I would only agree partially. There are counterexamples that are not illustrative, but it is fairly common that in thinking about how to construct a counterexample you gain a more thorough understanding of the original problem and at least one fundamental issue which prevents the conjecture from being true.

Counterexamples are literally the only way to show a "for all" statement is false. (Non-constructive proofs by contradiction work by showing a counterexample must exist.)

Also, 'brute-force' style attacks where one simply feeds the input into the computer and it yields a solution are nothing new and certainly predate LLMs: https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjec...

Hell, one could even go further into the past and refer to the thankless work of pre-computer era mathematicians who sweated over manual calculations in order to disprove various prime related conjectures: https://en.wikipedia.org/wiki/Mersenne_conjectures

You seem to have an objection to non-intutionist mathematics in general, a position that was once held by many an illustrious mathematician but is relatively fringe in the contemporary academic community. Mathematical facts don't have to be intellectually satisfying or make sense to you, the human, rather it is up to you to wrap your mind around discovered mathematical facts.

Are you thinking of proof by contradiction, which is rejected by constructionism?

[Dis]proof by counterexample is the most straightforward way to show a statement to be false. What better way is there to disprove a general statement like 'all x are y' than finding an 'x' that isn't 'y'?

You could spend the rest of your life coming up with conjectures that look elegant but are ultimately false. Disproof by counterexample only works if it's false, and we shouldn't be satisfied with a false conjecture to begin with.

The counterexample in the news cycle today helps better understand how the math works. I can't think of one that doesn't.

> and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians

Considering ChatGPT was released only three and half years ago, and LLMs could do high school math only less than two years ago, I think this "for now" will not last very long.

Elegance may not remain exclusively human forever but usefulness probably requires more than correctness

Not much worth in understanding a statement that is wrong and has been shown wrong.

Unless you want to spend time "proving" that 2 * 2 = 1.

Seems like a breakdown on the incentives / imperatives in the field? I hope that's not an over bold guess from a non-mathematician.

Couldn't people in principle continue to study a problem that's only been shown to break at one point? Prove something adjacent, or slightly weaker, or elaborate the counter example into a powerful explanatory framework?

Maybe I’m just not pure enough but I find the whole concept of proof by counterexample to be elegant, and I don’t see why proving that something must be true is superior to proving that it can’t be false.