A Primer on using AI and Mathematics

The arrival of artificial-intelligence systems that can relate mathematical ideas has provoked fascination, fear, and dismissal. This essay introduces the geometric notion of an informational space as a framework for understanding the strengths and limits of AI in mathematics, and the work that remains for people. I argue that mathematics is a closed informational system: once its axioms and rules of inference are fixed, its possible content is determined. Heuristically, I picture its geometry as a hyperbolic funnel, in which assertions with more complex proofs lie deeper and true assertions unprovable in the chosen informational system occupy a limit that no proof-path reaches. Physics, by contrast, is open: its funnel has a similar shape, but its rim widens as new observations arrive. This distinction helps explain why AI performs so well in mathematics, where proposed results can be checked internally, whereas in physics nature remains the ultimate arbiter. The two sciences also share something: in both, expert knowledge forms around patterns observed in examples. In mathematics, however, the examples can be generated, and the patterns proved, inside the system, so whether AI can do new mathematics depends largely on how well it explores examples and recognises patterns, as human researchers do. I go on to consider the technical and social consequences for research and education, describe practical safeguards for using AI, and identify open questions. I conclude that AI shifts what is scarce in mathematics away from producing results and toward choosing questions and definitions, and understanding the local and global structure of the informational space.