Monumental Proof to Torment Mathematicians for Years to Come
August 5, 2016 5:35 PM   Subscribe

Nearly four years after Shinichi Mochizuki (previously, previously, previously) unveiled an imposing set of papers (1, 2, 3, 4) that could revolutionize the theory of numbers, other mathematicians have yet to understand his work or agree on its validity — although they have made modest progress.

Some four dozen mathematicians converged last week for a rare opportunity to hear Mochizuki present his own work at a conference on his home turf, Kyoto University's Research Institute for Mathematical Sciences (RIMS).

Mochizuki’s theorem aims to prove the important abc conjecture, which dates back to 1985 and relates to prime numbers.

The glimmer of understanding that has started to emerge is well worth the effort, says [University of Nottingham's Ivan] Fesenko. “I expect that at least 100 of the most important open problems in number theory will be solved using Mochizuki’s theory and further development.”
posted by stinkfoot (44 comments total) 33 users marked this as a favorite
 
Pictures of the summit
posted by stinkfoot at 5:40 PM on August 5, 2016 [1 favorite]


Here is what I don't understand. How can you claim a proof is valid, if you don't understand it? If you do understand it, how is it justified to drop it on people and just say "you figure it out"? Isn't the point of publication to enable others to understand and build on your work? It's a scholarly journal, not the fucking Lament Configuration.
posted by Punkey at 6:08 PM on August 5, 2016 [4 favorites]


I would think that in some ways, forcing others to figure it out on their own strengthens our confidence in the approach's correctness. It's independent verification.
posted by sbutler at 6:18 PM on August 5, 2016 [3 favorites]


Mochizuki didn't drop it on people and say "figure it out". He dropped his understanding of it on people in the form of several papers that, to him, constitute a proof; but to other mathematicians, understanding his proof is itself a years long undertaking, which might be wasted if he made a mistake. That's the nature of bleeding edge math, and the dilemma for mathematicians.
posted by fatbird at 6:21 PM on August 5, 2016 [40 favorites]


Okay, then that takes us back to my questions. If he doesn't understand it fully, how can it be considered a proof? If he does understand it, how can you justify leaving it just hanging out there while people try to untangle it?

Just for perspective, my background is in hard science, where if you have some really complicated theory, it's very helpful - and some would say ethical - to actually explain what you're on about. That's the point of publication - for others to replicate, verify, and build on what you've done. This seems to defeat all three of those purposes. If you can't understand the logic and complexity of it, you can't replicate, you can't verify, and you definitely can't build on it.
posted by Punkey at 6:25 PM on August 5, 2016 [1 favorite]


The articles are somewhat sensationalized. He has been trying to explain the work, it's just very very difficult.
posted by dilaudid at 6:35 PM on August 5, 2016 [8 favorites]


Okay, then that takes us back to my questions. If he doesn't understand it fully, how can it be considered a proof? If he does understand it, how can you justify leaving it just hanging out there while people try to untangle it?

Mochizuki believes it's a proof and is participating in the workshop at RIMS. The nature of math is that having someone there to explain it to you doesn't mean you'll understand it immediately. Math talks (though I have no background in number theory) are frequently example-driven precisely because there's not a whole lot of value in standing there delivering a proof the audience isn't going to fully follow without sitting down with the paper anyway. You aim to convey enough of the idea to make reading the proof easier.
posted by hoyland at 6:38 PM on August 5, 2016 [17 favorites]


Okay, then that takes us back to my questions. If he doesn't understand it fully, how can it be considered a proof? If he does understand it, how can you justify leaving it just hanging out there while people try to untangle it?

I don't see where you're getting the idea that Mochizuki doesn't understand his own proof. He undoubtedly does understand it (although of course until there's independent verification it's possible that there is a flaw in his reasoning or understanding).

And he has explained his proof. The explanation is in the papers he published. It just happens that they are very complex and time consuming to understand. Maths is like that especially at the frontiers. Mochizuki invented a new field of maths in producing the proof. It's going to take others some time to get their heads around that. Understanding at that level doesn't come from listening to a lecture, it comes from working through the material by yourself. And maths can be very dense, even for the very best mathematicians it can take days or weeks to digest a couple of pages, particularly in a field you have never encountered before.
posted by roolya_boolya at 6:40 PM on August 5, 2016 [24 favorites]


If you can't tweet it, it ain't worth saying. You must not understand the idea well enough to explain it, right? People just use lots of words to show off.
posted by saulgoodman at 6:47 PM on August 5, 2016 [27 favorites]


There seems to be some degree of understanding, there was an error found in one of his papers and Mochizuki made a correction and re-submitted.

I would expect at some point some mathematicians with a gift of reinterpretation will make analogies that will be easier to grasp, but just reviewing what the abc conjecture means takes some head scratching. Remember most college math is over a hundred years old. (I talking about calculus here)
posted by sammyo at 6:53 PM on August 5, 2016 [3 favorites]


Okay, so it's...he understands it, I get that. Maybe it's just that the impenetrability and "eccentricity" of the whole thing is being overhyped. It just always seemed irresponsible to put something so potentially groundbreaking out there and then...not work as hard as you can to unpack it. If it's so enormous that it can't be explained easily, then how can you say that it's complete? I'm not a high-end mathematician, so this is more out of confusion than anything.
posted by Punkey at 6:58 PM on August 5, 2016


It does have the coolest most sci-fi sounding name, totally going to let us make a time machine:

Inter-universal Teichmüller Theory

we are going to Alpha Centuri with this!!!
posted by sammyo at 7:00 PM on August 5, 2016 [10 favorites]


I think it's fair to say that writing proofs so concrete and ironclad they are mechanically checkable is a narrow subfield of mathematics (if it's considered mathematics at all), and most advanced work is done at a level that requires the reader to fill in the gaps with their intuition about how certain constructions behave. And it's always been that way — calculus was on fairly shaky foundations and the subject of much mud-slinging for almost 200 years after it was invented.
posted by mubba at 7:06 PM on August 5, 2016 [20 favorites]


It just always seemed irresponsible to put something so potentially groundbreaking out there and then...not work as hard as you can to unpack it. If it's so enormous that it can't be explained easily, then how can you say that it's complete?

Some things are complicated. Maybe there is an easier way to understand it, but he didn't discover that one, he discovered this one. He wrote several comprehensive papers and he has participated in several efforts like this one to explain his work personally, so I don't think it makes sense to say that he is not working as hard as he can to unpack it.
posted by value of information at 7:16 PM on August 5, 2016 [22 favorites]


Well, that's the thing, it might just be that he is, but that's not how it's being portrayed, which is...unfortunate?
posted by Punkey at 7:20 PM on August 5, 2016


From the comments:

Two examples mentioned at the conference were Galois theory and Class Field Theory, each of which took about 40 years.”
posted by sammyo at 7:37 PM on August 5, 2016 [7 favorites]


(as in many respects, math blogs invert the rules, DO read the comments :-)
posted by sammyo at 7:39 PM on August 5, 2016 [9 favorites]


Okay, so, it's just in how it's being written about? Because I remember when this first came out, and every other time it's mentioned it's the same kind of breathless "the master comes down from the mountaintop" coverage. I mean, if it's just meant for those in the field, who know that proofs of this scale take decades to figure out, cool, but for those of us not versed in the very bleeding edge of mathematics, it can look a little...obtuse, the way it's being talked about.
posted by Punkey at 7:45 PM on August 5, 2016


(Pun half-intended.)
posted by Punkey at 7:46 PM on August 5, 2016


Damn, I didn't realize there were things in math that people don't get figured out for decades
posted by gucci mane at 8:14 PM on August 5, 2016


I see what you mean, Punkey. Like in these articles "experts have slowly discerned a strategy in the proof that the papers describe" absolutely makes it sound like Mochizuki was hiding some of the details about what he was doing. I do think that it's just breathless reporting, though, maybe combined a little bit with the fact that at first (apparently) no-one really knew who Mochizuki was so the informal networks had nothing to prime the pump with. It seems unlikely that he was being intentionally obscure or even uncooperative.
posted by No-sword at 8:19 PM on August 5, 2016 [6 favorites]


I think there's a bit of emotional silliness on both sides. Fesenko was upset about the Nature article's title "Torment mathematicians for years to come" and the fact that they illustrated the article using copies of some pages of the proof. That's just uptight and ungenerous.

On the other hand, Woit is blurting a kind of sloppy social relativism: "The ethos of the field is that it’s not a proof until it’s written down (or presented in a talk or less formal discussion) in such a way that, if you have the proper background, you can read it for yourself, follow the argument, and understand why the claim is true." If you have the proper background. So it's fair to suppose that other mathematicians don't understand the claimed proof, because don't have the mental models and internal language that took a mathematician a decade to build by himself and was necessarily very specialized and probably had to re-frame many of conventional ways of thinking that were taken for granted. The symptom is, why ignore such an obvious counterargument?

So professional mathematicians can be a little silly and petty about this. The media may distort or misrepresent their culture, from their point of view, but they fail to recognize that the public actually loves and appreciates hearing about these sorts of interesting ideas. It can be inspiring, even through all the noise.
posted by polymodus at 8:23 PM on August 5, 2016 [12 favorites]


gucci mane: The task of making a square the same area as a circle with compass and straightedge was posed by Oenopides around 500 years before Christ: it was proved that this was impossible in 1882.
posted by hleehowon at 10:16 PM on August 5, 2016 [13 favorites]


No-sword, this article seems to suggest that some other mathematicians initially thought Mochizuki was being kind of obscure and uncooperative. The word someone used was "overengineered" -- maybe meaning that Mochizuki developed a much more complete and abstract version of the tools that strictly speaking would have necessary to just get to the solution? (idk!)
posted by en forme de poire at 10:34 PM on August 5, 2016 [1 favorite]


It's a little like outsider art, except in this case it's outsider science. Instead of being part of a community of peers, developing new ideas one piece at a time with frequent feedback, Mochizuki toiled away at his theories for decades and then presented them all at once. You just don't see that anymore.
posted by Kevin Street at 11:18 PM on August 5, 2016 [12 favorites]


The word someone used was "overengineered" -- maybe meaning that Mochizuki developed a much more complete and abstract version of the tools that strictly speaking would have necessary to just get to the solution?

From my reading around this, it seems like maybe that's a possibility, but we won't really know how good an assessment it is until the relevant community have had enough time to properly get to grips with and check the validity of Mochizuki's proof.

I agree that the reporting isn't helpful for most of us. Being fair to the writers, though, I think it's very hard to report on very specialised areas at the best of times, and when you're dealing with with the very bleeding edge of such an area, the problem is compounded by the reporter's own (entirely reasonable) limited understanding of the specifics.

If it's so enormous that it can't be explained easily, then how can you say that it's complete?

Thinking out loud now; maths is kinda like this all the way through, isn't it, though? Even starting, to a limited extent, with the basic issues of number, it seems to me. If you think of any number over about 25 (?) or so, any explanation or conceptualisation of that number, other than in purely numerical terms, is something of a wooly analogue. You can get a sense of the relationships between numbers using spacial or kinaesthetic analogues, but those can't actually capture the significance of the number 73 (for example). Only the number and its mathematical context can actually give a proper account of 73. So if someone without our shared mathematical context asked you to explain 73, the only way you could satisfactorily do it would be to teach them a whole lot of things about numbers and maths that are actually quite complex and involved, and which might involve you working lots of things out or looking them up. But that doesn't mean that you're being obscure when you say that (e.g.) 42 + 31 = 73, even though that claim isn't very informative except on its own terms.

It's one of the fascinating things about maths, that it is a technique for massively expanding our ability to conceptualise that world, far beyond what would be possible without it - even at its seemingly most basic level. And that that introduces some profound challenges to talking about it using natural languages. So when you get to this level...well...yeah, tough reporting gig, even for a mathematician!
posted by howfar at 11:43 PM on August 5, 2016 [4 favorites]


It could be worse. He could go full Fermat. oops no space! I'm going to leave you hanging! for three hundred years! :) :) :)
posted by E. Whitehall at 11:49 PM on August 5, 2016 [3 favorites]


Punkey: It just always seemed irresponsible to put something so potentially groundbreaking out there and then...not work as hard as you can to unpack it. If it's so enormous that it can't be explained easily, then how can you say that it's complete? I'm not a high-end mathematician, so this is more out of confusion than anything.

This is a difference in values, and a different idea of the purpose of a publication. It comes in part from a fundamental difference between science and mathematics. With some apologies, I'm going to intentionally oversimplify the difference here. I'm going to overstate the difference because it makes the difference easier to communicate. The real difference is not quite this extreme, but the use of more extreme language can help someone to visualize the unfamiliar.

Scientists study the world. They look for patterns and construct explanations for those patterns. Mathematicians* do not study the world. Mathematicians study systems of rules. They create systems of rules, play games with rules, and study the consequences of the rules they made up. From that perspective, a proof is NOT an explanation. It's not supposed to be an explanation. At the most extreme edge of this perspective, there isn't even a thing to be explained.

So mathematicians aren't interested in explanations*, they're interested in the rules and the games. So while a scientist might ask what the proof means, that's not what a mathematician is interested in*. All* that matters to a mathematician is whether or not the proof is valid. In this case, the mathematical question is: Did Mochizuki play his game using entirely legal moves?

*remember: intentionally overstating the difference.
posted by yeolcoatl at 12:08 AM on August 6, 2016 [15 favorites]


It could be worse. He could go full Fermat.

From the Nature link it's seems if the Mochizuki is correct it could be possible to derive another, independent proof of that theorem of Fermat's! It took Andrew Wiles around 7 years to develop his proof in the late eighties/early nineties.

Speaking of Fermat and Wiles the 1996 BBC Horizon documentary on the Wiles' proof of Fermat's Last Theorem is worth seeking out for those interested in the process and personalities of maths at this level. It's informative and more entertaining, dramatic and emotionally affecting that one might imagine for a documentary about maths.
posted by roolya_boolya at 12:59 AM on August 6, 2016 [5 favorites]


Recreational math is one of my occasional hobbies, just reading texts about things like group theory or topology and proving results for myself, and so I've gotten pretty used to seeing terms of the form "[Proper Noun] [Compound Neo-Greekism or Just a Regular-type English Word Utterly Divorced From its Usual Context]" and just sort of accepting that the conceptual space involved is too broad and varied to be able to give things truly descriptive names, and while the greater part of my consciousness knows that I'll never really have the inclination or ability to absorb enough number theory to even begin approaching the subject, there's a part of me that does now and will forever yearn to understand just what the fuck a "Hodge theater" is.
posted by invitapriore at 1:00 AM on August 6, 2016 [17 favorites]


These four papers on Inter-universal Teichmüller theory total almost 600 pages, so we're talking about a book here.

Just some wikipedia links to give background :

If you were a math major in university, then you can probably understand the definition of a Teichmüller space. There are usually deep complexities in generalizing such analytic or topological techniques to arithmetic or algebraic settings. Alexander Grothendieck was a master of this via his "relative point of view" (see Étale morphism for an example).

Inter-universal Teichmüller theory is "an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve", making it very much in line with Grothendieck's relative point of view. There are lots of new mathematical objects that appear along the way though, each of which requires some considerable effort to understand on its own, both in terms of tricky algebraic gymnastics and of subtle analytic or topological inspiration.

Appears Mochizuki has been building these new objects his whole life, seemingly making understanding this similarly difficult to understanding Grothendieck's work, except it requires already understanding much of Grothendieck's work. And he even needs quite desperate fields like set theory and algorithms. It's hard.
posted by jeffburdges at 3:26 AM on August 6, 2016 [9 favorites]


Also, there is a FAQ by Mochizuki. One could even glance as his earlier paper, but you'll sometimes find more comprehensible explanations by reading someone's PhD students' theses. In this vein, Mono-anabelian Reconstruction of Number Fields by Yuichiro Hoshi looks interesting, although not a PhD thesis.
posted by jeffburdges at 3:57 AM on August 6, 2016 [4 favorites]




I understand that this guy understands his proof, has put the explanation out there in the form of four papers, and I understand why it takes years for anyone else to check the proof.

Can someone explain how peer review works in math, though? I mean these papers were published, right? So some reviewers had to read and understand them I would think, even if they weren't ready to stamp 100% correct on them. So how does that work?
posted by If only I had a penguin... at 4:37 AM on August 6, 2016


Can someone explain how peer review works in math, though? I mean these papers were published, right? So some reviewers had to read and understand them I would think, even if they weren't ready to stamp 100% correct on them. So how does that work?

They've not been published in a journal. It's common in math to circulate and work from preprints. (I assume this is common in any subject covered by the arXiv.) That's the stage when people catch mistakes or tell you about related work you didn't know existed or whatever. It looks (and again, I'm not a number theorist, and no longer academia) like they're at the point where the first people who attacked these papers are pretty sure there's something there, so it's worth getting more people together to understand it. Some subset of these people are going to be the refereeing the first paper. (I don't know if they've assigned referees for the later ones or if these conferences are basically expanding the pool of potential referees.)
posted by hoyland at 5:02 AM on August 6, 2016 [3 favorites]


One reason for Mochizuki's drop-it-on-us delivery is that he apparently developed Inter-universal Teichmüller Theory as a kind of hobby which he never thought would have a practical application, which is why he never published anything for the ten years he was developing his own notation. Why would he go to all this work on something he really thought was useless? Because it's fun! It's like exploring a cave nobody else has ever visited or knows about. It's the human version of what Iain M. Banks portrayed the Minds doing for entertainment in Excession. And this is how geometry was invented; nobody in ancient Greece thought geometry had practical applications. They thought the secrets they were exploring were important because they were sacred.

Then, one day as Mochizuki is playing in his private cave, out pops what appears to be a proof of the abc conjecture. There's treasure in that cave! These surprises happen in math, and are of course among the reasons we do math. He realizes that this is a big deal but that the only way he can reveal it is to also reveal the rest of his work on I-UTT and wait for others to catch up to his level of understanding. He has to open his private cave to his peers and let them explore it.

There are a mix of reactions. Some people are peeved that he kept his cave a secret for so many years. Some don't think there is really any treasure down there and don't want to waste their time going after it. Some worry that the treasure is fool's gold. Some (mostly non-mathematicians) think it's an elaborate joke. But some know that Mochizuki is the real deal and he has no reason to cloud his reputation by trolling his field with something that obviously took years of hard thought to develop. The treasure might be fool's gold if Mochizuki made a mistake, but if he really has proven the abc conjecture then it's very likely that I-UTT is the key to dozens of other famous problems. That would be a treasure worth claiming.

We may or may not live to see that possibility realized. As noted upthread, things like this have sometimes taken generations to sink in.
posted by Bringer Tom at 5:40 AM on August 6, 2016 [30 favorites]


What kind of university system fostered this amazing development? What a fascinating conundrum!
posted by eustatic at 5:45 AM on August 6, 2016


Math, not unlike other academic fields, does have a bit of a problem with readability of research papers. Structurally, academics get job security, promotions, and grant money based on number and impact of publications, not on how well-written those publications are. How well written an article is affects how much it gets cited and thus its impact factor, of course, but is generally less important than the actual mathematical content of the article in this respect. That said, some math is just dense, and requires the reader to spend a lot of time with the ideas in order to understand and develop an intuition for them.
posted by eviemath at 11:38 AM on August 6, 2016 [1 favorite]


As a student of math: the proof IS the explanation. It's not like he's not explaining it; he is, the proof is just really fucking hard.
posted by pcrsweetness at 12:40 PM on August 6, 2016 [1 favorite]


A particular problem with the readability of Mochizuki's thesis is that he invented his own notation, essentially an entire new mathematical language, in order to express it. So before you can even begin to treat his proof, you have to learn this language and establish that it does what it purports to do as far as expressing useful concepts and relationships. I think this is the main reason there was so much shock when he dropped the whole thing on us; it's a very high bar to clear just to be able to begin to evaluate the actual proof.

And I think I can personally understand his reticence about being too public. This was his personal playground for a decade and now he's had to open it to all manner of criticism and evaluation. And the standard in peer review is that papers should explain themselves. He probably feels that if you aren't willing to learn the notation, you have no business judging the conclusions he reached using it. He has given us the equivalent of a fat early Stephen King novel worth of documentation about all that. That is really more than custom required him to do.

I am personally glad that some of his colleagues have accepted the challenge. It might end up being a really big deal one day.
posted by Bringer Tom at 1:25 PM on August 6, 2016 [1 favorite]


I have actually studied advanced math. I hold a bachelor's degree in it. I am incredibly out of practice, but I can say from experience that math is just plain intellectually hard at the high levels. Number theory is particularly esoteric; there were basically no practical applications until the seventies. That's 1970s. It's not at all surprising that his work may take years to verify. Number theory ain't called the queen of mathematics cause it's easy.
posted by axiom at 10:57 PM on August 6, 2016 [2 favorites]


There's precedent for this kind of thing in math - the proof of the Poincare conjecture was revealed in a similar manner. Grigori Perelman posted a series of papers to the ArXiv in 2002 and then a group of mathematicians puzzled over them for a while. It wasn't until 2006 that the validity of the proof was verified.

For people outside of mathematics it's hard to convey just how difficult it is to truly check a paper. As an example, when I peer reviewed well-written papers in my field, on subjects I knew quite well, it could take weeks to thoroughly vet 10 pages. Now imagine that someone has created their own field of research that nobody else knows and that their proof is hundreds of pages long. This literally takes years to get through. This isn't a failing of the author or of peer review, it's just the nature of pure math.
posted by Frobenius Twist at 10:26 AM on August 7, 2016 [3 favorites]


There's precedent for this kind of thing in math - the proof of the Poincare conjecture was revealed in a similar manner. Grigori Perelman posted a series of papers to the ArXiv in 2002

And like Mochizuki there was almost identically breathless and overdramatic journalism about Perelman's work. I think it's good that journalists bother at all to write about advances in mathematics, but I wish they didn't have to act like they were writing the hacking scenes from Criminal Minds.

And he even needs quite desperate fields like set theory

I've never studied advanced maths exactly (I had a non-traditional education and taught myself calculus from a scan of Spivak but never went further), but have done a lot of formal studying (and, it boggles the mind, teaching of undergraduates) of logic and set theory. What's desperate about it, exactly?
posted by dis_integration at 8:25 PM on August 7, 2016


Disparate, I think.
posted by ectabo at 8:51 PM on August 7, 2016 [2 favorites]


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