Mind the gap: the Mochizuki controversy

If we were to carry out a survey about what is most characteristic of Mathematics, what distinguishes it most precisely from other branches of human knowledge, we would most probably find that the concept of proof is, in one way or another, associated with almost all the answers. Not surprisingly, for a long period of time, mathematics has been synonymous with Exact Sciences. Whether we focus on one speciality or the other, even those who dedicate themselves to more applied research recognise in proof the unequivocally mathematical feature which, for the same price, also allows us to progress as a discipline, free from discussions of the kind that constitute the day-to-day life of the so-called Experimental Sciences.

Naturally, even the most innocent of us who dedicate ourselves to this field will surely agree that mathematics, as a human activity, is not free of clashes and friction between those who work on it. The most common and also the most famous cases historically have come from paternity problems. The well-known conflict between Newton and Leibniz with the birth of differential calculus as a backdrop, or the bitter confrontation between Gauss and Legendre over the invention of the method of least squares are some of the best known.

But they were not the only examples. The early 20th century environs also witnessed a major debate about the foundations of mathematics, which were viewed for the first time in the perspective of the emerging logical rigour. The pioneering works of Cantor, Frege or Russell gave way to the discussion about axiomatisation, the acceptance or not of the continuum hypothesis, Gödel’s Theorems and many other great findings and diatribes in a field where, at times, it seemed that the steady pace of the Exact Sciences was somewhat slippery after all.

But you can’t argue about demonstrations, can you? After all, an argument is either a valid proof, or it is not.

Well, not exactly.

Today we are going to talk about a recent and, to a certain extent, very unusual case. Because it is not a question of discerning who proved a result first. Nor is it about traversing the fuzzy border between Mathematics and Philosophy, or arguing about infinity or the apeiron. The problem we bring you today has a fascinating context and is also of a truly unusual and surprising kind, which can be summed up in one question: is this proof right?

Let’s start at the beginning. And the beginning is, of course:

 

1. abc

No, we are not talking about a nostalgia-stricken Spanish newspaper, or the Motown theme popularised by The Jackson Five. We are talking about the Oesterlé-Masser conjecture, also known as the abc conjecture. The idea is very, very simple. If we have three natural numbers \(a\), \(b\) and \(c\) with no common factors, verifying \(a+b = c\), and we call \(d\) the product of all prime factors of \(a\), \(b\) and \(c\) (all with exponent \(1\)), then \(d\) cannot be in general smaller than \(c\).

The precise way of stating this is as follows. We define the radical of a natural number \(n\), noted  \(\mbox{rad}(n)\), as the product of its prime factors. For example, \(\mbox{rad}(180) = 60\) or, if \(p\) is prime, \(\mbox{rad}(p^r) = p\).

Then, the Oesterlé-Masser conjecture states that, for all real \(\varepsilon >0\), there exists only a finite number of natural ternaries \((a,b,c)\) such that:

1. The numbers \(a\), \(b\) and \(c\) have no common factors.
2. We have that \(a+b = c\).
3. We have that \(c >  \mbox{rad}(abc)^{1 + \varepsilon}\).

So said, it may not be obvious, but the conjecture has a list of consequences that resembles that of side effects of the Saridon, almost all of them focused on the area of Arithmetic Geometry, that is, the part of Number Theory that interacts with Geometry. Among these consequences there are some very famous results already proved, such as Roth’s Approximation Theorem, Faltings’ Theorem or Fermat’s Last Theorem, but also a remarkable number of open problems that would be solved; such as the Erdös-Woods conjecture (except for an eventual finite number of counterexamples), the Fermat-Catalan conjecture, the Granville-Langevin conjecture or the Lang conjecture, one of the most sought-after results in elliptic curve theory today.

It is by no means the first time, nor will it be the last, that in the face of a complex and apparently unassailable problem someone has raised the stakes and, instead of attacking the conjecture in question, deduced it as a corollary of something more general and abstract. Here is that someone:

 

2. Our man in Kyoto

Our first hero, Shinichi Mochizuki (Tokyo, 1969) is no newcomer or misguided amateur; in fact he has an enviable research career. Raised in the United States, he did his PhD at Princeton under the direction of Gerd Faltings (Fields Medal 1986) and at the age of less than 30 proved the Grothendieck conjecture in non-Abelian geometry (which is as complicated as its name suggests). Shortly afterwards he was invited as a lecturer at the International Congress of Mathematics, he appeared in several pools for future Fields medals and, in fact, before the age of 35 he was already a professor at RIMS in Kyoto, a world-leading centre.

In 2012 he published a preprint, i.e. a paper that had not yet been duly corrected and reviewed by expert researchers, but whose results (and proofs) he was convinced enough to announce publicly. The preprint actually came in four parts, with the generic title Inter-universal Teichmüller Theory (IUTT). These parts are:

• Inter-universal Teichmuller Theory I: Construction of Hodge Theaters.
• Inter-universal Teichmuller Theory II: Hodge–Arakelov-theoretic Evaluation.
• Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice.
• Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations.

Together, they run to some 500 pages. They are all available on Mochizuki’s website, as are all his earlier and later works. According to its author, the IUTT involved many far-reaching consequences, among others the proof of the Szpiro conjecture and the Vojta conjecture for curves (both very important results). But, beyond this, it provided a proof of the Oesterlé-Masser conjecture, thus closing one of the star problems of Number Theory.

The international community was taken by surprise. Although it was known that Mochizuki, being as he was a highly respected authority in his field, was working on the abc problem, a general theory of such enormous consequences was not expected. Neither a leak, nor a talk at an international conference, nor a veiled reference in a previously published paper hinted at what was to come. And as in these cases, a result of this magnitude had to be sifted through the research community before it could be accepted as valid.

This was no easy task. For one thing, Mochizuki was willing to cooperate in facilitating the understanding of his work – as long as he did it from Kyoto, refusing to leave his residence. For another, the work itself seemed impossible to understand, even for the most respected names in the field. In fact, the Clay Institute and Oxford University organised in 2015, three years after the publication of the preprints, and amidst an atmosphere of collective frustration, a conference focused exclusively on the IUTT. The aim was not to validate the proof of the abc conjecture as correct, or to point out a fundamental error, but to better understand the general theory created by Mochizuki.

The list of participants at the conference is practically a who’s who in Number Theory at the time, including Mochizuki himself (via Skype, of course). The work was distributed and the presentation of the results was staggered in the hope that they could be presented coherently. To no avail. The summary by Brian Conrad, a Stanford professor and one of the attendees, illustrates very well the feeling of collective failure that gripped the participants at the end of the conference. This inspiring metaphor can be found on the aforementioned page:

It felt like taking a course in Linear Algebra in which one is repeatedly told “Consider a pair of isomorphic vector spaces” but is never given an interesting example (of which there are many) despite repeated requests and eventually one is told “You have been given examples.”

As things stood, the situation was at an apparent impasse. On the one hand, Mochizuki and some experts who validated his results, took the IUTT with all its conclusions and corollaries for granted. On the other hand, most of the mathematical community (of Number Theory / Arithmetic Geometry, to be more specific), was simply unable to understand the work, not even in some substantial part, which could have helped to corroborate or discard the results. We do not know for sure to which journals Mochizuki sent his work, but it is certain that none of them took the step of publishing it.

And the years passed from the birth of the IUTT without any noteworthy advances, until 2018. That was just when it really got interesting.

 

3. That escalated (moderately) quickly

In March 2018 Mochizuki received in Kyoto (where else) a visit that would mark a turning point in this story. Two German mathematicians, Peter Scholze (of whom we have already spoken here) and Jakob Stix, both acknowledged experts in Arithmetic Geometry, set themselves the task of unravelling the Gordian knot that the IUTT and the proof of the abc conjecture had become. To do so, they went to the source, armed with many questions, and there they spent a week of discussions with Mochizuki and Yoichiru Hoshi, the latter’s collaborator in the development of the IUTT.

In this link you can find a summary (with reports added as links at the end) of the many disagreements that resulted from that week. On the one hand, Scholze and Stix made their their position clear: not only was the proof of the abc conjecture not correct, but there was also a gap (an argument gap) whereby it was impossible to arrive at such a proof using Mochizuki’s arguments. The reason: a very recent result in Arithmetic Geometry that we owe to… Shinichi Mochizuki. Well, that apple did not fall far from the tree, definitely.

The response from Mochizuki, and from the select (which is a polite way of saying small) community that supports his claims, was not to be long in coming. However, it was not expressed in the commonly expected smooth style, but rather in an unusually aggressive display of rebuttal. This was followed by a counter-reply from Scholze-Stix and again a response from Mochizuki (all documents are linked above). The discussion moved to the public forum, and Mochizuki eventually stopped referring to Scholze and Stix as “the other participants” and started calling them RCS (acronym for “the redundant copies school”), which also included those who publicly doubted his claims, such as Kiran Kedlaya or Peter Woit, whose blog Not Even Wrong became a lively discussion forum whenever the subject came up. Not always in good manners, it must be said.

However, one piece of the puzzle was still missing. Mochizuki’s articles had not been endorsed by any scientific journal, which could objectively be considered a problem for the credibility of the work. Finally, in February 2020, an established journal agreed to publish Mochizuki’s four papers. Several unusual events surround this publication, however. The journal involved is Publications of the Research Institute for Mathematical Sciences, or Publ. RIMS, as it is often abbreviated. It is not the journal one would expect when it comes to publishing such large-scale results, of course, but it is a journal with a long tradition in Algebraic Geometry and Arithmetic, published by the European Mathematical Society.

The problems, however, start when you consider that the Research Institute for Mathematical Sciences that sponsors the journal is the same RIMS in Kyoto where Mochizuki is a professor and, in fact, Shinichi Mochizuki is the editor-in-chief of the journal, i.e. its highest scientific authority. The publication was announced in April 2020, at a press conference (!!!) where two members of the editorial board defended the decision to publish the articles, claiming that the author had been completely unaware of the review and acceptance process. It cannot be denied, however, that even if these claims were valid, the credibility and prestige of the journal, as well as possibly Mochizuki’s work, have been permanently damaged by this decision.

So much for the summary of a controversy that is sure to continue for some time to come. In a few years’ time, the tools needed to better understand the IUTT’s approach will probably be more readily available, and perhaps then we will be able to say with certainty whether Mochizuki’s vision really succeeded in overcoming the challenge of the abc conjecture. Because, after all, a reasoning is either a valid proof, or it is not. Right?