Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472
Terence Tao explains why Navier-Stokes may require a liquid Turing machine to resolve and why AI-assisted formal proof is the next inflection point in mathematics.
- Tao’s 2016 averaged Navier-Stokes blowup paper rules out entire classes of proof strategies, showing any regularity proof must exploit features his artificial equation lacks.
- His road map for a full Navier-Stokes blowup involves constructing a self-replicating von Neumann machine made of water — smaller, faster copies powering down the parent before iterating to a singularity.
- Navier-Stokes is “supercritical”: at small scales transport terms dominate viscosity, which is precisely why weather prediction fails beyond ~2 weeks and why 2D has a proof but 3D does not.
- The parity barrier caps prime density inside sieve sets at 50%, blocking twin primes and Goldbach; Tao calls breaching it one of his long-term dreams.
- Twin prime conjecture is fragile — removing 0.1% of well-chosen primes can make it false while passing all statistical tests — whereas arithmetic progressions survive deleting 99% of primes.
- DeepMind’s AlphaProof and formal proof systems like Lean are, in Tao’s view, converging toward a genuine shift: he moved from advocating to actively formalizing his own work to justify the claim.
- Kevin Buzzard has a five-year grant to formalize Fermat’s Last Theorem down to objects known by 1980, with the remaining foundational gap left for future work.
- Universality (why simple macro-laws emerge from micro-chaos) is mathematically understood only in toy cases; the 2008 financial crisis showed the cost of mistaking Gaussian models for universal ones when systemic correlation breaks the assumption.
Guests: Terence Tao, UCLA mathematician, Fields Medal winner · 2025-06-14 · Watch on YouTube
