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."

Comments

In poor jest

[Anonymous Coward]

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

Re:In poor jest

[sconeu]

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

Re:In poor jest

[lgw]

The ABC theorum is a bit hard to explain. The best I can do is taken from Wikipedia: 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:The people wrong must be banned from Math

[bobbied]

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.

Re:Isn't this one of AI's applications?

[Anonymous Coward]

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.

Numberfile explains the ABC conjecture

[swm]

Numberfile gives a good intro to the ABC conjecture

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

actually

[thePsychologist]

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.

Re:Isn't this one of AI's applications?

[lgw]

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.

Re:How is any of this relevant

[gweihir]

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.