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Titans of Mathematics Clash Over Epic Proof of ABC Conjecture (quantamagazine.org)
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."
I guess it isn't as easy as 1,2,3...
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.
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.
"File to fit, pound to insert, paint to match" - Aircraft Maintenance 101
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.
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...) 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.
https://www.youtube.com/watch?...
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?
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.
"What lies behind us, and what lies before us are tiny matters compared to what lies within us." Ralph Waldo Emerson
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.
Socialism: a lie told by totalitarians and believed by fools.
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.
Slashdot, fix the reply notifications... You won't get away with it...
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.
Most ACs are not even worth the keystrokes to insult them. Be generically insulted by this and ignored otherwise.