Shinichi Mochizuki's ABC Conjecture and Replication Crisis in Maths

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There is a fascinating story that I'm sure a lot of you have followed.

In 2012, a top mathematician, Shinichi Mochizuki[1], has claimed to have solved the ABC conjecture[2] (an important longstanding problem in number theory), using his own very unique, complex, and abstract Inter-universal Teichmüller theory[3]. [4]:

To complete the proof, Mochizuki had invented a new branch of his discipline, one that is astonishingly abstract even by the standards of pure maths. “Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” number theorist Jordan Ellenberg, of the University of Wisconsin–Madison, wrote on his blog a few days after the paper appeared.

His Theory and proof has finally been accepted for publication in 2020.[5][7] Despite this, it is still not widely accepted, primarily because hardly anyone in the world is able to understand his theory and proof.

In 2018, Peter Scholze who directs the Max Planck Institute for Mathematics, and Jakob Stix, made an attempt, and attended a Clay Mathematics Institute workshop dedicated to getting to better understanding the proof in order to work towards determining whether it is correct. While unable to reach a full understanding of the proof, they claimed to have isolated one part of it that they determined was a gap, and that the proof was therefore at least not complete. In response, Mochizuki claimed that they had simply misunderstood the theory.[9] Mochizuki wrote in his rebuttle[8]:

Indeed, at numerous points in the March discussions, I was often tempted to issue a response of the following form to various assertions of SS (but typically refrained from doing so!):Yes! Yes! Of course, I completely agree that the theory that you are discussing is completely absurd and meaningless, but that theory is completely different from IUTch!

There is at least one peer, Ivan Fesenko, who claims to fully understand and accept it. In response to Peter Scholze and Jakob Stix's report, he wrote some pretty harsh words[11]:

3.1. On reaction to IUT from some mathematicians. Mochizuki’s work includes fundamental contribu-tions in numerous directions: Hodge–Arakelov theory, anabelian geometry, mono-anabelian geometry, com-binatorial anabelian geometry, Grothendieck–Teichmüller group, p-adic Teichmüller theory, inter-universalTeichmüller theory. Except for the last direction, none of his work has ever been criticised because it was read and appreciated by experts in the subject area. ‘Love of knowledge, without a love to learn, finds itselfobscured by loose speculation’.14‘You can lead a horse to water but you can’t make her drink’. Few mathematicians chose to talk in abenighted way about IUT and its study, while being fully aware they simply do not have any authority in the subject area. Talking exclusively with non-experts, who have very weird ideas about IUT, can only produce weird outcomes.

They made public their ignorant negative opinions about a fundamental development in the subject area where they have empty research record, with no evidence of their serious study of it, and without providing any math evidence of errors in the theory. Non-expert negative opinions about IUT seeded a pernicious mistrust of this rare breakthrough and pioneering math research in general. Their behaviour contributes to the erosion of professional norms. In particular, there are no active US researchers in anabelian geometry of hyperbolic curves over number fields, but most of irrational negative comments about IUT originated from a tiny group of mathematicians in that country. Some chose to spread a malicious distortion of the math truth or false rumours. One of them is talking about some kind of controversy about the status of IUT. This is not an argument that can hope to be accepted: in order to have a controversy about a mathematical work there should be genuine experts on both sides ofthe argument able to provide valid math arguments which can pass peer review. This is plainly not the case for IUT: not a single expert in IUT is known who sees mistakes in the published version of IUT and none of internet critical remarks about IUT can pass peer review. This also explains why not the trial of serious math peer review but the choice of shallow posting is the only venue for non-expert public chats about IUT.

Whatever the case about the validity of the proof, I am fascinated by the issues it brings to light in theory and proof complexity and communication of mathematical ideas. Some say it is a taste of what has long been coming, a replication crisis in mathematics and a point where the peer review system breaks down due to practical difficulties in third party verification. Vladimir Voevodsky writes [12]:

Seeing how mathematics was developing as a science, I understood that the time is approaching when the proof of yet another conjecture will change little. I understood that mathematics is facing a crisis ... [it] has to do with the complication of pure mathematics which leads, again sooner or later, to articles becoming too difficult for detailed proofreading and to the start of unnoticed errors accumulating. And since mathematics is a very deep science, in the sense that the results of a single article usually depend on results of very many previous articles, this accumulation of errors is very dangerous.

And Bordg writes[12]:

One reason for this unfortunate state of affairs is mathematical research currently relying “on a complex system of mutual trust based on reputations.”

and questions whether computer verification could ever solve the problem.

It is a bit sad to me to imagine dedicating your career to solving a major problem in mathematics, only to end up writing a proof that nobody is willing to take the time to read and understand. Ultimately, if Mochizuki was not a world famous mathematician, I doubt his paper would have received any attention. It would have simply vanished. It seems likely in fact that a large number of breakthroughs have vanished. P vs NP may already be a solved problem along with the rest of the millennium prize problems. The implication is that only celebrities can work using completely new approaches and the rest are relegated to incremental work that is easy enough to review that others will be willing to give it a chance and check it out. It seems like an unacceptable problem to me. Similar problems stem from the use of AI and it's interprebility.

What do you think about the topic? What to do when proofs becomes too complex to verify? Should we develop an international system for writing more verifiable (perhaps computationally verifiable) proofs? Is it even feasible? At some point, if proofs take the form that a computer can check, then they likely are in a form that people can't feasibly check, and vice versa. In order to be deterministically verifiable, Mochizuki's 400 pages would likely expand to many thousands of pages, with explicit encoding of all of the prerequisites/corollaries and previous results which are relied on. Otherwise, futuristic AI could try to verify it as is, or in some incompletely explicit form. But it would likely then, like us, be prone to mistakes, or at least not provably immune from them, and we would have to trust the AI (essentially replacing human expert's as authorities).

[1] https://en.wikipedia.org/wiki/Shinichi_Mochizuki
[2] https://en.wikipedia.org/wiki/Abc_conjecture
[3] https://en.wikipedia.org/wiki/Inter-universal_Teichmüller_theory
[4] https://www.nature.com/news/the-big...-mochizuki-and-the-impenetrable-proof-1.18509
[5] https://www.nature.com/articles/d41586-020-00998-2
[6] https://en.wikipedia.org/wiki/Edward_Frenkel
[7] https://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html
[8] https://www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf
[9] https://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html
[10] https://en.wikipedia.org/wiki/Ivan_Fesenko
[11] https://www.maths.nottingham.ac.uk/plp/pmzibf/rapg.pdf
[12] https://link.springer.com/article/10.1007/s00283-020-10037-7
 
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  • #2
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Historically, many ideas in mathematics had a hard time being accept as a mathematical fact or theory until there is a person that can see the beauty in the proof. There were a major push back of complex numbers, Galois Theory, non-Euclidean Geometry, and Naïve Set Theory. According to Wikipedia article on Galois Theory, it was poorly understood, but eventually, it was understood and modified for the next generation of mathematician to understand the basics concepts and practical uses.
 
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His Theory and proof has finally been accepted for publication in 2020.
In a journal published by his home institute, where he is the leading editor. As far as I know he didn't sign off his own paper, but the conflict of interest is still a concern.
Ultimately, if Mochizuki was not a world famous mathematician, I doubt his paper would have received any attention. It would have simply vanished. It seems likely in fact that a large number of breakthroughs have vanished.
Breakthroughs rarely happen in a vacuum. You have some intermediate result here, you develop some new tools there, you prove some interesting other properties on the way. You discuss the problems with colleagues on conferences. That's a track record others can follow.
The chance that a 2 page vixra upload by someone completely unknown proves some famous conjecture is essentially zero.
 
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  • #4
martinbn
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My impression is that the consensus is that there is a sirious gap.
 
  • #5
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Another problem is his denial to explain his theory. He wrote a very thick paper, made a claim, and left the world alone with it. With something out of the blue, others can hardly read, let alone understand. The only argument we have is, that he is an ordinary professor. Everything else is typical crackpot behavior.

It has been said, that Wiles's proof could have only been understood by a handful of mathematicians in the world when it was published. But he played by the rules. Explained it, discussed it, and fixed a gap.
 
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Is there any evidence that he hasn't been trying to explain it. Or what is the basis for arguing that?
 
  • #7
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Is there any evidence that he hasn't been trying to explain it. Or what is the basis for arguing that?
I read articles, that he doesn't leave his university for travel in general, and that people who visited him didn't get very far either. People who have a completely new theory are normally expected to convince others, not the other way around.

E.g.
Mochizuki himself hadn't even come to the workshop at the famous Clay Mathematics Institute. He is now reluctant to leave Kyoto, where he is doing research at the Research Institute for Mathematical Sciences (RIMS) at Kyoto University. At least the Japanese answered questions from the workshop participants via Skype, but without really contributing to the fact that his evidence could finally be verified.
https://www.spiegel.de/wissenschaft...matiker-verstehen-beweis-nicht-a-1068399.html
 
  • #8
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At some point would it be an unreasonable thing to expect though? Because it is not as though a talk will be enough. I've read that there is previous work that you need to learn first, and getting to that point could require up to 10 years.

In the first place there needs to be someone to meet him with evidence that they have what it takes and will make a serious attempt to make the effort.
 
  • #9
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Is there any evidence that he hasn't been trying to explain it. Or what is the basis for arguing that?
Not trying, or not trying very well. That's a complaint you can find in many places.

https://www.quantamagazine.org/hope-rekindled-for-abc-proof-20151221/
Compounding the difficulty, Mochizuki turned down all invitations to lecture on his work outside of Japan. Most mathematicians who attempted to read the papers got nowhere and soon abandoned the effort.
[...]
The three are among a small handful of people who have devoted intense effort to understanding Mochizuki’s IUT theory. By all accounts, their talks were impossible to follow.
And as far as I understand that meeting was already a big improvement over the situation before.

Here is an article by Frank Calegari’s

He also has a section about breakthroughs not happening in a vacuum:
Usually when there is a breakthrough in mathematics, there is an explosion of new activity when other mathematicians are able to exploit the new ideas to prove new theorems, usually in directions not anticipated by the original discoverer(s). This has manifestly not been the case for ABC, and this fact alone is one of the most compelling reasons why people are suspicious.
Terence Tao replied with a similar comment, including
It seems bizarre to me that there would be an entire self-contained theory whose only external application is to prove the abc conjecture after 300+ pages of set up, with no smaller fragment of this setup having any non-trivial external consequence whatsoever.
In the first place there needs to be someone to meet him with evidence that they have what it takes and will make a serious attempt to make the effort.
People have made serious efforts. And they gave up. Do you expect them to spend the rest of their career on a topic no one can understand?
 
  • #10
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It seems bizarre to me that there would be an entire self-contained theory whose only external application is to prove the abc conjecture after 300+ pages of set up, with no smaller fragment of this setup having any non-trivial external consequence whatsoever.

What about Ivan Fesenko's work?

The third development is a higher adelic study of relations between the arithmetic and analytic ranks of an elliptic curve over a global field, which in conjectural form are stated in the Birch and Swinnerton-Dyer conjecture for the zeta function of elliptic surfaces.[pub 15][pub 16] This new method uses FIT theory, two adelic structures: the geometric additive adelic structure and the arithmetic multiplicative adelic structure and an interplay between them motivated by higher class field theory. These two adelic structures have some similarity to two symmetries in inter-universal Teichmüller theory of Mochizuki.[pub 17]
...
Fesenko played an active role in organizing the study of inter-universal Teichmüller theory of Shinichi Mochizuki. He is the author of a survey[pub 19] and a general article[pub 20] on this theory. He co-organized two international workshops on IUT.[pub 21][pub 22]

https://en.m.wikipedia.org/wiki/Ivan_Fesenko
 
  • #11
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People have made serious effort. And they gave up. Do you expect them to spend the rest of their career on a topic no one can understand?
No, but I'm not sure whether I can dismiss Fesenko's comments that Scholze and Stix are simply not experts in the subject matter and have made no adequate effort to understand the proof. But it sort of seems that it is their remarks and their failure to be convinced that is usually referenced. It is also argued that S snd S's criticisms didn't hold water, and they dropped out of serious discussions upon the response to their report that pointed it out.

I can understand why Mochizuki would be annoyed, since to him, his work has been discredited illegitimately.

But if it is true that S and S are not qualified to understand the paper, don't have specific valid criticism, and have simply moved on already, then clearly they are not very relevant.

Except, I think it is uncomfortable to accept that there may be proofs to famous conjectures, which even the director of the Clay Institute can't understand with a reasonable amount of effort.
 
  • #12
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The real point of proving the abc conjecture isn't to know that it's true, it's to know why it's true. People are happy to write lots of papers where they say if they abc conjecture is true then xyz, with the hope that it will be proven true at some point. We do this because we think it's true and are pretty confident we can just keep progressing without a proof for now. If God came down and said yeah that's total true little humans, but the explanation why is too complicated, that would be.... Useful, but not as useful as you might imagine. If a whole new branch of mathematics was invented, it sounds like it's also going to die with him, so it's not that impactful on the course of human history.
 
  • #13
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No, but I'm not sure whether I can dismiss Fesenko's comments
Why not? You seem to have no problem dismissing comments by numerous world-class mathematicians without a second thought. Basically everyone apart of Mochizuki and some of his friends gave up understanding it or says there is a flaw at the place Scholze&co pointed out and it can't be understood because it's wrong. Publish a smaller non-trivial result of IUT? Nothing. Break down this lemma into smaller steps? Nothing. Find some other application of IUT? Nothing.
 
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  • #14
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But if it is true that S and S are not qualified to understand the paper, don't have specific valid criticism, and have simply moved on already, then clearly they are not very relevant.
This is not at all the case. They are very qualified- Scholze is a fields medalist(!) working in arithmetic geometry/algebraic number theory. They identified a specific lemma that they thought was incorrect. They wrote up a manuscript explaining the issue. They met with Mochizuki and and did not get their concerns addressed.
 
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Why not? You seem to have no problem dismissing comments by numerous world-class mathematicians without a second thought. Basically everyone apart of Mochizuki and some of his friends gave up understanding it or says there is a flaw at the place Scholze&co pointed out and it can't be understood because it's wrong. Publish a smaller non-trivial result of IUT? Nothing. Break down this lemma into smaller steps? Nothing. Find some other application of IUT? Nothing.
I'd personally just withhold dismissal of either side. Like Voevodsky alludes to, at this point we are simply interpreting a complex network of expert opinions. In this case it's made more complex due to the bitterness/feuding. S and S are criticized because they didn't just say they had concerns, they said there was no proof. In response their opponents said they made a a set of particular mistakes, and S and S didn't respond back, except to basically just say, oh well we're out, but we stick with our original argument. Spectators now just see experts in two camps asserting themselves and insulting eachother.
 
  • #16
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This is not at all the case. They are very qualified- Scholze is a fields medalist(!) working in arithmetic geometry/algebraic number theory.
It was claimed that an understating of the work in Anabelian geometry was the missing qualification for Scholze.

I have no idea.
 
  • #17
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Publish a smaller non-trivial result of IUT? Nothing. Break down this lemma into smaller steps? Nothing. Find some other application of IUT? Nothing.
It's a little tough though, because one of the problems now is the length of the 4 main paper, not to mention the other related works, and addition developments of IUT. If you break it into smaller steps, and explain it with more detail, it will be much longer.

I believe Mochizuki has asserted the gap is a communication gap, which is non-trivial in itself to solve.

People are blaming Mochizuki for not trying hard enough to explain it so that others can understand it. I'm not sure if he actually is just lazy and arrogant as people imply, or if it really is non-trivial to explain. And it isn't clear how much is on him to explain, rather than for the others to simply learn from the existing literature.

Maybe it comes down to asking for a shortcut to understanding the proof so as to save time. First you would need to discover the shortcuts. How do we even know that there are significant shortcuts? Are we sure it isn't like asking someone to explain GR to in a forum post, so that you can skip reading your textbook and doing the exercises?

And how do we know the things that we get Nothing's about are things we can reasonably demand or expect in the time frame, or that those things aren't being worked on as we speak?
 
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  • #18
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Like Voevodsky alludes to, at this point we are simply interpreting a complex network of expert opinions.
This is always the case with new theorems. Few can understand Wiles's or Perelman's proofs, or even have read them. Nevertheless, we rely on the community, the few who said they were correct. This is how science goes, you can't start at Peano every single time. It is therefore the author's duty to convince the community, not the other way around. If you can't or won't, what's the difference from any other crackpot?
 
  • #19
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If you can't or won't, what's the difference from any other crackpot?
I think the big question is: can't or won't, or maybe can and is trying but it will take time and patients? I actually cannot tell which is the case.

You can't really blame him for giving up teaching IUT to S and S, because they've also seemed to have given up trying to learn it. Maybe it will take years for a trustworthy authoritative community to develop around his work, or maybe it will never happen.
 
  • #22
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Maybe the problem is that pure mathematicians cannot longer think like physicists. The proofs in mathematics are so difficult because mathematicians have very high standards of rigor. They want absolute certainty that their claims are true, but the consequence is that often only a small number of people can convince themselves that a claim is true. With loosening standards of rigor, the proofs (or arguments, if you like) become much easier, so that a lot of experts can understand them. Sure, in that case there is no longer absolute certainty, but is that really such a problem? Physicists can live with their non-rigorous proofs without absolute certainty, so why can't mathematicians? Perhaps absolute certainty is overrated.

Don't get me wrong, I am not proposing to completely give up rigor and absolute certainty. I am proposing a peaceful coexistence of rigorous and non-rigorous math. The latter gives intuitive understanding, the former gives certainty. Both have a value. It would make sense if first non-rigorous guys gave an intuitive non-rigorous proof, and then later rigorous guys refined it by making it rigorous. That's how it works in physics. But in pure math, the opposite is common: First rigorous guys give the rigorous proof that only a few can understand, and then one of those few explains it non-rigorously so that more people can understand it. But there is something perverse about this opposite strategy, the easy intuitive idea should come first. And it seems that this opposite strategy does not work in the ABC case. Maybe the physics-inspired non-rigorous-first strategy would be better in the ABC case. But modern mathematicians are not used to that, that's not how they are trained. Maybe that should be changed.

But I'm a physicist, so I don't expect that mathematicians take my advice how math should be done. :oldbiggrin:
 
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  • #23
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With loosening standards of rigor, the proofs (or arguments, if you like) become much easier, so that a lot of experts can understand them. Sure, in that case there is no longer absolute certainty, but is that really such a problem?
If things become "easier to understand" by skipping details then you don't actually understand them.

If you build mathematics on statements where 10% are incorrect then you run into all sorts of nonsense later. And then you need to backtrack and figure out what's actually right and what is not. If some application relied on that mathematics it might break - and you don't know why, because you thought there was a proof that it works.
 
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  • #24
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If things become "easier to understand" by skipping details then you don't actually understand them.
It depends on definition of "actual understanding". What do you understand better, 4-color theorem or your best friend? Category theory or your favored sport? My point is that, for some purposes, intuitive vague understanding may be more powerful than formal understanding.

If you build mathematics on statements where 10% are incorrect then you run into all sorts of nonsense later. And then you need to backtrack and figure out what's actually right and what is not. If some application relied on that mathematics it might break - and you don't know why, because you thought there was a proof that it works.
Perhaps you missed the 2nd paragraph in my post. We need both kinds of understanding, they are complementary. The problem with modern mathematics is not that it uses rigorous formal understanding, but that it undervalues intuitive understanding.
 
  • #25
martinbn
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... The problem with modern mathematics is not that it uses rigorous formal understanding, but that it undervalues intuitive understanding.
I don't think that last statement is true. The intuitive undesrtanding and arguments are important and used all the time. The only requirement is that at the end one has to be able to provide a complete prove, all the details and rigour, no and waving.
 
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