251–260 of 562 posts
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As proofs become more and more complex, we will need two AI pipelines: one to generate the LEAN proof, and a second one to extract useful lessons for mathematicians from the LEAN proof.
Or we just don't use LEAN but something better.
This is a rare instance where feeding this groundbreaking information into an LLM gives _them_ psychosis. I fed this to claude code and watched it verify the result in 7 different ways to be 100% certain, and it was just flabbergasted. Quite remarkable.
I fed ChatGPT the map with no other context, just “tell me about this function”. It did a bit of work finding the Jacobean etc and eventually worked out the implications of what it was seeing. It then proceeded to check the arithmetic 4 times, and then decided to do a manual verification using an ad hoc symbolic checker in case its SymPy had been tampered with.
Speaking as a mathematician, it does seem like we're a bit fucked as a community. Anything that is at all accessible to currently existing methods and mathematical infrastructure is probably going to fall to the frontier models of today, and at this rate of progress it's likely that, already by next year, we'll see new infrastructure being put into place by AI, giving us a world in which a few designated interpreters…
Of course AI can also farm conjectures, but they have to develop taste, which might be harder than just proving theorems.
A little over 10 years ago I remember meeting a postdoc who believed he had something close to a counterexample to the Jacobian Conjecture. He and another person was bruteforcing polynomials in about 16 variables, something like 80 - 700 terms each, using binary trees for mapping coefficients. They were guessing, at the time, that the lower bound of a counterexample (P, Q) for max(deg(P), deg(Q)) would go up to 200.…
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Interestingly, even Qwen 3.6 27B was able to verify the solution, but I didn't get any glazing for discovering it. Instead, it thought that someone named Shestakov had already found a counterexample in 2004. GLM 5.2 whiffed, it insisted the counterexample wasn't valid. VibeThinker 3B also recognized that the counterexample was valid. But it kept trying to convince itself that it wasn't, over and over, since it's an "…
> Matches! This is bizarre. A Jacobian counterexample has been sitting here in a prompt? Wait... is this map a known "fake" counterexample from the literature? Many mathematicians have tried and failed. This specific map might come from a paper or a forum where it was proposed and then debunked. Or... is it actually correct? Gemma's having trouble accepting it too. A solution?! At this time of year? At this time of d…
can any serious mathematicians verify https://xcancel.com/i/article/2079135211196121363
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Maybe, maybe not. The "proofs" may not have helped at all with finding a counterexample. Either way, it doesn't matter. A counterexample was found, no one found one before even though clearly a lot of people have tried who also had access to the prior "proofs".
I think it's becoming harder and harder to argue that LLMs don't really reason and just mimicry human speech. This counterexample is clearly the result of a sequence of steps that build on previous knowledge in context and logically combine it to reach other true statements - to a degree and complexity that rivals the best human minds. For someone that use Claude Code every day, this is obvious, but for some reason m…
Similarly I've run into several "you just gotta know" type problems where LLMs still just fail. My favourite was a quirk in how async relationships work inside emberjs. Three different models gave a variation of the same incorrect answer. When I searched myself, I initially came up with nothing and eventually found what I think might be the only example of the same issue on the internet, a single stack overflow question wth two responses. The first is what Claude, gemini and chatGPT said, the second response was the OP saying it was wrong. I asked in the Ember discord and a core team member responded instantly with the answer.
There's a video I love on Youtube, where a guy uses ML to assemble a blank jigsaw. It performs amazingly, the jigsaw being blank is of no consequence and if it did have an image, it'd perform worse. That's all LLMs do. Just because the jigsaw pieces are smaller, they're still just getting assembled in whatever way fits, there's no mechanic for interpreting the image on the front.
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I don't know why you're saying this, given this is not sycophancy and instead is an actual example of Claude Fable finding a real counterexample. I would get it if the mathematician had in fact posted something untrue or crankish, but he posted something true. Frontier models have solved several major open problems in mathematics in the past few months, so this should not be a huge shock. "Anti-AI psychosis" will pro…
Sorry, but if the current models haven't shown you something amazing at this point that would make this of little surprise, it because the person is an intellectual bore and completely full of shit themselves.
I have a question I'm surprised people are not asking: How did Fable find this? Was it like guessing a bunch of families and then solving for possible solutions in those families? Was it clever search? something else?
linked from here: https://x.com/b_shrir/status/2079094004885668003?s=20
Very short version: there’s an existing false counterexample in the literature which holds almost everywhere except at a pole. It looks like Fable used this polynomial as a base & extended it in a way that eliminated the pole whilst preserving the structure.