The Collapse of the Theorem Economy

The best theorem I ever wrote was actually the one I didn't write. In an old preprint on Garside categories, what mattered about Theorem 0.5 wasn't the proof but the language that made it possible to even state the theorem. The conceptual framework built by Definition 2.4 and 9.3 was the real innovation, and this language was later used to resolve the K(π,1) conjecture as well.

So I see both formalism and Platonism as insufficient. Mathematics deals with meaningless formal symbols, but it operates closer to conceptualism, where human neural structure projects meaning onto them. What actually drives research isn't the proof so much as the definitions and intuitions that let us see the problem, along with the art of explanation.

• Official mathematics: axioms, proofs, verification, publishing theorems
• Secret mathematics: intuition, definitions, explanation, communal understanding

From this perspective, AI shakes up the theorem-centered reward system more than mathematics itself. The tradition of undervaluing explanation, as G. H. Hardy did, is an old problem, and Fields Medal winner Bill Thurston redefined the product of mathematics not as theorems but as clarity and understanding. Mathematics isn't a solitary list of accumulated proofs but a living community that expands each other's understanding.

In the end, the collapse of the theorem economy isn't the end of mathematics, but the weakening of an order that rewarded only proof and authorship. Even if AI automates proof production, the core work of creating concepts and sharing meaning isn't easily replaced. The real value of mathematics lies not in the number of theorems, but in the infrastructure of understanding that humans build together.