Cheap Feedback

AI progress in mathematics is better classified by feedback cost than by the usual distinction between combinatorial and structural work. The relevant distinction is between tasks with cheap feedback and tasks without it. An external checker, a short counterexample, a formal proof assistant, a benchmark, or a simulation loop turns intellectual work into a search problem with a scoring rule. In that regime, more sampling, better search, and more inference-time computation have a direct path to improvement.

This explains why AI systems look strongest on competition mathematics, finite combinatorial search, code, and theorem proving in formalized environments. The model can generate many candidates and keep the ones that pass a verifier. Human difficulty is a weak predictor. The limiting variable is the cost of detecting failures and certifying successes.

The same distinction also explains the negative cases. There is no comparable verifier for “this abstraction will be useful”, “this conjecture is deep”, or “this proof is the one worth understanding”. Those judgments require taste, compression, analogy, and knowledge of where a field can move. The labels are often unavailable rather than merely expensive.

For use, the practical rule is simple. AI is safer when it checks, attacks, searches, or fills in material whose correctness can be independently verified. It is less safe when it generates structure beyond the user’s ability to evaluate. The failure mode is an unfavorable cost ratio: error detection becomes more expensive than generation.

Blog Use

Use this near the beginning. It gives the essay its basic split: work with cheap checks, and work where the check is slow, expensive, or missing.

Source Trail

  • Jeremy Avigad, “Mathematicians in the age of AI”, arXiv:2603.03684.
  • Tanya Klowden and Terence Tao, “Mathematical methods and human thought in the age of AI”, arXiv:2603.26524.
  • Noga Alon et al., “Remarks on the disproof of the unit distance conjecture”, arXiv:2605.20695.
  • ../ai_views.md, sections on proof generation, verification, digestion, and red-team/blue-team use.
  • See reference_inventory.md for full source status.