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Every time OpenAI releases a batch of mathematics results, I find myself asking the same question as @Raemon. Are these results the product of some important kind of “creativity,” or something different? Raemon’s framing was that models “still seem to be missing some basic sauce that lets them actually form new concepts on the fly and apply them.” I agree, and in this post, I want to propose a way to detect the basic sauce.
There are two ways we evaluate frontier models on mathematics, and neither suffices for uncovering the basic sauce.
The first is math benchmarks. As benchmarks saturate (partly from newer models training on public ones, partly from models improving), newer ones appear that can make stronger claims about model performance. But that’s exactly the problem. Benchmarks evaluate accuracy on some task. Good benchmark performance rarely reveals anything about how the model got there. At worst the model had excellent recall, and at best, we see that it succeeds but we can’t really explain why.
The second is real-world results, from @dynomight's scattered thoughts and other mathematicians’ experiences, to the recent OpenAI result. At minimum, human mathematicians feel clear productivity gains from using AI, and at most, we are watching AI disprove the Exponential Time Hypothesis. But the models still struggle with abstraction, their proofs are hard for humans to read, and the secretive nature of frontier model releases leaves so much room for doubt, amongst mathematicians and non-mathematicians alike. Are the models actually exhibiting important creativity? Or are these results coming out of emergent properties of model swarms? Or is mathematics (or a subset of it) actually a lot more brute forceable than previously thought?
I like Raemon’s term “basic sauce” because it perfectly captures the fact that even mathematicians don’t quite know what to call the thing that makes them good at math. We could say taste or creativity or judgement or basic sauce, but