Mathematicians are increasingly turning to artificial intelligence and formal proof assistants to verify and accelerate research, with Fields Medalist Terence Tao emerging as one of the most vocal advocates for the trend. In recent months, Tao has used the Lean theorem prover and large language models to tackle problems that would have taken months of solo work, signaling what many researchers describe as a structural shift in how mathematics is done.
A New Workflow for Pure Mathematics
Tao, a professor at UCLA and one of the most decorated mathematicians of his generation, has documented his experiments with AI tools throughout 2024 and into 2025. He has used Lean, a programming language and proof assistant developed at Microsoft Research, to formalize complex theorems and check them line by line for logical errors. In parallel, he has paired the system with general-purpose chatbots like ChatGPT and GitHub Copilot to draft proof sketches, suggest lemmas, and translate informal arguments into formal code.
In a widely circulated post on Mathstodon, Tao described how a project that he initially estimated would take a year of solitary effort was completed in roughly three weeks with a small team using Lean and the collaborative platform Blueprint. The project involved formalizing a recent result on the polynomial Freiman–Ruzsa conjecture, a problem in additive combinatorics that had stood as a major open question until its resolution in late 2023 by Tao together with Tim Gowers, Ben Green, and Freddie Manners.
Why Formalization Matters
For most of the discipline’s history, mathematical proofs have been written in natural language and verified by human referees. This system has worked remarkably well, but it is not infallible: errors slip into peer-reviewed papers, and some long proofs — such as the classification of finite simple groups — are so complex that few people on Earth can claim to fully understand them. Formal proof assistants address this by requiring every logical step to be checked by a computer against a small, trusted kernel of axioms.
The appeal of tools like Lean and its mathematical library, Mathlib, lies in their combination of rigor and reusability. Once a theorem is formalized, it becomes a building block that anyone can invoke without having to re-verify it. Mathlib has grown to include more than a million lines of formalized mathematics, covering areas from category theory to measure theory, and it is maintained by a global community of volunteer contributors.
The Role of Large Language Models
While Lean provides the rigor, large language models supply the speed. Tao has noted that LLMs are particularly useful for the tedious work of suggesting which lemma to apply next, fixing syntactic errors in Lean code, and bridging the gap between a mathematician’s intuition and the formal grammar required by the proof assistant. Researchers at DeepMind have pushed this further with systems like AlphaProof, which reportedly achieved silver-medal performance at the 2024 International Mathematical Olympiad by combining reinforcement learning with Lean. Coverage by Nature and other outlets has framed these advances as early but meaningful evidence that AI can contribute to original mathematical discovery, not merely pattern-match against existing literature.
Skeptics caution that current systems still hallucinate, struggle with genuinely novel ideas, and require expert oversight at every step. Tao himself has been careful to describe AI as a collaborator rather than a replacement, comparing it to a tireless but inexperienced graduate student who needs constant supervision. Still, he has argued that the productivity gains are real and that the sociology of mathematics — historically a solitary craft — may shift toward larger, more software-intensive collaborations.
What to Watch Next
The next test will be whether AI-assisted formalization can move beyond verifying known theorems and contribute to genuinely open problems. Several groups are now formalizing parts of the proof of Fermat’s Last Theorem in Lean, a project led by Imperial College’s Kevin Buzzard, while DeepMind and OpenAI continue to release benchmarks aimed at olympiad-level reasoning. If these efforts succeed, mathematics could become one of the first scientific disciplines in which machine-checked rigor and AI-driven creativity are standard parts of the research workflow — a development whose consequences for education, publication, and even the philosophy of proof are only beginning to be explored.
