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Math

Titans of Mathematics Clash Over Epic Proof of ABC Conjecture (quantamagazine.org) 105

Posted by msmash from the closer-look dept.
Two mathematicians have found what they say is a hole at the heart of a proof that has convulsed the mathematics community for nearly six years. Quanta Magazine: In a report [PDF] posted online Thursday, Peter Scholze of the University of Bonn and Jakob Stix of Goethe University Frankfurt describe what Stix calls a "serious, unfixable gap" within a mammoth series of papers by Shinichi Mochizuki, a mathematician at Kyoto University who is renowned for his brilliance. Posted online in 2012, Mochizuki's papers supposedly prove the abc conjecture, one of the most far-reaching problems in number theory. Despite multiple conferences dedicated to explicating Mochizuki's proof, number theorists have struggled to come to grips with its underlying ideas. His series of papers, which total more than 500 pages, are written in an impenetrable style, and refer back to a further 500 pages or so of previous work by Mochizuki, creating what one mathematician, Brian Conrad of Stanford University, has called "a sense of infinite regress."

Between 12 and 18 mathematicians who have studied the proof in depth believe it is correct, wrote Ivan Fesenko of the University of Nottingham in an email. But only mathematicians in "Mochizuki's orbit" have vouched for the proof's correctness, Conrad commented in a blog discussion last December. "There is nobody else out there who has been willing to say even off the record that they are confident the proof is complete." Nevertheless, wrote Frank Calegari of the University of Chicago in a December blog post, "mathematicians are very loath to claim that there is a problem with Mochizuki's argument because they can't point to any definitive error." That has now changed. In their report, Scholze and Stix argue that a line of reasoning near the end of the proof of "Corollary 3.12" in Mochizuki's third of four papers is fundamentally flawed. The corollary is central to Mochizuki's proposed abc proof. "I think the abc conjecture is still open," Scholze said. "Anybody has a chance of proving it."
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Titans of Mathematics Clash Over Epic Proof of ABC Conjecture More

Titans of Mathematics Clash Over Epic Proof of ABC Conjecture

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  • In poor jest (Score:3, Funny)

    by Anonymous Coward on Thursday September 20, 2018 @04:19PM (#57350670)

    I guess it isn't as easy as 1,2,3...

    • Re:In poor jest (Score:4, Funny)

      by sconeu ( 64226 ) on Thursday September 20, 2018 @04:34PM (#57350742) Homepage Journal

      What about Do-re-mi? Or you and me?

    • Re:In poor jest (Score:5, Informative)

      by lgw ( 121541 ) on Thursday September 20, 2018 @04:50PM (#57350826) Journal

      The ABC theorum is a bit hard to explain. The best I can do is taken from Wikipedia: [wikipedia.org] If:
      * A, B, and C are co-prime
      * A + B = C
      * D = the product of the unique prime factors of A, B, and C
      Then D is usually not much smaller than C.

      Or, put a different way, if A and B are high powers of primes, C probably isn't. For example:
      * A = 64 = 2^6
      * B = 81 = 3^4
      * C = 145 = 5*29
      * D = 870 = 2 * 3 * 5 * 29

      In that example, the prime factors of C were to the first power, so D was a multiple of C. That's pretty normal.

      This apparently has much broader consequences when generalized broadly to number fields, but I've never gotten my head around "primes" in fields other than integers.

      • Re: (Score:1)

        by Anonymous Coward

        My roommate at Caltech showed how you can "almost" prove Fermat's last theorem (only finitely large number of solutions possible for A and B being co-prime) using this conjecture and that is the time I learned the power of this conjecture. I am not a mathematician but still it felt very interesting.

      • The ABC theorum is a bit hard to explain.

        Not as hard as it is to spell "theorem". *rimshot*

      • Re: (Score:2)

        by rtb61 ( 674572 )

        To put in another way maths geeks often get lost in the unreality of numbers. Maths might seem all hard and real like physics, but really it is not and for the most obvious reason. It is the number of 'somethings' that make math real, where there is no something, than maths becomes purely relative, an empty dance of numbers. It is the something that makes math real, applying the math to the actual real world, theoretical math, tends to drift off into a world of patterns. When you go there, as deep as you ca

        • Re: (Score:1)

          by Anonymous Coward

          Math is a priori, pure reason. It has no and needs no connection to the evidence which makes up reality.

          Math is not the language of the universe. Math is one language to describe the universe.

      • Re: (Score:2)

        by gringer ( 252588 )

        I've never gotten my head around "primes" in fields other than integers

        That might be because the integers aren't a field [wikipedia.org].

        • Re: (Score:2)

          by lgw ( 121541 )

          Thank you captain pedantic. Does "primes other than integers" make you feel better?

      • Re: (Score:2)

        by acvh ( 120205 )

        I'm sure your explanation works for some; but the linked article in the summary actually has a great layman's explanation that is extremely clear.

    • Comment removed based on user account deletion

  • I thought theorem checking was one of the applications that AI was being touted for. Just doing a quick check, there seems to be a large number of articles (like this one, which goes back a bit: http://www.dtic.mil/dtic/tr/fu... [dtic.mil]) written about this very topic.

    Rather that rely on a limited number of mathematicians, all of whom seem to know Professor Mochizuki, how about running his proof through these AI tools to see if they can validate the proof?

    • Considering how obtuse mathematician's found the proof, rewriting it into something a proof assistant could parse was almost certainly a mammoth task. Trying to nitpick errors in the proof was almost certainly a better use of time.

      • Re: (Score:1)

        by Anonymous Coward

        Seems like rewriting it to something that could be followed is exactly what is needed.

    • Re:Isn't this one of AI's applications? (Score:5, Funny)

      by Anonymous Coward on Thursday September 20, 2018 @05:06PM (#57350890)

      I thought theorem checking was one of the applications that AI was being touted for. Just doing a quick check, there seems to be a large number of articles (like this one, which goes back a bit: http://www.dtic.mil/dtic/tr/fu... [dtic.mil]) written about this very topic.

      Rather that rely on a limited number of mathematicians, all of whom seem to know Professor Mochizuki, how about running his proof through these AI tools to see if they can validate the proof?

      Hi, My name is Euclid Pascal-Poincaré, Professer of Mathematics at the Nigerian Institute of the 409 Theorems. Nobody has ever had a thought as brilliant as yours my friend. And I should know, since I have received the fields medal three times, as the youngest (age 7), most successful (age 22) and oldest (age 57) awardee. The idea of applying an AI proof machine which could obviously solve the problem to a proof that is obviously too easy for it would be something that our institute would pay dearly for. Your place is guaranteed.

      I have a research lab and $1,500,000 (One billion and ifty million dollars) and twelve beautiful virgin assistants waiting for you in Nigeria. All you have to do to claim your position is to wire $432 + $71422 (four hundred thousand and twenty two pounds to) to UK Bank: Nat West, Sort code: 60-16-03 Account number: 73754900.

      I am looking forward to greet you at our newly built facility with it's four hundred swimming pools and banks of tens of mechanical calculating machines.

    • Re: (Score:2, Troll)

      by 110010001000 ( 697113 )
      You do know that AI isn't real, right? What they call "AI" is just "neural" networks which have been around for decades.

    • Re: (Score:2)

      by gweihir ( 88907 )

      Only works for fully formalized proofs and a human has to explain the proof to the system in detail. That means it takes a lot of time to do and needs a human that fully understands the proof.

      Also, this is not actually AI in any meaningful sense, it only gets in there because of the AI hype.

    • Re: (Score:2)

      by pjt33 ( 739471 )

      As with much of AI, there has been progress but there is still a way to go. In particular, the input format for theorem checkers is not yet the mathematical paper. I'm not aware that anyone apart from Mohan Ganesalingam and his collaborator Thomas Barnet-Lamb have worked on parsing mathematical papers into something which could be supplied to a theorem checker; Ganesalingam's 2013 book The Language of Mathematics: A Linguistic and Philosophical Investigation gives an idea of the challenges and limitations o

  • That's true even for simple things: you look at the Pythagorean theorem and at some point the proof of the theory "clicks" somewhere inside you and you say yes this is true. A genius friend at the university argued with his mathematics professor on some advanced course as he didn't give my friend the full credit on some very complicated proof, and he said "see here, colleague" (they call all students "colleague"), "the mistake is you wrote this orientation here is clockwise but it's counter-clockwise". The

    • "Now for extra credit: explain this to someone a couple levels dumber than either of the two of us."
      "Clockwise vs counter-clockwise, I can handle. But that? Now you're just being screwy."

      • Re: (Score:2)

        by cb88 ( 1410145 )
        Sometimes you establish a convention to do a calculation or proof, and that apparently was not well defined in that case but apparently it was used consistently otherwise the professor would not have conseded.

        And example is the right hand rule (a convention), and the + or - nature of electrons (we actually got it backwards from reality but since it's just a convention it doesn't matter in practice we just had to choose a polarity and go with it consitently).

    • Re: (Score:2)

      by mlyle ( 148697 )

      Plenty of proofs have a very strong objective basis. E.g. all of the mathematics that has been proven (and verified) within systems of simple symbolic manipulation like Whitehead established in the Principia Mathematica.

      • Re: (Score:2)

        by pjt33 ( 739471 )

        I think that GPPs point can perhaps be reworded as "We write proofs to convince other humans, not to convince theorem provers". There are some theorems whose proofs have been verified automatically, and others which only have computer-assisted proofs (e.g. the four colour theorem, or the theorem formerly known as the Kepler conjecture), but from a philosophical point of view a proof is considered a proof if it convinces the experts in the field.

  • Humbling (Score:3)

    by ebonum ( 830686 ) on Thursday September 20, 2018 @04:40PM (#57350768)

    Understanding what the abc conjecture states takes effort. Proving it...
    A reminder of just how different a real mathematician's mind is from the rest of us.

    • Re: (Score:2)

      by sycodon ( 149926 )

      Is this problem an academic one or are there real world implications for this turn of events?

    • Re: (Score:2)

      by neoRUR ( 674398 )

      Yes, even Geniuses don't always know how to get the answer, sometimes they just know the answer.
      Watch this movie, it's a good movie about a good Genius who's mind was faster then his though process.
      https://en.wikipedia.org/wiki/... [wikipedia.org]

      Its a movie about Ramanujan.

  • Seriously, how is this relevant to anyone on the planet besides the author himself, and the 10 other people who've read his work.

  • Numberfile explains the ABC conjecture (Score:4, Informative)

    by swm ( 171547 ) <swmcd@world.std.com> on Thursday September 20, 2018 @05:33PM (#57351028) Homepage
    Numberfile gives a good intro to the ABC conjecture

    https://www.youtube.com/watch?... [youtube.com]

  • actually (Score:5, Interesting)

    by thePsychologist ( 1062886 ) on Thursday September 20, 2018 @05:51PM (#57351116) Journal

    I think the title here is misleading. Outside of Mochizuki's friends (and perhaps even including them), every mathematician involved has had serious doubts about this purported proof since the beginning. That's simply because the papers are written very different than the usual math paper ---- that is to say, leaving very many things not explained or explained poorly.

  • In their report, Scholze and Stix argue that a line of reasoning near the end of the proof of "Corollary 3.12" in Mochizuki's third of four papers is fundamentally flawed.

    I am definitely incapable of reading Mochzuki's proof, but it would have been interesting if the article had cited the line in question.

    • Well, to comprehend page 500 you have first to understand the 499 previous pages, which is why the flaw "near the end" is hard to spot.

  • Have NBC and CBS confirmed it?
    How about CNN?

  • So, if in a statement (theorem) nobody spots an error, it's considered proven and true? I wonder how many such theorems are out there in which we have faith?

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