Key Takeaways

  • Justin Solomon, associate dean of engineering education at MIT, points out that machine models operate inside the convex hull of existing human math rather than generating new theories.
  • Private labs with billions in compute can search vast proof spaces in formal languages like Lean, leaving universities behind on brute-force derivation.
  • Proof generation is largely mechanical verification: Solomon notes that mathematicians usually already know a theorem is true before spending months writing the formal proof.
  • The human moat in research and software engineering is shifting from solving equations to question formulation and mathematical taste.

The Convex Hull Trap

Silicon Valley treats mathematics as the ultimate benchmark for machine intelligence. If an algorithm can generate a rigorous proof, solve an Olympiad problem, or parse complex fluid mechanics, engineers assume creative thought has been solved. Solomon pushes back on that view.

He argues that current models simply rearrange pieces from an established library. “When you look at a lot of the proofs and the new results that are coming out of these AI systems, they're arguably in what mathematicians might call the convex hull of existing knowledge,” Solomon explains. “There's sort of a bunch of stuff we already know, and then a lot of these proofs are really intricate and interesting ways to mix and match those things.”

Mixing and matching existing facts can solve hard homework problems. It can check lines of code in formal verification languages like Lean. It can even help private AI labs search for boundary cases, such as OpenAI's work on counterexamples to the Navier-Stokes equations. But remixing known inputs is not the same thing as inventing a new branch of geometry.

Taste Is the Only Scarcity Left

Calculations used to be expensive. Then calculators automated arithmetic. Today, machine learning models automate symbolic manipulation and line-by-line derivation. If finding an answer requires only compute, private industry will always beat individual researchers by throwing thousands of GPUs at the problem.

That shifts all economic and intellectual value to the start of the pipeline. In Solomon's view, the technical mechanics of a proof are the least creative part of the job. “By the time we get to writing a proof, we probably already know it's true,” Solomon says. “It's more double-check situation, but the really interesting part is on the modeling and figuring out which questions to ask and what insights they bring us to the universe.”

Tracy Alloway observed during the conversation that people outside the field rarely associate mathematics with personal aesthetic choices. Yet taste dictates what deserves your attention. Anyone can point an automated prover at ten thousand arbitrary lemmas and verify every one of them. Ninety-nine percent of those lemmas will be completely useless. Deciding which relationship matters, which system models reality, and which question opens a fresh line of inquiry requires judgment that algorithms do not possess.

“A lot of mathematics is more about taste than it is about just proofreading and verifying some other conjecture that somebody wrote down,” Solomon notes. “Can a mathematical AI system actually come up with a truly surprising, new result that's like not just mixing and matching facts we already know... but making an entirely new theory? I don't think we've seen a whole lot of examples of that so far.”

What to Do With This

Take your company's product roadmap or technical spec backlog this week and split every project into two columns: derivation and question formulation. If a project merely executes a known optimization inside the convex hull of your industry, hand the boilerplate drafting and testing to automated coding agents immediately. Spend your saved hours interrogating whether the feature question you are answering is even worth solving.