Comment by QuesnayJr
2 months ago
I think in the long run mathematicians are probably fucked, but in the short run it's not that bad. All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good chance they would have solved it. (This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample.)
At this point the advantage of AI is that it's read the entire mathematical literature, and it doesn't have to worry about wasting its time. The solved problems have all turned out to be surprisingly easy, so the real lesson is that we're bad at judging how hard problems are.
Assuming this state of affairs lasts, the medium-term problem is that you learn something when struggling with a problem, even if you don't solve it, and if mathematicians become too reliant on AI the skills they develop through struggle will erode.
The long-term problem, of course, is that it seems much more probable that a future model will make mathematicians all obsolete. But so far Fable hasn't. (Anthropic has probably burned a billion tokens on the Riemann hypothesis already, without telling anyone.)
> This example, you could have given an undergraduate good at programming and computer algebra and told them to come up with a counterexample
please try go try it. There's no way someone didn't do massive computer algebra searches before today.
> All three of the big conjectures solved the answers were at the level where if you had given a grad student the questions and the right background reading there's a good chance they would have solved it.
You cannot be serious... why didn't they solve it before then? Do you think no one tried it? What background do you give the double cycle conjecture student after the flow reduction? a linear algebra textbook???
Why would I try it to win an argument on HN? That's a bizarre suggestion. Just look at the degree. If it were degree 47 in 17 variables then it wouldn't be surprising, but here it's surprising.
Of course people tried hard to solve them all, which is why it's so surprising that they were open. If anything, the solutions have gotten easier. The unit distance graph solution relied on a famous theorem remote from graph theory. The cycle double cover solution relied on a standard theory in graph theory. The solution of the Jacobian conjecture required nothing beyond knowing the definition of the Jacobian.
We're just surprisingly bad at judging the difficulty of problems. It's probably something psychological. It's even a known phenomenon, where someone will be stuck on a proof, someone else will announce the result, and the first person will suddenly get unstuck on their proof and produce an independent proof of the same theorem.
Sorry last comment was a bit emotional from me, but I do not think it's findable like you say--during my PhD I tried to find some ideals I knew existed in char 2 in 5 variables and low degree and I didn't think I ever got close. 3 variables, 7 degree, coefficients up to 6 is like 6^100 possibilities. You've got to narrow it down somewhat no? Even sparse is intractable I would guess.
I think the solutions which rely on the least amount of theory are the most telling of the AIs being higher in intelligence than humans today already. There's almost no theory to teach someone to understand the cycle double cover conjecture as you say, yet no one finds it. I don't think the conclusion is that it was "easy", but that it was in fact irreducibly difficult in a way that proofs developed with theory are not. Theory gives the human brain abstractions to simplify complex proofs to be understandable at our capacity--I think there are many proofs which probably are not of this form.
But I think our differences hinge on how hard we perceive these solutions to be--I think they are very hard to find!
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You can use that retroactive logic about any hard problem though. Unsolved murder cases, math, theoretical physics.
If tons of smart humans try for years and fail and then an LLM tries for a few weeks or hours and succeeds, the implications are clear. And these are by far the dumbest LLMs will ever be.
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