Scholze, Stix don't have the magic power to veto arbitrary proofs

The \(abc\) conjecture is a proposition in number theory somewhat analogous to Fermat's Last Theorem. If three relatively prime (possibly negative) integers obey \(a+b=c\), then some inequality holds\[

\Large \max (\abs a, \abs b, \abs c) \leq C_\epsilon \prod_{p|(abc)} p^{1+\epsilon}.

\] In 2012, Šiniči Močizuki (this website is written in Czech English, some of you appreciate it) presented his alleged proof and now, over six years later, the validity of the proof remains disputed and its status is therefore uncertain.

I find the inequality above rather contrived and uninteresting – which very well may be just because I haven't studied those corners (and most corners) of number theory intensely enough – but Močizuki claims to have a whole profound theory, Inter-universal Teichmüller (IUT) theory [=arithmetic deformation theory], which can generate proofs of many number-theoretical propositions. I feel that the broader theory attracts me more than the \(abc\) conjecture itself.



The Japanese guy's discovery would be rather profound if his claims are true. His proposed theory is grand, needs a lot of time to be understood, builds on some Grothendieck's spirit that remains incomprehensible to most, extends the Teichmüller theory of moduli spaces into arithmetics, and returns topological groups to the analyses of number-theoretical problems. The theory depends on anabelian arithmetic geometry recently developed in Japan (1990-2014).



The Hodge theaters, miniature representations of schemes or rings in which the number field is separated from the multiplicative structures, play an important role in the theory. All such theaters are isomorphic to each other but they may also be mapped to each other in a different way. All pre-Mochizuki mathematics may be said to take place in a single theater. But in his theory, one may move in between the theaters (with "translation", "reconstruction", and "quantification of distortion" which is required to be small in some precise sense) which adds some seemingly useless redundancy, but because one may return to the original theory after a detour, it's possible to find out new facts about objects in a single theater.

I do feel that the theaters are morally analogous to different observers' interpretations of some processes in quantum mechanics and some people's inability to even consider the numerous theaters as a useful tool (and their obsessive instinct to get rid of this "redundancy" from the beginning) is analogous to the inability of many people to even consider the basics of quantum mechanics and its refutation of the classical objective existence of observables.

There are various reasons why not too many mathematicians have tried to absorb everything that Močizuki wants to say. It's too new, the papers are too large, depend on many things, the spirit disagrees with the toolkit that (not only) junior people in the field are supposed to repeat all the time. In particular, while Robert Langlands has expressed his desire to understand Močizuki's theory, the "Langlands program" is a much more popular paradigm outside Asia and could be interpreted as a competitor to Močizuki's IUT. To some extent, these regional preferences may be compatible with the health of the world's mathematics community. But some of the reasons for this calm reception might be rather political and pathological.

Even though I am not a clear fan of Močizuki or his ideas, I was shocked when I saw the top Quora answer to a straightforward question:

Did Peter Scholze and Jakob Stix really find a serious flaw in Shinichi Mochizuki's proof of \(abc\) conjecture?

OK, Scholze and Stix made these negative claims, before and after some very tense March 2018 meetings with Močizuki. Especially Scholze is sort of a darling of certain community and the most often talked about fresh winner of the Fields Medal. Let me admit that my guess is that Močizuki is a far deeper mathematician than Scholze. So are the Europeans' criticisms valid?

What I've heard and seen from these criticisms seems shallow to me. It may be right but it doesn't look persuasive. People on Močizuki's side – who basically say to have verified the proof – may have been less famous with the media and they're less independent of Močizuki. But it seems to me that they have done much more genuine work than Scholze and Stix.

And I seem to have lots of empathy for the kind of feelings that Močizuki had while reading the criticisms, e.g.:

I can only say that it is a very challenging task to document the depth of my astonishment when I first read this Remark! This Remark may be described as a breath-takingly (melo?)dramatic self-declaration, on the part of SS, of their profound ignorance of the elementary theory of heights, at the advanced undergraduate/beginning graduate level.

Well, I don't really understand IUT but the sentiment above does indicate that he is fighting against a similar kind of pop-science and P.R.-driven superficial nonsense – and reacts very similarly as I react to kilotons of breathtaking melodramatic crackpots who often fail to understand advanced undergraduate or rather rudimentary graduate school material.

(Incidentally, if you care about the skeleton of the SS-vs-Močizuki disagreement, it's all about the SS' additional assumptions under which Močizuki's strategy yields vacuous results. He agrees but says it's really a straw man argument because if the additional assumptions aren't made, the results of his arguments are non-trivial, which is at least logically possible, and SS don't seem to have anything to counter this Močizuki's criticism of their argument. So unless I have overlooked some other part of the SS' criticism somewhere, their claim to have found a gap in the Močizuki proof is just a feeling combined with faith, not a fact.)

You may dislike any "personally sounding" comments like Močizuki's but at some moment, they simply are absolutely appropriate and needed. It's when a discussion becomes completely meaningless – lowered to a much lower level than the level where the actual scientific work was being done. And maybe it is the case here. Sometimes, we only scream that the emperor has no clothes and what the other side generates – whether it's polite or not – is just rubbish.

I am not able to settle the question whether Močizuki is right. But even with this uncertainty, I think that certain societal phenomena that are happening around mathematics – and it's obviously analogous to many things happening around physics and other fields – are certainly incredibly pathological.

OK, the Quora question about the validity of Scholze's and Stix's criticism of Močizuki's proof has two answers now. One, hugely read, upvoted, and insane one; and one, almost invisible and even less upvoted, sensible one.

The invisible answer is by Xiaowei Xu, one young mathematician, and points out that the fate of the proof and the possible new field of number theory is being crippled by street rumors spread by people who demonstrably haven't digested everything they need to digest in order to have a justified opinion. Xu, who links to a very wise and similarly frustrated essay by Ivan Fesenko (II), is disappointed by the rotten conditions in the field. After all, according to some rather standard rules that once worked, the proof could have been "officially validated" by a limited number of reviewers who give an approval to Močizuki's paper – and this has arguably happened. Xu's answer was read by less than 300 people and only has 4 upvotes including mine.

The much more widely read answer is by Senia Sheydvasser, a postdoc at CUNY who seems rather invisible to me in the world of actual papers. He is clearly one of the guys who became extremely influential on Quora – perhaps in some way analogously to the physics crackpot Richard Muller. OK, Senia's answer has over 2,500 views and over 100 upvotes. Dr Senia, have they found a gap in the proof?

Already the first short paragraph made me say Wow:

I would put it thus: they certainly found a serious flaw in the proof. Whether or not the flaw is the one that they actually identify is not entirely clear.

Scholze and Stix have "certainly" found a serious flaw in the proof but it may be a different flaw than the flaw they are actually talking about! ;-) Holy cow, how can you be "certain" that they have found a "serious flaw" if the serious flaw is something that they haven't even "identified"? Is it possible to find something without identifying it?

Senia is basically saying that even without saying the proposition of the form \(XY\), we must be "certain" that Scholze and Stix have "found" the proposition of the form \(XY\) and it is a very important one. Unbelievable. Not even Isaac Newton was able to "find" groundbreaking scientific results without talking or writing about them, without "identifying" what they are supposed to be. So I was curious how this Senia justified his extraordinary claims:

What has happened with the \(abc\) conjecture is that Mochizuki has a manuscript that he claims is a proof—however, no one (other than maybe some of his students) seems to be able to understand how it is a proof. This is a problem: [...]

The inability of most people including the bulk of mathematicians to understand Močizuki's proof may be considered a "problem" in some emotional or subjective way. But it is not a proof that Močizuki's proof is wrong. All state-of-the-art proofs of complex enough propositions in mathematics are incomprehensible to most people, including mathematics PhDs, especially to those who haven't dedicated an appropriate amount of time, aren't they? Senia seems to be aware of similar things:

[...] in practice, nobody ever writes out the whole proof with all of its gory logical details (as it would make the whole thing completely unreadable—imagine trying to understand how a computer program worked by reading machine code, or if you want to get really sadistic, binary). However, it should include enough information that any competent mathematician working in the area can reconstruct the full proof if needed.

Senia admits that proofs by professional mathematicians aren't supposed to be pedagogically perfect textbooks capable of teaching cutting-edge mathematics to the most retarded schoolkids. It clearly isn't even possible. It isn't even possible to guarantee that a proof of a complex enough proposition will be understandable to most mathematics PhDs.

But according to Senia, proofs are obliged to contain enough detail that any competent mathematician in the area may reconstruct the full proof if needed.

In reality, the history makes it totally clear that deep enough game-changing results in mathematics – but also in physics and all natural sciences – often face the violent misunderstanding not only by "some" professionals in the area but, rather frequently, "most" professionals in the area if not an overwhelming majority. So the condition Senia "demands" would have made many – if not most – of the truly important advances in mathematics and science impossible.

But let us ignore the fact that true revolutionaries were often "misunderstood" by the "peers".

There is an even deeper problem with Senia's condition. Even if you assume that the condition is legitimate – that the proofs should be reconstructable by specialized experts if needed – it isn't demonstrably relevant to the present situation and most other situations. Why? Because there's no evidence that Scholze and Stix really "needed" to reconstruct Močizuki's proof.

Instead, there is quite some evidence that they haven't tried hard enough to read all the texts and reconstruct the proofs. It seems rather likely that they're not even motivated to do so. As we can see, their simply saying that the proof is wrong is just fine for their personal careers and even fame. "Maybe" that's enough for what they care. And they will always find lots of people who trust them. So why should they reconstruct it?

Senia gets really explicit about his philosophy of a mindless belief in some chosen practitioners in his third paragraph:

Scholze and Stix are competent mathematicians, and have spent a protracted amount of time studying the proof and the underlying mathematics. If they don’t understand the proof, there is no proof.

Oh, really? All the world's hyped and celebrated geologists have spent a protracted amount of time – some 30 years – by studying Alfred Wegener's proof of the existence of continental drift. They said they didn't understand it and they said much worse things about it. Nevertheless, all sane people agree that what Wegener had was a proof of continental drift. We use more or less the same one when teaching the stuff today.

The Schutzstaffel guys are just two people. Four main U.S. mathematical societies have 35,800 numbers in total. Clearly, most of them are less competent in this branch of number theory than Scholze and Stix. But the number of those who are competent is still much higher than two. To say that "there is no proof because two particular people don't see a proof" is just plain insane. Even if a great majority of people similar to Scholze and Stix failed to get the proof, it would still fail to show that "there is no proof". It would still not be a solid reason for a professional number theorist not to study Močizuki's papers.

Senia's last paragraph starts as follows:

Now, Mochizuki’s objection is that Scholze and Stix are misunderstanding his proof. That might be true. However, if this is true, Mochizuki or one of his students could remedy the situation by writing out some of the details he is leaving out. [...]

As you can see, Senia – and those 100+ idiots who upvoted his answer – would like to turn mathematics (and probably also science and many other fields of human activity) to a platform of sociological rituals enforcing group think. After the group think is enforced, there is a vote, you know what the outcome of the vote is in advance, and that's how the truth is established. They don't want the substance to matter at all. Instead, what they want to be decisive is what someone says about the substance, and that someone should better be powerful and your ally. By having such allies, you don't need to read any "heretical" papers at all.

In fact, one commenter under Senia's answer makes this point rather clear. He celebrates that mathematics should turn into a social science – he praises Senia for highlighting the sociological mechanisms to decide about the truth value of proofs. Well, it isn't even a social science. It would be a mindless irrational cult.

We may describe this sick attitude to the truth as a generalization of the standard left-wingers' fight against hard and exact sciences, against the independent thinking, and for mindless group think revolving around grievance studies and adjacent social pseudosciences. But we may arguably pick much more specific analogies for this Senia-style thinking. The paragraph I quoted seconds ago is analogous to #MeToo. Why? Because the accuser is always claimed to be right.

If a disgusting and untrustworthy witch or bitch claims to have been raped by a successful man, everyone is bullied into believing these mostly ludicrous and insulting claims – in many cases, the successful man wouldn't even touch such an individual. Similarly, if a Scholze and a Stix say something bad about a proof by a Japanese man, they must be trusted. There is no proof!

This is not how mathematics can work, this is not how physics may work, this is not how courts may work, this is not how a civilized society may work. The accused person may also be right and this possibility mustn't be eliminated at the very beginning by some bizarre screaming and demagogic sociological would-be arguments. And criticized proofs may be solid, too. When a Scholze and a Stix fail to understand a proof, it doesn't have to imply a flaw in the proof. It may be a sign of their flaw, too. Even in the absence of dishonesty or laziness, a simple reason why they may fail to understand Močizuki's papers may be simply because they're far weaker, less creative, less imaginative mathematicians than he is – who simply can't learn anything truly new that is outside their mental boxes. If people are threatened so that they are even afraid to make this trivial point, it means that there are powerful pressures that want to eliminate all meritocracy and rational thinking from the process.

What we're seeing is that our institutions are being filled by dishonest and incompetent NPCs who claim to be competent but who always prefer to politically endorse an opinion that says that "there is even no proof to be read". It's so much easier for this lazy dishonest scum – a scum which praises itself, however – to join such a bandwagon than to study what they should study if they were actually fulfilling the moral duties that a mathematician really has if he or she is a real mathematician.

Needless to say, the critics of string theory, quantum mechanics, or natural sciences are analogous to one extent or another. They're lazy mediocre pseudointellectuals but because of the dropping standards and also affirmative action, they got to the system and they simply defend their political interests instead of doing honest scholarly work. They are building an infrastructure that makes sure that melodramatic self-declarations aligned with the collective interests of this lazy majority can politically beat genuine scientific arguments, including the most solid and profound ones. Incidentally, I think it's no coincidence that the target (Močizuki) is Asian – Asians (including people with names similar to Motlzuki and Nakamotl) are generally the main victims of affirmative action which is largely a new institutionalized racism. And it just happens that all the anti-Močizuki criticisms comes from the U.S. and Germany, two top politically correct countries with minimal experience in arithmetic anabelic geometry.

There is a controversy about Močizuki's proof and the "general recipes" proposed by Senia how to resolve such situations are clearly unacceptable and incompatible with any progress in mathematics or any discipline requiring mental powers. Scholze and Stix say that Močizuki doesn't have a proof of the \(abc\) conjecture, Močizuki says that they don't have any proof that there is a mistake or gap in his proof. The situation is really symmetric and whoever claims to be "certain" that Scholze and Stix are right is simply a dishonest charlatan who places irrationality, group think, fallacies of sociological arguments, and laziness above the merit.

Sadly, the number of such people who completely suck has increased dramatically, even in the Academia – and maybe especially in the Academia.