Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472

In this deeply philosophical episode, Lex Fridman sits down with Fields Medalist Terence Tao to explore some of the hardest unsolved problems in mathematics and their implications for physics and artificial intelligence. Tao begins by discussing the challenge of defining what makes a problem truly difficult, distinguishing between problems that are merely computationally hard versus those requiring genuine conceptual breakthroughs. One of the central topics is the Navier-Stokes singularity problem, which concerns whether solutions to the equations governing fluid motion can develop singularities in finite time. This problem remains one of the most significant unsolved challenges in mathematics and has direct applications to understanding turbulence and fluid behavior. Tao explains the subtle distinctions between partial regularity results and complete solutions, illustrating how mathematicians make incremental progress on seemingly intractable problems. The conversation then shifts to Conway's Game of Life, a deceptively simple cellular automaton that generates complex, unpredictable behavior from basic rules. Tao uses this as a lens to discuss the boundary between discrete and continuous mathematics, and how simple deterministic systems can produce apparent randomness and complexity. This leads naturally into a discussion of infinity, where Tao explores how mathematical intuition often fails when dealing with infinite sets and structures. He discusses different levels of infinity and the philosophical implications of working with concepts that transcend human intuition. A significant portion of the episode addresses the fundamental relationship between mathematics and physics. Tao articulates how mathematics appears to be a pure abstract discipline yet somehow perfectly describes the physical world, raising questions about whether mathematics is discovered or invented. This discussion touches on the nature of reality itself and why mathematical structures seem so effective at predicting physical phenomena. The conversation explores whether a theory of everything is possible and what such a theory might look like, including perspectives on general relativity and quantum mechanics. Toward the end, Tao discusses how mathematicians approach solving difficult problems, emphasizing the importance of understanding the landscape of a problem before attempting solutions. The episode concludes with discussion of AI-assisted theorem proving, where Tao expresses cautious optimism about how machine learning and AI tools might accelerate mathematical discovery while acknowledging the unique human insights required for truly novel breakthroughs.