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Titans of Mathematics Clash Over Epic Proof of ABC Conjecture

• drenvuk 8y ago
  I think this would count as esoteric if you ignore the drama portion of it. Just to get to the point of understanding what they're talking about would probably take months of dedicated uninterrupted work. Neat. I have no idea where to even start. https://en.wikipedia.org/wiki/Anabelian_geometry https://en.wikipedia.org/wiki/Anabelian_geometry
  • strikelaserclaw 8y ago
    I don't profess to understand the mathematics behind these papers but i have been following the debate for a couple years now. It is weird for such eminent mathematicians to not be able to reach a conclusion as to whether mochizukis arguments are true or false. I think this is the first time a really eminent mathematician like scholze has put in his two cents that the proof might not be right. At least it looks like work is being done at a serious level to prove or disprove these papers.
  • xxpor 8y ago
    I think what makes these number theory arguments so interesting to me is how simple the theorems are, but how impenetrable the proofs are. For example, Fermat's last theorem is so easy to understand, but there's no chance in hell I'll ever understand Wiles' paper proving it.
    • dagw 8y ago
      You might not understand all the details of the paper, but you can definitely follow the thrust of the argument and see the elegance in it without advanced math skills. The really hard part (and main part of Wile's contribution to the proof) was proving something called the Taniyama–Shimura–Weil conjecture. However if you skip that bit and just accept that the conjecture is true the rest of the steps of the proof are both elegant and relatively easy to follow (if you gloss over some of the details) for anybody with a decent grasp of basic math. The book Fermat's Last Theorem by Simon Singh is a pretty great read and does a good job of outlining the basic structure of the proof for anybody with a decent grasp of high school mat.
  • throwaway080383 8y ago
    Years even. And really, that's kind of the issue at hand. This stuff is so difficult that very few mathematicians really understand it, and Mochizuki's work is so much more esoteric than that, that basically no one understands it, even recent Fields Medallists.
• sajid 8y ago
  https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ https://galoisrepresentations.wordpress.com/2017/12/17/the-a... Terence Tao and Peter Scholze (and others) join the discussion in the comments section.
  • avip 8y ago
    Tao's commentary is especially interesting for me as a non-mathematician: >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. Likely such is the understatement-ish way of a mathematician to say "This is clearly wrong, but I have not the time to dig in and find where exactly"
    • bsder 8y ago
      > Likely such is the understatement-ish way of a mathematician to say "This is clearly wrong, but I have not the time to dig in and find where exactly" This is a statement of "If this thing has no other application, it isn't worth my time to dig through what is likely an impenetrable pile of garbage." Basically, proof of the abc conjecture should have a bunch of follow on consequences. If those aren't materializing, that makes taking the time to understand the proof quite a bit less enticing (ie. the Bayesian prior on "true" goes down a lot).
    • AnimalMuppet 8y ago
      I read it more as "This is probably wrong, because it differs in this important way from every other true groundbreaking proof."
    • neuronexmachina 8y ago
      I think it might be more akin to a programmer complaining about a monolithic/black-box chunk of code that can't easily be split into understandable and/or unit-testable functions.
      • 8y ago
• jim777 8y ago
  > When he told colleagues the nature of Scholze and Stix’s objections, he wrote, his descriptions “were met with a remarkably unanimous response of utter astonishment and even disbelief (at times accompanied by bouts of laughter!) that such manifestly erroneous misunderstandings could have occurred.” Is this normal in advanced Math or is this guy kind of a jerk?
  • Jweb_Guru 8y ago
    He is being a jerk. Also, not responding to the actual argument, as far as I can tell (which is technical, but quite clear). There are some other damning statements in the rebuttal, too--for example, they say that all the new machinery developed for anabelian geometry lies unused in the case to which Corollary 3.12 applies (meaning most of the effort mathematicians have been spending in order to try to understand Mochizuki's new mathematics has been wasted). They also mention other incidental complexities that serve to obfuscate the argument but don't seem to buy anything (for example, not choosing a specific isomorphism at one point in the proof, but instead asking the reader to consider all isomorphisms with particular properties, for reasons he has been unable to justify). Someone in his position should be aiming to clean up his proofs to be as straightforward as possible, not making them deliberately harder to verify.
  • xc10001 8y ago
    New Law: Any Internet discussion will be reduced to jerk/non-jerk within 30 minutes.
  • Soundest 8y ago
    Well, let's lay aside the fact that there clearly are plenty of jerks in academia. The entire article seems to layout a sort of string of unusual and problematic situations. Whether it's people who hit upon the problem conjecture and chose not to speak out for fear of causing trouble, or the journal publishing the proof being edited by the author, or the large number of 'delegates' who seem to be pushing the proof without being able to explain the proof, it seems this is a mix of problems that are arising from an unhealthy situation.
  • a-nikolaev 8y ago
• matthewbauer 8y ago
  I wonder if language is a factor in this at all? I know mathematics is written entirely in English but most likely discussions still happen in local languages. Could part of the proof be lost in translation?
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  • pishpash 8y ago
    It sounds like what is lost in translation is somebody's face and/or reputation, purely judging the presented situation socially. That is all we can do as non-mathematicians.
  • doall 8y ago
    I think Mochizuki won't have any problems with English, since he spent his school days, up to PhD in the US. https://en.wikipedia.org/wiki/Shinichi_Mochizuki https://en.wikipedia.org/wiki/Shinichi_Mochizuki About Hoshi, I have seen him in a conference video before, and have an impression that he isn't good at English, which can make a bit difficult in discussion. I don't know about Scholze and Stix.
    • laingc 8y ago
      Scholze has a strong accent, but speaks flawless, sophisticated English.
• Moodles 8y ago
  I have a MSc in mathematics so I am by no means an expert in mathematics proofs but I get the gist of it. To me a proof is literally a logical argument that you can follow to "believe" that a theorem is true. I do worry how many people actually understand or verify mathematics proofs. How many people have actually read and verified Perelman's or Wiles' proofs (I'm only singling them out because they're famous, not because they're particularly sketchy). We seem to think that once a proof is published in a reputable journal then it is definitely true, but really we should only think "a small number of qualified people have read it and think it is fine". The foundations actually seem a bit shakier to me than people seem to appreciate. Perhaps machine-verified proofs will be the answer one day.
  • bsder 8y ago
    > We seem to think that once a proof is published in a reputable journal then it is definitely true, but really we should only think "a small number of qualified people have read it and think it is fine". The foundations actually seem a bit shakier to me than people seem to appreciate. I think you are understating the certainty of fundamental proofs like this. Once such a fundamental proof is considered "true", it almost always has applications beyond what the original proof was used for. It is often used to try to prove things that we already know are "true" in other ways. So, any "true" proof generally gets tested from multiple directions. Wile's proof is a good example of this. It doesn't just prove Fermat's Last Theorem. Quoting Wikipedia: "Wiles' path to proving Fermat's Last Theorem, by way of proving the modularity theorem for the special case of semistable elliptic curves, established powerful modularity lifting techniques and opened up entire new approaches to numerous other problems."
    • Moodles 8y ago
      True, but even then, how large do you think the field is? And how many people verify these proofs? I have a PhD in an esoteric field myself. There's literally <50 people worldwide who write/verify proofs in it. I've spotted errors in my own work or other's work that got past peer review. And I don't think I'm particularly special. There's just not enough eyeballs, time and incentives sometimes.
• zokier 8y ago
  So what would this imply over the larger body of Mochizukis work? As far as I have understood the situation, the abc proof is more of a sideproduct of him developing this completely new system(?) for maths rather than the main thrust of the works. Is it all falling down like house of cards, or is there still workable and useful bits there?
  • contravariant 8y ago
    Somewhere in this comment section someone quoted the following line by Terrence Tao: >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. so it's not so much that the main result is called into question but there are some useful and workable bits left over. Rather the main result is called into question because there are no other useful and workable bits.
• peq 8y ago
  Mathematicians should do more computer verifiable proofs to avoid discussions like this. > Definitions went on for pages, followed by theorems whose statements were similarly long, but whose proofs only said, essentially, “this follows immediately from the definitions.” This sounds perfect for machine checked proofs, but I guess the proofs are actually a lot more involved than they are presented.
  • cycrutchfield 8y ago
    Per Peter Scholze (from https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/#comment-4619 https://galoisrepresentations.wordpress.com/2017/12/17/the-a...): >One final point: I get very annoyed by all references to computer-verification (that came up not on this blog, but elsewhere on the internet in discussions of Mochizuki’s work). The computer will not be able to make sense of this step either. The comparison to the Kepler conjecture, say, is entirely misguided: In that case, the general strategy was clear, but it was unclear whether every single case had been taken care of. Here, there is no case at all, just the claim “And now the result follows”.
    • trevyn 8y ago
      Well that's sort of the point, isn't it? If you can't make the computer make sense of it, you have no proof.
    • MaxBarraclough 8y ago
      > The computer will not be able to make sense of this step either. What does that mean, then? That a leap is being made that depends on the reader's fuzzy intuitions, rather than the established axioms? If that's the case, it's not a formal proof at all, no? Or am I way off here?
      • konschubert 8y ago
        Maybe he says that the computer won't help understand the proof since the proof is missing a step. He may be saying the equivalent of "A computer won't help you to prove that 1+1=3“
• S4M 8y ago
  Coincidentally I recently finished a MOOC about prime numbers where the ABC conjecture was introduced, I believe in order to show how mysterious prime numbers still were. The MOOC is here if someone is interested: https://courses.edx.org/courses/course-v1:KyotoUx+011x+2T2018/b46e533438fc4cb98eeef569abbd837a/ https://courses.edx.org/courses/course-v1:KyotoUx+011x+2T201... Apart from that, the article makes me curious about the state of automated mathematical proof. I remember having read an article posted here about a mathematician (I think he was a Field medalist) who was claiming that automated proof were the future of mathematics as they would enable mathematicians to collaborate much more easily, removing problems of trust in others' proof.
• spydum 8y ago
  I do have to wonder, if the proof is so complex, and yet doesn't offer insights to any other proof or theories.. does it matter? Older threads for reference: https://news.ycombinator.com/item?id=15971802 https://news.ycombinator.com/item?id=4502856 https://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture
• vertline3 8y ago
  Peter Scholze is too valuable to be lost to the hall of mirrors that is Mochizuki's conjecture. :P He is young and at the peak of his powers. Even I've heard of him.
• Obi_Juan_Kenobi 8y ago
  Slightly OT: it was very nice to read an article that made such wonderful use of links. Just about every case where material could be directly linked, it was done. I particularly enjoyed the link to Dr. Calegari's blog post, as it was interesting to read the comments in 'real time' and compare that with the author's synthesis. Very good article!
  • timb07 8y ago
    Even further OT: I was good friends with Frank Calegari's older brother (also a mathematician) at school, and used to play bridge with Frank at university. I'm not a mathematician but I do sometimes read articles when they're posted on HN, so it was a surprise to see someone I know mentioned. :p
• westoncb 8y ago
  > “The abc conjecture is a very elementary statement about multiplication and addition,” said Minhyong Kim of the University of Oxford. It’s the kind of statement, he said, where “you feel like you’re revealing some kind of very fundamental structure about number systems in general that you hadn’t seen before.” I can kind of see that, but there's something in it that gives me the feeling of arbitrariness. It's looking at the relationship between prime factors of A, B, C in A + B = C, where no prime factors may be shared between A, B, or C. The specific relationship in question is between the magnitude of C and the unique prime factors of A, B, and C multiplied together. To take an example from the article, 5 + 16 = 21 meets the basic requirements since the prime decomposition looks like (5) + (2 * 2 * 2 * 2) = (7 * 3) —no factors are shared. But, the quantity we're supposed to relate to C is (5 * 2 * 7 * 3), since it is the product only of the unique primes. Does that not feel arbitrary, to drop the repetitions? So it ends up being that the sought out smaller products arise because they had large numbers of repetitions, so we removed more when forming the product. But I suppose there are probably just some deep number theory mysteries behind it which give justification, so it only feels arbitrary to someone like myself who is basically ignorant on the subject :)
  • jetrink 8y ago
    > Does that not feel arbitrary, to drop the repetitions? If the conjecture turns out to be true, then that apparently arbitrary step turns out to be meaningful, because it uncovers a hidden relationship between the numbers A, B and C and their factors.
  • edflsafoiewq 8y ago
    Another way to think about it is to consider what needs to happen for rad(abc) to be much less than c. This can only happen if the "dropping" you mention drops a whole lot of factors. In other words, it can only happen if a, b, and c all contain mostly large powers of primes. The prime factorization of a number is essentially "random". That makes numbers that are large powers of primes rare; it's like rolling a die and continuously getting the same face. Now if you add two numbers, the prime factorization of the sum does not appear to be related to the prime factorization of the summands, so adding two very rare numbers and getting yet another very rare number seems unlikely. And that seems to be roughly what the abc conjecture says: it is not likely that all of a, b, and c are rare.
• lacker 8y ago
  I remember an instructor explaining during the math olympiad training program: "A proof doesn't have to be in any particular format. A proof is an argument that can convince other mathematicians." At this point, it seems clear that we do not have a proof of the ABC conjecture. Perhaps someone will be able to improve this to make a real proof - after all, there were some errors in the first Wiles proof of Fermat's Last Theorem, but they ended up being minor errors that were fixed up once they were found. But what we have is not a proof.
  • westoncb 8y ago
    It's interesting that this article doesn't touch on this aspect of the situation at all, but when reading about Mochizuki's proof in the past it was presented as clear that proving the ABC conjecture was one application of a new general theory—not that all of Mochizuki's innovations are tied specifically to that problem. I was given to understand that his work was more like Grothendieck's, where he's developing this entirely new, super general system. But you can imagine if Grothendieck for example had done all of his work for years and years in isolation and then presented Schemes, Topoi, and Motives to everyone all at once, using them in combination to prove something familiar, and expecting other mathematicians to just learn and understand them all in order to verify a single proof. I think in that case, folks would not have been immediately so interested in the new constructions Grothendieck had come up, viewing it initially as mountains of unnecessarily alien concepts built up just to give other mathematicians a hard time ;) (Not to identify the two mathematicians overly—I'm not sure how appropriate that really is—but I remember that being my impression when first digging into Mochizuki's work, and I haven't seen it mentioned here yet.)
    • Sniffnoy 8y ago
      This is where Terry Tao's comment from December seems relevant: https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/#comment-4563 https://galoisrepresentations.wordpress.com/2017/12/17/the-a...
    • emileokada 8y ago
      I believe the article explicitly steers away from this because nearly all the discourse on this proof has been precisely about this (c.f. remark Scholze that 'The whole discussion surrounding the proof had gotten “too sociological,”').
• CodeSheikh 8y ago
  My God! This Titans' website needs urgent attention: http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html
  • dan-robertson 8y ago
    Why? It might be worse than a <p> and a <ul> with links to papers but it loads ok and it’s basically a point of pride amongst academics to have an ancient or amateurish looking handmade website. Perhaps it is a signal that time is spent on academic work or a that the site was made by the academic (and so has useful materials for academics) and not by some publicist who does not provide useful academic materials.
  • Jach 8y ago
    Why? At least it's not hot pink or otherwise eye-blinding. I still find it charming that many Japanese sites seem stuck in 90s web design. But recently I've been running into more Japanese sites that seem to have advanced to mid-2000s web design, with proper CSS and designs that use curves and angles and don't force everything into a box shape (visible or invisible)! That's my favorite era so I hope it's a while before they join the rest of us in bloated SPA hell.
  • ssmmww 8y ago
    No no no! His website is a treasure! I think the world needs many more garish, lurid, irreverent websites like this.
  • amiga-workbench 8y ago
    Why? It actually performs well.
• onemoresoop 8y ago
  http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-...
  • dvdkhlng 8y ago
    Thanks for the link. As that seems very relevant to the discussion, let me add some context: This is Mochizuki's answer to the recent criticism titled "Report On Discussions, Held During The Period March 15 – 20, 2018, Concerning Inter-Universal Teichmuller Theory (IUTC)" For TL;DR see around page 40 in that PDF: > Indeed, at numerous points in the March discussions, I was often tempted to issue a response of the following form to various assertions of SS (but typically refrained from doing so!): Yes! Yes! Of course, I completely agree that the theory that you are discussing is completely absurd and meaningless, but that theory is com- pletely different from IUTch! > Nevertheless, the March discussions were productive in the sense that they yielded a valuable first glimpse at the mathematical content of the misunderstandings that underlie criticism of IUTch (cf. the discussion of § 3). In the present report, we considered various possible causes for these misunderstandings , namely: > (PCM1) lack of sufficient time to reflect deeply on the mathematics under discussion (cf. the discussion in the final portions of § 2, § 10); > (PCM2) communication issues and related procedural irregularities (cf. (T6), (T7), (T8)); > (PCM3) a deep sense of discomfort ,or unfamiliarity ,with new ways of thinking about familiar mathematical objects (cf. the discussion of § 16; [Rpt2014], (T2); [Fsk], § 3.3). > On the other hand, the March discussions were, unfortunately, by no means sufficient to yield a complete elucidation of the logical structure of the causes underlying the misunderstandings summarized in § 17.
• auggierose 8y ago
  A few years ago I asked Mochizuki via email if he would be interested in organising with me an effort to mechanise his proof. At the time I was developing a collaborative interactive theorem prover, for which this would be an ideal application. He politely declined, and I am kind of happy about this now, because I need a few more years anyway before my theorem prover could be used for something like this without being a pain for its users :-D
• graycat 8y ago
  The math community can't accept a new result is true without a good proof. As part of good, the proof has to be readable by peer reviewers. Net, if the writing is tough to read by some of the best qualified peers, then the writer has more work to do, in particular, to learn to write better. We have lots of examples of good writing in math from Bourbaki, W. Rudin, P. Halmos, H. Royden, and more. Sorry 'bout that: No one wants math to miss out on a great, new result, but the writer has to do their job first, in particular, do good math writing.
• notoriousjpg 8y ago
  We need Yasantha on this
• hedvig 8y ago
  So Mochizuki has had years to revise and restate things right? Knowing that it hasn't been well received wouldn't one want to clarify things? Did he move on to a different project, or is he so arrogant that once he submits a paper he walks away from it?
• ur-whale 8y ago
  To me, this article highlights two things: - the sore need for a way of automatically verifying proofs. - the sore need for a formal, agreed-upon language in which math proofs can be written. The language of math has always been much more mushy than mathematicians are willing to concede, and the chickens are coming to roost: 21s century math has become so complex and sophisticated that very few people can actually even read the content of proofs, much less understand them. Shinozuki's case is an extreme example of that: after almost ten years, even he other experts in the field aren't sure of what he's saying. There is a clear need for formalizing the language of mathematics in a way that allows machine to verify he validity of a proof.
• deleted 8y ago
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• swerveonem 8y ago
  Will others fork his proof to fill in the gaps? Is there a github for maths proofs?