Learning Interestingness in Automated Mathematical Theory Formation

We take two key steps in automating the open-ended discovery of new mathematical theories, a grand challenge in artificial intelligence. First, we introduce $\emph{FERMAT}$, a reinforcement learning (RL) environment that models concept discovery and theorem-proving using a set of symbolic actions, opening up a range of RL problems relevant to theory discovery. Second, we explore a specific problem through $\emph{FERMAT}$: automatically scoring the $\emph{interestingness}$ of mathematical objects. We investigate evolutionary algorithms for synthesizing nontrivial interestingness measures. In particular, we introduce an LLM-based evolutionary algorithm that features function abstraction, leading to notable improvements in discovering elementary number theory and finite fields over hard-coded baselines. We open-source the $\emph{FERMAT}$ environment at this URL(https://github.com/trishullab/Fermat).