I don’t really have time to write seriously about this, and there’s a very good argument that this is a topic anyone with any sense should be ignoring, but I just can’t resist linking to the latest in the abc saga, the REPORT ON THE RECENT SERIES OF PREPRINTS BY K. JOSHI posted yesterday by Mochizuki.

To summarize the situation before yesterday, virtually all experts in this subject have long ago given up on the idea that Mochizuki’s IUT theory has any hope of proving the abc conjecture. Back in 2018, after a trip to Kyoto to discuss in depth with Mochizuki, Scholze and Stix wrote up a document explaining why the IUT proof strategy was flawed. Scholze later defended this argument in detail and as far as I know has not changed his mind. Taking a look at these two documents and at Mochizuki’s continually updated attempt to refute them, anyone who wants to try and decide for themselves can make up their own minds. All experts I’ve talked to agree that Scholze/Stix are making a credible argument, Mochizuki’s seriously lacks credibility.

The one hope for an IUT-based proof of abc has been the ongoing work of Kirti Joshi, who recently posted the last in a series of preprints purporting to give a proof of abc, starting off with “This paper completes (in Theorem 7.1.1) the remarkable proof of the abc-conjecture announced by Shinichi Mochizuki…”. My understanding is that Scholze and other experts are so far unconvinced by the new Joshi proof, although I don’t know of anyone who has gone through it carefully in detail. Given this situation, an IUT optimist might hope that the Joshi proof might work and vindicate IUT.

Mochizuki’s new report destroys any such hope, simultaneously taking a blow-torch to his own credibility. He starts off with

.. it is

conspicuously obviousto any reader of these preprints who is equipped with a solid, rigorous understanding of the actual mathematical content of inter-universal Teichmüller theory that the author of this series of preprints isprofoundly ignorantof the actual mathematical content of inter-universal Teichmüller theory, and, in particular, that this series of preprintsdoes not contain, at least from the point of view of the mathematics surrounding inter-universal Teichmüller theory,any meaningful mathematical content whatsoever.

and it gets worse from there.

**
Update**: A commenter points to a response from Joshi here.

**Update**: Scholze has a comment on MathOverflow indicating precisely where Joshi’s attempted proof runs into trouble.

**Update**: Mochizuki and those around him award themselves \$100,000 (this is the IUT Innovator Prize described here).

Do I understand this correctly? Joshi worked hard in an attempt to patch up the perceived deficiencies in Mochizuki’s proof, and Mochizuki shot him down? I guess because … in Mochizuki’s eyes there are no “perceived deficiencies”?

Yes, that’s about it. He’s basically, in a very unprofessional manner, attacking as an incompetent the one person outside his circle who believes his IUT methods are worth pursuing.

A while ago I attempted to comprehend the plain English content in an earlier such communication. I forgot how much an onslaught of italics and bolding could make my eyes throb.

Among other weirdnesses in this report one thing sticks out to me: in this report as in his previous ones Mochizuki steadfastly refuses to call Scholze and Stix by name, always using euphemisms like “RCS” or “redundant copies school”. I cannot but notice a strange similarity to Vladimir Putin’s steadfast refusal to ever mention by name his political opponent Alexey Navalny, until the latter’s death.

And speaking of unprofessional manner, I think Mochizuki hits the new lows. He basically tells that to him the papers of Joshi look like they were produced by ChatGPT, although he graciously gives the poor guy the benefit of the doubt!

Well, calling Scholze (not by name) an ignoramus that makes bad undergrad-like mistakes sounds positively like a compliment by comparison…

I hope the experts will tell us if there is any meaningful mathematical content in this report, but given his tone, and the hard-to-read style that Mochizuki adopts, I am not holding my breath.

Joshi has a Theorem 9.11.1 in one of his papers. Mochizuki comments:

“[where we note that it is not clear whether or not the number “9.11…” assigned by the author to these key results in [CnstIII] was purely coincidental or a consequence of some sort of sense of rhetoric or humor that lies beyond my understanding].”

I don’t know whether Mochizuki is connecting Joshi’s theorem numbering to the September 11 terrorist attack, the 911 emergency number in the US, or something else. In any case, I fear that more than anger may be at work in Mochizuki’s mind.

Next time I really want to insult someone I’ll call them “ignorant of inter-universal Teichmüller theory” and that will certainly do the job.

This is painful to look at. It does not seem to me that Prof. Mochizuki is well.

Perhaps Mochizuki was also irked by Joshi’s papers because he wants all the glory for an abc proof.

Besides a consensus that the proof doesn’t exist, do experts consider that there is any interesting or useful mathematics in the IUT papers? If so, one sign of hope is that I think there’s a real possibility that this could be settled via a formalisation effort within, say, 10 years. If the proof doesn’t compile as Lean code, Mochizuki will have to argue that computer CPU’s are incompetent at mindless elementary computational steps.

My admittedly cheap shot about formatting was kind of hinting at that suspicion: This does not read like the product of an entirely well mind. At minimum, a dearth of independent editorial feedback to moderate the tone (and the font) has allowed a tendency toward self-sabotage to run amok.

@Mike: The expert consensus is that there is essentially nothing of interest in any of the IUTT papers. This is well-reflected in the discussion hosted on this blog back in April 2020.

It is also clear that there is zero chance of settling all of this by a computer formalization of Mochizuki’s work, since at the heart of things (i.e. in the proof of Corollary 3.12) there is nothing actually there to feed into the computer. Something which is not already well-understood by humans cannot be formalized!!

It seems that there are fewer mathematicians thinking this includes useful ideas. I think they, who insist that the theorem was proved and want that it is accepted by mathematicians, should deal with computerization by themselves. Although it was unrealistic a little ago, it is prospering more and more these days. Well, if it doesn’t work well, then they can’t avoid conceding the fact that the proof is incomplete, so they will hesitate to do it.

The growing weirdness in Mochizuki’s rebuttals, his peculiar sentence construction, his obsession with identifying non-existent hostile patterns in the criticized texts, and even his polarization between the Western and Eastern mathematical communities are probably not indicatives of arrogance and single mindedness, but rather of some form of cognitive decline, not much different from what happened to several prominent mathematicians (needless to say which one comes to mind first). In this case, this might not be the appropriate forum to discuss the situation or criticize a once prolific mathematician, who apparently is not able anymore to defend himself and his work in an articulate and objective manner.

Right or wrong, Joshi’s work be judged on its merits and not be intermingled with the circus that surrounds Mochizuki..

As a very gentle admonition: I feel it is bad form to speculate about someone’s mental health, and such comments may be hurtful to those struggling with mental health issues and to their loved ones. Let’s try to be kind.

Arpex/momerathe,

I agree that such speculations are not a good idea, but recognition that there is a problem is important. Those surrounding Mochizuki in Kyoto should be doing something about this, not enabling his behavior but staging an intervention. Even if they believe he’s completely right about the mathematics, why isn’t anyone telling him that writing documents like this is destructive to his own interests?

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Although IUT has potential, Mochizuki is extremely arrogant (with his refusal to attend proposed conferences on IUT and with regards to clarification of the theory to non-experts, and that could be one of the reasons why most mathematicians have given up on it.

Prior to the Comments against Joshi and his paper, from what Prof. Mochizuki wrote in his blog in Japanese this January, I have had serious concerns, not about the IUTT itself: https://plaza.rakuten.co.jp/shinichi0329/diary/202401020000/

I would never recommend using a translation tool with the expectation that meaningful content would be written. Only the translation of the title of the post here: Deification, Ethnocentrism, and Suppression of Rationality of the West based on Christianity and Judaism

The problem is already out of mathematics. I believe we should not help it get any worse.

Peter,

> why isn’t anyone telling him that writing documents like this is destructive to his own interests?

Well,

> ‘In light of the blatant obviousness underlying (Reac1), many consultees were not interested in investing the time and effort necessary to discuss Joshi’s series of preprints in detail and indeed strongly encouraged me to simply ignore them as well’

This is a kind of crash-test on how the maths community is healthier than the hep-th one. Although Mochizuki is a well recognised mathematician (as far as I understood), when he speaks no sense, then some prominent mathematicians say so. At the end of the day, only Mochizuki’s pride is wounded.

When recognised physicists speak non sense in high energy theoretical physics, the all field is ruined…

I’m convinced that Scholze and Stix’s objection to Mochizuki’s Corollary 3.12 is valid, as Mochizuki hasn’t been able to defend himself by spelling out all the details in all those years after the objection was made. Assuming the objection is valid, then nothing would save IUT, including any attempt to make IUT even fancier by incorporating perfectoid fields, because nothing prevents the real numbers to have nontrival automorphismes as ordered abelian groups.

But in any case Mochizuki’s response to Joshi’s hard work is unacceptable. In doing mathematics we pursuit the ultimate consensus that is unaffected by any personal status, political power or even the law of physics. Nobody is entitled to shut another mathematician up by personal attack or political hints.

Millstone beside bath,

That’s another document that would argue for an intervention from those close to Mochizuki.

fwiw,

The right advice to Mochizuki would have been not to ignore Joshi’s work, but to respond professionally to the mathematical content.

Peter, I feel your “right advice” to Mochizuki presupposes he has the ability to respond professionally to mathematical critique, which seems an unrealistic expectation at this point. The people who work with him surely realize he cannot respond in a professional manner, so telling him not to respond at all is probably the best advice they can offer.

I don’t see an intervention forthcoming. If they’re willing to publish his papers in Kyoto then they are very unlikely to back down on support, not only due to this action demonstrating an unwillingness to challenge him when they should, but also due to the escalation of commitment. Admitting he needs an intervention would be tantamount to an admission that the IUTT publications are evidence of a serious systemic error. This entire phenomenon is about saving face and the longer it persists the more entrenched the actors become.

A. Wolf,

I fear that you are right. Concerning Martibal’s comment above, I think the difference between this and the string theory case is that here the problem has remained localized in Kyoto, while in physics it spread through the entire community.

A response by Kirti Joshi to Shinichi Mochizuki’s recent comments on his papers are here: https://mathoverflow.net/questions/435110/consequences-of-kirti-joshis-new-preprint-about-p-adic-teichm%c3%bcller-theory-on-th/467851#467851

Stix and Scholze ultimately did read Mochizuki’s mathematical content and pointed out the first thing they thought wasn’t correct (3.12).

It would be very nice if someone could do the same for Joshi’s work. Mochizuki’s document appears to be mostly disparaging and (I need to review it again) but he doesn’t seem to say something like “I follow Joshi’s work until paper ___ page ___ proposition ___” which Scholze and Stix HAD done for Mochizuki.

Amused that Mochizuki insists on writing “Fontaine-Fargues curve”, while the universally accepted name is Fargues-Fontaine curve. It’s a small detail, but coming from someone who insists so forcefully on accuracy and carefulness etc., it is rather ironic.

By the way did anybody manage to make any sense out of the Oberwolfach report

of Mochizuki?

It reminds me of the hallucination of ChatGPT (ops).

More seriously, I might be missing something, but that report aims to give a completely elementary description of his “main idea”.

Perhaps I am limited, but by reading it, there seems to be no fucking way he has a proof of abc. It seems more or less accurate that he has gone senile from that report.

If not, how is it possible that such an institution as Oberwolfach has accepted to lower their standard to the point of accepting this trash in their reports? Isn’t it enough that RIMS has already decided to trash its name for good? Shall we have this guy sabotage the prestige of other important mathematical institutions too?

I would love if some person involved with that publication could step in here and justify this choice, by making some explanation for this. To me this document makes no fucking sense!

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Some might think that this is a trivial detail, but keep in mind that Mochizuki is 54 years old; i.e. not a senile old man exhibiting what has been labeled “cognitive decline”.

Recall that he maintains a “Safety Confirmation Information for Shinichi Mochizuki” webpage; Internet Archive has crawled it as early as 01 April 2012. (Sorry but the date is what it is.) He seems to update this page often but not daily. (Weekly, perhaps?) I hope that the impulse that lead to such a webpage doesn’t further develop.

Current developments are, simultaneously, surprising, and somehow, not surprising.

Very sad.

Enraged Number Theorist,

I assume you’re referring to

this

This was just one contribution to this workshop

https://ahgt.math.cnrs.fr/activities/workshops/MFO-RIMS23/

organized as part of this

https://ahgt.math.cnrs.fr/

There’s supposedly an eight-member committee supervising that, and Benjamin Collas seems to be the main person responsible for promoting IUT this way. He and that committee might want to discuss what they should do about the situation Mochizuki has created.

Hey Peter,

Noticed this odd comment on page 7 of Mochizuki’s report:

>>[where we note that it is not clear whether or not the number “9.11…” assigned by Joshi to these key results in [CnstIII] was purely coincidental or a consequence of some sort of sense of rhetoric or humor that lies beyond my understanding].

Any idea what this means? Is this a 9/11 reference?

Dear Peter,

Yes, exactly that report, indeed. Thanks for pointing out the name of the responsible person.

I hope Oberwolfach, he and the other organizers are aware that the number theory community is watching this, and nobody is amused, to put it mildly.

Given the public attention that this guy is drawing with his inflammatory bullying with the mathematical community at large, and given how little sense that report makes (and how much at stakes there is with these claims!), it is clear to many in the community that Oberwolfach and the organizers should do something to distance themselves from this person.

Accepting that nonsense as a report is a terrible testament of low standards of academic quality. And for an institutions like Oberwolfach, where reputation is pretty much everything, that’s a very dangerous move.

I trust somebody will do something about this!

@Enraged number theorist

that example of his in section 3.2 (the Riemann sphere being the gluing together of two affine complex charts), feels to me like either linguistic sophistry or a total unwillingness to learn what people mean when they talk about colimits like this.

I went through the process of calculating this colimit in an elementary way from the definition, in a pdf linked this blog post. And showed that doing the kind of apparently wicked identification Mochizuki condemns really doesn’t matter. It’s the kind of conceptual mistake you might expect an undergrad to make when learning about category theory.

Or for that matter, physicists 100+ years ago worrying about the “hole paradox” because they are sufficiently mathematically sophisticated to think about diffeomorphisms of manifolds and charts and whatnot correctly.

One can even do this gluing together of P^1 in a skeleton of the category of complex manifolds, or complex varieties or whatever, where you don’t have the freedom to find a new, distinct copy of the affine line.

The affine line itself can be literally unique (not even up to isomorphism, but the only representative of its isomorphism class), but there are two different embeddings into the projective line, and hence we can still distinguish the coordinates on the two charts.

I really don’t understand why this particular example is so important to Mochizuki, when it really portrays an unfamiliarity with how people speak in modern geometry.

Dear David,

Thanks for the thoughtful reply.

What now brings particular gravity to the situation is that such futile discussions on his side are implied to be the “main idea” behind the proof of such a deep statement as ABC. As a report of an institute like Oberwolfach and to top it up now, from somebody that twice has shown extremely bullying behaviour with other professionals, drawing a lot of attentions from the larger public, which must be thinking something particularly clownish is going on with the number theory community: hard to refute that!

What is particularly sad about this, is that it might give to the outsiders the misleading idea that whether somebody has proved ABC or not could really rest on such futile discussions as the one of his report. That’s really a sad unfortunate turn of events, since Diophantine Geometry is that one subject where the biggest gems (which are logically weaker results than ABC) have all came with tour-de-forces of absolute brilliance, from masters like Siegel, Bombieri, Faltings and Vojta (to name a few), with no empty trivialities, but one difficult idea after the next.

It is of the utmost importance that as a community we distance ourselves from this crank. Personally anybody can use their time as they wish (and I hope that as a community we will be able to address Joshi’s investment and its intellectual value on its own ground). But as a community it is of utmost importance that this clown/bully does not anymore have a space in our most revered institutions.

I am pretty sure the German tax payers wouldn’t be amused to know on what kind of farse Oberwolfach is wasting their time on. This sounds like a joke or an exaggeration. It is not:

As a community the bare minimum we can do to keep our credibility from a society that invests in our intellectual pursuits, is to keep such clowns at bay.

Like everybody, I’ve found Mochizuki’s work and behavior on this baffling. But I’ve also found the Scholze/Stix reply uncompelling–it is, at best, a sketch of what’s wrong with the IUT program. In a way this is not surprising; if the target of your objection is indeterminate, your objection will be too. So when Peter says that “all experts [he’s] talked to agree that Scholze/Stix are making a credible argument”, that is exactly correct: it’s a credible argument, not a compelling one, and certainly not a disproof of IUT ABC.

In contrast, Joshi and Dupuy have attacked this problem the right way: comprehensively and with full rigor. Whether there are better things they could be doing with their time is another story . . .

On MO there’s a question about a purely mathematical claim of Mochizuki stated in the note Peter links to, but apparently nobody knows either what it means or how to prove/refute it https://mathoverflow.net/questions/467696/global-character-of-abc-szpiro-inequalities

What does it mean to “sum up local inequalities at each prime number of a number field”?

I surely don’t know how to prove a statement when I don’t even know what exactly is the statement!

Can somebody please clarify the precise meaning of this claim? I believe the MO’s question does not do any job in making this precise.

This might be actually helpful. If Joshi is indeed “summing up local inequalities” (whatever that means) and if one can show that this can never give a proof of ABC through those triples as Mochizuki suggests, then at least we can get a cheap verification that Joshi’s “proof” is (also) flawed/inexistent (like Mochizuki’s one) and avoid waste time in empty discussions for 12 more years.

Peter Scholze is participating in the discussion on MO.

@GeoMan

I think Scholze has better things to do than spend his time responding to a person who insults him (and others) and who is incapable of producing an understandable mathematical text, let alone a credible counterargument to his objections.

It is not up to Scholze to disproof IUT ABC. He has already done his part. It is up to Mochizuki to produce an irrefutable and clear proof. But, given the last 12 years, that probably won’t happen.

If Joshi’s demonstration proves incorrect, it’s time to move on and leave Mochizuki alone in his delusions.

It is interesting to see that the disproof of Joshi’s proof that Mochizuki added in his revision is actually readable and has a fairly standard layout, at least relative to the surrounding text. I think there are several ways in which this is humorous, but let me just point out that the part where he seems the most like a normal, functioning mathematician is exactly the part which he says an AI could write.

Kirti Joshi has briefly responded to Peter Scholze’s concerns about his proof in this MathOverflow response: https://mathoverflow.net/a/468180/472518 . Joshi mainly states that he is working on a full response to the points that Scholze raised, and thanks Scholze for his comments.

There are several linked MathOverflow responses on this topic now, they might be interesting reading.

A fuller response to various questions has been posted by Joshi on MO and his site https://mathoverflow.net/a/468180/472518

https://www.math.arizona.edu/~kirti/qa.pdf

Apparently Mochizuki &Co awarded themselves $100,000 every year for 10 years: “The IUT Innovator Prize will be awarded annually for 10 years”

The same page states that “The IUT Challenger Prize of $1,000,000 will be awarded to the first mathematician to write a paper on the IUT theory that shows an inherent flaw in the theory”. Why do I have a feeling that Scholze and Stix won’t get it even if they publish their findings in a journal?..

Leaving aside the non-mathematical part of the discussion, is it fair to say that Mochizuki’s mathematical objection to Joshi’s paper is now seen by other experts as fundamentally correct?

Where do they got this money from ? Did they manage to convince some rich guy to spend money on that (like the breakthrough prize) or did they mage to convince the japanese governement that this question was worth $1.000.000 ? In both case it is a bot worrying for the health of the math community.

@Curious Observer

It seems Joshi has not yet responded to Scholze’s objection restating Mochizuki’s objection, but promises to do so in the future. From what I can tell the pdf you linked does not address this objection at all. Yet.

@AO

I have the same question, but from reading the comments on MO it certainly does seem like Scholze agrees with the specific mathematical objection that Mochizuki makes of Joshi.

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New responses from Joshi to Mochizuki and Scholze.

https://www.math.arizona.edu/%7Ekirti/local-global-issue.pdf

https://www.math.arizona.edu/%7Ekirti/response-to-Mochizuki.pdf

Will Sawin has already pointed out a flaw in Joshi’s response, here https://mathoverflow.net/questions/467696/global-character-of-abc-szpiro-inequalities#:~:text=I%20believe%20the%20claim,dealing%20with%22%20is%20wrong.