Grant Sanderson, the creator of the YouTube channel 3Blue1Brown, has made a case for mathematicians to celebrate explanations over proofs. His argument, published as a guest post, centers on the idea that solving problems and proving theorems have always stood in for the real goal of mathematics: furthering human understanding. But when proof-generating machines can produce results without that understanding, the proxy breaks down.
Sanderson wants to replace the proof as the sole measure of mathematical achievement. He proposes that “motivated explanations” — clear, intuitive accounts of why a result matters — deserve the same academic credit as new proofs of open problems have historically received. He also sees this shift as a way to help outsiders grasp what mathematicians actually contribute.
“If outsiders believe that proof-generating machines render mathematicians obsolete, while insiders see that as a misconception of what researchers add, it’s incumbent on this community to better project its true values through the kind of work that it rewards.”
Proof vs. Motivation
The distinction is sharp. In a proof, definitions come first, and every statement follows strictly from what came before. In a motivated explanation, definitions arrive in the middle, after the problem has been clearly stated. A proof is binary: it exists or it doesn’t. A motivated explanation is messier — it can start with an idea that isn’t quite right, one that needs fixing along the way.
Sanderson cites a genre he calls “discovery fiction,” a term coined by Michael Nielsen. The idea: begin with a simple but wrong solution to a problem, watch where it breaks down, fix that flaw, find a new problem, and keep going. The scope of a proof is narrow — it explains why a theorem is true. A motivated explanation asks broader questions: why is this the right theorem to pose, and how does it fit into the larger picture?
| Feature | Proof | Motivated Explanation |
|---|---|---|
| Definitions sit at the start | Yes | No |
| Definitions sit in the middle | No | Yes |
| All statements must be correct | Yes | Sometimes not |
| Starts with an idea that’s not quite right | No | Often |
| Scope is binary (exists or doesn’t) | Yes | No |
What Counts as Motivated
Sanderson defines a motivated explanation by what it avoids. It doesn’t begin with a new construction and analyze its properties. Instead, it waits until the problem has been established before bringing in new terms. It allows ideas to be not quite right, as long as their origins are relatable. And it reaches beyond the theorem itself, asking why it matters in the first place.
He admits that this standard is harder to measure than a proof. “There will never be Lean for motivated explanations,” he writes, meaning there’s no automated system that can verify them the way proof-checking software handles formal logic. But he argues that the fuzziness is the point. Defining human understanding is inherently messy, and avoiding messier metrics shies away from the more human aspects of the field.
The Princeton Companion
Sanderson points to one of the best collections of motivated explanations he knows: Part IV of the Princeton Companion. That section covers over two dozen active fields of research, each introduced by an expert with a talent for clear communication. Whether it’s Andrew Granville on analytic number theory or David Ben-Zvi on moduli spaces, these articles offer a level of intuition and motivation more typical of one-on-one conversation at a blackboard than of formal papers.
The book’s editor, Timothy Gowers, has discussed it in an interview on the Numberphile Podcast with Brady Haran. Asked about the impact of the Fields Medal on his life, Gowers noted that winners often feel freer to pursue slightly different projects. He cited the Princeton Companion as an example of something he took on precisely because he could afford to do so.
Why the Shift Matters
Sanderson’s own career shows the tension. He runs a channel focused on producing videos about mathematics, a non-traditional path that emphasizes explanation and intuition rather than solving outstanding problems. Some might see this proposal as self-serving. But he insists that his work exists outside academia, with no stake in what the community assigns credit to.
He also distinguishes his idea from popularization. A motivated explanation, in his view, targets anyone with deep expertise to appreciate — it’s not just for the public. The goal is to answer the question “how would you think of that?” rather than to make something accessible.
The Limits of the Proposal
Sanderson acknowledges that his proposal is modest. He’s not arguing for a radical overhaul of academic recognition. He’s asking for clearer definition and higher status for a kind of work that already exists in practice. He points to discovery fiction as one example of the genre, and to the Princeton Companion as a repository of it.
He also notes that proof-generating machines aren’t the only threat to the field’s perceived value. The rise of automated tools means mathematicians need to project their true values through the work they reward. If outsiders believe machines make mathematicians obsolete, the community has a responsibility to correct that impression.
The argument is persuasive on its own terms. It reframes a familiar debate — proof vs. intuition — as a practical problem about recognition. Mathematicians already do this work; they just haven’t celebrated it as a distinct contribution. Sanderson’s proposal is a modest step toward giving it its due.
The full post, titled “If math is more than proof, we need to better celebrate the rest of it,” is available online.
See the video the story is built around at terrytao.wordpress.com.
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