Comment by jboggan
19 hours ago
I'm still digesting the proof and translating a bit from the dual case back to the primal in which I most commonly thought about it. I don't think it was a brute force proof in the sense that it combined every possible paper and commentary. It's rather odd because I feel like most of the work on the conjecture was focused on an induction proof based around graph reductions, and this proof avoided those issues entirely by offering a concrete constructive proof of finding a Hamiltonian cycle. Rather, it explicitly selected the edges not in the Hamiltonian cycle, which is in line with previous attempts via the dual.
The "aha" insight for this is actually f**ing wild, it involves a complex valued exponential sum on the edges. I've seen a lot of clever counting arguments before in graph theory but this is the first time I've seen complex roots and annihilating terms like this, the symbolic manipulation tricks in this look like things out of quantum physics. I don't understand where this trick originated, I need to really digest this.
You should try asking an LLM to look for previous papers using similar ideas. The current/frontier generation of math AI is unfortunately very bad at citing the relevant literature for techniques its using.
I asked GPT here: https://chatgpt.com/share/6ac5fd7d-0390-83ed-a02a-6d80fc64f6... and it says:
> the exact Barnette argument appears quite novel, but nearly every ingredient in its cancellation trick has a recognizable ancestor.
> The closest precedent is much closer than I expected: in fully packed O(n) loop models, people have been assigning complex phases to the two orientations of a loop and making them cancel for decades. At n=0, the phases are literally +I and -I. And the n->0 limit has specifically been used to extract Hamiltonian cycles/walks.
You can judge better than me. But it's definitely worth it having a research assistant AI with you when reading these papers.
So much about LLMs can be framed as Information Retrieval, Compression, and Search. Computers have always been good at ruthlessly hammering through a huge but finite set of possibilities. The wild thing now is that you can define that set of possibilities as "all the ideas ever published in mathematics journals."
It makes solving advanced math problems feel like cracking a hash. If it's possible, it's just a matter of compute time.
BTW, reading your last paragraph reminds me of how Lee Sedol felt after move 37.
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.
And we’re only a bit more than halfway through this current one. Exciting/terrifying.
I just revisited this to make that exact comment.
I'm sympathetic to the mathematicians who are worried about the future of their field, but as an outsider I wonder if they couldn't learn from the go community's "recovery" after the introduction of an alien intelligence.
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.).
Complex roots and annihilating terms -- is it something like the derivation of Fourier / Laplace transform?
WOW
Why would the trick have any "origins", isn't this model creating new techniques never before seen or imagined?
There is a chance that someone from a completely different field came up with a solution for a tiny part of your problem.
If you can remember the content of any scientific publication and any book in the world, you are able to make use of this knowledge in every step of you proof.
However, this does now answer how the model came up with the specific route it has taken for the proof.
LLMs don't have super memory like that. I mean I don't know what this internal OAI model is, but at least for other LLMs, they aren't databases of training data with a smart search on top.
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As I understand it it's undetermined yet whether LLMs can actually come up with anything novel or are instead pulling from their incredibly deep corpus of knowledge to present solutions that were there but we didn't realize it because our brains aren't libraries of almost all human writing.
Synthetic data allows them to train well past the limits of human writing.
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Only in the same sense it's not yet determined about humans, either.
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No this is not an issue. As long as their is a way of verifying things, they do the same thing with creating novel things as humans: Searching through an infinite space of possibilities opitmized by knowledge.
They combine things, verify it and if it works and progresses the problem, they created something new.
Let me introduce you to 'obscure Russian mathematicians'.