The speaker reflects on the idealized view that mathematics is a flawless edifice built solely from self‑evident axioms through pure deduction, noting that this “fantasy” is far from reality. He discusses how modern proof assistants such as Lean formalize mathematics by storing verified results and checking each logical step, acting like a relentless colleague that forces mathematicians to clarify vague parts of their arguments. While these tools excel at verification, the challenge remains to get computers to generate novel proofs rather than merely follow human input. The talk raises philosophical questions about what counts as a proof, the role of primitives, and how increasing reliance on AI might reshape mathematical training and the value we place on deep, human‑crafted proofs. Ultimately, he wonders whether, in a few decades, the point of doing mathematics will change as machines take over more of the proof‑generation process.
1. Andrew Granville works in analytic number theory.
2. Before it was proved, he worked on Fermat's Last Theorem.
3. He currently works on L‑functions and multiplicative functions.
4. He is interested in computational and algorithmic questions in mathematics.
5. He has long been interested in popular writing about mathematics.
6. His sister is a writer.
7. Together they developed a graphic novel about mathematics.
8. Michael Hallett, a philosopher of mathematics, read the graphic novel and was interested in its portrayal of how mathematics is done.
9. Aristotle said that to prove something true, an argument should rest on primitives already known to be true.
10. Eventually there must be some axioms that serve as the foundation for all arguments.
11. In geometry, point, line, and plane cannot be defined; instead, axioms are laid down about them.
12. The phrase “these truths are self‑evident” is often associated with such axioms.
13. Traditionally, mathematics is communicated by publishing papers in journals or books, which are stored in libraries for verification.
14. Artificial intelligence used for proof stores information within the program; systems like Lean maintain a library of already‑proved statements based on axioms.
15. A mathematician can input a proof into Lean and have the system check each logical step.
16. Lean behaves like an obstinate colleague that repeatedly asks for clarification when a step is not understood.
17. Peter Schulzer used Lean to verify a difficult proof; Lean prompted him to explain the parts where he felt uncertain.
18. The current question is whether machines will change mathematics and how to get computers to lead in a proof rather than merely follow or suggest.
19. Much of the work on computer‑generated proofs to date has not produced significant consequences.
20. Some researchers are optimistic that new ideas may lead to the generation of interesting new proofs by computers.
21. If machines can reliably handle most proof details, the role and training of human mathematicians could change.
22. Without needing to think about proof, mathematicians might not be trained in proof‑thinking.
23. It is unclear what limits exist to what computers can eventually do in mathematics.
24. Predicting the purpose of doing mathematics in 30–40 years is difficult.