Earlier quoted context omitted.
Would this counterexample not be included in their search space?
based on the approach described they were probably in dimension 2, this counterexample is 3 dimensional
[deleted]
Claude Fable produced a counterexample to the Jacobian Conjecture
341–350 of 562 posts
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#342I’ve been math-vibe coding a few months now. It’s surprisingly easy to do with AI. The hard part has been manually verifying and validating the results. I took one of the smaller findings (disproving a conjecture) and wrote a paper as my first endeavor into publishing. Because the next few findings i have in the pipeline are substantial in the field of quantum topology and physics im taking some time to publish them…
I'm a phyicist by training and have published in the past but I left academia and so haven't for a number of years now.
I tried out using Claude to do some physics problem solving - mix of maths and simulation - and ended up with it getting in quite a mess. It's incredible at setting things up, suggesting approaches you might not have considered. I found it much much worse at interpreting things.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#343Earlier quoted context omitted.
Or we just don't use LEAN but something better.
Does anything truly better exist? I'm not a mathematician but I did use Rocq and Lean during university. And I found lean to be better.
pay attention to this one https://higherorderco.com/ and wait for bend2 announcements
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#344The 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…
Hilariously absurd cope. They're still not very good at writing, but you've flipped the takeaway. The correct conclusion is writing style isn't very relevant to intelligence.
I think the real tragedy is that they are good at writing, just by default are tuned to have a kind of bland corporate tone. If you give the LLM a few pages of writing you like and tell it "Continue, but using this style" it will do a pretty good job of it. Most people just .... don't bother to do that.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#345Earlier quoted context omitted.
OK, so the two options are: A) Claude really produced this counterexample B) A mathematician working for Anthropic solved a problem mathematicians have been working on for more than a century, and then credited it to Claude for PR purposes If you believe B is more likely, why would you then believe a proof in the form of a chat log, when said chat log could itself have been faked by Anthropic way more easily than sol…
The concern is that there could have been expert knowledge input, whose importance/worth we are unable to evaluate. I don’t believe a mathematician produced the counter example secretly, but how much did they contribute to the result? AI isn’t magic, so to evaluate the value delta, you need to know the value of the input.
While I agree that we need the inputs to properly evaluate what this means for LLM capabilities, I don't really believe that the amount of knowledge input matters much for the overall significance of the result.
These kinds of results are interesting for LLMs because mathematicians have been working on them for decades. If the result doesn't already exist, there's no way it's in the training data, and if mathematicians have been unsuccessfully tackling the problem for decades, it is believable that the use of a new tool made the result possible, even if guided by a great mathematician.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#346Earlier quoted context omitted.
In my view, it's theoretically possible for a combination of the author's iterative prompts + evaluation with Wolfram Alpha to activate the weights that encode the language that describes the constraints on these polynomials (from the faulty proofs) in such a way that the author eventually arrives at this: > ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3
How is that not like claiming that a "combination of iterative prompts and evaluation with Wolfram Alpha" would produce my Google Mail password? I also have to note that this response abandons the original claim, that the counterexample was found by exhaustively searching prior failed attempts.
I can write a program right now that will print your password.
It will just take a while.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#347Earlier quoted context omitted.
The retrospective view is important, though. In retrospect, these problems weren't that hard. (The unit distance graph problem was the hardest.) There are some problems that still seem hard, even when we know the answer. Nobody thinks that Fermat's last theorem is easy, even though now know it to be true. Before AI, it was pretty rare that a problem that turned to be unexpectedly easy, so mathematicians thought they…
If some LLM is able to come up with a simple proof of Fermat's Last Theorem (a solution that Fermat himself could come up with) in the future, would you still say that Fermat's Last Theorem is hard?
No. I would say that it was easy, and that something had gone terribly wrong with math research that we missed it.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#348Using LLMs to generate piles of code and/or proofs of dubious quality is very questionable thing, and I understand these non-stop debates about it. But in this case, as using plain brute force is already quite a common thing in searching for counterexamples, using LLMs as a sort of more advanced brute force seems to be just the right thing to do, so I struggle to understand so much hostility to this approach.
> I struggle to understand so much hostility to this approach.
It's going to be really rough for a lot of folks as machines get better and better at domains that the human brain was exclusively useful for. I tend to view the hostility as a mix of both "unless we have proof this could all be hokum" and "we're going to lose a lot of what we consider makes humans amazing".
Compassion is going to be very, very important in the next few years. We're going to hurt, both internally and externally.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#349> The incredible Yitan Zhang ( https://newyorker.com/magazine/2015/02/02/pursuit-beauty ) worked on proving this conjecture for 7 years. Moh, his advisor, wrote that Zhang "failed miserably" in proving the Jacobian conjecture, "never published any paper on algebraic geometry" after leaving Purdue, and "wasted seven years of his own life and my time". https://x.com/aminkarbasi/status/2079129649830137989 https://en.wik…
Zhang, Yitang’s life at Purdue (Jan 1985-1991) T.T.Moh - https://www.math.purdue.edu/~ttm/ZhangYt.pdf kind of a wild document to exist...
The world doesn’t need advisors like Moh that are bitter enough to gossip about a family funeral.
Re: Claude Fable produced a counterexample to the Jacobian Conjecture
#350What’s more Impressive you got Fable to do anything useful.
The model is still useless for real data to day work, no matter how many parlor trucks it performs.