Sunday, April 07, 2019

Category theory as an egalitarian religion

Several TRF essays have discussed the controversies around the Mochizuki proof of the \(abc\) conjecture, most recently in November 2018. The conjecture states that whenever integers obey \(a+b=c\), then the maximum number, let's assume it's \(c\), isn't parametrically larger than a (multiple of a) power of the product of all primes in \(a,b,c\). So it's some inequality linking both the additive relationship between \(a,b,c\) with some multiplicative one.

Šiniči Močizuki's solution is a corollary of a whole new ambitious theory in mathematics (possibly a flawless theory, possibly a flawed one at some point) that he has developed, the "Inter-Universal Teichmüller (IUT) theory" or "arithmetic deformation theory", these terms are synonymous. He claims to study some permutations of primes and integers etc. as if these permutations were analogous to continuous deformations.

Equivalently, he claims to disentangle the additive and multiplicative relationships between the numbers by looking from many perspectives, by using new terms like "Hodge theaters". I've read and watched many texts and promotional videos and they look incredibly creative and intelligent to me. I am of course far from being capable of verifying the theory up to the applications – one needs to master at least 500 pages plus some 500 more pages of the background etc. I am not motivated enough to go through, in particular because I don't really see why the \(abc\) conjecture should be important in the grand scheme of things.

But I am very interested in the general complications that great minds often seem to face – and things don't seem to be getting better. In the recent issue of Inference, I read the thoughtful essay by David Michael Roberts,
A Crisis of Identification.
Roberts' writing is highly impartial – after all, Adelaide, Australia is "just" 8,000 kilometers from Japan. He sketches some history of the proof, similar proofs in the past, the Grothendieck approach as a driving engine of many mathematicians on both sides, the social dynamics, and the philosophy of the category theory and its predecessors since the era of Hilbert.



I want to focus on two things: on his statements about the "responsibility for misunderstandings", which I totally disagree with, and on his apparent explanation why there seems to be such a continental gap between Močizuki and the Euro-American mathematical "consensus" of the professionals. I think that Roberts' comments have confirmed what I thought, from a new and rather detailed perspective, so I think that I should share the theory that has passed a new test.



First, my disagreement with Roberts about the "misunderstandings". In 1952, Kurt Heegner, a private scholar in Germany, published this paper. The group think of the field has declared the paper "fatally flawed". These days, the proof is considered correct and Heegner's contributions are well-recognized.

Optically, the first two pages of the paper look just fine. Harold Stark and Alan Baker have helped to complete the proof and Stark has admitted that Heegner's proof was almost complete, except for a minor gap. At any rate, in 1952, Heegner ("almost perfectly") proved that there were exactly "nine complex quadratic fields of class-number one", and those were in one-to-one correspondence with the Heegner numbers,
1,2,3,7,11,19,43,67,163
It's no coincidence that the largest number involves 163 which appears in \[

e^{\pi \sqrt{163}} = 262\,537\,412\,640\,768\,743.999\,999\,999\,999\,25\ldots

\] which is almost integer. This numerological coincidence was known since 1859 (Charles Hermite) and in 1975, Martin Gardner made an April Fool's prank and claimed that the result was actually integer, in SciAm – SciAm was still a very intelligent journal with the entertainment done by very intelligent people. Well, it's no coincidence that it's so close to an integer. A proof involves the \(q\)-expansion of the \(j\)-invariant, something you know from tori (in string theory etc.).

Try the same thing with the smaller numbers than 163 in the Heegner list. It should have been obvious that Heegner was no idiot – well, I think that he was self-evidently brilliant – and he had found something. But some mathematicians clearly didn't like the idea that a "private scholar" finds something this important, so they abused their political power to mock his work and himself personally. Isn't it basically a proven fact that this is what was happening? And isn't it obvious that this is exactly one of the worst things that mathematicians or scientists can make with their institutionalized influence?

It just happens that Heegner has such a similar surname to Alfred Wegener, the discoverer of the persuasive body of evidence backing the continental drift – i.e. the first scientifically serious proponent of this correct theory. He was also mocked by "geologists" many of whom just couldn't remotely compare to him. But they had quite some power and that was the reason why the scientific community only began to "see" what Wegener already knew – some 30 years after Wegener's death. This delay seems so terrible: the collective "consensus science" is so often shown to be so much dumber than the best people's science. Already during Wegener's lifetime, people should have focused on careful observations that would complete the details of plate tectonics etc. But 30 years? Just sad.

Heegner and Wegener are similar names but unlike Heegner, Wegener was Alfred, not Kurt. That's a small difference but Wegener actually had a brother, Kurt, who completed an expedition of his.

But let me return to the sociological and moral questions. Roberts asks about Heegner:
A poignant question remains. “[W]as it a disgraceful scandal that his contribution was not recognized in his lifetime?” [27] Van der Poorten thinks not: "[A] recognized mathematician, had best have clear arguments written in the language of the majority—the language expected by other mathematicians—if her surprising arguments are to get a proper hearing. That’s not unfair; it’s our playing the odds."
Such statements simply make me immensely angry because what I read in the quote above is "it's great to be a dishonest immoral jerk – who can urinate into your face – and it's especially great when large mobs urinate in this way". I just can't understand where this amount of misunderstanding of the purpose of mathematics comes from.

Was it a disgraceful scandal that Heegner was mocked in this way? Of course it was a disgraceful scandal – and as far as I can say, it is still one. How it could not be one? We know that his result was correct and his proof was basically complete, up to some minor gap. There is no actual doubt today that the statement "the proof was fatally flawed" was simply a lie. And when something is a lie, it simply has to matter.

In the feudal era, scientists who would turn out to be this wrong and corrupt would probably be hanged by their king. I don't claim that we need to preserve all the details of the protocol but there has to be a punishment for such demonstrable failures that are harmful for the system – and someone's being in a majority by some counting shouldn't change anything about the need for the punishment.

Yes, when things look incomprehensible, people "play the odds" and have a higher probability to denounce a proof (or physical theory or anything) proposed by someone else. That's completely normal. But what is not normal is the suggestion that by "playing the odds", people should get rid of their responsibility for torpedoing correct and important results. People just shouldn't be able to get rid of the responsibility. You may "play the odds" and almost everyone does it in one way or another. But the history must still judge you according to results, not according to your own interpretation of the "odds"!

Heegner was brilliant and the people who mocked him were failures. To suggest anything else means to place the collective personal interests of some malicious, mathematically inferior jerks above the mathematical truth. Roberts adds
It is one thing for a proof to be correct; quite another, for a proof to be comprehensible.
Right. But it's only the truth that should matter for the history – at least in science and mathematics. After all, the adjective "comprehensible" is incredibly subjective and consequently, and that's even more serious, it may be used in a corrupt way, as an unjustified insult to suppress someone else. It's really hard to "prove" that something (like Močizuki's papers) is incomprehensible. People generally realize that it's hard which is why they don't demand a proof. That's the reason why the accusation that "something is imcomprehensible" may be used to suppress people like Heegner. And maybe Močizuki, too.

Maybe his papers were "incomprehensible" but we must always add "incomprehensible to whom". They were incomprehensible to the people who weren't really as good mathematicians as himself – because while he could have found those cool results as the first man, they were not even capable of following a paper where the reasoning was describe in detail, with almost no missing pieces. The paper looks reasonably comprehensible to mathematicians today.

So indeed, there may be a breakdown in communication and papers may be "incomprehensible" but the fault may often be – and I actually think that in most cases, it is – on the side of the people who fail to understand. It's just totally unacceptable to claim that the "failure of comprehension" is always the speaker's fault. To codify this rule would mean to place masses of incompetent and/or lazy and/or dishonest listeners above the creative activity forever. With this rule, the dumb masses of listeners would have the opportunity to comprehend "less and less" and effectively demonize if not criminalize an increasing portion of the creative people's activities.

OK, I am switching to the second topic which is that some people apparently don't want to comprehend Močizuki.

Again, I am not certain that his whole theory is correct, including all the steps that are needed to apply it in a proof of the \(abc\) conjecture and other things. But despite my uncertainty, I think that I can understand some misbehavior that is actually causing the communication problems – and it's a misbehavior of the European folks.

Roberts often discusses that the key problem is Theorem 3.11 and Corollary 3.12 of Močizuki's work. The misunderstanding has been localized quite clearly – I think that all of us may understand it. Močizuki is using a bunch of objects that are isomorphic to each other (they have the same "shape") but they are considered not equal to each other, a point that he emphasizes both verbally as well as by the attachment of a rich collection of subscripts and superscripts.

As far as I can see, the likes of Peter Scholze and Jakob Stix behave as they were completely deaf, they ignore the superscripts, subscripts, and verbal disclaimers saying that the objects aren't equal to each other. Instead, they keep on assuming that the objects "should" be equal according to something that should really be called an irrational faith or an ideology. And the ideology is an extremist application of the assumption
objects that are isomorphic to each other are equal.
Roberts discusses this particular philosophy in some detail. This "structural thinking" goes back to guys like David Hilbert and Richard Dedekind. They emphasized that you don't need to know what a point or a number "is". You may imagine that numbers "are" animals, like the mouse and cat are 1 and 2 in the SJW Hejný method in Czechia. What matters are just the relationships and structures. Great.

Roberts identifies "category theory" as the modern codification of this philosophical thinking. You know, during the high school, I won a book by Birkhoff and MacLane, Algebra (after a mathematical olympiad I won or something), which was already written with the "category theory" philosophy in mind. I was around 17 years old or so and while the book was useful in some respects, I was mostly disgusted by it and it helped to push me in the direction that "I didn't want to have much to do with the professional mathematicians' culture". The book was too formal, like Bourbaki, and it did lots of things for non-existing reasons. Also, I had hoped that a book with "algebra" in its title had to explain the Lie groups and their representations to me (needed for grand unification etc.) – but this thick book did nothing of the sort. A disappointing prize, indeed.

Of course, even afterwards, I have asked an incredible number of professional mathematicians, including some very famous ones, and including on trips done in the last summer, to convince me that the existence of "category theory" as a field is really useful, it makes sense. I have learned various snippets many times, some of them have been useful in string theory etc., but I have never been persuaded that there's a good reason for the existence of a real "theory" with all these ambitious claims of a unifying power.

We normally talk about sets, relationships, and maps – and isomorphisms etc. that respect the operations and other structures – but it's still not "category theory" yet, is it? What are the actual steps – and reasons for the steps – that make you "upgrade" to "category theory"? What is the actual purpose of "functors" and why should one believe that it's useful to try to reorganize the mathematical knowledge according to this Ansatz? No clear answers have ever been given to me. Category theory is a rigorous expression of some intuition about "analogies", I was repeatedly told. Great. But why would one believe that a seemingly unavoidably fuzzy concept such as "analogies" should stand on a rigorous package of definitions at all? It seems that none of the mathematicians has ever asked a similar question. None of them could give a coherent answer. As far as I can say, all of them were just blindly repeating some theory they heard from someone else. I think that there's a lot of mindless indoctrination in mathematics.

At some moment, Roberts says something that makes sense:
A category does not and cannot distinguish between isomorphic objects. Anything that can be specified about a given object, using only the language and structure of a category, is true of any other isomorphic object. This idea is immensely powerful, ...
You know, I understood this idea as a kid, well before I got that huge book by Birkhoff and MacLane. For example, \(SO(3)\) may be defined as the set of \(3\times 3\) matrices obeying a certain condition, as a set of formal transformations acting on an abstract space represented in a certain way, or the real space in a physical model of our world, or as \(SU(2) / \ZZ_2\) acting on spinors etc. And it makes sense to say that all these groups are "the same group" because its internal structure is what we care most.

So of course I always appreciated – usually much more than almost anyone around – that it's "right" to consider complex enough mathematical structures to be "the same" as soon as they are isomorphic. But what I see that many mathematicians are doing is something stronger. They effectively say:
If two objects are isomorphic, you are obliged to consider them equal immediately.
If "category theory" may be framed as a rationalization of this thesis, then "category theory" is a fundamentally counterproductive ideology because the quote above is just an unjustified extremist ideological slogan. For it to be a good foundation of mathematics, the following would have to be true:
If you always insist on labeling two objects as equal as soon as they are seen isomorphic, you will never overlook any interesting structure or result or possibility in mathematics or mathematical physics.
OK, call me an infidel or a heretic but I believe that the faith in the claim above is completely wrong. There is no reason why it should be an "obligation" to consider isomorphic objects to be equal. And there is no reason to believe that this attitude will always lead to "better mathematics".

In physics, we have very good examples of situations in which it's still important not to consider isomorphic things equal. In particular, you may act with symmetries on physical states. There are global and local symmetries. Local symmetries aren't real symmetries – you only allow states that are invariant under local symmetries (i.e. singlets).

On the other hand, global symmetries are real symmetries. They map \(\ket\psi\) to \(G\ket\psi\) but in general, the states \(\ket\psi\) and \(G\ket\psi\) are not equal to each other. They are physically analogous, including quantitative measurements, but they simply must be considered different. Well, even points in a regular 3D Euclidean space are "equivalent to each other" (you may map one to another by rotations or translations) but we must still consider them "not equal to each other", right?

Clearly, there is nothing mathematically inconsistent about assuming that two objects are isomorphic yet not equal. If the likes of Scholze and Stix are unable to "hear" that two objects are unequal even though Močizuki makes this point super-comprehensible, both at the level of notation and words – because they assume some "isomorphic must be equal" ideology – that's too bad and as far as I can say, it is absolutely obvious that the cause of the breakdown in the communication is the Europeans' fault. If they claim that in order to maximize the progress in mathematics, it's right to assume that "all good proofs must always immediately consider isomorphic objects equal", then I first want to see a proof of this bold assumption. I think it's completely wrong to make far-reaching assumptions of this type without a good reason.

But you know where I am going now, right? I think that this philosophy that "isomorphic things must be considered equal" is no longer just a purely mathematical, impersonal, socially neutral meme. It is correlated with some other political and ideological movements that are increasingly ruining the Western societies. Well, look at the statements:
Mathematical objects that are isomorphic must be considered equal.

All people and their groups – defined by sex, nation, race, sexual orientation, and more – must be considered equal in all circumstances and unequal outcomes must be considered a proof of someone's malice.
The second slogan is clearly an umbrella slogan for identity politics – producing things like "reverse" sexism ("feminism"), "reverse" racism ("multiculturalism"), and related pathologies. These pathologies make common sense, ordinary discussions, and rudimentary meritocratic choices increasingly impossible in the West.

But the first slogan is somewhat analogous and it seems rather plausible that its proponents – and proponents of "category theory" – are well aware of this similarity. After all, Roberts' text is titled
A Crisis of Identification
so aside from the clearly left-wing "equality", we also have a word with the "ident*" root, something that has an obvious proximity to "identity politics". What is your identity? Can two isomorphic mathematical objects discussed by a Japanese men accepted to have two different identities, or is it politically incorrect? So it has seemed increasingly likely to me that the likes of Scholze and Stix "don't want" to understand what Močizuki is saying because it conflicts with some ideology that they place above everything else – and the ideology, while completely unjustified, is fundamentally inseparable from the politically ideological delusions of many contemporary Western academics, too.

In this sense, it looks very plausible that "identity politics" may also be blamed for the Westerners' incapability of catching up with the Japanese "arithmetic deformation theory", a topic that you would normally believe to have zero links with any politics or ideology!

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