Grant Sanderson argues math should reward explanations, not just proofs

Grant Sanderson, the creator of the 3Blue1Brown mathematics channel, argues in a new essay that the mathematics community should grant real academic standing to work that explains and motivates ideas, not only work that proves theorems. The piece runs as a guest post on Terence Tao's blog — with a note from Tao that it was converted from another file format using AI — and it reached Hacker News' front page, where it drew wide discussion.

Sanderson's premise is a concern now circulating in the field: proving theorems has long served as a proxy for the discipline's actual aim, furthering human understanding. Once proofs can be produced without that understanding — the machine-generated case — the proxy weakens, and the community needs other ways to recognise progress.

Elevating the motivated explanation

His proposal is to define a category he calls the "motivated explanation" and to reward it with the kind of credit historically reserved for new proofs of open problems. He is careful to say this is not a call to reward popularisation. What he means is any work whose primary goal is showing how a result could have been arrived at, even when the material requires deep expertise to follow.

The contrasts with proof are structural. A proof begins with definitions and proceeds through claims that must each be correct; a motivated explanation introduces definitions only once the problem they address has been established, and may deliberately start from an idea that is wrong but relatable before correcting it. A proof establishes that a theorem is true, while a motivated explanation also covers why that theorem was worth posing and how it is used in the surrounding field. One style he singles out is "discovery fiction," a term coined by Michael Nielsen, in which an argument unfolds as a sequence of plausible attempts, failures and repairs.

Sanderson discloses his own bias up front: his career consists of making videos about mathematics rather than solving research problems, so a cynic could read the essay as self-promotion. He responds that his career and funding sit outside academic credit systems, leaving him nothing to gain from how the community assigns status.

The verification gap

He also admits the obvious weakness. A proof's validity is binary; an explanation's is not, and he does not expect anything like the Lean proof assistant to exist for judging explanations. Advancing understanding is inherently fuzzier than establishing truth, and he argues that accepting softer, partly subjective measures is the price of taking the field's real goal seriously. His practical test is whether each new idea in an explanation has a clear origin — whether a reader could picture themselves arriving at it.

Existing exemplars

Sanderson treats this kind of work as something mathematicians already do informally, and points to Part IV of the Princeton Companion to Mathematics as the best existing collection of it, citing Andrew Granville on analytic number theory and David Ben-Zvi on moduli spaces as pieces that carry intuition usually passed along in person between researchers. He recalls Timothy Gowers, the book's editor, describing the project as taking up about half his working hours for around five years — something Gowers said he felt free to take on because he had a Fields Medal. It is a shame, Sanderson writes, that standing of that kind is needed to justify such work.

The essay's other reference point is Bill Thurston's essay On Proof and Progress in Mathematics, written roughly three decades before large language models. Thurston reframed the question of what mathematicians accomplish as a question of how they advance understanding, and read the controversy around the computer-assisted proof of the four-colour theorem as a desire for comprehension rather than doubt about correctness.

Why it matters

If automated systems become reliably good at producing correct proofs, a reward system that measures mathematicians by theorems alone will both misstate what the profession contributes and make it look replaceable to outsiders — a misunderstanding Sanderson explicitly wants the field to pre-empt. Naming and crediting explanatory work is a concrete proposal for what to value instead. The same question now faces many disciplines: when the most easily measured output can be automated, communities have to decide what their human core is and build incentives around it.