alt Hacker NewsHuman mathematicians have been being out-counterexampled for at least two decades. The main difference, as I understand, is that (A) we now have a lot more compute to throw at such things, and (B) it is currently trendy to do so. But the sizes of counterexample we're seeing are around about what I'd expect pre-generative-AI counterexample search systems to be able to find.
It's not easy to find a counterexample to the Jacobian conjecture, by any means – by which I mean to say that naïve brute-force search will take too long – but the scope of existing searches listed on Wikipedia[0] suggest that many tricks are already known, and that people just hadn't looked, systematically, for a counterexample in three variables before. Wikipedia writes:
> Tzuong-Tsieng Moh checked the conjecture for polynomials of degree at most 100 in two variables.[17][18]
where reference 17 is from 1983, and reference 18 is a preprint with no given date. Knowing very little about this problem, my impulse is to side with the unnamed faculty member cited in the article:
> [who] said to me that the fact that the counterexample was so easy to find just indicated that humans had not spent enough time thinking about the problem,
For context, the auto-generated counterexample is in three variables, has degree 7, and was discovered in 2026.
Replies
The word "just" is the mark of a coward.
AI is only as good as a human mathemetician, which we don't have enough of? "just"?
https://en.wikipedia.org/wiki/Jacobian_conjecture
> The conjecture was first stated for two variables by Ludwig Kraus in 1884 [...] an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus.
> The Jacobian conjecture is number 16 in Stephen Smale's 1998 list of Mathematical Problems for the Next Century. It was notorious for the large number of published and unpublished false proofs that turned out to contain subtle errors.
> about what I'd expect pre-generative-AI counterexample search systems to be able to find.
The difference today (and the reason why everyone is excited about it) is that the same system that does advanced math can write poetry, play an above average game of chess, code frontend/backend stuff and do cybersec. These are not "expert systems", nor are they trained for each task individually. That's the catch.
> just indicated that humans had not spent enough time thinking about the problem
Heh, this is a weak excuse. We've seen variations on this theme every time something cool gets solved by the models.