kashiwara and Tamagawa chaired the committee that decided to publish the papers, and appeared at the press conference last year announcing this. Mochizuki’s main argument at this point is to pit their reputations against Scholze’s, and politically this may work well in Kyoto, less well elsewhere.

    Kashiwara is not an arithmetic geometer, but Tamagawa is, and he should be in a good position to evaluate Scholze’s argument that the proof is flawed. That he made the decision to publish implies that he understands why Scholze is wrong and how the proof overcomes Scholze’s objections. Tamagawa is at RIMS, has collaborated with Mochizuki, and should be very interested in Mochizuki’s IUT ideas and how they prove abc. As far as I know though, he has not spoken at any of the various IUT workshops, or written anything about this.

    The one thing that has been missing here has been an explanation of how the proof overcomes Scholze’s objections from one of the people other than Mochizuki who supposedly understands the proof. This is something that Tamagawa should be able to provide, and it would be very helpful if he were to do so.

  30. Timothy Chow says:

    lroberth: You used the term “logical disproof,” but the Scholze-Stix argument falls just short of that. As I understand it, they are not explicitly claiming that the notorious Corollary 3.12 is provably false. What they are saying is that Mochizuki’s argument fails to establish Corollary 3.12. That is, the objection is that there is a glaring gap in the argument. In theory, the question of whether there is a gap in a mathematical argument that B follows from A is an objective one, because someone who says there is no gap should be able to fill in all the intermediate steps in the chain of reasoning from A to B on demand, to the point where even a machine could check that each step is correct. The catch is that this process might require, say, one million intermediate microsteps, so in practice, mathematicians don’t actually produce all the intermediate steps explicitly. An expert might be able to see how to get from A to B without any further explanation; a less expert mathematician might require ten intermediate macrosteps spelled out; a graduate student might require a hundred intermediate mesosteps, and so forth.

    What happens if mathematician X claims that B follows from A but mathematician Y does not see it? If X and Y have a healthy relationship then X will spell out some intermediate steps for Y’s benefit, and will keep increasing the amount of detail until either Y is able to see how to supply the remaining details, or X realizes that there is a problem with the argument. But if X and Y do not have a healthy relationship, then X might lose patience at some point and stop cooperating. That is basically what has happened here, except that a lot more than two mathematicians have gotten involved, and a sizable sub-community has become alienated from the rest of the community.

    These kinds of rifts are rare, and it is not just laypeople who are perplexed. Most mathematicians are watching the IUTT fiasco from the sidelines with amazement. But perhaps what we should really be surprised at, and grateful for, is that such rifts don’t happen more frequently. Human nature being what it is, sociological rifts are to be expected, and it is a wonder that the mathematical community functions as well as it does.

  31. @W

    Luckily Peter Scholze confirmed in the previous extended discussion that he and Stix had internalised and compensated for Mochizuki’s hangups that I describe around “copies”. My observation is only a very tiny contribution trying to unravel what’s going on.

    There’s stuff in the new document that utterly mystifies me (I’ve not read or even looked at the whole thing). Section 2.3 for instance is … Not Even Wrong.

  32. André says:

    The 65 page document has a clear message: to understand what IUT is, you need to think about it for 6 months or 3 years (section 1.6), come to Japan (sections 1.6 and 1.9), talk to “M” for a long time, and think about it, discussing with “M”, until you understand it. As a result, the worldwide number of people who understand it is of the “order of 10” (section 1.6), centered around Kyoto.

    The math examples that “M” gives are so simple that nobody will ever believe that Scholze and Stix stumbled over such issues.

    “M” clearly states, page over page, that Scholze and Stix are stupid. In fact, the document states that all his critics are stupid. Above all, the document shows that “M” is abusive towards people of different opinion.

    The whole paper is clearly written by an aspiring leader of a sect. There is no need to come to Japan to understand this. I have lived in Japan, and this type of sect leaders is common there. It is sad to see that such people exist in mathematics, and especially sad to see that they exist at Kyoto University, one of the best in Japan.

  33. Peter Woit says:

    André,

    I’m not sure that “sect” is the right name for the weird phenomenon going on here, but if it is, this “sect” has not been very successful at gaining adherents. Back in 2018, according to Fesenko, 12-18 people had studied the proof sufficiently to understand it and believe it correct, see
    https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/
    Two and a half years later, Mochizuki now describes the number of people who understand IUT as “roughly 10”.

    A weird aspect of this story is that almost no one admits to actually understanding the proof themselves. Back in 2014, Mochizuki described the main people working on checking the proof as Yamashita, Hoshi and Saidi. Many years later, the only documents not from Mochizuki that I’m aware of explaining the proof and addressing the Scholze-Stix argument are from Yamashita. Hoshi has also written something pre-Scholze-Stix in Japanese, and participated in discussions with them. But as far as I can see, there’s no evidence that anyone besides Yamashita and Hoshi can publicly vouch for the proof and answer questions about it. PRIMS had some undisclosed number (“several”) of referees (quite possibly including Hoshi and Yamashita), but it’s unclear how many of them could vouch for the whole proof (as opposed to just checking some piece of the IUT papers).

    If this is a “sect”, it’s a small one, arguably getting smaller.

  34. Peter Woit says:

    Timothy Chow,
    I don’t think your portrayal of the current situation as due simply to a lack of detail is accurate.

    Scholze-Stix not only say that there is a gap, but give an argument for why the gap cannot be closed using Mochizuki’s methods. The debate hosted here last year mostly revolved around whether that argument was bullet-proof or whether there might still be some way to get Mochizuki’s methods to work. What did not show up here (or anywhere else) was anyone claiming they knew how to do this and could provide details.

    It’s not the case that Mochizuki has lost patience and stopped engaging with the Scholze-Stix argument. He just put out 65 pages supposedly doing this. The problem is that what he is doing is classic behavior of someone who has lost an argument but won’t admit it and wants to fight on. I think those 65 pages will do a lot to convince mathematicians that he does not have a proof. Scholze and Stix are the ones who now seem to have stopped responding, for the good reason that Mochizuki has no serious counter-arguments, just insults.

    By the way, I just noticed this from Scholze, in case anyone thinks that the publication might have changed his mind:
    https://t.co/0zbGxpgAfy?amp=1

    There has also been a long discussion of this at Reddit, see
    https://www.reddit.com/r/math/comments/lz4ccm/mochizuki_strikes_again/
    I recommend the comments by “whisperfiends”
    https://www.reddit.com/user/whisperfiends/
    who makes an attempt to follow Mochizuki’s claims that Scholze-Stix are confused about something basic, is suspicious that it is Mochizuki who is confused about this.

  35. W says:

    @Charlie G and André:

    There exist other works which apply Mochizuki’s language to other questions of anabelian geometry. However, my understanding is that this language is not essential, and the works would be shorter and clearer without it.

    @The Vlad:

    If you look at his old papers like “The local pro-p anabelian geometry of curves”, when he was doing work that everyone agrees is correct (and brilliant), you can see some strange formatting, but much less (mostly a lot of italics – also the equations aren’t numbered, but that isn’t too strange in the more algebraic fields of mathematics).

    @Peter Woit

    I don’t think there will be so much of a problem with people trying to cite the papers but avoid the problematic parts, simply because there doesn’t seem to be much worth citing in the unproblematic parts. But there certainly could be a fork in the mathematics community regardless.

    @David Roberts

    My interpretation of what Peter Scholze says there is a bit different. I just bring your observations up because they are very helpful to me!

    Isn’t 2.3 also the exact kind of thing you discussed – gluing topological spaces is a colimit of a diagram, and if you define diagrams correctly you can’t get into trouble based on whether you identify things or not?

  36. JE says:

    As of now, the English-speaking media have turned their backs on the publication of Mochizuki’s papers. In fact, one can hardly find any mention of it other than on this blog or reddit. The situation vastly differs from last year’s, when many articles quickly announced their publication. Be it the result of poor communication strategies on the part of the EMS or exhaustion, Mochizuki’s attempted proof of the ABC conjecture seems to be a dead issue in Western media’s terms. Coupled with his 65-page manuscript, containing plenty of arguments from authority, implicit ad-hominem attacks and appeals to herd behavior, the damage he is inflicting on his reputation by either refusing to accept that the proof is flawed or being able to provide valid counter-arguments is enormous, as Peter said.

    And this is really unfortunate, because he has proved to be an extremely talented mathematician. The fact that no senior mathematician or anabelian geometer in Kyoto has been willing or able to convince him otherwise may be hard to grasp by Western standards. Mochizuki is a highly respected mathematician in Japan, who was appointed as editor-in-chief of PRIMS few months before his papers on IUTT were released (and from being excluded from the editorial committee, paradoxical as it may seem), and one of the most prominent figures at an institution with a strong background of success, including Fields medalists Hironaka and Mori, among many other respected figures.

    At this point, I seriously doubt that this affair may even create a fork in the math literature. Any figures, other than Mochizuki, who have implicit or explicitly sided with him may slowly backtrack and even take advantage of the media impact of this affair to propel their careers in Japan or elsewhere. The number of mathematicians who have published explanatory reports on his proof, like Yamashita and Hoshi, has even decreased in the last few years. And the highly-respected figures who have been involved in the process (as co-editors-in-chief) have solid mathematical careers and are not facing a reputational risk in any way comparable to Mochizuki’s.

  37. @W

    The idea that having an equivalent subcategory of Top where there is only a single terminal object means that the somehow [0,1] (called \mathbb{I}) becomes the same as S^1 (called \mathbb{L}) is laughable. The same argument would imply that every single topological space collapses to a single point, since every one-element subspace is the “same”, according to Mochizuki (as in: why are the endpoints of [0,1]—in more proliferation of useless notation, called \alpha and \beta—singled out? you could replace them by any pair of points of [0,1], or any space). The argument confuses taking a colimit with taking a skeleton, as far as I can tell around the verbiage (why do we need to be reminded of the standard topology on [0,1], and that it is a topological manifold with boundary?)

  38. Timothy Chow says:

    Peter, I pretty much agree with what you say. All I was trying to do was explain to lroberth that “logical disproof” isn’t quite the right term. We have seen how Taylor Dupuy has argued that the Scholze-Stix argument isn’t, as you put it, bullet-proof. I have privately asked other experts for their opinion, and a common response I get is a shrug and a comment that Mochizuki hasn’t provided a clear argument. I think that this behavior (as well as the behavior of Mochizuki’s quiet supporters) is more puzzling if there really is a “logical disproof,” but it’s easier to understand if there is a gigantic gap.

    As for “lose patience,” perhaps I should have said “lose his cool.” Also I didn’t say “stop engaging”; I said “stop cooperating.”

  39. S says:

    I’ve noticed an (understandable) resistance to taking Scholze/Stix’s word for it, among interested onlookers (mostly non-mathematicians, I think). If I can’t engage with Mochizuki’s material or Scholze/Stix’s objection to it, I should remain neutral, goes the thinking. To these people, I would emphasize the fact that it’s been 8 years, and if IUT were actually right, it’s overwhelmingly likely that someone would have written it down in an understandable and uncontroversial form by now. The specific objection of Scholze and Stix brings this point into greater focus, because we know exactly which aspect of the theory must be zoomed in on, in order to clarify matters. There is just no plausible explanation for why this hasn’t happened yet, except that it can’t be done.

    Things could change in the future if Mochizuki or someone else provides new mathematical content. But that wouldn’t retroactively change the fact that there’s no valid proof now.

  40. Jim Eadon says:

    Peter,
    The publishing house must surely be aware of the explicity identified error(s) in the proof, identified due to Scholze and Stix.
    If they do publish the paper, and they fail to acknowledge the identified error(s), then are they not fraudulent, in some way, for selling fraudulent documents with false advertising?
    Could they be accused of (even illegally) misleading consumers? There are legal consumer-rights acts in many countries, where you’re not allowed to mislead consumers.

  41. Timothy Chow says:

    I just noticed that on March 18, Mochizuki is conducting an “Interactive Q&A Session on the Essential Logical Structure of Inter-universal Teichmüller Theory.” It seems that this session is not open to the public, since the List of Participants page says, “In order to ensure a cohesive and focused working group, note that participation is restricted to a limited number of participants, and that registration is done by invitation only.” I wonder if the session will be recorded and made available to the public later. Over 30 people are listed as participating; I wonder if any of them will dare to ask a difficult mathematical question? Failing that, will anyone request that the “mathematicians with whom I [Mochizuki] have had technically meaningful discussions concerning the assertions of the RCS” (saying that Scholze and Stix’s objections are “obviously completely mathematically inaccurate/absurd”) identify themselves?

  42. Peter Woit says:

    Jim Eadon,
    I know for a fact that the EMS Press is aware of the issue, am somewhat mystified by the fact that they promoted this issue of PRIMS and have so far not apparently done anything to address the problem. Likely they have contractual arrangements such that they can’t interfere in editorial decisions of PRIMS, but I find it hard to believe there is nothing they can do here.

  43. The Vlad says:

    @W

    I took a look at the M paper you mentioned, and even there I find his style and the paper structure highly unusual (e.g. no abstract, acknowledgements section in the introduction, the unnecessary quotation marks and italics everywhere). He also makes some sweeping meta-mathematical statements, such as:

    “Moreover, it is the feeling of the author that,more than the technical details of the state-ment of TheoremA, it is this fact – i.e., that the Grothendieck Conjecture for hyperboliccurves is best understood not as a global, number-theoretic result, but rather as a result inp-adic Hodge theory – that is the central discovery of this paper.”

    Putting all this together, it paints a picture of M abandoning or disregarding standard math communication conventions in favor of his own idiosyncratic style (this also fits with his refusal to travel overseas to convince others of his work). As PW points out, this seems to indicate an inability of mentors, colleagues and.or reviewers to dissuade M from behaving in a way that may harm his reputation.

  44. Peter Woit says:

    The Vlad,

    Mochizuki’s typographical idiosyncracies are easily ignored, and his not liking to travel isn’t unusual. I actually like the “sweeping meta-mathematical statements” that tell you what the author thinks the “big picture” is, often find them helpful to understand what the author is doing. When the papers first came out, they were long and technical, beyond my abilities or patience to try to read. Later Mochizuki came out with
    http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf
    which did give a readable overview of what he was trying to do.

    After talking to some experts it became clear that they could understand what he was trying to do, they just couldn’t understand whether what he was trying to do really worked, because crucial parts of the argument were unclear and not well explained. Scholze changed everything by pointing to specific place where the argument could not work for general reasons, and most experts are now convinced by that.

    So, the real problem here is not Mochizuki’s idiosyncracies, it’s his behavior when faced with someone pointing out a flaw in his argument.

  45. For what it’s worth, I think my previous comment was not quite right. I thought through what was going on in this weird example a bit more in a blog post here. Now I think it’s just a straw man argument, at best.

  46. mahmoud says:

    Timothy Chow,
    as Scholze repeatedly said himself in prior comments to this blog Mochizuki could easily clear up the issue regarding the Scholze-Stix objection by pointing out where in their argument a non-commutating diagram shows up that in fact will commutate provided one replaces it with one of his “mutually alien copies” diagrams. Instead, however, he keeps putting out new documents with increasingly bizarre analogies (the latest where he substitutes a real number for the integral sign is a new record).

    Talking about a “gap” in the proof or the S&S objection as a “logical disproof” muddles what is a quite simple situation: there just is no proof that has been communicated in an intelligible fashion. Thus I also take issue with Woit saying Mochizuki’s idiosyncracies can be ignored; in fact his highly idiosyncratic style (of communication and – apparently – thinking about perfectly conventional mathematical topics) has prevented any progress on understanding his purported proof from the very beginning.

  47. DrB says:

    Has someone taken a look at this recent (Nov 2020) “RMS preprint” by Mochizuki et. al.: http://www.kurims.kyoto-u.ac.jp/~motizuki/Explicit%20estimates%20in%20IUTeich.pdf

    It is stated that it contains a proof of Szpiro’s conjecture and an alternative proof of Fermat’s last theorem.

  48. Peter Woit says:

    DrB,
    This depends on the flawed proof in the IUT papers, so is also not a proof. It will be interesting to see if any journal other than PRIMS is willing to publish it.

  49. John says:

    I understand where Timothy Chow is coming from. Someone not familiar with academic mathematics might wonder why a complex paper can take so long to verify. The reason is that most papers will have a dozen gaps in them, with the understanding that an expert can easily fill them. So when a mathematician encounters corollary 3.12 his first instinct is not ‘this argument is non-sensical’ but rather ‘I don’t have the background required to fill in the gaps’. So, how do errors ever get found? The clearest but most difficult way is to prove that one of the statements in the paper is wrong, this is not something Scholze and Stix did, and it can’t find out wrong proofs of true statements either. Another way is to take an example of the things in the statement, one where you understand how to fill in the gaps better, and show that some of the logical implications does not follow in this example, this is less decisive but often does point to a genuine gap. The way Scholze and Stix did if I understand correctly is to introduce modifications to the theory, so they are talking about different objects, but these objects have enough properties in common with the original theory that the arguments seem to follow, but the conclusions are false. There are two ways this can happen, either the argument is wrong in some way it can’t be filled easily somewhere in one of those gaps, or the modified theory is different in some important ways (contained in those gaps in understanding). My understanding is that Mochizuki claims the second of course, but he hasn’t pointed any ‘important’ difference, important is a bit of a value judgement, so instead you could say he hasn’t pointed to one that helps understand the gaps better. So most people at this point believe the argument to be either wrong or at least not convincing.

    What I find surprising about this is that several serious mathematicians seem to be convinced of Mochizuki’s proof in ways that they have a lot of trouble communicating, I don’t recall anything like that happening before.

  50. Peter Woit says:

    John,
    About
    “several serious mathematicians seem to be convinced of Mochizuki’s proof in ways that they have a lot of trouble communicating”
    I’d put it a bit differently. There is literally nothing written down by a mathematician other than Mochizuki engaging with the Scholze-Stix argument and showing that it is wrong, and Mochizuki’s engagement with the argument has no credibility.

    As far as I know, Yamashita and Hoshi are the only two people besides Mochizuki supposedly understanding the proof and able to provide a counter-argument to Scholze-Stix, and Hoshi has not written such a counter-argument down. Yamashita has written a long survey of the proof, but it makes no explicit reference to Scholze or Stix or their argument.
    Mochizuki’s own attempts to engage with Scholze-Stix involve a large amount of absurd ad hominem attacks on their competence, coupled with a refusal to directly reference and deal with the core of their argument. This is the sort of thing you expect from someone who has no real counter-argument. As a non-expert trying to read the more technical parts of Mochizuki’s supposed counter-argument, I can’t follow the details, but I also can’t find anywhere where he engages with the arguments Scholze has made here. He goes to a lot of trouble to not use Scholze’s name and not refer directly to anything he has written. Checking with people who are experts to ask if they see something I’m missing I’m told it’s not there.

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