- #1

- 659

- 538

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