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Claude Fable produced a counterexample to the Jacobian Conjecture

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Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#381

Post by ev0lv »

tristanj wrote: post #354

The poster works at Anthropic, so they likely have internal access to the next generation of Fable. Their internal model is probably an absolute beast at mathematics, and the upcoming benchmark results will likely set a new record for maths performance. I suspect this is what happened, because the poster is coy about sharing the actual prompt / reasoning trace used to reach this result. That would be covered by an ND…

I'm not very excited. Access to the best AI is not a party I was invited to. And the people who are at that party, well, they don't exactly reflect on my best interests.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#382

Post by muldvarp »

tacomonstrous wrote: post #138

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…

Anybody that has to work for a living is fucked and not on the "can't do mathematics which they would find fulfilling"-level but on the "can't afford food, because human intelligence is simply not required anymore"-level.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#384

Post by muldvarp »

pfdietz wrote: post #253
tacomonstrous wrote: post #138

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…

Proving theorems will have lower payoff, but posing new questions (for AI to chew on) will have higher payoff. Math will go from theorem proving to conjecture farming/exploration. In a way this could be even more fun. Of course AI can also farm conjectures, but they have to develop taste, which might be harder than just proving theorems.

> Of course AI can also farm conjectures, but they have to develop taste, which might be harder than just proving theorems.

Do you have any argument why you might think this would be true?

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#385

Post by dist-epoch »

Lockal wrote: post #365
dist-epoch wrote: post #334

Earlier quoted context omitted.

> all previous chats are loading fine, but the only "Counterexample to the Jacobian Conjecture" chat is not loading. This happened multiple times to me with Gemini. For the most trivial of requests, like translating a video into English. > "ground-breaking results that could be attributed to Gemini, if act quick" This would be such a dumb thing to do, and so easy to get caught with...

That is not dumb - erasing copyrights is just their business, and even when caught has zero consequences (for them). Providers can use any users input for improving their models, either by consent, or by flagging any dialogue for safety review (nonconsensually), or by training on whatever content they want anyways (obtained via torrents from pirate sites with U.S. court approval). On top of that, I retried the same q…

Why stop at Gemini then. They could also scan everything passing through Gmail for emails with math/science breakthroughs and then claim Gemini found them.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#386

Post by piombisallow »

hliyan wrote: post #17

I suspect the LLM was able to synthesize a counterexample because of the availability of a lot of prior work: > The Jacobian conjecture is notorious for the large number of published and unpublished proofs that turned out to contain subtle errors. https://en.wikipedia.org/wiki/Jacobian_conjecture#cite_note-...

I mean yeah, maybe, but this is how everything works

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#387

Post by QuesnayJr »

meowface wrote: post #367
QuesnayJr wrote: post #347

Earlier quoted context omitted.

No. I would say that it was easy, and that something had gone terribly wrong with math research that we missed it.

This would seem to lead to absurd implications. What if in, say, 20 years, an AI is able to independently prove nearly everything important in under an hour, including the Riemann hypothesis, with no contamination from other proofs? (Let's say it's also free to write and run arbitrary code, as well.) Whether it's 5, 10, 20, 50 years, obviously the takeaway cannot be "something had gone terribly wrong". The takeaway w…

That's not at all the scenario you proposed. You proposed a simple proof that Fermat himself would understand. We have developed a tremendous amount of math since Fermat, and if none of it was relevant that would be damning. If there's a simple proof of the Riemann hypothesis that Riemann would understand, then I would say the same thing.

That doesn't rule out an AI that makes a genuine breakthrough. If there's some new branch of math that no human has even imagined that answers the Riemann hypothesis, then that is exactly how I would expect it to go.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#388

Post by matrix87 »

SubiculumCode wrote: post #113

For all thehubbub, as far as I know, all the math breakthroughs via AI that I've heard about have come from Anthropic and OpenAI, not Chinese models. I could have missed those announcements, but one might think that between close to frontier performance plus cheap tokens, that they'd be leading the way on these things.

> as far as I know, all the math breakthroughs via AI that I've heard about have come from Anthropic and OpenAI, not Chinese models

Maybe because dishonesty is more normalized in SV business culture

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#389

Post by pfdietz »

muldvarp wrote: post #384
pfdietz wrote: post #253

Earlier quoted context omitted.

Proving theorems will have lower payoff, but posing new questions (for AI to chew on) will have higher payoff. Math will go from theorem proving to conjecture farming/exploration. In a way this could be even more fun. Of course AI can also farm conjectures, but they have to develop taste, which might be harder than just proving theorems.

> Of course AI can also farm conjectures, but they have to develop taste, which might be harder than just proving theorems. Do you have any argument why you might think this would be true?

For theorem proving, once the statement is formalized, there's an oracle for correctness of the proof. For deciding if something is interesting, well, de gustibus non est disputandum, you know?

Experience with Lenat's AM decades ago had it go off making all sorts of uninteresting hypotheses. That's very weak evidence, of course.

This suggests people also have role for fundung "beautiful" or "the best" proofs, since that also involves taste. More generally, perhaps the role of people is to reveal their preferences, and that requires people be in the loop somehow. Maybe "math criticism" becomes the job. And if AI is to serve people in general, it needs to know these preferences.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#390

Post by pfdietz »

muldvarp wrote: post #379
pfdietz wrote: post #250

Earlier quoted context omitted.

The Erdos problem solutions have been accelerating, including a $250 problem and a $100 problem. It's been fun watching the cope collapse from day to day. No one told me a slow takeoff Singularity would have so much schadenfreude.

> It's been fun watching the cope collapse from day to day. No one told me a slow takeoff Singularity would have so much schadenfreude. I'm constantly surprised at how much schadenfreude there is on hacker news about LLMs. Like, do you guys (and girls) not have to work for a living?

I don't. I'm retired.

I'm also not going to live all that much longer, most likely, so it's kind of annoying I'm not really going to see any upsides to an AI world either. I'm discounting the possibility of our AI overlords figuring out a miracle like reversing aging.

But it is really f-ing cool that automated math is now a thing and we are seeing it. Eat your heart out, past me.

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