Comment by Syzygies | Hacker News Reader

Comment by Syzygies | Hacker News Reader

I'm a math professor who has lost years to this question. I've met various authors involved in successful proofs. My 1976 grad school application from the University of Illinois came postmarked "Four Colors Suffice" (Appel, Haken). I've also been inundated with well-intended crank proofs of every tar pit like this. One can tell the difference without focusing one's eyes.

These authors have long, established careers, with 2,000 MathSciNet citations between them. Waterloo is a hotbed of combinatorics; I doubt that they posted this before passing the paper around a bit.

Graph theory as a branch of mathematics is like photography as an art form: Anyone can point a camera, and one sees a lot of incredibly weak graph theory if one digs deep enough into the regional conference circuit.

This work is deep, relying on an understanding of generating functions that many casual practitioners don't possess. If the proof has a flaw, the issue will be technical and difficult to uncover. I love how people take the obvious bet that this is likely wrong because so many proof attempts have failed. In a prediction market, I'd bet on this proof being right.

I caught a rumour of this paper about 6 months ago, so I'm pretty sure that in the meantime it's been evaluated and assessed by colleagues. So I'm pretty sure it's going to be right, with only small fixable things that need tidying up.

Just as the incorrect proof by the amateur mathematician or crank is likely to be in a predictable form - a bunch of drawings of graphs with increasingly obtuse notation, reference to special cases within special cases, etc., isn't the incorrect proof by working combinatorialists likely to look exactly like this paper? An argument from complex analysis relating to either generating functions, Tutte polynomials, or some similar concept, together with some difficult arguments about how the asymptotics should be correctly used?

The fact that this includes arguments which casual practitioners would not be able to confidently use in a crank proof does not evidence its correctness. It just shows us that it's from a different type of person, susceptible to making a different type of error.

> In a prediction market, I'd bet on this proof being right.

Let's create a question on Metaculus!

How about Manifold? https://manifold.markets/ZviMowshowitz/will-the-new-nonconst...

well put.

Your post feels like a dispatch from bizarro-world.

There is a good amount of second-order evidence that the proof is wrong. Further, I skimmed the introduction, and it seems the indicated approach cannot possibly work. My guess is that both authors are senile. (They're quite old.)

If you give me decent odds, I'd be happy to bet against you regarding the proof's correctness. Would you take 1:1?

The tone of your comment is pretty harsh. What kind of 2nd-order evidence do you see? And why did you learn from the introduction that makes you so sure? It's only an introduction, which might only give you an oversimplified summary of their ideas.

Are both of them senile?

Yeah, my guess is that both are senile if they're putting this on the arxiv.

Second-order evidence: Old mathematicians (~80). Famous problem. Weird, imprecise writing style. No mention of the proof by mainstream mathematical news sources (breakthroughs are usually accompanied by excited blogging/tweeting). No acknowledgements directed at other mathematicians who have checked or commented on the proof.

[deleted argument here; replaced by more precise comment below]

My tone is harsh because I think the best thing to do is to quietly ignore it, similar to how the community treated Atiyah's claims of a RH proof at the end of his life.

> The argument looks like it's based on large n asymptotics, so even assuming everything works correctly the strongest statement they can hope to show is that the theorem is true for all n > n_0, where n_0 is some large constant. But there is no mention of this fact. The theorem is claimed for all n.

Have you seen this comment? https://news.ycombinator.com/item?id=34083099

Yes. How does it bear on what I wrote?

You're claiming:

> the argument looks like it's based on large n asymptotics, so even assuming everything works correctly the strongest statement they can hope to show is that the theorem is true for all n > n_0, where n_0 is some large constant. but there is no mention of this fact. the theorem is claimed for all n.

They're claiming:

> Theorem 7. If there is a map L which cannot be 4-coloured then only an exponentially small fraction of the maps with n edges can be 4-coloured.

These claims appear mutually exclusive.

See above clarification about the bad writing.

The point is that if there were a small graph that was a counter-example, then there would be large graph counter-examples (and the percentage of them would increase with graph size), so proving that the 4 color theorem is true for large graphs implies that it is true for all graph sizes.

Some counterpoints: 2 mathematicians greatly increases chances of senility. Blogs/twitter also correlate much stronger with author age than truth of statement.

>The argument looks like it's based on large n asymptotics, so even assuming everything works correctly the strongest statement they can hope to show is that the theorem is true for all n > n_0, where n_0 is some large constant. But there is no mention of this fact. The theorem is claimed for all n.

This is completely wrong. A proof for all n>n_0 is a proof for all n, since any counter-examples have to exist as subgraphs of arbitrarily large graphs.

I don't think asymptotic estimates of that form suffice to treat this problem. (Where else in combinatorics has an argument of this form succeeded? What intuitive reason is there to expect it to succeed here?)

Specifically I think section 4 is basically nonsense. (I see Sniffnoy has already pointed this out below.)

(Re: your comment, Theorem 7 is going to fail below the smallest counterexample, right? This is bad, imprecise writing - a red flag.)

> I think the best thing to do is to quietly ignore it

If you think that's the best thing to do then why not do it?

This is a bizarrely hostile attitude to have. Critique their work, not their age.

It's not bizarre at all. The math community has unfortunately been down this road many times before. When an 80-year-old announces a 7 page proof of a famous problem, the smart money is on the proof being wrong. As the comments elsewhere on this story indicate, this heuristic turned out to be correct. The only new twist is that we have two 80-year-olds this time, not one.

To be clear, I don't like this state of affairs. As suggested above, the best course of action seems to be to ignore the posting.

It is not appropriate for you to speculate publicly about the mental health of people you've never met.

It's so inappropriate that in the field that's actually trained for this, they're disallowed by compact, even with extensive evidence.

Please stop.

>It's so inappropriate that in the field that's actually trained for this, they're disallowed by compact, even with extensive evidence.

The field trained for this is not refraining from this because it's hard to get right, but because it undermines the privacy promise they give their clients! not an argument applicable to people who do not have that professional reputation to uphold.

Of course you should speculate about mental health of people when it's relevant to the topic - it's a factor heavily shaping many people's behaviour!

* No discussion of how these techniques were developed.

* No discussion of why they were not found by other people in the past.

* No discussion of how these techniques could be used on other problems.

* A lot of calculations which would be left out as trivial by most graph theory papers (for example, calculations about the edge counts of subgraphs)

Hey, could you please make your substantive points without swipes and personal attacks? If you know more than others, that's great, but then share some of what you know (without putdowns) so the rest of us can learn. If you don't want to do that, not posting is also a fine option.

I have no idea whether the proof is right or wrong, and I'm not so sentimental as to believe that the harsher person is more likely to be wrong. (Besides, naysayers are usually right, just like they are when predicting that any given startup will fail.) However, the HN guidelines (https://news.ycombinator.com/newsguidelines.html) ask you not to post like this and that's important too. Other users are making their case against the OP while following the guidelines just fine. Please be more like them.

I am going to finetune a content moderation model based only on the principle that it should, at all times, ask itself: “what would dang do?” and then do that

Hi Dang,

There's no personal attack here. I'm sorry if that comment comes across as curt, but the post I responded is (in my opinion) grossly misleading and deserves pushback. For example, consider the statement: "If the proof has a flaw, the issue will be technical and difficult to uncover," made by someone claiming years of experience as a math professor thinking about this question. This is easily identifiable as wrong, because the purported proof is just 7 pages (actually less than that due to extraneous material) and does not use any particularly technical mathematics (a strong undergraduate major probably knows enough to comprehend what's going on). Indeed, people found the error in (less than) 12 hours after that comment was made.

The comment about senility is not a "swipe." It is a very real problem in the mathematics community. Mathematicians get old and are sometimes afflicted by dementia, and this unfortunately can manifest as hopeless attempts at famous problems. Atiyah's "proof" of the Riemann Hypothesis right before his death is perhaps the most well known example, but there are others that (thankfully) aren't disseminated publicly. In the pre-internet age, journal editors were able to make such things quietly disappear. But now we have the arxiv. This is valuable context for understanding why two mathematicians with distinguished publication histories might be posting an incorrect proof, which had not been provided by other comments.

That being said, I will endeavor to phrase such comments more carefully in the future. Thank you for your message.

I believe that you're talking about something real, and when you provide the fuller description, it no longer feels like an attack.

Their LaTeX compiled, though.

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