211–220 of 562 posts
I'm very excited for the impact of this effect in science and medicine and other disciplines too.
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…
next year is a long time away friend
(-(1+xy)^2 z - y^3(1+xy), 2x(1+xy)z + (1+xy)^3 w + y^2(7+12xy+4x^2y^2), 2x^2z + 3x(1+xy)^2w + 2y(1+10xy+6x^2y^2), 2x - 4x^2y - x^3w): C^4 → C^4 has Jacobian determinant 4, and sends (-2,0,1,0) and (-1,0,1,-2), (1,−2,−7,14), (2,−1,0,3) to (-1,-4,8,-4)
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?
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.
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.
Some quotes from a day ago, https://news.ycombinator.com/item?id=48957779 : > I hold my stance that LLMs are stochastic parrots... Making the parrots ever more complex and training > Except solving problem is probably the least (even though it's important) interesting thing in research. > Can we use AI to get a cure for cancer yet? Or is math-turbation the only thing these things are good for? > Train on enough examp…
Yes-ish. Custom mRNA treatment used to reverse the progress of a dog's cancer.
https://www.theaustralian.com.au/business/technology/tech-bo...
I have a question I'm surprised people are not asking: How did Fable find this? Was it like guessing a bunch of families and then solving for possible solutions in those families? Was it clever search? something else?
I’m doing this by working all logical steps into lean (formal verification) the quick feedback loop between the AI prose and the Lean verification errors and warnings ensures that its logically consistent. The issue that remains are two things, ensuring the idea of the proof is actually the thing you want to prove and the interpretation of the results you get. But besides that, everything inside of the kernel checked…
Could we maybe get more information about the problem from the LLM trace itself here?
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.…
Would this counterexample not be included in their search space?
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…
Thanks that's a very nice summary.
If the determinant were zero, then (for basically the same reason as why you can't divide by zero) the matrix wouldn't be invertible. You wouldn't expect the polynomial to be invertible in that case, either.
If the determinant had free variables left over, then there is some combination of variables that will make it zero (e.g., if it came out to x^2-1, then it would be 0 for x=1 or -1). So maybe inversion won't fail everywhere, but it will fail in those spots, which would also lead one to think the polynomial is not invertible.
But if it's constant and non-zero, then there is no place where you can't invert the Jacobian. So it seems very plausible that this would apply to the polynomial as well. That's essentially what motivated the conjecture in the first place (leaving out some details, of course).