Mehtaab Sawhney & Mark Sellke on why AI reaches the results humans gave up on
OpenAI's models are producing short, human-like proofs for open math problems, shifting the bottleneck from proving results to understanding and organizing them.
The Renaissance of Reachable Results
Human mathematicians abandon uncertain approaches because their time is scarce; the model has no such cost, so it executes on the exact ideas people gave up on, opening a class of results that were always within reach but never worth the human gamble.
you're kind of gambling against the problem. You're like maybe I should try this approach but it seems really unlikely and just not worth my time.
It Reasons Like a Colleague, Not a Brute-Forcer
The released chain-of-thought shows the model pruning the search tree with judgment rather than trying everything, reading, in the mathematicians' words, shockingly like the notes of an expert human.
it's reasoning kind of shockingly like an expert human would.
A Polluted Context Traps the Human, Not the Model
A human who spends weeks on a wrong path gets stuck in it, because the failed attempt is wired into how they now see the problem, while the model can simply start a fresh session that never saw the dead ends.
as a human if you have some like wrong path you go down for a while it can be hard to like rewire your brain to like start over and like try a different path.
The Sphere-Packing Bound, Pinned Down Exactly
On the decades-old question of how densely spheres pack in high dimensions, the model did not solve packing outright but proved that a classic linear-programming bound is asymptotically the best that framework can give, an equality that explains a limit earlier work had only guessed from numerics and tightens the known bound on density.
I had actually thought about this problem for about six months at some point when I was a graduate student and yeah just I remember making like absolutely zero progress on it.
One Model for Taste, One for the Grind
Because today's models are task-oriented and stop when the task is done rather than when the mathematics is exhausted, the mathematicians floated splitting the work: one model to supply judgment and direction, another to grind out the long-horizon proof.
one model that's responsible for taste and one model that's responsible for going out and like you know working for a long time at solving a hard problem
Short and Elegant, Not Thousand-Page
The fear was that machine proofs would be unreadable thousand-page monsters; instead they have been short and elegant, with the non-sofic-group counterexample running about 15 pages where the related human disproof took 250 pages on top of 200 more.
you're kind of like afraid that they're going to like generate all these thousand page things and you're just like never going to be able to understand it. But it's been kind of the opposite.
A Ceiling That May Never Fall
Even granting exponential improvement, the hardest problems may stay out of reach; the mathematicians think it is plausible AI will never solve something like P versus NP, which could push the field toward its deep mysteries and away from the routine ones.
it's plausible we'll never solve something like P versus NP
The New Scarce Skill Is Understanding
When proving a result was the hard part, understanding it came along for the ride; now that models can produce the proof, the scarce and valuable human work becomes absorbing, organizing, and explaining the mathematics they generate.
of course models are going to help us produce exponentially more mathematics but they also make it much easier to absorb it.