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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

#61

Post by tptacek »

rowbin wrote: post #55
rvz wrote: post #48

Earlier quoted context omitted.

I'd rather wait for independent seasoned mathematicians to verify such claims first before someone at said AI lab posting a claim about solving a proof online. Let this be a lesson to those who fell for such AI psychosis and to not believe everything you see on the internet as real.

Maybe have a seasoned mathematician check if the counterexample is really bogus before saying people fell for ai psychosis.

I'm pretty sure you can just check this for yourself. The Sage POC is like 8 lines.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#63

Post by urams »

rvz wrote: post #48
kmaitreys wrote: post #42

Maybe not? https://en.wikipedia.org/w/index.php?title=Jacobian_conjectu...

I'd rather wait for independent seasoned mathematicians to verify such claims first before someone at said AI lab posting a claim about solving a proof online. Let this be a lesson to those who fell for such AI psychosis and to not believe everything you see on the internet as real.

It's a Princeton math PhD who posted. The verification is quite straightforward and was posted by the tweet author. Wolfram would have to also be producing incorrect outputs for the counterexample to be false. The counterexample works as claimed and conjecture has been proven wrong.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#64

Post by ignoramous »

rowbin wrote: post #24
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 The Jacobian conjecture is notorious for the large number of published and unpublished proofs that turned out to contain subtle errors.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-...

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".

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.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#65

Post by sciyoshi »

rvz wrote: post #48
kmaitreys wrote: post #42

Maybe not? https://en.wikipedia.org/w/index.php?title=Jacobian_conjectu...

I'd rather wait for independent seasoned mathematicians to verify such claims first before someone at said AI lab posting a claim about solving a proof online. Let this be a lesson to those who fell for such AI psychosis and to not believe everything you see on the internet as real.

Looks like this has already been formalized: https://github.com/deancureton/jacobian

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#66

Post by SyzygyRhythm »

tptacek wrote: post #52
SyzygyRhythm wrote: post #49

Earlier quoted context omitted.

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…

Nice! (It's gotta be a nonzero constant, right, a nonsingular matrix).

Yeah, the only possible counterexamples would have a constant but non-zero determinant (in this case it was -2).

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#68

Post by slopinthebag »

mathisfun123 wrote: post #44

The 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…

I would think it's more that some aspects of math (like anything really, including programming) don't require creativity and can be solved through "brute force" or whatever you want to call what LLMs do. But it's pretty obvious that LLMs are not capable of solving the vast majority of problems in math (or programming) at this time. Eg. Google went 9/353 on Erdos problems. If a more powerful LLM is capable of solving those or if they require a certain je ne sais quoi of the human variety is still up in the air at this point. In either case it seems like they require a long and detailed prompt from a domain expert (ie. human) regardless.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#69

Post by [deleted] »

tptacek wrote: post #20
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-...

How does that work in this case? What do those proofs do to help find this counterexample?

[deleted]

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#70

Post by gmadsen »

mathisfun123 wrote: post #58
ViscountPenguin wrote: post #51

Earlier quoted context omitted.

I dunno, this screams of goal post moving. Even if LLMs lack whatever nebulous definition of "creativity" that someone favours, there's no inherent reason for "creativity" to be required to solve any problems at all, "creativity" could just be a human method for solving problems that evolved because of it's broad applicability but is suboptimal at any given task.

> could just be a human method for solving problems that evolved because of it's broad applicability but is suboptimal at any given task. Ya sure let's just posit another random hypothesis about evolutionary biology in order to substantiate the claim that LLMs are intelligent. Or (bear with me) you can recall your (likely) experience proving stuff like SAS triangle identities and reflect on whether that required inte…

Why is your example SAS triangles and not something like the modularity theorem?
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