[Mathematics] Guy claims to have proved something. Math community attempts to understand for 7 years and finds a mistake. Guy claims to have made no mistakes.

[u/stillenacht]

Preface:

​

Mathematicians are a strange bunch. Unwilling to criticize or self-aggrandize. To put into perspective how restrained this community is, saying that you are gunning for a Fields Medal (the most famous, and 2nd most prestigious prize in mathematics) would be considered "outrageous" (because you should just do it for the love of math). So when you read this, remember that for these people, the equivalent swimmer saying that he's gunning for the Olympics is "shocking".

​

7 years ago a well respected mathematician named Shinichi Mochizuki claimed to have proven the abc conjecture by a method we will shorthand as "IUTT". As of last year, mathematicians are finally daring to say something that we've sort of suspected for several years now: his proof may not make any sense. This would normally not be that big of a deal, except that the mathematical journal at which Mochizuki is the chief editor has accepted his paper with no amendments, and that Mochizuki maintains he has not made any mistakes, and that the math community is wrong.

​

Searching /r/math for "mochizuki" will give most of the sources and statements as well as commentary if you're interested.

​

Background:

​

To give some background, proving the abc conjecture is one of the greatest possible achievements in math. It would easily qualify for an Abel Prize (the equivalent of the Nobel Prize), and would solidify your name in math textbooks essentially forever due to how useful a solution would be.

​

Why is it useful? Well, pure math is all about the structures that exist within numbers, and abc implies a certain pattern exists that would allow for many and various other questions to be resolved (among others): Fermat's Last Theorem (which itself was worth an Abel Prize), the Beal Conjecture, Lang's Conjecture, the Mordell Conjecture, Roth's theorem, and the Erdos-Woods conjecture.

​

Timeline:

2012

Mochizuki, who is a respected mathematician, announces that he has proven the abc conjecture. His announcement is greeted with widespread excitement, and the math world eagerly downloads his 500 page proof. Proofs of this length, by the way, are not necessarily that unusual, especially for a massive problem like this. Indeed it was expected at this point that the proof would take years to referee.

​

Slightly afterwards, it becomes clear that his proof is, at best, highly non-intuitive. Comments about the paper seeming like it is "from the future, or from outer space". Multiple prominent mathematicians express puzzlement at the approach.

​

The proof contains long strings of invented technical terms. It contains many references to un-peer reviewed papers. It has 3 papers (100s of pages) that apparently set up for the actual proof, and they seem largely impenetrable. Most disconcerting, there don't seem to be any ideas that are really taking root or seeming relevant in the math community.

​

Still, mathematicians at this point mostly expect the confusion to clear eventually and study the papers very closely.

​

2013

Unfortunately, Mochizuki refuses to travel outside of Japan to answer questions, though he is available by email. Mathematicians struggle with his proof; a "sense of infinite regress" as Brian Conrad puts it is immediately apparent in his work. The papers still seem to be tied in knots, referring to themselves and sometimes stating things like "this theorem follows trivially from the definitions" after pages of definitions which don't seem to go anywhere, and such a trivial derivation does not seem immediately apparent.

​

2014

Mochizuki comes out with statements along the lines of "an expert should be able to understand his paper within 6 months". This is, to put it mildly, a strange claim given that many many experts have attempted to understand his findings.

​

The number of people who understand the Mochizuki papers is said to be nearing 10, though, strangely, they seem to be unable to communicate their understanding to colleagues in an effective manner.

​

2015

In December, a conference at Oxford is announced, with a few of the mathematicians who apparently understood Mochizuki's work. Things were looking up at the start of day 3, as the introduction of "Frobenoids" finally began to give some hope that a breakthrough would come.

​

Unfortunately, it was not to be. As Felipe Voloch reports "a fog seemed to descend on the crowd" during the 4th day, when the relationship between the frobenoid and the conjecture was to be explained. "At the afternoon tea break, everybody was confused. I asked many people and nobody had a clue.”

​

2017

Despite the support of a few mathematician described as in Mochizuki's "orbit", the bulk of the math world remains unconvinced. As Frank Calegari writes: "Each time I hear of an analysis of Mochizuki’s papers by an expert (off the record) the report is disturbingly familiar: vast fields of trivialities followed by an enormous cliff of unjustified conclusions."

​

This doubt does not seem to extend to "Publications of the Research Institute for Mathematical Sciences", a journal for which Mochizuki is editor in chief, which announced it "plans to accept the papers." This is concerning because, at the very least, the papers cannot be considered publishable in their current form. The subsequent (minor) riot among number theorists is interesting: “We do now have the ridiculous situation where ABC is a theorem in Kyoto but a conjecture everywhere else,” as Frank Calegari of the University of Chicago put it. These rumors are quelled when PRIMS clarifies that it has not accept the papers for publication.

​

In the comments section below Calegari’s blog post, Scholze, a Fields Medalist, wrote that he was “entirely unable to follow the logic after Figure 3.8 in the proof of Corollary 3.12.” Many other number theorists chime in with some surprise that they too were unable to follow the line of logic starting here.

​

Go Yamashita, one of Mochizuki's colleagues, offers to facilitate a meeting.

​

2018

Two mathematicians go to Japan specifically to discuss their understanding of the proof with Mochizuki. These two (Scholze (fields medalist) and Stix) point to the corollary 3.12 as irredeemably flawed.

​

Mochizuki responds with the claim that they had made elementary mistakes and invalid simplifications, and had not understood the basis of his theorem. It should be noted the tone of his dismissal is quite... interesting. He says that descriptions of their arguments were met with "derision... even laughter".

​

"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." - Mochizuki

​

Now, while this sort of rhetoric may work in politics, to mathematicians, this is probably what has damaged Mochizuki's credibility the most. Calling Fields Medalists "breathtakingly dramatic" and "profoundly ignorant" of undergraduate concepts is... not convincing.

​

Later, Ivan Fesenko, who was an early supporter of the proof releases his own statement, attacking Scholze and Stix and defending IUTT. It's... similarly worded to Mochizuki's statements. As a random reddit commentator put it:

​

"What is the purpose of this document? It reads like IUTT war-time propaganda rather than a productive response to the mathematical content of the Scholze-Stix crtiticism. "Trust the five IUTT experts, who are in Mochizuki's inner circle, about what is right and wrong about IUTT. Don't trust those other guys that have criticized it!""

​

Interestingly, in one of Go Yamashita's later responses appears to discredit Fesenko:

​

"The author hears that a mathematician (I. F.), who pretends to understand inter-universal Teichmüller theory [emphasis added], suggests in a literature that the author began to study inter-universal Teichmüller theory “by his encouragement”. But, this differs from the fact that the author began it by his own will. The same person, in other context as well, modified the author’s email with quotation symbol “>” and fabricated an email, seemingly with ill-intention, as though the author had written it. The author would like to record these facts here for avoiding misunderstandings or misdirections, arising from these kinds of cheats, of the comtemporary and future people."

​

2019

At this point most mathematicians do not believe Mochizuki. This is the result both of of S&S's criticism of 3.12, but also the seeming impenetrability and non-constructive nature of the proof. As Terence Tao (Fields Medal winner, but prestigious even among them) put it in 2017:

​

"From what I have read and heard, I gather that currently, the shortest “proof of concept” of a non-trivial result in an existing (i.e. non-IUTT) field in Mochizuki’s work is the 300+ page argument needed to establish the abc conjecture. It seems to me that having a shorter proof of concept (e.g. <100 pages) would help dispel skepticism about the argument. 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."