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:

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.


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”.

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

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78 Responses to ABC is Still a Conjecture

  1. DrB says:

    Perhaps part of the difficulty is that the unwritten rules for acceptance of a proof within mathematics seem to be adhered by. These rules state that the elders need to accept the proof as correct. The elders usually means experts in the field. Unfortunately, as is often the case, the field of Anabelian Geometry is very small in terms of the number of experts. Scholze works in Arithmetic Geometry. Fields medal or not, he is not an expert in Anabelian Geometry. It seems to me people like Fesenko, Yamashita and the anonymous referees that accepted the paper count as the experts. Their verdict counts by the unwritten rules. Who can really say that it is necessary for the experts within a field to address any kind of criticism raised, except to state that the criticism is unfounded based on an insufficient understanding of the theory? If they were in fact obligated to do this it would change the entire paradigm of peer review. Experts would be obliged to address endless amounts of misunderstandings by mathematicians that do not work in the field in question. Publication could be protracted indefinitely as a result. It is pretty clear by now that the system is completely broken. We need objective or objectifiable methods to validate results in mathematics.

    Btw that preprint of Mochizuki et. al. that I have mentioned is already being cited in published work:

  2. Peter Woit says:

    Anabelian geometry is a subfield of arithmetic geometry. The “Scholze is just too ignorant to judge work in anabelian geometry, and too incompetent to realize he’s out of his depth” argument isn’t plausible, and if you do believe that, he was joined in this by Stix, who is an expert in that subfield.

    The paper you mention evidently is based on old, pre-IUT work, just mentions the new IUT-based claims, but does not use them. It will be interesting to see if any reputable journal will publish a paper claiming to prove something assuming the flawed IUT-based stuff.

  3. Jens Franke says:

    Dr. B,

    while it is obviously reasonable to cover the identity of referees who decide against a paper (eg, to protect them against retaliation), it is less obvious why the same thing should be done for the referees who give their OK to a paper. So maybe this is what at least some journals should do?

    For me, someone who OKs a paper should be able to explain it at least to experts of neighboring fields. Also, it makes sense to mention the referees of a published paper to credit them for the work they had to invest. In some cases this is known (eg, Herbert Federer being the referee for the hard case of the Nash embedding theorem). In the case of the series of IUT papers by Mochizuki, the amount of work required to go through it must be huge, even for experts in anabelian geometry. Why not credit them for this effort?

    Outside the field of mathematics/theoretical physics, I know of at least one example (Organic Syntheses) who seem to do just that. They did so when they started in the days of world war I and seem to have stuck to this habit ever since.

  4. Peter Woit says:

    Jens Franke,
    In this case, having your name made public as a referee who approved the IUT paper might be problematic.

    On the other hand, the names of the committee at PRIMS that approved the paper were published with the paper. Those mathematicians agreed to publicly put their reputation behind an approval of the proof. It seems to me that they’re the ones who need to provide an explanation. They have decided based on some argument that Scholze-Stix are wrong and the consensus of the arithmetic geometry community is wrong. What is that argument? Is it Mochizuki’s argument that Scholze-Stix are incompetent? Is it something else? If it’s based on a report from a referee, they should be able to provide the substantive part of that report, stripped of the identity of the referee.

  5. Jens Franke says:

    Peter Woit,

    yes they give the names of the committee members but they just say “Several mathematicians kindly accepted an invitation to referee the papers; …”. So it does not list anyone who claims to have made his way through the proof. This, of course, is the usual way of doing things. For me, it does not make that much sense, but the only journal I know of which does not have this approach is Org. Synth.

    I do not see why it would be problematic to be named as the person who gave his OK to the IUT series of papers, provided that you are able to defend your opinion. After all, it would be a big honour and for most mathematicians it would be the most important thing they have done in their lives.

  6. Peter Woit says:

    Jens Franke,
    I think it’s extremely implausible that there is an anonymous referee who has a convincing counter-argument to Scholze-Stix, but that everyone involved in this is keeping it secret.

    On the other hand, it’s unfortunately plausible that the referees agreeing to approve the proof were people who might have reason to fear significant career implications if they took a stand that the proof was incorrect. The proof should have been refereed by an independent journal and independent editors, not by Mochizuki’s colleagues at RIMS.

  7. HM says:

    What could be the long-term consequences of such a fork in the mathematics literature? Does it become a political statement whether one assumes ABC?

  8. Jens Franke says:

    Peter Woit,

    They speak of “several mathematicians”, so there are several persons possibly involving Fesenko as well. But if PRIMS had the policy of publishing the names of referees giving their OK to a paper, they would not be able to make an exception for the IUT papers and we would know rather than merely suspect.

    I now think your “having your name made public … might be problematic” applies to the hypothetical case of a journal J run by persons close a powerful person X publishing an article by X choosing a poor devil D working under X as the referee, with D being practically forced to OK the paper. Obviously, it would be embarrassing for D to be named as a referee. But then, it would be even worse for the editors of J as it would force them to openly admit they picked someone close to X. Would they still dare to do so?

  9. Peter Woit says:

    Since there are so few mathematicians who accept that Mochizuki has a proof, the impact of this is very limited, essentially a small number of people cutting themselves and their work off from the rest of the math community. It seems likely this is how things will stay. A larger scale split in principle is possible, hard to imagine how that would evolve, I don’t think we’ve seen anything like in modern mathematics.

  10. Peter Shor says:

    Jens Franke:

    Occasionally, papers get published despite one terrible referee report. In this case, you really shouldn’t make the names of the person responsible for the negative report public. But I suppose it would possible to make the names of referees who gave positive reports public.

  11. Jens Franke says:

    Peter Shor,

    Yes, its the positive ones which should be made public.

  12. Peter Woit says:

    I’d like to discourage this turning into a general discussion of refereeing practices. This is an extremely unusual case, with general issues of referee anonymity I don’t think particularly relevant. In particular, if the referee’s names were known I doubt that would change anyone’s mind about the viability of the proof. What would change minds would be a serious mathematical counter-argument to Scholze-Stix, from any source.

  13. Mizan R Khan says:

    A result published in a good journal does not necessarily mean that it is accepted by the general mathematical community. A good example of this phenomena is the work of Heegner on the the class number 1 problem. He published his proof in 1952, but it was only in the late 60’s that the number theory community acknowledged that the proof was essentially correct. This was through the efforts of Birch and Stark. Without their intervention, it is conceivable that Heegner’s work would still be viewed as a failed attempt at proving the class number 1 problem.

  14. David J. Littleboy says:

    FWIW, a quick Google revealed a short article in the Asahi Newspaper* March 5, 2021 digital edition on the Mochizuki question. A quick and dirty translation:

    “A paper by Prof. Mochizuki of Kyoto University that is said to prove the abc conjecture has been published on March 5th after a consecutive period of 8 and a half years since its submission. The publisher, the EMS publisher, has responded to Asahi Newspaper inquiries. It will be published first in electronic form and then in printed form a month later by PRIMS, an international mathematical journal that publishes mathematical research”

    “The paper is titled (title here) and since it comes to a total of 720 pages of English text, PRIMS will dedicate a single issue, which normally would consists of five or more papers, to this paper. Since the paper is extremely difficult and there are mathematicians who doubt the correctness of the proof, EMS is taking this opportunity to make this work more widely known.”

    I find that last sentence rather interesting. (The Japanese does not use quote marks, and uses a Japanese abbreviation for EMS (without explaining it), but it sure looks to me that EMS told the Japanese newspaper that there were doubts about the proof.)


  15. AZ says:

    About the Asahi article, the part about mathematicians doubting the proof, it says that RIMS (数理研; not EMS) is taking the opportunity to make this work more widely known. The part about a response from EMS Press in the first paragraph only concerns the fact that it will be published electronically one month before the printed version.

    The article is actually longer. In one part it says that the papers are so novel/eccentric and hard to understand that even mathematicians say “they don’t understand where they don’t understand” them, as if the papers “came from the future”. A well-known mathematician has also said “there is an uncorrectable gap in the way the proof proceeds”, therefore the verification took the unusually long time of seven and a half years to complete.

    This is only the public part; the rest of the article is behind a paywall and I haven’t tried to access it.

  16. Dan Winslow says:

    From my read they are definitely still saying that it’s considered ‘proved’, it just took longer because of that pesky mathematician.

  17. TonyG says:

    Actually, the May 3, 2020 issue of the same newspaper had a story quite a bit more critical of the claim than the recent one:

  18. Per Östborn says:

    Found a Japanese article which says (in Google’s translation):

    ‘Some overseas mathematicians are skeptical about the content, but Professor Akio Tamagawa of the institute, who was involved in the editing, said, “The counterarguments have been exhausted and may remain parallel in the future.” I hope that young researchers will read the treatise seriously and that subsequent research such as improvement, generalization, and application will appear.’

    As a layman unable to make a judgement of my own, I wonder why nobody seems to discuss the technical third part of Mochizuki’s document. If the treatment there is correct, then the simplifications made by Scholze and Stix are invalid, and their objections irrelevant to the validity of Mochizuki’s claimed proof (if I understand the structure of the arguments correctly).

    Is this because the content of the third part is 1) wrong, 2) meaningless/irrelevant, 3) incomprehensible, or 4) nobody thinks it’s worth the effort trying to understand it?

  19. Peter Woit says:

    Per Östborn,
    I don’t want to speak for others and I’m not an expert, but when I look at Mochizuki’s third part all I see is what seems to be a repetition of the argument of his papers, with no attempt to engage at all with Scholze-Stix. He doesn’t refer anywhere in that section to them, to anything they’ve written or said to him explaining the problem. In particular Scholze has repeatedly asked for something very specific that Mochizuki should be able to produce if their argument is wrong (a diagram with certain properties). No one else has been able to produce this, and Mochizuki just ignores the question (along with all the rest of the substance of what Scholze is saying).

  20. Greg Price says:

    In that Asahi article from last year, it’s also interesting that they give it quite an explicitly geographical framing. The headline is:

    ABC conjecture: “Is the proof real?”
    In the West, objections one after another

    And that accurately captures the framing in the first paragraph, which says that on the subject of the paper, “mainly in Western countries, the discussion has become ‘has it really been proved?'”.

    (Or one might read “Europe and America” what I’ve translated as “the West” or “Western countries” — the original says 欧米, which literally means the former but commonly means the latter.)

  21. Taro Nakano says:

    In the April 3 issue of the Japanese business magazine Weekly Diamond, Professor Fumiharu Kato of the Tokyo Institute of Technology wrote about the reasons for the “misunderstanding” surrounding IUT. He is a researcher who has regularly held seminars with Mochizuki to discuss IUT and is known as an evangelist of IUT in Japan.

    This article is a column for the general public and has no technical content, but I have translated it into English for your reference.

    The researchers around the Kyoto University group who support the Mochizuki paper do not often mention that the paper has been questioned overseas. I think this column is a rare example of a supporter of the Mochizuki paper speaking out about the controversy.

  22. Peter Woit says:

    Taro Nakano,
    Thanks. Kato is just following Mochizuki in his claim that Scholze and Stix are just incompetents making elementary mistakes. This is not plausible or a serious response to their argument.

  23. Taro Nakano says:

    FYI, I translated into English two articles published by the Asahi Shimbun at the stage when the Mochizuki paper was accepted in 2020 and published in 2021. I have translated the entire articles including the paid part.

    – May 3, 2020, Asahi Shimbun article, “ABC Conjecture: ‘Is the Proof True?’”

    – March 5, 2021, Asahi Shimbun Article, “Proof of the ABC Conjecture Finally Published”

  24. Winnie Pooh says:

    @Peter Woit:

    You’ve complained several times already about Mochizuki failing to refer to Scholze & Stix by name in his paper. I think there’s a misunderstanding here about Western vs. Asian culture that needs to be cleared up. Source: I’m Asian.

    In Asian culture, it is considered impolite to point your finger at somebody, or even gesture in their general direction. Similarly, when you want to point out that somebody else is wrong (as Mochizuki does with Scholze & Stix), it is considered more polite to do so in an oblique, indirect way.

    Ultimately, this is all about “saving face”, an important concept in Asian culture. What Mochizuki basically does with his “no names” criticism is (1) point out that Scholze & Stix are wrong (or so he believes), while still (2) minimizing the number of people who know who exactly he’s criticizing, and thereby minimizing the damage to the “social standing” of Scholze & Stix. At least that’s the theory and rationale behind “saving face” and “no names”. Whether it works in this particular case is of course another matter.

    An example that might be easier for Western people to wrap their heads around is when you criticize somebody in public (which will damage their public reputation) vs. when you criticize somebody in private or at least in a small group (= minimal damage to their public reputation).

    To summarize, Mochizuki’s “no names” criticism is probably a form of Asian politeness, not an attempt to deprive Scholze & Stix of deserved recognition.

  25. Peter Woit says:

    Winnie Pooh,
    Many people seem to want to attribute the problems with Mochizuki’s response to criticism of the abc proof to cultural differences, but I don’t see that at all (although maybe you could argue that culturally-determined unwillingness to embarrass a colleague explains why people at RIMS went along with publication of the defective proof). Looking at everything Mochizuki has to say about this, the problem is that he refuses to actually engage with the mathematical arguments Scholze and Stix are making, to the extent of claiming that everyone he has talked to agrees with him that they have no substantive argument. I don’t want to psychoanalyze him, but it looks to me like this refusal to engage with their argument (because he has no counter-argument) has a lot more to do with why he refuses to write their names or link to the document with their argument than any cultural explanation.

    I sometimes see people suggest that the problem here may be Mochizuki’s unfamiliarity with the English language or Western culture. For those unaware of his background, note that from age 5-25 he lived in the US, and was educated at Philips Exeter and Princeton (and was a postdoc at Harvard).

  26. Winnie Pooh says:

    @Peter Woit

    Fair enough, and thanks for your answer. I only commented to provide an alternative angle, not to start a pointless psychoanalytical debate. In the end, who knows what the man is thinking.

    Another angle that occurred to me after I wrote my first post is that maaaaybe Mochizuki feels deeply offended by how Scholze & Stix handled the situation, and now does what the stereotypical Japanese person seems to like doing when offended, which is to clam up and passive-aggressively refuse to acknowledge the other person’s existence. I think this is more in line with how you view his behavior.

    As to how Scholze & Stix could have managed to offend Mochizuki, maybe he felt he wasn’t given enough time to explain himself before Scholze left Japan? Of course that would be Mochizuki’s side of the story. Whether he has any right to feel offended is another matter.

    Anyway, both feeling offended and the ensuing passive-aggressiveness strike me as fairly “stereotypically Japanese” (and also somewhat typically Asian, for that matter), so I still feel there’s a cultural component to all this.

  27. Winnie Pooh says:

    @Peter Woit

    After giving your reply some more thought, I now realize that we seem to fundamentally disagree on one important point:

    If I understand your position correctly, what you are suggesting is that deep down Mochizuki knows he’s screwed or at least in a very weak position, and now he and his buddies are trying to cover it up / pretend it didn’t happen. That would make this a case of deliberate deception, and would make Mochizuki the Donald Trump of mathematics (if you squint hard enough).

    But honestly, I don’t see this at all. After rereading the Mochizuki passages you quoted, I get the impression that, even if his attempted proof turns out to be a dud in the end, and even if he has no substantive counter-argument for Scholze & Stix, right now he still seems genuinely convinced that he’s right, or at least “essentially” right. So under this interpretation the worst one could accuse him of is being a delusional cult leader, but not one who’s actively lying.

  28. Peter Woit says:

    Winnie Pooh,
    I don’t think this is deliberate deception on Mochizuki’s part. In my (extensive) experience with very smart people unwilling to admit that an idea they’ve worked on for decades is flawed, all evidence I’ve seen is that the human capacity for self-delusion under such circumstances is an essentially infinite quantity.

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