Learning Interestingness in Automated Mathematical Theory Formation

Abstract

We take two key steps in automating the open-ended discovery of new mathematical theories, a grand challenge in artificial intelligence. First, we introduce $\text{FERMAT}$, 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 $\text{FERMAT}$: automatically scoring the $\text{interestingness}$ 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 $\text{FERMAT}$ environment at this URL(https://github.com/trishullab/Fermat).

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