Home > math, rant > The ABC Conjecture has not been proved

## The ABC Conjecture has not been proved

November 14, 2012

As I’ve blogged about before, proof is a social construct: it does not constitute a proof if I’ve convinced only myself that something is true. It only constitutes a proof if I can readily convince my audience, i.e. other mathematicians, that something is true. Moreover, if I claim to have proved something, it is my responsibility to convince others I’ve done so; it’s not their responsibility to try to understand it (although it would be very nice of them to try).

A few months ago, in August 2012, Shinichi Mochizuki claimed he had a proof of the ABC Conjecture:

For every $\epsilon > 0,$ there are only finitely many triples of coprime positive integers $a, b, c$ such that $a+b= c$ and $c > d^{(1+\epsilon)},$ where $d$ denotes the product of the distinct prime factors of the product $abc.$

The manuscript he wrote with the supposed proof of the ABC Conjecture is sprawling. Specifically, he wrote three papers to “set up” the proof and then the ultimate proof goes in a fourth. But even those four papers rely on various other papers he wrote, many of which haven’t been peer-reviewed.

The last four papers (see the end of the list here) are about 500 pages altogether, and the other papers put together are thousands of pages.

The issue here is that nobody understands what he’s talking about, even people who really care and are trying, and his write-ups don’t help.

For your benefit, here’s an excerpt from the very beginning of the fourth and final paper:

The present paper forms the fourth and final paper in a series of papers concerning “inter-universal Teichmuller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the log-theta-lattice, a highly non-commutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ell NF-Hodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θ-data. This data includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. Consideration of various properties of the log-theta-lattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGP-monoids”.

If you look at the terminology in the above paragraph, you will find many examples of mathematical objects that nobody has ever heard of: he introduces them in his tiny Mochizuki universe with one inhabitant.

When Wiles proved Fermat’s Last Theorem, he announced it to the mathematical community, and held a series of lectures at Cambridge. When he discovered a hole, he enlisted his former student, Richard Taylor, in helping him fill it, which they did. Then they explained the newer version to the world. They understood that it was new and hard and required explanation.

When Perelman proved the Poincare Conjecture, it was a bit tougher. He is a very weird guy, and he’d worked alone and really only written an outline. But he had used a well-known method, following Richard Hamilton, and he was available to answer questions from generous, hard-working experts. Ultimately, after a few months, this ended up working out as a proof.

I’m not saying Mochizuki will never prove the ABC Conjecture.

But he hasn’t yet, even if the stuff in his manuscript is correct. In order for it to be a proof, someone, preferably the entire community of experts who try, should understand it, and he should be the one explaining it. So far he hasn’t even been able to explain what the new idea is (although he did somehow fix a mistake at the prime 2, which is a good sign, maybe).

Let me say it this way. If Mochizuki died today, or stopped doing math for whatever reason, perhaps Grothendieck-style, hiding in the woods somewhere in Southern France and living off berries, and if someone (M) came along  and read through all 6,000 pages of his manuscripts to understand what he was thinking, and then rewrote them in a way that uses normal language and is understandable to the expert number theorist, then I would claim that new person, M, should be given just as much credit for the proof as Mochizuki. It would be, by all rights, called the “Mochizuki and M Theorem”.

Come to think of it, whoever ends up interpreting this to the world will be responsible for the actual proof and should be given credit along with Mochizuki. It’s only fair, and it’s also the only thing that I can imagine would incentivize someone to do such a colossal task.

Update 5/13/13: I’ve closed comments on this post. I was getting annoyed with hostile comments. If you don’t agree with me feel free to start your own blog.

Categories: math, rant
1. November 14, 2012 at 9:06 am

You are much closer to the gossip here than I. Doesn’t sound like an emerging view that there are some unpatchable holes, though I heard of some patches a while ago. Is verification bogging down? Will this be the Mochizuki-de Jong theorem in a year?

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2. November 14, 2012 at 9:49 am

I recently developed a keener appreciation for the theorem-proof style of publication while translating some papers for the project below. One author commonly stated results without proof in a couple that were published in Russian in the early 1980s. It got to the point where I’d be saying to myself “uh-oh, here we go again – there must be something false here – let’s find it.”

The project’s nowhere near to being finished but to all fellow ‘babers and fans of alliterative, Thanksgiving-themed titles, I hereby present Kuratowski’s Closure-Complement Cornucopia:

http://www.mathtransit.com/cornucopia.php

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3. November 14, 2012 at 10:04 am

Your description of the situation matches my impressions. However,…

1) Perelman’s proof, as he wrote it, was not enough to convince people that he had a complete proof of the Poincare conjeture. In fact, several people had to work rather heroically to fill in the details before everybody decided that what Perelman wrote could be turned into a rigorous proof of the Poincare conjecture. But despite this and the fact that his proof relies heavily on Hamilton’s ideas and theorems, all of the credit has been given to Perelman, as indicated by the decision to make Perelman the sole recipient of the Millennium Prize. Some of us still argue about whether this was the right thing to do or not.

2) I have been amazed, given the situation (as you described it above) by how seriously at least some top number theorists take Mochizuki’s ideas and papers, even though they admit they don’t yet understand much of it at all. One told me (while we were hanging out in the Columbia Math Lounge) that he intends to study Michizuki’s papers quite carefully.

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• November 14, 2012 at 10:09 am

Agreed. Credit is a funny thing in math, and you’re right that Perelman’s name being attached to that theorem certainly doesn’t tell the whole story.

It’s a problem because even the people who are interested in the ABC conjecture aren’t willing to offer up months if not years of their lives so they can be a footnote. After all, the experts who even can attempt to read this stuff could be off proving theorems of their own in those months or years.

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4. November 14, 2012 at 10:11 am

So the number theorist I mentioned in item 2) claimed to me that he *is* willing to devote a lot of his time and effort to Mochizuki’s papers. I have the impression that he believes that there are enough interesting new ideas and techniques in the papers to justify the time and effort.

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• November 14, 2012 at 10:12 am

So I don’t know who you’re talking about, but I’ve been in touch with a bunch of people who said that and whose stories have since changed. Just sayin’.

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5. November 14, 2012 at 10:14 am

Ah! So the situation has changed! Thanks!

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• November 14, 2012 at 10:17 am

Maybe! I don’t know who you’re talking about, seriously.

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• November 14, 2012 at 4:10 pm

Sorry but when I said the situation has changed, I was referring to you saying that the people you’re talking to (who I’m sure are also pretty high-powered number theorists) have changed their stories. I don’t know if the one I talked to has or not. There is a rather technical discussion of the proof, notably by a Yale graduate student named Vesslin Dimitrov, here: http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/

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6. November 14, 2012 at 12:17 pm

Culturally, mathematician’s don’t award much credit for tasks that are perceived as somehow routine (in the sense than many people could do them) no matter how much work is involved. Thus we wouldn’t call it the “Mochizuki and M Theorem” but just “Mochizuki’s Theorem” in some sense because the number of people who could decode the papers is larger than the number of who could create them.

I’m not arguing against calling it the “Mochizuki and M Theorem” — I feel that mathematician’s value system devalues things like good exposition to our collective detriment — but the example of the proof of the Poincare conjecture, as Deane accurately describes, shows that it probably wouldn’t happen unless there’s a gap in the proof. (If I recall correctly, the amount of time from Perelman’s posting his initial papers to anyone actually understanding them is more like a couple of years rather than a couple months.)

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7. November 14, 2012 at 12:23 pm

So how long will it be before the only theorems left to be proven are ones for which the proofs cannot be understood by anyone other than the prover?

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• November 14, 2012 at 3:32 pm

I wondered the same thing, albeit in a slightly different way: how long will it be until all of mathematics has been outsourced to machines? Mathematicians don’t have to worry anytime soon, but it seems inevitable (assuming humans are around long enough to provide the machines with their evolutionary spark).

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8. November 14, 2012 at 3:09 pm

It’s not my area at all, so I don’t know the details, but just to point out that de Branges’s proof of the Bieberbach conjecture was almost exactly the situation you describe.

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9. November 14, 2012 at 3:13 pm

One A-list mathematician who generously paused to take a look when presented with some borderline-unintelligible ingenuity is of course Hardy…and it’s a good thing he did!

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10. November 14, 2012 at 4:01 pm

As I recall, Wiles didn’t first announce, then hold lectures. Rather, he held lectures which built up to the announcement. After which he then held more lectures, to convince people more thoroughly 🙂 The story is in this excellent documentary (for those who haven’t seen it): http://www.youtube.com/watch?v=7FnXgprKgSE

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11. November 14, 2012 at 9:37 pm

The issue of “getting credit” does not seem to be (or at least should not be) the most serious obstruction to the investment of time: more problematic is the fear that one might be investing huge amounts of time on something which could collapse on page 472 without any other useful insights surviving. That is, even if one couldn’t care less about getting credit for helping in the decoding of the method (personally I couldn’t care less, and would invest the time if I didn’t have that fear), it is psychologically hard to convince oneself that the effort is worthwhile unless there is an amazing idea early on which can grab one’s imagination (enough to make it very plausible that something profound is likely to survive even if the eventual proof of the main result hits a snag, and which would moreover probably be useful to do other things later).

For example, I have heard that at the time of Wiles’ announcement, it was clear to experts very quickly that his method was going to at least prove modularity for infinitely many non-CM elliptic curves, which would be a huge advance even if some error deep in the work obstructed catching enough elliptic curves to prove FLT.

There are slides in the “What’s New” part of Mochizuki’s webpage for a survey talk he is giving in early December, and unfortunately those slides still fail to convey an idea that seizes the imagination; it remains too opaque. (Of course, one cannot expert slides on their own to reach the level of clarity of a written paper. Nonetheless, one might hope to at least see the germ of a brilliant idea.)

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12. November 15, 2012 at 12:36 am

The ideas of Galois were pretty hard for his contemporaries to grasp, but the people who first explained them to the rest of the world didn’t get much direct credit for that task; we still call it Galois theory.

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13. November 15, 2012 at 11:28 am

I think “Mochizuki Mathbabe theorem” has a nice ring to it.

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14. November 16, 2012 at 6:59 am

Mochizuki is also as weird as perelman?
Young students should peer-review.

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15. November 16, 2012 at 1:29 pm

I think there’s a distinction to be made that is missing from this argument. There are 2 cases. Either Mochizuki has managed to couch in incomprehensible terms something that’s easily expressed in conventional number theory, in which case his “interpreter” will deserve much credit. Or he’s invented a new, excruciatingly difficult language because that’s what the new result requires, in which case I don’t see why anyone else should share the credit, regardless of how long it takes number theorists to understand it. Grothendieck’s new language is an example of that: it was never “interpreted” to the world. Trained number theorists typically have had to spend a year of their lives going through SGA.

Maybe that’ll be true of Mochizuki’s “universal geometry.” I have no idea, but I see no a priori reason to rule that out, especially given Mochizuki’s reputation as a topflight number theorist.

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16. November 16, 2012 at 10:19 pm

An idea I had would be that it might be worthwhile to look for “local errors” in the papers. By “local error” I mean a mistake that isn’t too subtle and doesn’t require understanding everything that came before. It could serve as a not-so-laborious form of quality control. Naturally, typographical errors don’t count. Mistakes that are trivial to correct don’t show much. Is there a lot of the proof that is easy to read in small parts? Or is it more like even experts have trouble grasping 5 full pages, after say page 100?
So, it’s more a “search for gaps” I’m wondering about.

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17. November 17, 2012 at 10:51 pm

Weren’t there 2 guys who express-published their fleshed out version of Perelman’s argument and effectively claimed that geometrization is the “Perelman and X and Y theorem”? That didn’t work out so well.

“> But he had used a well-known method, following Richard Hamilton, ”

Perelman’s well-known, though hard to implement, method consisted of being a genius and using that ability to write down some quantities that magically and precisely control the problem, then adding several other breakthroughs. The number of important new ideas in Perelman’s work showed how far the other experts were from solving the problem.

” > and he was available to answer questions from generous, hard-working experts. ”

Mochizuki replied publicly, and reasonably quickly, to the known error reports that came his way. So far he has turned down some offers to lecture on the proof that involve travel, which is not unreasonable, and scheduled some lectures in December in Japan.

From the postings on MathOverflow one can see that some people (i.e., Vesselin Dimitrov and Akshay Venkatesh) were able to quickly parse enough of the material in Mochizuki’s papers to test the claims against known theorems.

A lot of algebraic complexity, new terminology, and possibly an extension of foundations is not that surprising in a problem like ABC. I would, however, expect a proof of ABC to contain a lot of analytic estimates visible in the paper and it’s hard to say how much of that is present in Mochizuki’s work without a closer reading.

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18. November 19, 2012 at 3:23 pm

It doesn’t sound to me like Mochizuki really did anything wrong. If his proof turns out to be correct, he should certainly get all of the credit, since not only did he come up with all of the ideas, he actually wrote them down in detail. (If the details are missing, then that’s another story.) If solving ABC required him to invent some impenetrable theory simply because of the nature of the beast, that would make his achievement even more monumental.

If, as you suggest, other people were to understand the proof well enough to present a *superior* presentation of the proof, that itself would be a nice achievement, and the mathematical community would show appreciation for their efforts (as it did in the case of the Poincare Conjecture), but it would still be Mochizuki’s theorem.

The usual smell test for a claimed proof of a big conjecture is always, “What are the big new insights here?” The problem here is that there is a whole new theory, so the logical question to ask experts here is, “Is there any reason to think that these novel *concepts* could add any value here?” And if so, one then has to understand the theory well enough to appreciate what the new insights are.

I think that abc makes a great point above. People don’t need “credit” to motivate them to try to understand the proof, but they do want to “get something” out of the effort (mathematical enlightenment at the very least), and if the whole thing is just wrong, then there’s nothing to be gained. If it is true that experts are reluctant to get involved with this thing at all, then that suggests to me that they are not merely skeptical of the proof, but skeptical of the whole idea that the new concepts add any value, which is perhaps a bit damning.

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19. November 22, 2012 at 10:13 am

When Hironaka came up with his proof of resolution of singularities, which won him a Fields medal, it was also very long and hard to understand, to the point where people weren’t comfortable using it. Now, 50 years later, there are much simpler versions of his proof. It is still very much considered to be his theorem and his proof… once you know the modern versions it is *much* easier to understand his proof.

I have no idea if Mochizuki’s inter-galactic whatever is correct, but if it ever does get absorbed and simplified by the mathematical community, it will very likely still be considered to be Mochizuki’s theorem since his papers will very likely be much more understandable. The interpreters will get some acclaim but no one would suggest they are responsible for the result, similar to with Perelman or Hironaka’s work. If a mathematician chops down a tree in the forest and no one is around, it still makes a sound…

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• November 22, 2012 at 10:45 am

…a sound that only that mathematician hears and reports back on. Another example: it’s still called the Lewis and Clark Expedition rather than say, the Jefferson Expedition.

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20. November 28, 2012 at 1:04 pm

Charles Babbage never could explain his computer invention to anyone. Using your logic, therefore, Ada Lovelace invented the first computer and was also the first computer programmer. Do you agree with that?

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21. December 21, 2012 at 1:00 am

I think “NotSoSure” has it most accurately. I quote wikipedia on tensors: “Tensors were first conceived by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.”

You and I know (roughly) what absolute differential calculus, manifolds and the Riemann curvature tensor are, plus maybe a bit of history about how that totally fucked Gauss’s labors up.

These extremely conceptually dense words required a lot of explanation. I have no idea if Mochizuki’s gone crazy, has re-named a bunch of easy to understand stuff, or has spent a huge amount of time creating (hopefully useful) novel, conceptually dense objects and words to prove what he wanted to. We will need generous experts to start getting our heads around which direction he went in.

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22. February 14, 2013 at 11:12 am

I am no math geek but I have to honestly disagree with what you are saying.
It does not make any sense, I almost want to say it is immoral, in the last part with your example.
If today someone would claim having discovered the Unified theory of everything and has a 6000 pages manuscript about it, ALL BASED ON THE CURRENT LAWS OF PHYSICS and you go on read them, don’t understand them, basically that guy’s work equals zero.
I would imagine those papers took MANY MANY years to develop and you come and say you want them translated to you almost like to a baby, otherwise they do not matter. I say this is unfair, immoral and pretty ignorant. Did you ever took your time to first think:
– Do I really want to understand this ?
– If yes, do I have what it takes (patience, knowledge, will) to look into this paper work.
– If yes again, will I blame the author for not making it more clear for me because I was unable to put a 6000 pages manuscript all together and understand the theory.
And if you will do that in the end, please just look into another conjecture to solve or research. There are so many interesting ones. You really don’t need to write articles like this, I personal, which I did not write a letter in those papers I just read about the and the models there, find it very offensive to find a maths’ person having this way of thinking and old mentality..
Good luck.

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• February 14, 2013 at 11:14 am

Keep in mind, math and math conjectures and all the delicate maths stuff do not necessarily have to be understood by everyone. They don’t have to be on the tonight show and presented and expect everybody to agree, comment and understand them. That is why they are called conjectures, unsolved problems, millennium problems etc..
They are not for everyone, not even for all mathematicians.

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23. February 22, 2013 at 7:50 pm

Mochizuki has posted a summary “A Panoramic Overview of Inter-universal Teichmuller Theory” on his website.

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24. March 9, 2013 at 5:59 am

Hard conjectures lead to hard proves, its not unusual that professional mathematicians cannot talk about their theories to fellows which work in another area of pure mathematics (and even more or less the same area!). You wrote about Grothendieck and let me tell you that his SGA-series was alike Mochizuki’s pre-prints: highly technical and enormous. Your article sounds a bit naive, be shure that there are mathematicians who are good enough to understand 6000 pages of theories even if you are none of them.

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• March 9, 2013 at 6:06 am

Incorrect. SGA was based on a seminar. By definition it was explained to other people as it was being written.

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25. March 11, 2013 at 10:41 pm

I’d say that if you have created a new idea, you are entitled to name it whatever you like. And if it is impractical to not name the idea (or ideas), it only make sense to name it. So I’d say what’s wrong with doing so?
Now, of course, the responsibility of the reviewers of the papers is to review them. And if the idea presented is faulty, the paper should be rejected. Or if the ideas are not explained sufficiently, the paper should be rejected. But the mere fact of the paper siting the ideas from other papers (perhaps of own work) does not make the paper insufficiently explained. Of course, reviewers may need to investigate the legitimacy of the arguments presented in the cited works…

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26. March 24, 2013 at 11:02 pm

Was the method of differential forms, as a systematic algorithm for solving all systems of PDEs, proved when Eli Cartan published it?

Apparently nobody used it for the next 50 years and it was rediscovered under a different name (“involution and extension”).

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27. April 21, 2013 at 10:55 pm
28. May 10, 2013 at 11:39 am

I agree that M should explain his proof in forums, lectures, letters, etc. It is his civic duty as a mathematician, especially one who works in academia.

However, I completely disagree with the idea that ‘it’s not a proof unless you convince others of its truth.’ It’s a proof if it is a logically correct argument, irregardless of who understands it, if anyone. Frankly it seems very self-centered to say that he must ‘explain it to me’, or someone like me, before we will even consider it to be a candidate proof. The fact of the matter is that M’s proof is either correct, incorrect, or incomplete, right now, a little like Schrodinger’s cat.

Recognition may be a matter of consensus but truth is not.

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• May 10, 2013 at 1:32 pm

abelardx, in principle we all agree with you (” It’s a proof if it is a logically correct argument”), but mathbabe is talking about the practical aspects of verifying this. Or even defining precisely what exactly a “logically correct argument” is, since no mathematical paper or book, no matter how carefully written it is, contains a logically complete proof.

The only practical way we have of verifying whether a proof is correct or not is for enough trusted people to read it and believe it is true. It is even better if over time people use the theorem or crucial parts of the proof and don’t run into any contradictions. But at no time are we able to say that a proof is definitely “logically correct”.

Another problem is the axiomatic foundation of mathematics, which most of us mathematicians prefer not to think about.

Some people naively propose computer verification of proofs, but that just leads to the question of how do we know that the software used is “logically correct”.

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29. May 10, 2013 at 5:58 pm

Actually, as I recall, Wiles’ work with his former student Richard Taylor came to nothing. It was only later that Wiles was able to find a way to solve the problem he’d had with the earlier proof.

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• May 10, 2013 at 6:00 pm

absolutely false.

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