Maybe instead of proofs, we should encourage the publishing of prompts and reasoning traces?
Claude Fable produced a counterexample to the Jacobian Conjecture
191–200 of 562 posts
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#192Maybe instead of proofs, we should encourage the publishing of prompts and reasoning traces?
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#193Speaking 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…
next year is a long time away friend
It's so soon that it may be rational to delay certain math and software projects until smarter models arrive
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#194Earlier quoted context omitted.
> This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample please try go try it. There's no way someone didn't do massive computer algebra searches before today. > All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good cha…
Why would I try it to win an argument on HN? That's a bizarre suggestion. Just look at the degree. If it were degree 47 in 17 variables then it wouldn't be surprising, but here it's surprising. Of course people tried hard to solve them all, which is why it's so surprising that they were open. If anything, the solutions have gotten easier. The unit distance graph solution relied on a famous theorem remote from graph t…
You can use that retroactive logic about any hard problem though. Unsolved murder cases, math, theoretical physics.
If tons of smart humans try for years and fail and then an LLM tries for a few weeks or hours and succeeds, the implications are clear. And these are by far the dumbest LLMs will ever be.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#195Anybody can ELI5?
Are you at least a little familiar with linear algebra? If so, you've probably heard of the determinant. It's a certain way of "summarizing" a matrix with one value. The determinant in this case is of the Jacobian, which is a matrix you can construct from a multi-variable function. Each term is the partial derivative with respect to each variable (x, y, z, etc.), with one line per output variable (vector element). Th…
Thanks that's a very nice summary.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#196I find it interesting that the counterexample uses C as a field. C is twisted and weird. Maybe the Jacobian Conjecture still holds for reals?
All the coefficients and evaluation points are rational, so it's a counterexample in all fields where 2 ≠ 0 and 3 ≠ 0, doesn't matter whether that field is the complex numbers, real numbers, rational numbers or even a finite field.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#197I've come to understand that while an LLM is a parrot, it's a parrot that's smarter than I am.
Hot take: even GPT-3 was not a parrot. Skeptics have never properly internalized that the fundamental operation is basically irrelevant to the gestalt. Humans are not parrots yet neurons likely also largely operate via predictive processing.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#198The funniest thing about LLMs is the cognitive dissonance they cause people. People clearly recognize (and bemoan) the fact that LLMs produce derivative breathless prose ie they fundamentally fail at "unstructured creativity" (something the might accurately labeled intelligence ) but are then shocked that the same LLMs can do math. It's reasoning from a flawed premise that math universally requires intelligence and c…
Hilariously absurd cope.
They're still not very good at writing, but you've flipped the takeaway. The correct conclusion is writing style isn't very relevant to intelligence.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#199This 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.
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 day? In this part of the country? Localized entirely within my own prompt?
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#200Earlier quoted context omitted.
This makes me wonder, what if anyone uses Fable-class LLM and passes of its novel results as their own work? There's no shortage of folks doing that in software, right now.
Currently there seems to be very little anti-AI bias in maths, and more of a "huh, cool new toy, let's see what it can do" vibe.
Strongly depends on the subbubble of mathematics. In some parts of type theory / formal proofs for instance, there is a rather strong rejection of LLMs (for moral reasons in addition to quality reasons). The proof assistant Agda was even forked for this reason: https://types.pl/@amy/116522250630340534