The Riemann hypothesis is one of mathematics' most famous unsolved problems, and if an AI proves it tomorrow, we will want to know whether the proof is correct. We will also want to know what it taught us, which is a different question: every step might check out while the proof leaves mathematicians staring at a thousand pages without a better way to think.
That possibility came up in Grant Sanderson's conversation with Dwarkesh Patel. A proof can settle a theorem without giving us a useful definition, a new object, or a bridge to another field. It counts as a real proof, but it may still be an intellectual dead end. Software has the same problem in miniature, because a patch can pass every automatic check, or unit test, while leaving the rest of the program harder to understand.
Galois did not answer the obvious question
In school, we learn one formula that solves every quadratic equation, so for centuries mathematicians searched for a similar recipe for equations whose highest power is five. Niels Henrik Abel eventually proved that no such recipe could be built from addition, subtraction, multiplication, division, and roots, which answered the headline question.
Évariste Galois did something harder to grade. Instead of hunting for another recipe, he looked at the possible answers to an equation and asked which ones could trade places without changing their relationships; those legal swaps formed a structure, and we now call this way of studying symmetry group theory. At the time, however, the vocabulary was not ready.
His manuscripts were rejected or went unreported, and in a preface written in Sainte-Pélagie prison, Galois complained that Poisson had not understood the work. He urged future mathematicians to look beneath long calculations and classify operations by their structure, but the importance of that shift took decades to become clear.
Galois died in 1832, long before Joseph Liouville announced that he had found deep results in the manuscripts in 1843 and published them in 1846. Camille Jordan's 1870 book later helped make these structures part of mainstream mathematics.
An immediate evaluator would have had a rough time with the original submission, which was hard to read, incomplete in places, and apparently late to a famous problem Abel had already solved. Most unclear work is simply unclear, yet the Galois story leaves us with an uncomfortable fact: a fertile idea can look like a mess until other people start using it.
Tests only know the future we wrote down
A unit test is a small contract with the future: given this input, return that output; if the request is malformed, fail in this particular way. A machine can check these promises every time the code changes, but the promises inherit the limits of our imagination.
Imagine two programs that turn text into structured data. One has a separate rule for every format it knows. The other has a small rulebook describing what all the formats have in common. Today they pass the same tests, and the first version may be shorter and safer, so an agent rewarded for closing today's issue will probably choose it. Then a sixth format arrives. At that point, the common rulebook might make the change trivial, or it might reveal itself as needless complexity. Yesterday's tests could not tell us which future we were in.
This is not an excuse to overengineer; it is a limit on what "all tests pass" can mean. The suite tells us that the behavior is acceptable now, but says very little about whether the code has found concepts that will remain useful when the next request arrives.
Test the aftershocks
Coding agents get a clean reward when a failing test starts passing, whereas the design of the whole program pays out later, perhaps when a boundary prevents a future bug or a good name reveals a distinction everyone had missed.
We could test more of that by giving two agents the same initial task, then revealing several follow-up requests they could not see at the start. We could measure how much code has to move and how often old behavior breaks.
Agents would eventually learn the shape of that benchmark too, and I do not know how to escape that problem entirely. Even so, it would reward patches that make the next change easier instead of only rewarding those that close today's ticket.
Mathematics has a much longer delay, because a definition may prove its worth only after it turns many separate arguments into one idea and gives other people questions they could not previously ask. By then, the original invention may have disappeared into the field's ordinary language.
Page count is not the point; a good idea gives the mind one handle for many facts, which is why Galois described mathematical elegance as grasping many operations "with a single glance." The strongest evidence arrives later, when other people use the idea to ask questions its author never wrote down.