Beyond Proofs: Can AI Invent Mathematics Humans Have Never Imagined?

Source abstract

Artificial intelligence is beginning to contribute important mathematical results, but a deeper possibility remains: might AI discover mathematical representations humans have never conceived? The history of science suggests that progress depends not only on solving equations but also on finding invariants and symmetries that make difficult problems transparent. Using Poincaré's notion of mathematical sameness, Klein's transformation-based geometry, and the Alice-Georgette thought experiment, I distinguish new results, new connections, and new mathematical languages. Studies of language models and AlphaFold illustrate how learned representations can contain useful structure beyond their explicit outputs, without establishing that AI has invented new mathematics. Statistical teleodynamics offers one proposed account of how organized semantic structure might emerge. I discuss tests involving generalization, causal interventions, comparison with established formalisms, and independent verification. The implications extend to biology, chemistry, and engineering, where previously unknown representations could reveal new organizing principles.