Mathematics as a search problem

Machine learning is usually presented as function approximation: in supervised learning, one aims to recover an unknown map from inputs to outputs, and results such as universal approximation theorems and generalization bounds explain why neural networks can, in principle, learn rich function classes. I will begin with this viewpoint, briefly reviewing the basic supervised setup and outlining three conceptual sources of error. But research mathematics is not mainly a function-approximation problem. It is a search problem: we search for proofs, for examples and counterexamples, and for the right definitions that reveal underlying structure. This perspective leads naturally to reinforcement learning, where efficient exploration is central. I will illustrate these themes with an example of exploration via OpenEvolve, applied to the planar isoperimetric problem.