Comment by porcoda
11 hours ago
They aren't, but the problem is that open problems tend to emerge when people are working on other problems. If fewer people are spending time deeply thinking about current problems since a handful of labs are solving them with AI without an eye towards understanding and only on verification, the pool of open problems won't be continuously growing. There is a fear that there will be a chilling effect on the community if people are disincentivized from trying to solve deep problems or study them for understanding as opposed to simply focusing on verification. It's more of a social and community problem than a fundamental problem with mathematics itself becoming "completed".
So we can let the ai generate some math problems based on the solutions found? Other fields (computer science, physics, ...) can generate math problems too.
There's an infinite number of possible math problems, but the things that make these open problems worthwhile is they're interesting to people who have worked in related areas.
They're good to give to new mathematicians, and they're good to help humans understand the shape of the problem space and relative difficulty with the tools we have.
Cheesing these problems with LLMs gets rid of both the training benefit and our ability to create good related problems. There's an aesthetic part of this, too, that LLMs do not capture.
This kinda reminds me of the guys who decided to industrialize digging up dinosaur fossils, in order to feed the dinosaur fossil collector market. They were amazed that paleontologists were so "inefficient" at finding and digging up dinosaur fossils.
But from paleontologists' perspective, they go out looking for dinosaur fossils when they have questions that digging up a fossil may answer. The metric they're focusing on isn't tons of fossil mined out of the ground, it's a developing understanding of extinct life.
These open problem solutions often reveal tighter bounds on prior conjectures. Even if the solutions produced are far from elegant and only machine verifiable, we do learn new information. But I agree that just like writing prose and code, brainstorming frontier math proofs is a perishable skill
That explains a lot on why his arguments always focus on the "social part"
tl;dr - it's content creation rather than process and understanding