jboggan

They absolutely have been the main force behind the timing of this push. It was the main topic of General Conference a few weeks ago and one of the most eyebrow-raising articulations of official church policy.

Given the way that the church has explicitly encouraged state legislators to shoot down recent laws it's not surprising to think they'd be pushing for this.

Aha, thank you, this makes more sense now.

I started typing out some specifics and realized it was honestly too weird and lascivious to describe in an HN comment section, shoot me an email and I'll send you the blog post about it. 2002 was wild in Budapest.

I'm probably the 1,000,000th ranked contemporary mathematician and I matter a great deal to the human species.

I would be more excited if the proof doesn't hold up because a) it would be the best and most complicated hallucination to date b) I could still solve the problem myself and c) I still learned some weird new counting methods.

Ah, cool. I'm still learning Lean - is there somewhere else in the repo with the Lean specification of the cycle construction for the full argument?

Thank you!

Igenis!

I would love to know the true unsubsidized cost of all of this. How many grad student-years did this cost?

Indeed!

I've been mentoring (or so I'd like to think) a very bright undergraduate mathematician, in fact he was the one who pointed out the final irreducible flaw in my proof last summer. And I am extremely curious to see what he does and if he even finishes his degree in mathematics. He had already expressed to me some dismay that his summer undergrad research program with several Ivy-league math majors got blown out of the water by a few hours of a frontier model. It's making everyone question what the future will look like and what education and training and certification will even look like.

But the future belongs to those who show up. Maybe this is the beginning of a mass democratization of scientific and math research, maybe we are going back to the gentleman-scholar model of amateur researchers and Twitter will be the new Journal of the Royal Society.

Thanks. I think AI is going to be a net benefit for people like me who have a surplus of ideas and too few hours to explore them. I may actually restart my graduate thesis research using AI, I did a survey of what has happened in the field since I left and about half of what I was working on back then has since been discovered and published by others, but there are some really interesting threads to pursue now that modern datasets are so much richer (this was computational biology research).

You may achieve far more than you plan on and it may come years and years after you think it should happen. You probably haven't met the right problem yet. You will.

Link? I need to show up and claim my reddit gold.

I think this is going to come down to personal philosophy and religion. And having a strong grounding in history to help us all through whatever changes we are rapidly living through.

I think the next test will be asking an AI to come up with a new branch of mathematics - just letting it rip and telling it to construct a system that doesn't reduce to combinatorics, group theory, graph theory, analysis, etc. Just get wild with it and don't start with any known problem as a jumping off point.

I think something like the Collatz conjecture will be solvable not as number theory or ergodic theory but some other completely wacky environment that humans haven't even sniffed at.

Right. And I'm kicking myself for not having the mathematical breadth to know about them.

Oh man, we should talk. I have been working on a patent with ChatGPT specifically to get around two complementary patents that are now together because of a corporate merger this year. I am not sure how much longer anything is going to be patentable with this kind of design assistance available to everyone.

Also, once upon a time I wanted to be a patent lawyer. It's incredibly hard to sit for the patent bar if you have a pure math degree and don't have an engineering degree. Thankfully New Hampshire lets anyone sit for the FE exam.

Look, I quit Google a decade ago and tried to make a ChatGPT-lite LLM in my living room (turns out 2017 and GTX1080ti era was a shade too early). I knew that this technology was eventually going to revolutionize programming and mathematics and everything else. I am still flummoxed on a daily basis watching it transpire.

But also I am excited to be living through this new era of programming and new era of mathematics. I'm still saddened that I couldn't be the one to solve this old problem, but now I realize that my personal approaches were really solving a level of this problem even stronger than the original conjecture, and I'm energized to tackle those (in my free time between being a solo founder and father of 3, etc.).

It's a bit aggravating that I cannot interrogate the session that yielded this result and ask it why and where it got the crucial calculation from, or why it went in that direction. It doesn't even rightly know even if it gives you a legible answer, that doesn't have any correlation with whatever happened under the hood.

Humans are the same way sometimes but I guess there's romance in that. If a human had solved it a la Kekulé and said "it came to me in a dream" I would at least understand that.

It's a grand experiment isn't it? Us senior programmers are pretty good at using AI (or so we think) because we have decades of grinding and problem solving to inform our intuitions. Is that really necessary? The next generation of programmers certainly will not have that level of desk-head interface. Maybe they'll be fine? Maybe the models will get so good it won't matter? Open question.

I imagine the same will be true of AI, but I'll say that in the short term AI is going to make mathematicians better because it solves the breadth problem. Again, I feel like this Barnette conjecture got solved (if it is solved) because of some clever partition function sums which are intellectually tractable but simply too far out of anything I'd seen before (I see the apparition of my GT combinatorics professor intoning gravely that "everyone knows that, Jake, you're an idiot"). Maybe AI will help identify common threads far greater than Google and journal search.

I think if I had ChatGPT when I was 20 and working on this problem for the first time I might not have solved it, but I would have learned every angle and facet of it far more quickly. But then again I would not have spent so many late nights staring at the Országház across the Danube and letting my mind drift and bump against the problem like spilled cargo in the river.

I was given this problem by Ervin Györi at the Alfréd Rényi Institute of Mathematics. I wasn't really at any school, it's a long and very bizarre story I should tell at length about being an illegal immigrant, getting kicked out of a graduate math program as a 20-year-old, and winning a grey-market apartment with my knowledge of Petöfi's poetry.

I did as I wrote it. I actually used that phrase often before it became an LLM-ism, just like how I rather enjoyed peppering my writing with em-dashes. Oh well.

I never lost a single hour thinking about this problem. Those were all hours that I gained.

I think this just has to be the problem statement, there's several lemmas I would expect to see in there. Granted I know very little about Lean but it seems like the question and not the proof outlined in the paper.

From my current understanding (and a lot of theoretical physics I'm having to Google because the sentences I'm reading from Fable's analysis are so bizarre I think they are hallucinations) there are possibly 3 neat symbolic tricks borrowed from theoretical physics that make the heart of this proof. Forgive me for posting LLM output but I find this darkly hilarious:

"it's a matrix-tree cancellation wearing Kasteleyn's planar signs, run as a Witten index over Penrose-lineage states, evaluated as a fugacity-zero loop gas in an infinitesimal magnetic field — and the reason it reads like physics is that every one of those tools was built for partition functions"

I thought this was pure slop when I read it but there are some clear analogues in these other areas of physics, really neat computational tricks, and a very interesting paper by Penrose calculating Tait colorings I never knew about previously (extremely relevant, actually related to a separate approach I had once taken on this problem). The problem is that the paper isn't saying "aha, we were inspired by the related problems of pairing excited states and creating spanning trees out of cancelled coefficients" it just defines the function apropos of nothing. Which is kind of like the Jacobian counterexample in that it works but doesn't really explain how exactly it got there.

I really think the load-bearing concept here is "prior work". If prior work is considered papers on this problem or graph theory, yes this has one huge subjectively unintuitive logical leap. If "prior work" is the entire corpus of neat computational tricks that physicists derived to make their equations spit out something other than zero or infinity, maybe it's not so crazy?

I went back to that Fable chat and showed it this new preprint. It coded up the new constructive algorithm and ran it against the existing test suite, that looks good at least.

It has been super helpful in delineating where the crucial concept came from. The proof is rather simple as graph theory proofs go, but it does seem to use some constructions that would only seem obvious if you had serious physics experience with partition function and calculating energy states that cancel out. It's not a wholly alien bolt from the heavens, but I can also see how there hasn't been a human being with the broad theoretical physics knowledge combined with the deep graph theory experience in planar graphs to come up with this idea. I don't know, I'm looking for precedents of this formulation and some old papers of Penrose counting the number of edge colorings of this same graph type are coming up, the line of argument at least rhymes.

But I agree with the thought that this sort of progress should not be siloed inside those companies. I propose a tax so that every slop cannon AI video pays for another hour of compute time for advancing mathematics.

I think you and I have fundamental disagreements about identity and consciousness.

Lol it should be, but it doesn't seem complete. Line 49 just says "sorry"

/-- Cubic bipartite three-vertex-connected plane graphs have a Hamiltonian cycle. -/ def MainStatement : Prop := ∀ (V : Type u) [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj], G.IsRegularOfDegree 3 → G.IsBipartite → Planar G → ThreeVertexConnected G → HasHamiltonianCycle G

theorem main : MainStatement.{u} := by sorry

Which is one reason I never really did. I probably should have but I always thought my attempts were too amateurish. Though I did manage to replicate some partial result papers that I didn't know about, lol. Writing openly would have saved me some years.

Well I'm sure some people (maybe me if I had time) will do a write-up of this proof. It treads familiar ground for most of the setup, it's mostly the disk lemma and cancellation calculations that need to be understood, it's a fairly short paper and quite tractable.

I think it helps that basically everyone thinks this conjecture is true, it's just been so darn weird to attack. There's this odd thing that the induction proofs of this problem kept running into, which is that the N+1 condition would work except for in one tiny case when it could fail, but it would be covered by a very slightly stronger version of the conjecture. But then that would fail on one tiny case in induction, but you could solve that with another slightly stronger version. Etc., etc. I almost wondered if there were some sort of structure to the increasingly strong conditions and wanted to prove something about the meta-induction between the stronger conditions and the N's that they needed the next level to remain true. But that failed after 5 steps I think (Fable actually helped me write a few hundred test cases to explicitly show that pattern didn't continue forever, thank God).

BTW my existing test suite from previous proof attempts jives with this new algorithm, so I haven't seen any evidence yet that it's incorrect. Waiting for a Lean proof obviously.

Ironic, as I remember staying late at the Google office to watch that match live. I didn't really understand anything going on but I knew enough to be excited. What a decade.

Shoot me an email, in my bio.