If math is more than proof, we need to better celebrate the rest of it

The way mathematical contributions are credited could expand beyond proofs to include explanations that foster understanding.

Grant Sanderson, known for his math videos, proposes that the mathematics community should redefine its values to include "motivated explanations" alongside traditional proofs. He argues that as AI becomes capable of generating proofs, the unique contribution of mathematicians lies in fostering human understanding. Sanderson suggests that explanations which clarify not just why a theorem is true, but why it is important and how it fits into the broader context, deserve academic credit. He contrasts "motivated explanations," which build understanding by introducing concepts as solutions to problems, with proofs, which start with definitions. Sanderson acknowledges that these explanations are harder to quantify than proofs but stresses their crucial role in conveying the essence of mathematical inquiry. He cites examples like Part IV of the Princeton Companion to Mathematics and Bill Thurston's work as embodying this spirit of deep explanation.

The proposal highlights a growing tension between AI's ability to generate proofs and the human element of mathematical discovery. By advocating for "motivated explanations," Sanderson suggests a shift towards valuing the narrative and intuitive journey of mathematical thought, not just the final, verified outcome. This could influence how mathematical research is evaluated and funded, potentially opening doors for educators and communicators whose work deepens public comprehension of complex ideas.

This re-evaluation of academic credit challenges the long-standing hierarchy that prioritizes novel proofs. It implies that work focused on clarifying "why" and "how" behind mathematical concepts—akin to Michael Nielsen's "discovery fiction"—could gain parity with traditional problem-solving. Such a change might redefine the perceived value of mathematical communication, impacting everything from graduate student training to the recognition of established mathematicians who contribute through exposition.