If math is more than proof, we need to better celebrate the rest of it
4 hours ago
- Proofs have been a proxy for human understanding in mathematics, but AI-generated proofs can bypass understanding, undermining their value.
- The author proposes elevating 'motivated explanations'—explanations that clarify why a theorem is true and how to think of it—as a new proxy for mathematical progress.
- Motivated explanations place definitions in the middle, allow starting with imperfect ideas, and aim to answer 'how would you think of that?' rather than just verifying correctness.
- Examples of motivated explanations include the Princeton Companion to Mathematics, Bill Thurston's work on sphere eversion, Timothy Chow's 'open exposition problems,' and a recent AI-assisted proof of Erdős's primitive sets conjecture clarified by human mathematicians.
- Practical suggestions include treating expositions as PhD deliverables, listing unsolved exposition problems, establishing journals focused on clarity, and evaluating hiring and tenure based on expository work.
- Visible cultural shifts can reassure young mathematicians and funders that human understanding remains central, making the field more attractive despite AI advances.