In case you haven’t been following this story, “abc” refers to a famous conjecture in number theory, for which Shin Mochizuki claimed last year (see here) to have found a proof. His argument for abc involves a new set of ideas he has developed that he calls “InterUniversal Teichmuller Theory” (IUTeich). These are explained in a set of four papers with a total length over 500 pages. The papers are available here, and he has written a 45 page overview here. One can characterize the reaction to date of most experts to these papers as bafflement: what Mochizuki is doing is just so far removed from what is known and understood by the experts that they have no way of evaluating whether or not he has a new idea that solves the abc problem.
In principle one should just be able to go line by line through the four papers and check the arguments, but if one tries this, one runs into the problem that they depend on a long list of “preparatory papers”, which run to yet another set of more than 500 pages. So, one is faced with an intricate argument of over 1000 pages, involving all sorts of unfamiliar material. That people have thrown up their hands after struggling with this for a while, deciding that it would take years to figure out, is not surprising.
Mochizuki has just released a new document “concerning activities devoted to the verification of IUTeich”. It explains the state of his efforts to get other mathematicians to check his work, a project that has been going on since last year, leading to many ongoing updates to the papers making up the proof. He explains that he submitted the four IUTeich papers to a journal last August, but will not have anything to say about the journal or the state of the submission process. This is the way mathematics is supposed to work: the papers should be refereed by experts who have agreed to go through and check them carefully (and confidentially). Given the unusual character of the series of papers, finding willing and able referees may be very difficult. It would of course be most satisfying if such referees can be found and can either identify holes in his argument, or vouch for correctness of the whole thing.
In the meantime, he has been working since October 2012 with Go Yamashita, who has carefully gone through the papers and is now writing a 200300 page survey of what is in them. Yamashita may also give a course on the topic at Kyushu University sometime after next April. As part of this process, three other mathematicians participated in a seminar in which Yamashita lectured on the papers.
Another mathematician working on this is Mohamed Saïdi, who devoted about six months to studying the papers, then spent three months visiting Kyoto and discussing them with Mochizuki. According to Mochizuki, he has said that he believes the theory to be correct. Mochizuki summarizes the current situation as
the issue of whether or not one should regard the verification of IUTeich as being, for all practical purposes, complete, i.e., as a result of the activities of Yamashita and Saïdi, is by no means clear, and any sort of “final conclusion” on this topic must be regarded as a matter that lies beyond the scope of the present report.
Mochizuki goes on to claim that, based on what he has heard from Yamashita and Saïdi, researchers trying to read his papers should find it possible to understand the theory if they work on it for roughly half a year. He warns that they do need to be aware though that an attempt to make sense of what he is doing by expecting “a similar pattern of argument to existing mathematical theories is likely to end in failure.” They also need to keep in mind that he’s not particularly focused on proving abc, that for him it is the IUTeich theory itself that is the object of interest.
This is a remarkable story, with little precedent. After more than a year, I haven’t heard anyone willing to bet either way on how it will turn out. Mochizuki is a talented mathematician and maybe he has a proof. Or maybe he has a complicated set of ideas which don’t do what he hopes. Perhaps someday one of these alternatives will start to emerge, but it doesn’t now look like this will be anytime soon.
“Mochizuki is a talented mathematician and maybe he has a proof. Or maybe he has a complicated set of ideas which don’t do what he hopes. Perhaps someday one of these alternatives will start to emerge, but it doesn’t now look like this will be anytime soon.”
An ideal candidate for the new MilnerZuckerberg math prize?
// Sorry, couldn’t resist… //
Best,
Marko
Putting aside for a moment the question of whether or not the supposed proof of abc is valid, does anyone have any indication that IUTeich will be useful for obtaining other results? In other words, is there a reason, other than wanting to verify the abc proof, for someone to study IUTeich? Or is this a case where if the abc proof turns out to have a flaw, any time spent on IUTeich would be a waste?
Perhaps IUTeich is a part of 22ndcentury mathematics that has fallen by chance into the 21st century.
Has Faltings (his Ph.D. supervisor) come out and said anything? I heard some rumors going around.
“is there a reason, other than wanting to verify the abc proof, for someone to study IUTeich?”
You could just be curious to know what these new ideas are about, and how they relate to known math.
This looks to me as a social problem, not a mathematical one. On one side, there are no “experts” in Mochizuki field, because he made it all. On the other side, the idiotic pressure to publish, which is imposed in academia (the legacy publishers being only opportunistic parasites, in my opinion), makes people not willing to spend time to understand, even if Mochizuki past achievements would imply that there might be worthy to do this.
To conclude, is a social problem, even an anthropological one, like a foreign ape which shows to the local tribe how to design a pulley system, not at all believable to spend time on this. Or it is just nonsense, who knows without trying to understand?
If this proof is accepted (with something between a linebyline verification and complete understanding), but nobody else ever learns the theory well enough to do anything with it, should it be considered a success or a failure?
On a more positive note, I expect that once people are more convinced that it’s correct, more of them will be willing to invest the amount of time need to understand what’s going on.
The problem is one of culture. Modern Mathematics has become so fragmented, so disconnected, that it is in fact possible for someone to write 1000+ pages on a completely new theory working in complete isolation and moreover to have the prospect of this work actually taken seriously by the wider mathematical community because of the way that minor field specialization has become the norm.
Another problem is what I like to call “simplicityfree mathematics”. Theorem’s are proved by a small wall of text peppered with multiple references to, at best, previous lemmas of theorems in the text, more usually not so well known theorems from increasingly obscure branch of mathematics in question, and at worst, references to outright conjectures that neither the author nor anyone else has yet proved(This last style goes back to Galois). Simple elements and frequently actual numbers are usually completely absent.
My gut feeling for mathematics is that you’ve either got a formula/algorithm, or you’ve gotten lost. Obviously this doesn’t apply to absolutely everything, but the lack of such hard results has real consequences in the form of the increasing obscurity of various fields, and the lack of feedback from these into the core discipline.
But, aside from that perspective; At the end of the day if you require 1000 pages of specalised argument to prove your result I think it is reasonable to conclude that you don’t really understand your results and neither will anyone else. Regarding such works as proved, and worse, as a foundation for future material is what has lead us to our present state of affairs, and it shouldn’t be tolerated.
OMF/chorasimilarity,
For some great wisdom on this topic, I urge everyone who hasn’t done so to read Bill Thurston’s “On proof and progress in mathematics”
http://arxiv.org/abs/math/9404236
For Mochizuki’s proof to be accepted, other members of the community are going to have to understand his ideas, see how they are supposed to work and get convinced that they do work. This is how mathematics makes progress, not just by one person writing an insight down, but by this insight getting communicated to others who can then use it. Right now, this process is just starting a bit, with the best bet for it to move along whatever Yamashita is writing. It would be best if Mochizuki himself could better communicate his ideas (telling people they just need to sit down and devote six months of time to trying to puzzle out 1000 pages of disparate material is not especially helpful), but it’s sometimes the case that the originator of ideas is not the right person to explain them to others.
>>>It would be best if Mochizuki himself could better communicate his ideas

In preface to ‘Three – Dimensional Geometry and Topology’ W. Thurston wrote:
“The notes were originally aimed for an audience of fairly mature mathematicians…
Some of the feedback from seminars and individuals convinced me that it would be
worth filling in considerably more detail and background; there were several places
where people tended to get stuck, sometimes for weeks…”

Last chapter of this book covers Teichmuller space ( BTW: Mochizuki likely did not read the preface).
I recently gave a course on the responsible conduct of research, and one of the main messages of the course is the necessity of publishing one’s results and methods so that others can verify them. To underscore the point, we looked at infamous cases of scientific misconduct in which proponents of great discoveries failed to share the details of their work, thereby making it impossible for others to reproduce and confirm their results. I emphasized that scientific research must be shared with the scientific community, else it (almost) isn’t science. Review and criticism by peers is a major part of the process of moving forward.
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From reading the comments above, one almost has the impression that mathematicians put less value on publishing and verification from peers than scientists do. Is it OK to be right but impossible to understand? Is it OK to have a great insight but not to record it and have it approved by the community? Are individual insights more important than widespread comprehension?
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Probably I am reading too much into some of the comments people have written…
This sounds like a problem for Nicolas Bourbaki’s group.
Michael,
Actually I think mathematicians put more value on publication and the refereeing process than other scientists do. It’s very important to them to know which results have been checked and are reliable, in order to use them to build reliable proofs of something else.
It is true however that mathematics research can be very specialized, very abstract, and so intensely demanding in terms of anyone being able to follow and check it. For typical work, often only a handful of people in the world have the right background to quickly understand a certain new piece of work and evaluate the arguments. The Mochizuki case is quite unusual: as usual there really are only a small number of experts in the field, but he has gone off by himself in his own direction a great distance, then reappeared to claim to have made a major discovery. The experts are faced with the problem of how to figure out whether he is right. Do they invest the six months he is asking from them? Do they tell him: forget it, you need to write this down in a more comprehensible form that won’t take us 6 months or more to understand? Do they wait and see if someone (like Yamashita) who has invested a lot of time in following this will report back with a more comprehensible explanation of what is going on?
Allan,
Except for the problem that they’re no longer active…
One of the motivations for Bourbaki was to produce a careful and complete set of arguments for the standard results used in different parts of mathematics, so that these could carefully be checked for reliability and be used to go further.
At least in math each investigator has pretty compatible (with overall progress) incentives in deciding whether to spend his time reading something obscure. If you figure out how to make better sense out of something that’s already been proven the community gives you some credit.
Across the whole span of disciplines I suspect that we reward writing (original stuff) too much and reading too little.
..first of all, the main problem in the reception of Mochizuki’s work IS indeed a cultural one, but obviously not quite in the sense it is discussed here; from a ‘western’ perspective, Mochizuki is perceived as the ‘great other’ and the ‘otherness’ of his work comes with a mutuallyamplifying ‘otherness’ of his culture and ‘race’, if he would be white and/or (at least) western his work would be considered not as ‘scientific misconduct’ but as a ‘great and visionary’ cultural achievement, his teachers would be hailed, his descendants would be famous and he would be ubiquously present at various lifetimeachievementawards still in 30 years. But the ‘otherness’ of his work is probably on the other hand exactly the result of being perceived as the ‘other’ on a sociological basis, which can mean exclusion, but also freedom, a certain space for creativity. In any case, all this exactly transcends western norms of ‘workforce’, his papers are more meditation than results and given this, they are among the most inspiring pieces of work I have read in the last ten years (and I know only a tiny piece of them).
Also from another, more substantial perspective: from my point of view, the discourse which is reflected here is wrong in essential aspects, that his, his contribution to a possible proof of the abcconjecture is certainly LESS, not more important than his general insights, in this sense, it is excactly not important whether his announced proof turns out to be true in a line by lineevaluation, not of course from a ‘factual’ point of view, but from a ‘philosophical’ one. ‘Millenium problems’ are catalysts for theorybuilders, often not so quite interesting in itself (again, from my perspective). As I already mentioned in a related discussion, his theory of ‘étale theta functions’ seems to reflect a deep aspect of the latter: these functions form a bridge between ‘purely arithmetic and topological data’ of a given object, or in his terms between ‘Frobeniuslike’ and ‘étalelike’ structures and in this sense his ‘etale theta functions’ have for instance great similarities to ‘flux mappings’ in symplectic topology. The ‘etalelike structures’ have ‘the ability to ‘penetrate walls’ (cf. his survey paper on p. 32), all these are eminently deep and important observations; the correspondence of certain rigidity structures. From my personal point of view, very similar constructions should be possible in diff. geometry and should for instance allow to relate rigidity of ‘classical Hodge structures’ and symplectic rigidity, again using certain thetalike structures. The existence of a ‘Galoistheoretic Kodaira spencer morphism’: from my point of view this is neither esoteric nor completely ‘out of the blue’, it HAS to exist. One should consider that mathematics can be a very lonesome and ‘individual’ endeavour, Mochizuki possibly represents a certain, very special ‘type’ of mathematician, who is as important as those types of mathematicians who will eventually evaluate and rework his ideas.
This is not the first case when mathematical and social issues have been raised and similar sentiments were voiced. The first one was the classification of finite simple groups. The only difference is that many (over 100) mathematicians worked on it rather than one, but still the mathematical community was skeptical. Since the complete proof takes over 10,000 pages it is fair to assume that no one understands it in its entirety.
mo
AndreasK,
It’s just not true that Mochizuki is “other” to Western mathematicians. He grew up in the US, went to Princeton, his advisor was Faltings (who is “hailed”). He’s someone personally known to most experts in his field, and they met him at pinnacles of the US establishment like Princeton and Harvard. The idea that he’s not taken seriously because he’s not white is absurd. No one thinks he is engaging in “scientific misconduct”, and everyone agrees his ideas are potentially interesting, possibly revolutionary and of great significance. Whether they can prove the abc conjecture, which our current best ideas don’t seem to be able to prove is a measure of the depth of these ideas, whether they tell us something new, or just repackage things we already know.
Is this a good candidate for computer verification of the proof?
CPV,
No, I don’t think Mochizuki’s argument is written in anything like a form that can be made ready for computer verification. Also, in general, if experts can’t understand the overall structure of an argument and which of its parts are the new and tricky ones which need to be carefully checked, I don’t believe a computer program exists that can do this for people.
Peter, I appreciate a lot these posts about mathematics and mathematicians. They help me, an experimental physicist, gain some notion of the wonderful world of mathematics (or at least the experiences of mathematicians). The article you referred to by William P. Thurston is extremely interesting and stimulating, although it concludes on a rather odd note. There is even something in Section 3 about the nearimpossibility of teaching students mathematics (or physics). Thurston’s commentary shows that mathematicians are very caught up in the issue of communicating their results – indeed, publication is only one of several important modes of communication. My earlier comment was indeed very naive!
I am wondering if Mochizuki can provide a “appetizer”, i.e. an important result, but
perhaps not as significant as abc, whose proof is relatively “quick”.
In Perelman’s case, the appetizer is the noncollapsing theorem, whose proof people
verified and accepted in weeks. This built up huge confidence in Perelman’s other
arguments.
This seems like the kind of problem which would lend itself well to the polymath approach, except that instead of discovering new results it would involve checking results (presumably) already obtained. That (apart from the scarcity of experts in the relevant field) may be enough for that kind of collaborative effort to become unlikely.
Personally, I think that if someone claims that that he or she has made a major discovery, the responsibility for explaining the work to the rest of the world in a manner that it can follow rests squarely with the claimant. Also, it may even happen that when one makes a concerted effort to explain an idea intelligibly to others one will find ways in which the idea simplifies, points to new connections or becomes clearer in one’s oen mind. Thus, even the original claimant may benefit from such efforts.
Let me second the motion on the value of the Thurston article you linked. It is a model of introspection, social observation, and writing clarity.
Peter, the words ‘scientific misconduct’ were taken from this very thread, even if they were not aimed ‘openly’ at him. Also, the subtext which I attacked was also taken mainly fom this very thread. In the mathematical community, things don’t lie ‘out in the open’, the acceptance of proofs is a sociologically and psychologically (apart from any mathematical question) not completely understood process and is intertwined with power discourses that without ANY doubt part south from north, white from nonwhite, established from nonestablished. I am not accusing here anyone personally, but it seems that certain processes or mechanisms might be not completely conscious to many people involved in certain discourses. My text above was meant as a polemic, I thought this would be sufficiently clear. Still, there are many odd questions in the reception of Mochizuki’s work, most of his ideas evolved since 10 to 15 years and obviously they are only interesting SINCE he claimed to have proven the abcconjecture, and, in between of many other questions, one could ask why this is the case. Why do experts expect to understand ideas which were obviously to a certain degree ignored for 15 years should lead to a line of argumentation understandable to anyone in time scales of some weeks? Can anyone here answer this simple question? Grothendieck’s ideas were famous long before Weil’s conjectures were finally settled. But again this simple fact: the discrimination of nonwhite mathematicians, of women in mathematics will not vanish just because one claims there wouldn’t be ‘proofs’ for it, and that to have grown up in the US is no guarantee not to be ‘othered’ in which (white, western) context ever is unfortunately one of these not ‘provable’ facts.
AndreasK,
Enough of this. You’re just trying to put this story into an ideological framework where it doesn’t fit. Mochizuki is as “Western” a mathematician as anyone, and no, he’s not a woman or black. The idea that the people having trouble figuring out what Mochizuki has done are having this problem because he’s racially East Asian is just ridiculous. As for the comparison to the reception of Grothendieck’s ideas, just look at the history. Grothendieck developed his ideas in active collaboration with others (they were often written up by other people he was working with).
If I remember correctly, when Andrew Wiles gave the expose of the first version (the one that turned out to be flawed) of his famous proof, he disguised it as an advanced class and asked his colleagues – who were checking the proof – to sit on it, over the course of a semester. Maybe this would be the right method here as well/
@AndreasK:
This way of interpreting the story does not sound very plausible for me either. The field of algebraic/arithmetic geometry is full of brillant Japanese mathematicians, with a long tradition going back at least to Shimura and Taniyama. The staff of Mochizuki’s home institution RIMS includes (among others) Morihiko Saito, Shigefumi Mori, Shigeru Mukai, Takuro Mochizuki, Masaki Kashiwara. Each of them has produced deep mathematical work, spawning whole new field, e.g. Saito’s mixed Hodge modules, the Mori (minimal model) program, FourierMukai transforms, Mochizuki’s irregular RiemannHilbert correspondance, Kashiwara’s crystal bases.
All of them have interacted productively with the “Western” mathematical community. Because of this tradition, quite a few arithmetic geometers are familiar to various degrees with Japanese culture, have learned some Japanese, etc.
I fully agree that it is a pity that few other mathematicians (including Japanese/Asian ones !) tried to keep in contact with Mochizuki’s ideas over the years.
milkshaken,
As far as I know, there was only one person (Nick Katz) in on the fact that the material being discussed in that class was a crucial part of a proof of Fermat. The case of the Wiles proof is a good comparison to the Mochizuki one. Wiles did work for many years alone and developed new ideas, but once he made his proof public experts very quickly were able to understand his ideas, then dig in and check things carefully. The problem was found (I believe by Katz) as part of the conventional refereeing process for the paper.
This story makes clear that for this kind of proof, even when an extremely careful mathematician like Wiles has checked things thoroughly with another expert, there can still be a flaw in the proof. The community is not going to accept that Mochizuki’s proof is valid until it has undergone a similar level of scrutiny. The problem now is that people seem stuck at an early stage of the process, that of understanding the structure of the proof and the techniques used well enough to be able to start checking carefully for subtle flaws.
I’am not qualified to talk about the correctness of these papers.
However I am suspicious. If I have to, I will bet there are unfixable problems.
* people don’t mix proof in number theory with ZFC axiomatic set theory.
* people don’t put out 5 papers in a series and revise all of them for others to check out.
* people don’t say “Go Yamashita gave a course, Mohamed Saïdi said he believes the papers to be correct”
“Wiles did work for many years alone and developed new ideas, but once he made his proof public experts very quickly were able to understand his ideas, then dig in and check things carefully. The problem was found (I believe by Katz) as part of the conventional refereeing process for the paper.”
To my information, none of these is totally right. First, Wiles is purported to have completed the Cambridge lectures with the words “I think this completes the proof of Fermat”. Next, Nick Katz – together with P. Sarnak, if I remember well – looked for it that a team of 812 mathematicians got involved in understanding Wiles`s proof. This was not part of the refereeing process yet; the material was distributed in pieces, and this is how the gap was found relatively fast. It is not easy to believe that the material of Mochizuki can be split in parts, since there are fundamental new ideas that everyone must understand, who wants to verify even a small part of the work.
The more I read about the issue, the more I gather the feeling that there is hardly anything that Mochizuki could have actively done in order to facilitate the verification process, and which he did not do – or even “refuse to do”, as one sometimes reads. We have at present two people who studied the material over a longer period of time, and had long discussions with him. It is to be hoped that this example will spread.
preda,
I don’t think there is anything at all inaccurate in what I wrote. See for instance Mozzocchi’s “The Fermat Diary”, page 19. He describes what happened with the Wiles proof was that Wiles gave it to Mazur, who was an editor of Inventiones. Mazur chose six referees and assigned each one of them a section of the paper. Katz was assigned section six, began work on it (with help from Illusie) and was the one to find the problem.