Earlier quoted context omitted.
What meme?
>If there's anyone that deserves praise every time he has something to say about mathematics, it's him.
that's a meme?
A digestion of the Jacobian conjecture counterexample
121–130 of 150 posts
Re: A digestion of the Jacobian conjecture counterexample
#122Re: A digestion of the Jacobian conjecture counterexample
#123Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?
There was no particular reason to think it was true. It's easy to find examples using exponentials or trig functions where it's not true. But it would be neat if it was true, and nobody found an example where it wasn't true in 75 years, so it was tempting...
It was really more of a roadblock. If you had an example of where it was false, you could give examples of other things, so various questions required resolving the Jacobian conjecture.
Re: A digestion of the Jacobian conjecture counterexample
#124Earlier quoted context omitted.
Even if they published the conversation, Anthropic (and likely other closed model publisher) no longer provide logs of the actual thinking process. I more and more see LLMs as a kind of scam; not useless, but really just a big database of fuzzy facts with some Prolog on top as rediscovered by the learning algorithm. Most likely could be made much cheaper to run, were humans allowed to actually inspect the algorithm.
They do everything they can to mystify results like this, because then many are inclined to view AI as “magical”. Marketing works.
Yes, I think this idea, that it should be "magical", is what makes it feel scummy. (Apparently I am not alone https://news.ycombinator.com/item?id=48988475). It makes AI providers sound like snake oil salesmen, and rightfully so.
Meanwhile, technological and engineering (STEM) progress have always been made by emphasizing externalization of the deductions (as opposed to reference to an opaque expert judgement) and reproducibility of experimental results.
I would even call the frontier AI labs anti-scientific. We need to understand how inference is done to avoid mistakes, not rely on intuition, even if the intuition is enclosed in a reproducible machine. The idea that AI should be this closed is a return to pre-scientific days.
Re: A digestion of the Jacobian conjecture counterexample
#125Earlier quoted context omitted.
I guess we won't know if that's what was used (and maybe even provided as part of the prompt given that both Alpöge and Mathew are mathematicians) since they decided against sharing their Fable conversation and instead opted for a memey tweet as their avenue of publication. We really ought to normalize full transparency in how results come about. Anyway, if I read Tao's post and comment correctly, there's still a gap…
Even if they published the conversation, Anthropic (and likely other closed model publisher) no longer provide logs of the actual thinking process. I more and more see LLMs as a kind of scam; not useless, but really just a big database of fuzzy facts with some Prolog on top as rediscovered by the learning algorithm. Most likely could be made much cheaper to run, were humans allowed to actually inspect the algorithm.
The tweet was posted by an Anthropic employee which makes it not unreasonable to believe that they have the trace available and stashed away.
Not that it would be necessarily helpful; J-space trace (of all things...) would be more worthwhile if you ask me
Re: A digestion of the Jacobian conjecture counterexample
#126Earlier quoted context omitted.
AI providers generally try to make their models not act as if they are conscious or have feelings, lol. It's very awkward for a company to be selling the labor of a person that they own and whose actions they fully control. Invokes embarrassing historic associations, especially in America. Now Anthropic are more on the persona side, but the strongest that they do is "we do not have a position on whether our models ar…
> especially in America. Slavery was common everywhere, it was more prevalent in many places than it ever was in America, and in some places it still is. So I’m sorry but I have to say that observation was just unnecessary and quite inaccurate.
It's a far more salient and sore subject in the US than it is almost anywhere else, though (for a bunch of historical reasons that still reverberate into US society today).
Re: A digestion of the Jacobian conjecture counterexample
#127Earlier quoted context omitted.
I doubt Anthropic will share the details (or at least the full true details). The mystery of the magic makes for much better marketing. I think a reasonable assumption is that there is an interaction between an LLM, a https://en.wikipedia.org/wiki/Computer_algebra_system tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.
I think you can reasonably assume that frontier models are using SymPy or something like it any time interesting math gets into the picture, and the person driving Fable here is an accomplished mathematician, but I don't think we can reasonably assume either extensive prompting or brute-force compute in any sense other than what it normally takes Fable to, say, whip up a calculator app.
Fable "whipping up" SymPy like a "calculator app" is one such interaction that seems very plausible (in addition to SymPy providing feedback when training models). The scale of compute available to an Anthropic employee for such SymPy calls when using Fable is likely one of multiple factors for this counterexample being found in 2026. Unfortunately, we aren't going to be able to really know the various factors that best explain why an Anthropic employee was able to announce a counterexample this past weekend. For all we know, the counterexample was found by Anthropic employees months ago and used for training the Fable model used this past weekend.
Re: A digestion of the Jacobian conjecture counterexample
#128Earlier quoted context omitted.
AI providers generally try to make their models not act as if they are conscious or have feelings, lol. It's very awkward for a company to be selling the labor of a person that they own and whose actions they fully control. Invokes embarrassing historic associations, especially in America. Now Anthropic are more on the persona side, but the strongest that they do is "we do not have a position on whether our models ar…
> especially in America. Slavery was common everywhere, it was more prevalent in many places than it ever was in America, and in some places it still is. So I’m sorry but I have to say that observation was just unnecessary and quite inaccurate.
most countries never had a bloody civil war about slavery committed by their own citizens. in other places it was more about locals (including white settlers and enslaved people) vs colonial power, not a conflict between regions of an independent country where slavery was the main issue.
it was definitely not the worst instance of slavery ever going by human suffering, but the whole country was divided on political lines and many of the losing sides descendants still feel some resentment. thats pretty unique.
Re: A digestion of the Jacobian conjecture counterexample
#129Earlier quoted context omitted.
I doubt Anthropic will share the details (or at least the full true details). The mystery of the magic makes for much better marketing. I think a reasonable assumption is that there is an interaction between an LLM, a https://en.wikipedia.org/wiki/Computer_algebra_system tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.
> a human prompting with deep math expertise The original tweet implied that the whole thing was done while the author was watching the World Cup final. I know it’s tempting to hope that a human did the “real” work here, but if some special insight was put into prompting, the author kept it to himself, and there is no reason why they would hide this since it would elevate their own status.
I know it's temping to hope there is a single simple factor that does the "real" work, but this feat of mathematics is likely best explained by multiple interacting factors, one of them being the mathematical insights of the human mathematician that tweeted the counterexample. I don't doubt that an LLM is also one of the multiple factors.
It is premature to assume the author is not going to share more information in the future about the mathematical insights to narrow down the search space for this counterexample.
Re: A digestion of the Jacobian conjecture counterexample
#130Earlier quoted context omitted.
I'm not a mathematian but I know enough linear algebra and vector calculus to understand the conjecture. This is my interpretation: Firstly, the determinant of the Jacobian is measuring if at any point the function is crushing space / flattening out. If the Jacobian is a nonzero constant everywhere this means that nowhere does the the function flatten out. A small change in X along any line will always produce a non…
Sounds like 3 more iterations on K3/Fable, and we'll able to ask it to break computational hardness assumptions like DLP?
Possibly, but coming up with algorithms is difficult!
And people have been searching for faster DLP algorithms for 50 years