It may have occurred in the past but I have never heard of such a thing. Yes, there have been incorrect proofs that have gone unnoticed usually because few have paid attention. But when they are discovered and pointed out, the originator, if he is not a crank, acknowledges the error.

But not in the case of the claim of a proof for the "abc conjecture" which has just been published. The Japanese mathematician who claims to have solved this simply stated number theory conjecture is a Field's medalist winner and he has world class colleagues who claim his 600 page proof has no flaws. The problem is that another Fields medalist and a larger number of his colleagues says there is. And this is occurring AFTER both groups have had discussions trying to make their case. I think this is unheard of when it comes to pure math.

Thus it is up to you guys to ferret out the answer.

Can't be the 4-colour problem, because that was a while ago.

https://en.wikipedia.org/wiki/Four_color_theorem

But that wasn't the same thing I don't believe. No one spotted the errors at first. In this case the error has been supposedly exposed and world class mathematicians are arguing about it.

Problem

https://www.quantamagazine.org/titan...ture-20180920/
-----------------------------
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.”

“Each time I hear of an analysis of Mochizuki’s papers by an expert (off the record) the report is disturbingly familiar: vast fields of trivialities followed by an enormous cliff of unjustified conclusions,” Calegari wrote in his December blog post.
…
“Those who understand the work need to be more successful at communicating to arithmetic geometers what makes it tick,” he wrote.
…
They had been able to read and understand the papers until they hit a particular part. “For each of these people, the proof that had stumped them was for 3.12,” Conrad later wrote.
Kim heard similar concerns about Corollary 3.12 from another mathematician, Teruhisa Koshikawa, currently at Kyoto University. And Stix, too, got perplexed in the same spot.

Gradually, various number theorists became aware that this corollary was a sticking point, but it wasn’t clear whether the argument had a hole or Mochizuki simply needed to explain his reasoning better.

But much more concerning to many number theorists was the fact that the papers were still, as far as they were concerned, unreadable.
“No expert who claims to understand the arguments has succeeded in explaining them to any of the (very many) experts who remain mystified,” Matthew Emerton of the University of Chicago wrote.

Calegari wrote a blog post decrying the situation as “a complete disaster,” to a chorus of amens from prominent number theorists. “We do now have the ridiculous situation where ABC is a theorem in Kyoto but a conjecture everywhere else,” Calegari wrote.

So in the comments section below Calegari’s blog post, Scholze wrote that he was “entirely unable to follow the logic after Figure 3.8 in the proof of Corollary 3.12.” He added that mathematicians “who do claim to understand the proof are unwilling to acknowledge that more must be said there.”

=================


PairTheBoard

Resolution:
https://www.quantamagazine.org/titan...ture-20180920/
-------------------------
The abc conjecture then boils down to proving a certain inequality between two quantities associated with the elliptic curve. Mochizuki’s work translates this inequality into yet another form, which, Stix said, can be thought of as comparing the volumes of two sets. Corollary 3.12 is where Mochizuki presents his proof of this new inequality, which, if true, would prove the abc conjecture.

The proof, as Scholze and Stix describe it, involves viewing the volumes of the two sets as living inside two different copies of the real numbers, which are then represented as part of a circle of six different copies of the real numbers, together with mappings that explain how each copy relates to its neighbors along the circle. To keep track of how the volumes of sets relate to one another, it’s necessary to understand how volume measurements in one copy relate to measurements in the other copies, Stix said.

“If you have an inequality of two things but the measuring stick is sort of shrunk by a factor which you don’t control, then you lose control over what the inequality actually means,” Stix said.

It is at this crucial spot in the argument that things go wrong, Scholze and Stix believe. In Mochizuki’s mappings, the measuring sticks are locally compatible with one another. But when you go around the circle, Stix said, you end up with a measuring stick that looks different from if you had gone around the other way. The situation, he said, is akin to Escher’s famous winding staircase, which climbs and climbs only to somehow end up below where it started.

This incompatibility in the volume measurements means that the resulting inequality is between the wrong quantities, Scholze and Stix assert. And if you adjust things so the volume measurements are globally compatible, then the inequality becomes meaningless, they say.

Scholze and Stix have “identified a way that the argument can’t possibly work,” said Kiran Kedlaya, a mathematician at the University of California, San Diego, who has studied Mochizuki’s papers in depth. “So if the argument is to be correct, it has to do something different, and something a lot more subtle” than what Scholze and Stix describe.

Something more subtle is exactly what the proof does, Mochizuki contends. Scholze and Stix err, he wrote, in making arbitrary identifications between mathematical objects that should be regarded as distinct. 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.”

Mathematicians will now have to absorb Scholze and Stix’s argument and Mochizuki’s response. But Scholze hopes that, in contrast with the situation for Mochizuki’s original series of papers, this should not be a protracted process, since the gist of his and Stix’s objection is not highly technical. Other number theorists “would have totally been able to follow the discussions that we had had this week with Mochizuki,” he said.
==================


Sounds like this will be getting sorted out as they concentrate on the crux of the disagreement. Sklansky should probably step in at this point with one of his insightful simplifications of the whole matter.


PairTheBoard

^ lol, one of the guys involved in this is the successor of my old supervisor!

As a precedence in the past, Cantor's theory of transfinite numbers comes to mind:

However, Cantor introduced a new proof method (his diagonalization argument) and concepts (i.e. that some sets are "more infinite" than others), so the need for some digestion of the mathematical community was to be expected. For the case at hand the controversy seems to stem from the sheer complexity of the proof, while Cantor's proof for his claim was essentially super-simple.

The controversy in that case was about which axioms to accept, as e.g. similarly with the proof of the Banach-Tarski paradox. These people seem to be on the same page with the axioms.

https://en.wikipedia.org/wiki/Banach...Tarski_paradox

FWIW Mochizuki does not have a Fields Medal. One of the guys critiquing his proof, Peter Scholze, does. And he was awarded the medal at 30 years old.

It looks like the only mathematicians vouching for Mochizuki are in his sphere of influence. And the criticisms of the proof all point to the same corollary; no one outside of Mochizuki and his friends can make sense of it, and the entire proof follows from that. But they can't explain it to anyone else. Mochizuki's summary of Scholze and Stix's visit is hilarious. He basically starts making fun of them at the end of his 45 page summary for not being smart enough (keep in mind one of them is probably the world's best mathematician in his age bracket) to understand his genius. It's straight up ad hominem personal attacks and political BS. Zero technical reasoning on why their criticism is wrong.

Like 0% chance his proof is correct at this point. Mochizuki is literally just trying to save face and won't admit an error. He looks like a clown right now.

Controversy, among Scientists/Mathematicians, burns away the superfluous and adds to truth and knowledge*.


*All part and parcel to the "Scientific Method".

I assume that once this is resolved that we will finally all have flying cars.

No, they all know it's true anyway.

The difference between the abc conjecture and Cantor's theory is that the latter isn't about what you might call a concrete "fact". It isn't just about whether a supposed proof has logical flaws. In the case of the abc supposed proof, if it has no flaws then we know there are a finite rather than unending, number of certain entities (if I understand it correctly). If Cantor's argument is flawless, I don't believe that it implies something quite as specific about real world objects.

I have no idea what this sentence was supposed to say.

I think it's too easy to gloss over history the way you've done it here. Cantor's arguments were a genuine mathematical dispute and a true controversy.

https://en.wikipedia.org/wiki/Contro...tor%27s_theory

At the time, mathematical reasoning was very "finite" in nature, with the concept of the "infinite" being some abstracted thing that maybe wasn't real. Cantor's arguments (eventually) ushered in a very different concept of the infinite as a tangible object that can be worked with in particular ways. While some mathematicians shrugged it off as an interesting side bar, others were quite adamant about how non-mathematical the ideas were. And the ideas were not initially well-received by many in the mathematical community. I think it's a very reasonable parallel for the present situation.

The axiom of choice was another one. It was thinking about that sort of thing that lead to the Banach-Tarski paradox (which was already mentioned). I think it can be possible to underplay how much debate/dispute there was in that. My understanding is that some mathematicians were convinced that the axiom of choice must be false because the logical consequences of it were too bizarre for it to be true. And it took 60-70 years before it got "resolved" (shown to be logically independent of the other axioms).

I'm pretty sure the tv remote I'm holding in my hand is a "real world object". I'm not so sure about prime numbers. I think I'd have trouble explaining to an 8 year old how they're the same kind of thing.


PairTheBoard

I think you're being generous here. They understand it and it's just flat out wrong. I mean from Scholze/Stix which is of course far more readable than the actual proof, Mochizuki is claiming "special" properties for the fundamental group of a hyperbolic curve over the P-adics that you don't see in the automorphisms of the curve itself. If that's somehow possible (even though it seems like it can't be) just give an example. I don't think one has so nobody is convinced. This doesn't seem to touch on any of the remote IUTT stuff that follows that maybe even Scholze/everyone not in the Mochizuki camp doesn't understand, so it's hard to see how the proof could possibly work.

Right. To make it clear, we can define a Turing machine that lists all the exceptions to the ABC conjecture. That machine either halts or it doesn't. We can disagree all we want about what axioms have the standard integers as a model and why we should or should not believe they are consistent. But none of that matters because the ABC machine either halts or runs forever and that's all we're talking about. Mochizuki claims he has a proof the machine halts, other people say he does not.

The arguments over Cantor were different. They were arguing about certain constructions/definitions and whether these are valid/consistent if we want to talk about things like the standard integers. Here we can't avoid talking about axioms/consistency and boring set theory stuff because that's what the whole discussion is about.

For a situation closer to Mochizuki, the best I can come up with off the top of my head is maybe Galois theory itself? I think there was a 50 year period after it's discovery where everybody was fumbling around with it or straight up ignoring it because people didn't understand it. This was largely because Galois never wrote up his findings properly, then died at 20 so wasn't around to help guide everyone. Of course Galois theory provided a valid algorithm for say deciding when a 5th order polynomial is solvable in the radicals while Mochizuki probably doesn't have a valid program for proving ABC.

I don't think that's right. ABC says that for any epsilon > 0 there are finitely many relatively prime a,b,c with a+b=c such that

(primes product)^[1 + epsilon] < c

For a Turning machine to check this for a given epsilon it would have to keep running forever to make sure there's not another a,b,c pair out there that's another exception. For the Turning machine to halt it would have to somehow know it had already found all the exceptions for that epsilon. For each epsilon, a Turing machine checking the conjecture has to run forever whether the conjecture is true or not. Then you need infinitely many Turing machines running at the same time for the infinitely many epsilons. Or one Turing machine checking infinitely many epsilons each time it expands its search out to another integer, which it has to do infinitely many times.


PairTheBoard

Cantor's result is not as "concrete" as what the abc conjecture states, but it is fundamental to mathematics itself. Like I don't think Measure theory would be possible without it, and the diagonalization argument lead the path to the realisation that there are uncomputable functions.

I'm not saying the machine blindly searches for counter examples. Of course that machine runs forever pretty much regardless of the conjecture. I just meant platonically, a machine that simply lists out all the counter examples either stops or it doesn't ie the abc conjecture is either true or it's false. If we are talking about proving whether this machine halts or not, there is a remote possibility that again we have to start talking about axioms and boring set theory stuff. But that doesn't impact whether the machine halts or not. It only impacts our ability to prove what the machine will do. This is not really the same as with Cantor because they were only arguing about set theory, foundations and proofs.

I'm not so sure about saying, "the abc conjecture is either true or it's false". And I'm not so sure about supporting that statement by appealing to a Turing machine doing an imaginary possibly impossible thing. But that's a derail to this thread and I may be a minority of one with my instinct about it.


PairTheBoard

You can have doubts. But Scholze and Mochizuki don't. And they are the ones arguing.

It's slightly interesting, though. A Turing machine cannot prove Goldbach, but would at least find a counter-example if it were false. As you observe, that does not appear to be the case with ABC.

A Turing machine could prove Golbach. Proof verification given a set of axioms is itself Turing computable. Another way (perhaps remote) is having a constructive algorithm that simply spits out for any even integer n, 2 prime numbers x and y such that x+y=n.

Not if it's impossible to prove.