Artifical intelligence and inherent mathematical difficulty

This paper explores the relationship of artificial intelligence to the task of resolving open questions in mathematics. We first present an updated version of a traditional argument that limitative results from computability and complexity theory show that proof discovery is an inherently difficult problem. We then illustrate how several recent applications of artificial intelligenceinspired methods - respectively involving automated theorem proving, Satsolvers, and large language models - do indeed raise novel questions about the nature of mathematical proof. We also argue that the results obtained by such techniques do not tell against our basic argument. This is so because they are embodiments of brute force search and are thus capable of deciding only statements of low logical complexity. Suppose... that we could find a finite system of rules which enabled us to say whether any given formula was demonstrable or not. This system would embody a theorem of metamathematics. There is of course no such theorem and this is very fortunate, since if there were we should have a mechanical set of rules for the solution of all mathematical problems, and our activities as mathematicians would come to an end.