Titans of Mathematics Clash Over Epic Proof of ABC Conjecture (quantamagazine.org) 105
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."
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."
In poor jest (Score:3, Funny)
I guess it isn't as easy as 1,2,3...
Re:In poor jest (Score:4, Funny)
What about Do-re-mi? Or you and me?
Re:In poor jest (Score:5, Informative)
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.
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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.
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The ABC theorum is a bit hard to explain.
Not as hard as it is to spell "theorem". *rimshot*
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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
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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.
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They wear a funny hat, keep one hand down their pants and like to conquer most of Europe?
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My virtual funny mod to you this day.
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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].
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Thank you captain pedantic. Does "primes other than integers" make you feel better?
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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.
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Isn't this one of AI's applications? (Score:2)
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?
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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.
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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)
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.
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Re:Isn't this one of AI's applications? (Score:5, Funny)
Neural networks are "really AI" in the same sense that chickens are really dinosaurs: the experts in the field get to define the terms. Except for Pluto, which remains a planet regardless.
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In addition, interactive proof assist systems do not even use neural networks.
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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.
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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
Re:The people wrong must be banned from Math (Score:5, Insightful)
Let's start defining some stakes.
If you come up with a proof and it's wrong, you're banned from Math.
If you say a proof is wrong and it turns out you are wrong, you're banned from Math.
Solving challenging problems can earn you "Unbanned from Math" cards, but they must be incredibly challenging.
So... Einstein would have been banned from theoretical physics using your rules. His original theory of Special Relativity was wrong in some cases, so we got "general" relativity as a correction... With your rules we would have banned him.
I wouldn't be too quick to "ban" anybody, unless they *should* have known better or they obviously violated the rules of math with their work and tried to hide it. You punish willful deception (those who are lying and know it), but mistakes and oversights are part of the human experience and why we have peer reviews. If you get found to have made mistakes or overlooked something, your reputation will suffer but you should be allowed to correct and proceed.
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Why is this voted up!?!?
General Relativity was not a correction to Special Relativity. They're about completely different things! Special Relativity is about the speed of light being a constant in all reference frames and the implications of that. General Relativity is about how mass distorts spacetime giving rise to gravity.
I agree with your point that bans are bad, but stop trying to sound smart using Einstein as an example until you learn more about him, OK?
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Special Relativity is about the speed of light being a constant in all reference frames and the implications of that.
If you actually read Einstein's original paper on special relativity, you'll find that it's about electrodynamics.
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All of it or none of it?
FTFY
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If you actually read Einstein's original journals, you'll find that it's about the subjugation of the working class.
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I could live with that, if folks where actually keeping score.
Personally, I'd give points for coming up with a theory and trying to prove it correct without resorting to deception, even if you are eventually shown to be wrong. Such things happen, folks propose ideas which turn out to be bad all the time; it's part of the process. My issue is when folks propose bad ideas, then insist they must be true and resort to lying and deception to gain notoriety. If you are honestly trying, great, if you are not b
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re quote "So... Einstein would have been banned from theoretical physics using your rules. His original theory of Special Relativity was wrong in some cases, so we got "general" relativity as a correction... With your rules we would have banned him." no theoretical physics is what it says on the tin theoretical , you could come up with a theory that the universe is just a magnified representation of a bacteria that resides in the gut , and it would still be valid theoretical physics , then its up to s
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This is not a published paper. It is being reviewed now to determine whether it should be published. That's partially why peer-review exists in the first place. Mochizuki just made his opus available to other mathematicians so that they can determine whether it makes sense to them. That's how research works.
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...and everyone wrong here should be banned on slashdot?
*tumbleweed*
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You have just created the best incentive ever to NOT say that something is wrong even if you're 99.9% sure it is.
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If you come up with an idea and it's wrong, you're banned from Ideas.
*BANNED*
Proofs are established subjectively (Score:2)
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
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"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."
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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).
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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.
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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)
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.
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Is this problem an academic one or are there real world implications for this turn of events?
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Until sometime down the line, someone comes up with a use for it.
Like taking prime number theory and making public private key encryption, that lets you safely conduct business online
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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.
How is any of this relevant (Score:2)
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Possibly to generating large prime numbers for cryptography?
Or has everyone moved on to elliptic curves anyway?
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10? Which optimism training school did you go to?
Re:How is any of this relevant (Score:4, Insightful)
You know what most of modern CS is based on? Mathematics, sometimes done thousands of years ago. This is relevant and that is why we keep these people around.
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Numberfile explains the ABC conjecture (Score:4, Informative)
https://www.youtube.com/watch?... [youtube.com]
actually (Score:5, Interesting)
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.
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This is not at all true. First off, mathematics itself has many different highly specialized sub-fields, many of which don't communicate effectively between each other. An complex analyst and a homotopy theorist speak very different mathematical languages, and may have difficulty communicating their ideas to each other. It is reasonable to suggest that this represents a different "cultural" background (as per Tylor's definition [britannica.com], these differences are differences in knowledge and belief, as well as differ
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Shinichi Mochizuki has a solid history of producing good mathematics. While it is possible that he is trying to pull a fast one, that seems quite unlikely, given his reputation. The most charitable explanation is that he has invented a new branch of mathematics (the "inter-universal Teichmüller theory") in order to resolve the ABC Conjecture, and that the "newness" of this approach is causing difficulty for outsiders.
A line of reasoning near the end of the proof... (Score:2)
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.
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ABC conjecture (Score:2)
Have NBC and CBS confirmed it?
How about CNN?
Faith? (Score:1)
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I think technically they are FCC 1984 phones (Reagan phones) but who cares, those who keep beating that drum are already dead for the most part.