Coining the right definitions may matter more than proving theorems
Proving theorems wins prizes, but coining the right idea may matter more for letting everyone else move forward.
Dwarkesh Podcast · Grant Sanderson (@3blue1brown) – AI disproved a famous math conjecture. Now what?Grant Sanderson, who makes the math videos at 3Blue1Brown, was talking about what artificial intelligence in math should do next, now that it can make headway on famous conjectures. Solving old problems, he argued, is no longer the most interesting test. The next one is whether a machine can pose a good problem, or invent a new kind of object that unifies a field.
In making the case, he repeated a line he said came from a person in a video by the math channel Polylog: "good mathematicians prove theorems, great mathematicians come up with conjectures, and the greatest mathematicians come up with definitions."
What Sanderson said he meant was the conceptual groundwork that makes later theorems possible. His example was Riemann's zeta function, which Riemann built on the hunch that the zeros of a particular function would say something about how prime numbers are spread out. Deciding that this was worth studying, and that this was the question to ask about it, was the creative act. Proof comes later.
Sanderson tied that to an argument by David Bessis, who wrote a blog post called "The Fall of the Theorem Economy," as Sanderson retold it. In that retelling, Bessis argues that theorem proving gets all the credit but lives off definition making. That mismatch did not matter much for credit in the past, in this account, because the person who coined a definition was usually the one who proved the first theorems about it.
If AI automates the proving, Sanderson argued, what is left is consolidation, turning scattered proofs into higher level insight. A machine that could generate good conjectures and then the definitions to frame them would be what he called the premium-tier mathematician.
He said he would be surprised if that ability ever showed up as a benchmark with a score for how good a conjecture is. Instead, he said, it would show up as a tone shift, with mathematicians saying conversations with a model were genuinely helpful when they stepped back to decide what their field should even be.
Questions people ask
What does a definition do in mathematics?
A definition names an object or property precisely enough that people can reason about it. Examples include functions, symmetries, or number systems. Once the definition sticks, many different mathematicians can prove theorems about the same thing.
What is the Riemann zeta function?
It is a function first studied in detail by Bernhard Riemann in 1859 to understand prime numbers. Its most famous open question, the Riemann Hypothesis, says its nontrivial zeros all lie on a particular line. It is one of the seven Millennium Prize Problems.
Why would AI find definitions harder than proofs?
A proof has a clear finish line, while a definition is judged by whether it proves fruitful over years or decades. There is no instant test for fruitfulness. Sanderson argues that judgment resists the kind of clean benchmark AI labs like to use.