Am: an Artificial Intelligence Approach to Discovery in Mathematics as Heuristic Search

A program called 'AM', is described which models one aspect of elementary mathematics research: developing new concepts under the guidance of a large body of heuristic rules. 'Mathematics' is considered as a type of intelligent behavior, not as a finished product. The local heuristics communicate via an agenda mechanism, a global list of tasks for the system to perform and reasons why each task is plausible. A single task might direct AM to define a new concept, or to explore some facet of an existing concept, or to examine some empirical data for regularities, etc. Repeatedly, the program selects from the agenda the task having the best supporting reasons, and then executes it. Each concept is an active, structured knowledge module. A hundred very incomplete modules are initially provided, each one corresponding to an elementary set-theoretic concept (e.g., union). This provides a definite but immense 'space' which AM begins to explore. AM extends its knowledge base, ultimately rediscovering hundreds of common concepts (e.g., numbers) and theorems (e.g., unique factorization). This approach to plausible inference contains great powers and great limitations.