Here is a 3-paragraph summary of Thomas Hales' blog post for mathopen.com:
Thomas Hales opens by making the case that mathematics derives its unique value from its consistency and reliability, qualities that have made it an indispensable foundation for science and civilization. With that standard in mind, he turns his attention to the Lean Theorem Prover, a powerful tool that allows mathematicians to write formal, machine-verified proofs. Lean represents a significant shift in how mathematical truth can be established and checked, moving beyond peer review toward a more rigorous computational standard.
Hales examines the question of reliability head-on, asking what it actually means to trust a proof verified by Lean. He explores the layers of trust involved, from the underlying logical foundations of the system to the correctness of the software itself. While no system is perfectly free of risk, he argues that Lean offers a level of verification that goes well beyond what traditional mathematical practice can guarantee, making it a genuinely valuable tool for building dependable mathematical knowledge.
The post also addresses the growing role of artificial intelligence in mathematics, particularly how AI tools interact with formal proof systems like Lean. Hales considers both the promise and the pitfalls of combining AI with theorem proving, noting that AI can assist in finding proofs but also introduces new questions about trust and verification. His overall message is that mathematicians should take Lean seriously as both a practical resource and a reflection of deeper questions about what it means for mathematics to be truly reliable.