Generative AI can produce the "miraculous" insights needed for formal proofs, like finding an inductive invariant, which traditionally required a PhD. It achieves this by training on vast libraries of existing mathematical proofs and generalizing their underlying patterns, effectively automating the creative leap needed for verification.

The standard for mathematical proofs is shifting from peer-reviewed papers to formally verified code. This makes math more like a large open-source project, where anyone in the world can contribute. Because the contributions can be computationally certified for correctness, collaboration becomes easier and the field becomes more accessible to amateurs.

Mathematician Terence Tao finds AI doesn't speed up his core problem-solving but makes his papers "richer" by adding complex plots and deeper literature searches. Tasks that were previously infeasible are now easy. AI expands the scope and quality of work rather than just shortening the timeline for existing tasks.

Formal proof systems like Lean provide a unique training ground for LLMs. Unlike natural language reasoning, a proof's correctness can be programmatically verified. This creates a strong reward signal for training long-horizon planning and coherence, skills that can generalize to other tasks.

Languages like Lean allow mathematical proofs to be automatically verified. This provides a perfect, binary reward signal (correct/incorrect) for a reinforcement learning agent. It transforms the abstract art of mathematics into a well-defined environment, much like a game of Go, that an AI can be trained to master.

Proving theorems is only part of math. Axiom is developing tools for the pre-conjecture phase, helping mathematicians find interesting examples and constructions (like graphs or sequences). This AI-assisted discovery builds the intuition necessary before a formal proof can even be attempted.

Verification isn't just a compliance tax or a fix for hallucinations. It's a tool to amplify genius, much like mathematical proofs enabled Ramanujan to scale his intuitive brilliance into theorems that future generations could build upon. Its purpose is to compound superintelligence.

Simply generating a mathematical proof in natural language is useless because it could be thousands of pages long and contain subtle errors. The pivotal innovation was combining AI reasoning with formal verification. This ensures the output is provably correct and usable, solving the critical problems of trust and utility for complex, AI-generated work.

We have formal languages like Lean for deductive proofs, which AI can be trained on. The next frontier is developing a language to capture mathematical *strategy*—how to assess a conjecture's plausibility or choose a promising path. This would help automate the intuitive, creative part of mathematical discovery.