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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

#171

Post by Davidzheng »

QuesnayJr wrote: post #164
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…

I think in the long run mathematicians are probably fucked, but in the short run it's not that bad. 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 chance they would have solved it. (This example, you could have given an undergraduate good at programming and computer algebra and told them to come…

> 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 chance they would have solved it.

You cannot be serious... why didn't they solve it before then? Do you think no one tried it? What background do you give the double cycle conjecture student after the flow reduction? a linear algebra textbook???

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#172

Post by cgio »

aizk wrote: post #112

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.

[flagged]

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#173

Post by [deleted] »

k1t wrote: post #150

Since I actually don't know math, maybe my ELI5 understanding can be helpful (or corrected). The conjecture says that you can always reverse (a process) to determine the original inputs. But this proof shows multiple inputs creating the same output - which obviously cannot be reversed to determine the input - thus falsifying the conjecture.

[deleted]

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#174

Post by oliculipolicula »

krackers wrote: post #155
oliculipolicula wrote: post #117

Earlier quoted context omitted.

Wasn't it proven true in general for n=2 (NGL I wanted to suggest someone to go for the JC using 5.6 after the CDC proof came out, but then on reflection felt I should neither waste people's time NOR contribute to the myth of AI :) My prediction is that the bubble will burst in 2031 Q4, one year after the Riemann Hypothesis is expected to fall (according to Demis) After 2031, I will suggest going for the JC for N=4 b…

>Wasn't it proven true in general for n=2 Assuming you mean C^2 -> C^2, Do you have a link? If so it would be good to add to the wikipedia page. Also I'm not sure, but does the fact that there's a disproof for n=3 imply that it's false in all n>=3, or could there be higher dimensions where it still holds (I'd guess not since you could probably trivially "embed" this in higher dimensions in some way)

Paper linked above?

Edit my bad. I was more familiar with Dixmier and doing the translation I assumed A1 -> n=2. A2 is open ofc. Maybe to fall in 2031

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#175

Post by simianwords »

cyphar wrote: post #28
aurareturn wrote: post #12

Earlier quoted context omitted.

The goal posts have moved. People generally stopped saying this stuff now. Even if you go to the ultimate anti-AI subreddit r/betteroffline, they've changed from "AI is useless" to "AI is good but the AI bubble will collapse soon" over the last 6 months.

I don't think the (fairly factual) description of these systems as stochastic parrots means that they will never do useful work, just that they are not intelligent in the way we believe animals to be (to "push back" on your anecdata, I've also heard fewer people claiming that LLMs are actually conscious in the past year -- maybe we're reaching the happy medium?). That was the point the stochastic parrots paper and Ch…

Classic case of Motte and Bailey applied.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#176

Post by koito17 »

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.

To think that Claude Fable was able to find a counterexample in degree 7 is insane to me. We are truly in a new era.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#177

Post by js8 »

krackers wrote: post #159
js8 wrote: post #158

I find it interesting that the counterexample uses C as a field. C is twisted and weird. Maybe the Jacobian Conjecture still holds for reals?

>C is twisted and weird. Why do you say this? I've admittedly never done a proper complex analysis course but I got the impression that that complex differentiability was a very strong condition that results in holomprhic functions behaving "nicely" in ways that real functions do not

Should have used quotes.. I didn't mean it in any formal sense. What I am saying the nature of unit in complex plane makes it difficult to intuitively imagine invertibility and determinants.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#178

Post by QuesnayJr »

k1t wrote: post #150

Since I actually don't know math, maybe my ELI5 understanding can be helpful (or corrected). The conjecture says that you can always reverse (a process) to determine the original inputs. But this proof shows multiple inputs creating the same output - which obviously cannot be reversed to determine the input - thus falsifying the conjecture.

It was known to be false in general. But there are many questions that are false in general, but true when restricted to polynomials.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#179

Post by slopinthebag »

Lerc wrote: post #146
meowface wrote: post #93

Earlier quoted context omitted.

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…

> "Anti-AI psychosis" will probably grow to outcompete AI psychosis by year's end. Everyone has their own experiences making it hard to judge what attitudes are prevalent. From what I have encountered the Anti-AI people have been more common and more outright scary. I have heard some ideas from the Pro AI crowd and thought to myself "Yeah, that's not going to happen anytime soon", There was enthusiasm leading to over…

I initially wanted to downvote you but I can’t because you’re right.

I’d probably be called an anti-ai luddite on this site despite using it all the time for programming. I naturally gravitate to anti-ai viewpoints, but only because the breathless hype of the AI boosters is so nauseating. I have no moral qualms about AI (although I’d like nothing more than Anthropic/OAI/Google dismantled), but I really hate the idea that AI can replace artists, or humans as a whole. I think it’s anti-human to its core.

But AI is obviously useful and we could harness it for good. We shouldn’t give up on it entirely. And the people who are militantly against all usage are scarier than the boosters for sure.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#180

Post by left-struck »

aurareturn wrote: post #12
scarmig wrote: post #11

"Any idiot could have done this, it's just high school calculus and just a counterexample anyway. Stochastic parrot, spicy autocomplete, AI psychosis. Wake me up when an AI does something real."

The goal posts have moved. People generally stopped saying this stuff now. Even if you go to the ultimate anti-AI subreddit r/betteroffline, they've changed from "AI is useless" to "AI is good but the AI bubble will collapse soon" over the last 6 months.

I think we’re not adapted well for this rapidly changing world. Here you have some people who were rightly skeptical about a new technology being shoved down their throats by giant tech corporations, and a technology that really was, and probably still is overhyped.

Yet it’s a technology which has rapidly grown in its capabilities.

So yeah now many of the people who thought it was useless before probably don’t think it’s useless anymore, but you’re holding them to their original words even though those words were about something completely different at this point.

If people aren’t saying it anymore it might be because they don’t think that anymore, and the people who have new goal posts might be entirely different people.

It’s like you’re looking at a different set of goal posts on a different field and saying, no! The goalposts have moved!

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