A growing sentiment in the mathematics community holds that proof-generation has long served as a proxy for mathematics’ true goal: advancing human understanding. As artificial intelligence becomes capable of generating proofs without that understanding, according to a guest post by Grant Sanderson on Terry Tao’s blog, the field must reconsider what work it values most.
Sanderson proposes elevating the status of what he calls “motivated explanations”—work that clarifies not just why a theorem is true, but why it’s the right theorem to pose and how it fits into broader mathematical context. This differs fundamentally from proofs, which begin with definitions and proceed through necessary implications. Motivated explanations, by contrast, can start with incomplete ideas that require correction, allowing readers to see how mathematical thinking naturally unfolds.
The distinction matters because outsiders may believe AI-driven proof generation renders mathematicians obsolete, while the mathematics community recognizes this as a misunderstanding of what researchers actually contribute. Sanderson argues that if the field continues to celebrate proofs while treating exposition as secondary, it will struggle to communicate its true values to the outside world.
One genre of motivated explanation Sanderson highlights is “discovery fiction,” a term coined by Michael Nielsen, which develops ideas narratively—beginning with a simple-but-wrong solution, identifying where it breaks down, and iterating toward understanding.
Sanderson acknowledges motivated explanations lack the binary certainty of proofs. Understanding itself is inherently subjective, making rigorous measurement difficult. However, he argues this subjectivity reflects mathematics’ fundamentally human aspects. The word “motivated” offers a practical, if imperfect, standard: for each new idea introduced, one can ask whether its origins are clear.
Examples already exist. Part IV of the Princeton Companion to Mathematics, edited by Fields Medalist Timothy Gowers, contains motivated explanations from experts like Andrew Granville on analytic number theory. Gowers noted he felt empowered to undertake this massive five-year project partly because his Fields Medal gave him professional standing to pursue work beyond traditional proof-generation.
Bill Thurston’s 1994 essay “On Proof and Progress in Mathematics” made similar points decades before large language models emerged, arguing that the core mathematical question should be “How do mathematicians advance human understanding?” rather than merely “What is proven?”
Key facts
- Sanderson proposes defining and rewarding ‘motivated explanations’—clear expositions that explain not just why theorems are true but why they matter
- Motivated explanations begin with relatable incomplete ideas and show how they evolve, contrasting with proofs that start with formal definitions
- The proposal aims to help outsiders understand what mathematicians contribute, as AI becomes capable of generating proofs
- Examples of valued motivated explanations already exist, such as articles in the Princeton Companion to Mathematics
- Bill Thurston argued in 1994 that mathematics should focus on advancing human understanding rather than solely on proof-generation