abc News

[For those not up to speed on this story, see blog posts here and here from last December, as well as comments to those posts.]

The last couple months I’ve heard reports from several people claiming that arithmetic geometers Peter Scholze and Jakob Stix had identified a serious problem with Mochizuki’s claimed proof of the abc conjecture. These reports indicated that Scholze and Stix had traveled to Kyoto to discuss this with Mochizuki, and that they were writing a manuscript, to appear sometime this summer. It seemed best then to not publicize this here, better to give Mochizuki, Scholze and Stix the time to sort out the mathematics and wait for them to have something to say publicly.

Today though I saw that Ivan Fesenko has put out a document entitled Remarks on Aspects of Modern Pioneering Mathematical Research. It refers in footnote 18 to:

two recent texts by Sh. Mochizuki, ‘Report on discussions, held during the period March 15–20, 2018, concerning inter-universal Teichmüller theory (IUTCH)’and ‘Comments on the manuscript by Scholze–Stix concerning inter-universal Teichmüller theory (IUTCH)’, July 2018

I haven’t seen these two texts, or the Scholze-Stix manuscript. What I have heard about them is that Scholze-Stix identify what they see as a specific, serious flaw in the proof, and that Mochizuki denies that this is a problem or that his manuscript needs to be revised. Presumably, after the two sides try and sort this out amongst themselves, at some point we’ll see something publicly available describing the details of their disagreement.

Fesenko’s document has a lot of unpleasant things to say about those who have written anything at all skeptical concerning Mochizuki’s claimed proof, mostly without naming names. He refers to journalists and “US bloggers” as having produced “ignorant absurd articles and posts”, presumably has someone other than me in mind since the information posted here about this I believe has been quite accurate and of reasonably high quality. The one negative reference to identified mathematicians is in the text with footnote 18 pointing to Scholze and Stix, which says:

Several researchers, who could have become potential learners of IUT and then progressed to become experts, declined invitations to participate in the IUT workshops. Some, affected by negative emotions, broke professional rules of conduct and made public their ignorant and sometimes intolerant opinions. Tellingly, the only questions produced were shallow and misplaced and they were communicated only after several years of requests to do so.

Peter Scholze is by far the most talented arithmetic geometer of his generation, a sure thing to receive a Fields Medal at the ICM in a couple weeks. That his questions about Mochizuki’s proof were “shallow” seems highly unlikely, to me at least.

Much of Fesenko’s article concerns the question of whether contemporary mathematical research is too narrow and unambitious, devoted to minor improvements and producing lots of publications. This is a serious issue, one though where other fields than arithmetic geometry (e.g. fundamental physics) are in a much worse state. Fesenko tries to make the difficulties mathematicians have had with Mochizuki’s claims about his IUT research an exemplar of this problem, but this seems to me misguided. There are quite good reasons for why experts have been skeptical about IUT and the supposed abc proof, reasons which will be conclusively vindicated if Scholze and Stix turn out to be right. Ironically, an excellent example of the kind of fundamental breakthrough that Fesenko is asking for is Scholze’s own ground-breaking work over the past few years.

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24 Responses to abc News

  1. mJ says:

    What a wonderful article! Thanks Woit!

  2. anon says:

    Fesenko also confirms that Mochizuki’s papers were indeed submitted to PRIMS.

    The whole IUT story has plenty of bizarre features. I find Fesenko’s extremely aggressive promotion of the theory very strange. I wonder if Mochizuki himself is happy with it…

  3. Thank you, Peter. It was very frustrating to have people winking and nudging each other around the internet when hinting at what you discuss, because they were in the extended personal circles of mathematical celebrities and got to hear it in person.

    Hmm, now I see totallydisconnected deleted their discussion of the rumour…

  4. No wish to be harassed says:

    Being criticized by Ivan Fesenko for breaching “professional rules of conduct” is the equivalent of being criticized by Trump for “not telling the truth” – it should be worn as a badge of honor. You do have to admire his political mastery to somehow have wrangled the biggest grant ever awarded to a mathematician in the UK to study “IUT,” a theory that at this point resembles nothing more the emperor’s new clothes. (Explanatory note: grants in the UK in pure math are decided by non-experts who may not even be pure mathematicians.) I can see plenty of reasons for someone in such a position to act in a manner not entirely aligned with the pursuit of truth. This is not a document to be taken seriously, and anyone with basic knowledge of the people involved will see through it immediately.

  5. Peter Woit says:

    I see no reason to doubt that Fesenko believes what he says and is motivated not by money, but by enthusiasm for the new mathematics that Mochizuki has produced. It would be a good idea for him to take a more charitable view of the behavior of others who, also out of love of new mathematics, have been doing the best they can to try and understand and check the proof.

  6. T says:

    As an outsider (physicist/mathematician) who has been watching this story with interest, I have to say I can’t completely disagree with these statements by Fesenko:

    “Unusually for mathematics, some mathematicians felt appropriate to publicly criticise IUT and its study without having studied it in any serious way. Negative online criticism went always in a very vague form without any single concrete mathematical evidence. Sometimes it was hostile.”

    Among others, this post comes to mind, which was eventually closed to comments with the note “If you don’t agree with me feel free to start your own blog.”

  7. Anon says:

    The is a typo in the article name: Remarks on Aspects of Modern Pioneering Mathematical ResEarch

  8. Also don't wish to be harassed says:

    “It seemed best then to not publicize this here, better to give Mochizuki, Scholze and Stix the time to sort out the mathematics and wait for them to have something to say publicly.”

    And the production of some document by Fesenko was enough to change your mind about this? Seriously?

  9. Peter Woit says:

    Thanks fixed.

    The point is that Fesenko has now chosen to publicize the Scholze-Stix story (in the form of an attack on them). Now that this is public I think it’s fair game to blog about, and it’s actually a good idea to make public a better explanation of what is going on than just what Fesenko has done.

    What might not be obvious about that blog posting is that Cathy O’Neil is both a number theorist herself, wife of an expert in arithmetic geometry and friend of many other experts in the field. What she wrote was based on detailed inside knowledge of the process leading experts in the subject were going through trying to understand the proof during the first few months it was available, and the problems they were running into. She waited to write this until it had started to become clear to experts how difficult it was going to be to understand and check this proof in conventional ways. In retrospect several years later, her analysis of the problems with the proof was quite on target.

  10. @Also don’t wish to be harassed:

    well, Fesenko did just out a manuscript due to Scholze-Stix and the existence of a written-up response by Mochizuki. Given the rumours that ‘two prominent and very well-regarded mathematicians have isolated a specific and serious error in Mochizuki’s proof of the abc conjecture’ were already documented online, one might put together the facts independently of Peter writing anything of what he had heard.

  11. ADWTBH says:

    Surely this blog post is going to publicize the authorship of the not-yet-publicly-announced [SS] manuscript massively more than footnote 18 on page 6 of Fesenko’s document ever would have. Perhaps what you say about it being “fair game” is true by some standard of journalism (though that same standard of journalism might have required asking [SS] for comment — did you?). But with respect to the collegial standard of not publicly broadcasting mathematical work that the authors have not yet announced, this post falls massively short.

  12. Peter Woit says:

    Again, I’m not the one who made public the existence of the Scholze-Stix manuscript. Fesenko did this, together with a characterization of it as raising only “shallow and misplaced” questions about the proof. If you’re unhappy about this, you should take it up with the person responsible.

  13. Sometime number-theorist says:

    I had not read Cathy O’Neil’s blog post before, but it is the exact argument of which I was wondering why there weren’t more mathematicians making it. To put it in the simplest possible terms: the burden of proof is on Mochizuki — no matter how often he claims he has proven ABC, or no matter the loftiness of the terms in which he expounds the correct manner to become an “expert” in his proof.

    The problem in the dialogue occurs when one party starts making assumptions that the other party does not accept, and predicating their arguments on these assumptions. A consistent failure to recognize the fact that one makes such assumptions, for me is a clear sign of intellectual dishonesty.

    Now the above statement may point to a reason why mathematicians by and large have been loathe to take a harsh stance on Mochizuki (O’Neil apparently excepted). It is not a nice thing to accuse another of intellectual dishonesty. The insult is hurled about far too much, especially on the internet, and it has a tendency to escalate matters.

    Perhaps the thing with accusations of intellectual dishonesty is not so much that they can’t be correct, but that they can’t be productive. No one can be expected to take such a qualification in good stride. So then the question becomes: what is the honest and correct way to make the same point? Because if you can’t call out this kind of trickery for what it is, perhaps out of a fear to be perceived as a troublemaker or a name-caller, then you are not taking responsibility for the integrity of your field.

    I think this is an important question, although I haven’t thought about it for long enough to pretend that I have any solid answers. It might even be one of the great questions of our time: from fake news, to fake physics, and now on to fake mathematics.

  14. Peter Woit says:

    sometime number theorist,

    I don’t think “intellectual dishonesty” is a good way to characterize the problem with Mochizuki’s proof, or a term that Cathy herself used or would use . I have no doubt Mochizuki was and is sincere in his belief that he both has a proof, and has written it down carefully so that others who apply themselves can follow it and check it. It’s not an unusual problem though that an author needs to be told to rewrite a manuscript since in its current form others cannot understand it or check it with some reasonable amount of effort.

    The new twist to this story is that evidently Scholze long ago had specific criticisms of the proof, and that these never got answered by Mochizuki or addressed by the referees. So at least one expert could follow the proof well enough to identify a possible problem. If you want to look for a past failure in how mathematics is supposed to be done, it seems to me that this is the place to look. It’s great though that things are now working as they should, with the mathematical issues getting appropriately discussed and debated by those expert in them.

  15. Sometime number-theorist says:

    @Peter Woit:

    I think it is exceedingly hard for both you or me to say whether Mochizuki is convinced whether he has a proof or not. More generally, if you’re defining intellectual honesty in terms of the mental contents of the individual in question, you run into the problem of having a subcategory without a feasible criterion for deciding whether an object belongs to it or not. We need to go by what we can observe as much as possible, I would say, even if we don’t want to be called behaviorists.

    (Consider this question: can an individual delude themselves? If yes, isn’t is possible for an individual to think on a conscious level that they have proven something, while being at a deeper level aware that they haven’t, or might not have? And isn’t an indication of such a state that such an individual reacts aggressively, evasively, or in any kind of obstructing manner, when the point of the integrity of the proof is pressed?)

    I really think that mathematicians who are reluctant to call Mochizuki out on intellectual dishonesty should ask themselves the question at what point (or under what circumstances) they would be prepared to do so. This has gone on for more than five years now, and the amount of rhetoric from the IUT camp hasn’t let up as of yet, and on top of that, substantial amounts of tax-payer money start to get involved as well.

  16. Peter Woit says:

    Sometime number-theorist,
    To be fair to Mochizuki, the over-the-top rhetoric generally hasn’t been coming from him. It’s quite possible we’re going to soon see a resolution of this, as experts will see that either Mochizuki has a convincing answer to whatever problems Scholze-Stix are raising or he doesn’t. At that point, people may have different takes on how to characterize a possible refusal by Mochizuki to recognize a problem, with “intellectual dishonesty” one of the possibilities.

  17. Sometime number-theorist says:

    @Peter Woit:

    Fair enough. But let me just add, I do think that the behaviour of Mochizuki’s defenders partly reflects on Mochizuki himself. And if people here are posting anonymously out of a fear of harassment, then I think that that is also a signal that something else than a pursuit of truth might be going on here.

  18. Interested says:

    Fesenko’s article mentions some recent papers by Mochizuki in the footnotes:

    “See two recent texts by Sh. Mochizuki, ‘Report on discussions, held during the period March 15–20, 2018, concerning interuniversal
    Teichmüller theory (IUTCH)’and ‘Comments on the manuscript by Scholze–Stix concerning inter-universal Teichmüller
    theory (IUTCH)’, July 2018”

    However, I am unable to find these anywhere on Mochizuki’s website. Does anyone know whether they are available anywhere?

  19. Peter Woit says:


    As far as I know, these and the Scholze-Stix manuscript have not been made publicly available, and the intention was that they would not be made public until each side had a chance to fully address the other side’s points. I don’t know why Fesenko chose to discuss them publicly.

  20. T says:


    FWIW, I’m aware (I’ve been aware) that C. O’Neil used to be a number theorist, and that she knows some experts. My (humble) disappointment stands.

    It appears that Mochizuki has been open to correspondence and technical questions about his work all along. It sometimes seemed (from a distance, to an outsider like me) that some of the tension/friction stemmed from Mochizuki’s not having visited various institutes in person to explain his theory. This implicit demand (or, again, perhaps my uninformed, outsider feeling of such) was surprising and disappointing to me.

  21. T,

    “It sometimes seemed (from a distance, to an outsider like me) that some of the tension/friction stemmed from Mochizuki’s not having visited various institutes in person to explain his theory. This implicit demand (or, again, perhaps my uninformed, outsider feeling of such) was surprising and disappointing to me.”

    I’m a physicist, not a mathematician, so I might be talking from another side of a cultural divide, but the demand that Mochizuki goes around explaining his theory seems entirely natural to me. Putting a paper on the arXiv is not enough, you definitely need to actively defend it, especially if people are interested in it and cannot understand it.

  22. Peter Woit says:

    T/Mateus Araujo,
    No one has ever “demanded”, explicitly or implicitly, that Mochizuki travel to their location. Many have, accurately, pointed out that his refusal to travel has made communication with other experts more difficult.

    In any case, this has nothing to do with the latest situation, where, exactly because of the importance of face-to-face communication, Scholze and Stix traveled to Kyoto to talk to him.

  23. Some observer says:

    ” I don’t know why Fesenko chose to discuss them publicly.” Since he cites full titles, I assume he got authors permission, and presumably will also have read the work. Now you can ask why Mochizuki has not released the paper and the answer will be similarly generic: He will wait for the Schulze-Stix one. Since it seems that from Mochizukis POV the Schulze-Stix objection is “mostly harmless” it might well be that the whole thing is a ‘non-starter’ and we will find ourselves in a few months at exactly the same state of affairs we had one year ago.

  24. Peter Woit says:

    Some observer,
    I’m having trouble seeing why, whatever he thinks of the Scholze-Stix manuscript, Mochizuki would approve of the idea of having it first characterized publicly as “shallow and misplaced”, before it or his response were made available.

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