Artificial Intelligence and Inherent Mathematical Difficulty

Abstract

This paper explores the relationship of artificial intelligence to resolving open questions in mathematics. We first argue that limitative results from computability and complexity theory retain their significance in illustrating that proof discovery is an inherently difficult problem. We next consider how applications of automated theorem proving, Sat-solvers, and large language models raise underexplored questions about the nature of mathematical proof — e.g., about the status of brute force and the relationship between logical and discovermental complexity. Nevertheless, we finally suggest that the results obtained thus far by automated methods do not tell against the inherent difficulty of proof discovery.