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.
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.
It makes solving advanced math problems feel like cracking a hash. If it's possible, it's just a matter of compute time.
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.
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.).
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.
I'm fairly sure your understanding is not fully accurate.
And then they are for sure able to fill their context based on 'smart search on top' to actually progress further.
But obviously, it adds up to something greater than went in; in aggregate, our contributions are something to awe.
But my point is, if you zoom in at the marginal, incremental contributions of any individual human in this process, it's really hard for me to say LLMs are not at the same level already.
On this topic, people like to compare LLMs to Einstein, but as far as I know, Einstein did not zero-shot special relativity in an afternoon. He built it up incrementally over time, it took him three times longer than the time between first ChatGPT release and today, and it depended on centuries of prior art, culminating in the right observation and right notation being available to him in his moment of greatness.
At what level LLMs are is then an entirely separate discussion, I think.
Name three.
So your view is that everything was there at the creation of the universe (it's a possible view, of course)? Or are there any "things" that can create ideas from scratch?
I think it's not impossible that words evolved as adaptations of the environmental sounds with which our ancestors lived. The human creativity producing DNA is also a remix of preexisting molecules formed under evolutionary pressure, so the view that it's turtles all the way down, unintuitive as it is, may not be so indefensible after all.
They combine things, verify it and if it works and progresses the problem, they created something new.
Loaded question. A "brand-new insight" is still built off the work of others. A possibly better way to frame it would be in how many subjectively unintuitive logical leaps have been made from prior work.
"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?
Makes me wonder how the patent space will be disrupted when that inventiveness step becomes obsolete because of LLMs. Given your example above, it seems like a combination of different methods from many different sources. This would be regarded as inventive, clearly. If eligible patents can now be brute-forced, the bottleneck becomes only selecting the most promising ones and paying for the patent.
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.
Are there no loads left to be borne?