Terence Tao: AI proofs make mathematical understanding more urgent

Tyler Cowen welcomes UCLA mathematician Terence Tao, author of Six Math Essentials, and asks what different mathematical ideas make him feel.

Tao finds Euler’s identity beautiful because it connects constants and expresses rotation as multiplication. Multiplying at right angles links algebra, geometry and motion.

Quantum mechanics requires unfamiliar mathematics and a change in intuition: superpositions and interference describe behavior better at molecular scales, even though the idea is difficult to picture in daily life.

The Riemann hypothesis concerns the distribution of primes. Tao says a proof would strengthen many conditional number-theory results and have value beyond solving a puzzle.

Tao reports recent progress but says a full resolution remains distant. He describes a new difficulty: AI can produce ideas or answers that mathematicians must work to understand and communicate.

Gödel’s incompleteness theorem challenged Hilbert’s hope of settling every mathematical question within fixed rules. Tao says mathematicians have adapted to considering different foundations and systems.

He describes a pluralistic view: there is a standard mathematics, but alternative geometries and number systems are also useful. He tries to update his philosophy when new evidence changes the field.

Tao says the history of mathematics gives the subject stories and context that classes often omit. Those stories can make equations and methods come alive.

He compares mathematics to the soil of the science and engineering ecosystem: it provides foundations and a common language, even when applications appear only decades later.

Information theory can establish a limit on a communication channel, not merely demonstrate a better system. Knowing that an approach is near its theoretical limit can tell engineers when further optimization is unlikely to pay off.

Tao says string theory has inspired geometry, while cautioning that he is not a physicist and that decisive empirical confirmation of it as a model of nature remains elusive.

Although Cowen senses a special love of geometry in his book, Tao says analysis is more natural to him. Geometry often requires writing and visual aids, which makes the ideas he does understand especially satisfying.

Asked why real analysis developed relatively late, Tao discusses approximation and limits. Treating a fluid as continuous, for example, can be simpler than tracking every particle.

Zero required a conceptual leap because absence is not an object one can point to. Arithmetic rules also led people to negative numbers, fractions and roots that cannot be counted as ordinary objects.

Tao finds economics useful for thinking about models and reality. Game-theory language can clarify mathematical arguments, while comparative advantage helps him decide how to spend his own time.

Unlike very well-tested physical models, economic models may predict some situations well and others poorly. He says using a model includes understanding where its assumptions break down.

Tao calls Black-Scholes a beautiful example of a model with specific assumptions, and warns that applying it beyond those assumptions can be dangerous.

On the simulation argument, he says Bayesian reasoning compares possible models but can miss an explanation no one considered. Intellectual humility matters because imagination limits the hypotheses on the table.

Newcomb’s paradox illustrates that more apparent freedom or calculation can sometimes be penalized in a particular game. The two discuss how credible commitments can change incentives in related games.

On US math education, Tao stresses the limits of his own K–12 evidence. Small experiments can look promising but fail when scaled because teachers, institutions and communities differ.

He supports more statistics, probability and linear algebra as data-intensive work grows, but values math above all as a low-cost place to practice modeling, uncertainty and problem solving.

Advanced options for interested children can work where teachers and institutional support exist, Tao says, but forcing every student into them can make the experience miserable.

Quantitative finance provides jobs for mathematicians beyond a small academic market and can improve pricing and liquidity. Tao also notes that badly chosen metrics can contribute to instability.

Tao describes a shift from scarce proofs toward abundant machine-generated proofs. Instead of treating a few landmark problems as the only unit, he expects researchers may study much larger collections of problems.

He is excited about possible new mathematics but disturbed by the speed and disruption of the transition. He argues that moving too quickly can destroy skills and cultural value before their importance is recognized.

A human paper traditionally came with a trained author who could explain the result, mentor students and form collaborations. Tao says AI can unbundle the PDF from that surrounding human expertise and communication.

He expects papers to show more of the process and research to move toward connected, queryable databases and larger collaborative projects involving humans and AI.

Exposition may become a primary output: an interactive talk lets people ask questions and use visual aids in ways a static paper cannot. Tao thinks human-to-human explanation becomes more valuable amid a flood of papers.

The conversation turns to personal interests. Tao likes Stephen Chow’s Kung Fu Hustle and discusses music, observing that easy access to a large catalog can sometimes mean listening less.

His parents arranged an unusual education: he advanced in math and science while taking other subjects with his age group, and attended nearby Flinders University while still living at home.

He recalls failing classes when he refused to learn Fortran or complete an essay, and says the small, supportive environment suited him before he moved to Princeton for graduate study.

Failing an unexpected oral qualifying exam changed his habits: he worked harder on both the subject and explaining what he knew.

Tao says he is learning to delegate to junior collaborators and AI. Giving motivated people responsibility can produce ideas he would not have imagined, and delegation also helps him manage fatigue.

Asked whether the Collatz conjecture is true, Tao says he believes it is but sees no proof. The rule repeatedly triples an odd number and adds one, or halves an even number; the question is whether every positive integer eventually reaches one.

Many cases and statistical results support the conjecture, but a rare exceptional sequence cannot yet be ruled out. Tao doubts he will see a proof in his lifetime, even with AI.