3Blue1Brown: As AI Cracks Proofs, Math Must Reward 'Motivated Explanations'

In a guest post on Terry Tao’s blog, Grant Sanderson (creator of 3Blue1Brown) argues that proofs have always been a proxy for math’s real aim—advancing human understanding—and that AI systems capable of churning out proofs without any understanding are exposing the limits of that proxy. His proposed alternative metric is the ‘motivated explanation’: work whose primary goal is answering ‘how would you think of that?’ He wants the field to grant such work academic credit comparable to what solving open problems has historically earned, and stresses this means rigorous, expert-level exposition, not mere popularization.

Sanderson contrasts the two modes directly. A proof front-loads definitions and demands every statement be correct; a motivated explanation places definitions only after the problem motivating them is clear, and may deliberately start with an appealing-but-wrong idea before refining it (a ‘discovery fiction,’ in Michael Nielsen’s phrasing). The catch he concedes: unlike a proof, an explanation’s validity isn’t binary—there will never be a Lean-style verifier for it—but he argues ‘motivated’ is verifiable enough to serve as a practical standard, since you can ask for each new idea whether its origin is clear.

The stakes are partly about public perception: if the community keeps celebrating proof-generation above clarification, outsiders will reasonably conclude that proof-writing machines make mathematicians obsolete. Sanderson points to exemplars like Part IV of the Princeton Companion to Mathematics and Bill Thurston’s essay ‘On Proof and Progress in Mathematics’—which noted the Four Color Theorem controversy stemmed from a desire for human understanding, not doubt about the result—and laments that Timothy Gowers felt a Fields Medal was needed to justify spending years editing such a book.