Terence Tao on AI and the Future of Math Proof

Two days before this conversation was recorded, something happened to one of the oldest open problems in mathematics that unsettled even its most eminent living student. Terence Tao is a Fields Medal winner and professor of mathematics at UCLA, often described as the most versatile mathematician alive, and he arrived with news: an AI system had produced a significant advance on the Riemann hypothesis, a century-old conjecture about how prime numbers are spaced. Primes are the numbers divisible only by themselves and one, and they underpin the cryptography that protects financial transactions (03:30). The Riemann hypothesis would give mathematicians a precise way to describe how randomly the primes behave (03:48). The trouble is that nobody, including Tao, fully understands the new idea yet. "Unfortunately, the ideas were generated by AI, and we don't yet understand them," he says. "We are getting answers without understanding, which is a very strange era to be in" (04:41).

That sentence is the spine of the whole episode. For most of mathematical history, understanding and proof were the same thing: you could not claim to have solved a problem unless you could explain, to another human, why the solution worked. Tao says that link has now been "decoupled quite severely" (08:01). Machines can hand over correct answers without a trail of reasoning a person can follow. It is a little like inheriting a working bridge with no blueprints: useful, even beautiful, but unsettling if you are the one responsible for knowing why it stands.

Tao places this moment inside a longer pattern of upheaval. A century ago, the German mathematician David Hilbert believed every open mathematical question could eventually be settled, declaring, in a phrase Tao quotes in the original German, that mathematicians must know and will know (05:39). Then the Austrian logician Kurt Gödel proved otherwise. His incompleteness theorem shows that no fixed set of logical rules can ever answer every question that can be posed within it; some statements will always be neither provable nor disprovable from those rules alone (06:02). The field absorbed the shock over decades and came out the other side more comfortable with multiplicity. Tao describes modern mathematics as having moved toward what he calls a pluralistic view, with different number systems, different geometries, even different foundational rule sets coexisting the way chess has standard rules and also sanctioned variants (06:39). There is no longer one true mathematics, only a family of consistent ones, each useful for different questions.

From Scarcity to Abundance

The AI moment, in Tao's telling, is the next installment of that same story: another forced update to what mathematicians think proof and understanding even mean. He frames the shift in a single line: "In one sentence, we have moved from an era of proof scarcity to proof abundance" (29:08). For decades the field organized itself around a small number of landmark open problems, like the seven Millennium Prize Problems that carry a bounty for their solution (29:18). Scarcity made sense when every proof was hard-won; a handful of famous unsolved questions could serve as shared goals that guided generations of careful, slow work. Now machines can brute-force through many of the old landmark problems at once, and Tao thinks the individual problem is losing its old status as a unit of progress. He imagines a coming "Math 2.0" built around enormous shared datasets of thousands or millions of problems, tackled in bulk rather than one trophy at a time (30:01).

He is honestly ambivalent about this. Asked whether he is happy about the change, he resists the question's binary: "That's not the right word... I'm excited by the possibility of completely new types of mathematics that didn't exist before, but I'm also actually quite horrified by the disruption, which I think was much more than necessary" (30:41). His analogy is the Industrial Revolution: transformative and, on net, good for living standards, but traumatic in its speed, with entire careers and bodies of craft knowledge cut short before anyone could transition. He worries mathematics is being pushed through that same compression in years rather than decades, with less time to preserve what might later turn out to matter.

One casualty, he suspects, is the research paper itself. A paper used to be bundled with everything that came from writing it: a trained author who could give talks, train students, and make new connections at conferences. AI can now generate the paper without any of that penumbra (32:48). Tao thinks the bottleneck in mathematics has quietly shifted from producing results to explaining them in ways another human can absorb, since the flood of machine-assisted output already vastly outpaces what anyone can read. "The bottleneck is now what are the results that can be explained human-to-human," he says (35:42), adding that printed papers were always a clumsy format compared with a conversation, where a listener can ask questions and read body language. He has started a venue that accepts recorded talks instead of written papers, betting that exposition, once treated as a secondary byproduct of research, is becoming one of its most valuable outputs.

The Soil Beneath the Ecosystem

If AI is reshaping how proofs get made, Tao is just as interested in why mathematics mattered before any of this. He describes science, engineering, and mathematics as one ecosystem, with mathematics as the soil: it rarely produces anything a consumer touches directly, but it sets the limits everything else grows within (09:33). His example is wireless communication. Information theory, a branch of mathematics built by Claude Shannon, can calculate the absolute maximum number of bits a channel can carry. If an engineering team's wireless protocol is already operating at 98 percent of that theoretical ceiling, no amount of further tinkering or spending will meaningfully improve it (10:28). Only mathematics can tell you such a ceiling exists at all; engineering alone can only tell you what has already been built and shown to work.

That same respect for quiet, structural ideas shows up in Tao's account of zero. The concept did not arrive until roughly the seventh century in India, startlingly late given how simple it now seems to a schoolchild. His explanation is almost anthropological: naming follows seeing. People will name a sheep, or two sheep, because they can see them, but there is no visible object called "zero sheep" to point at (14:58). Zero, negative numbers, and square roots all had to be inferred from the logic of arithmetic operations before anyone could see a use for them, which is why he compares the discovery of concepts like infinity to handling radioactive material: powerful once understood, but something that took centuries to learn to use safely (14:22).

Tao also treats economics as a genuine intellectual playground rather than applied arithmetic. He uses the economist David Ricardo's law of comparative advantage, originally devised to explain why nations should specialize and trade rather than each try to do everything, as a personal rule for deciding which tasks deserve his own limited time (16:48). He finds the Black-Scholes options-pricing model beautiful for a narrow mathematical reason: it works because of an algebraic coincidence that only holds when market fluctuations follow a Gaussian, or bell-curve, distribution; apply it where that assumption fails, especially compounded through layered derivatives, and the same elegant formula becomes dangerous, as the 2008 financial crisis showed (18:02). And he is drawn to Newcomb's paradox, a thought experiment in which an apparently all-knowing predictor rewards people who commit to taking only one of two boxes rather than both. Its lesson, he says, is that more rationality and more free will are not always advantages: sometimes visibly giving up a choice, the way a driver in a game of chicken might throw the steering wheel out the window, is the winning move precisely because it removes the other side's reason to gamble (22:26).

The conversation closes on something more personal: failure as a tool rather than a verdict. Tao calls mathematics "a good playground to think where the cost of failure is cheap," unlike civil engineering or surgery, where a mistake can be catastrophic and irreversible (25:16).

"I view math as kind of a good playground to think where the cost of failure is cheap." - Terence Tao [25:16]

He speaks from experience. He skipped five grades in math and science as a child and entered university at 13 (38:15), yet failed two undergraduate classes out of sheer stubbornness, and later failed his graduate oral examination at Princeton so badly that his advisor could not be convinced he understood his own field (39:56). He says that failure changed his life, pushing him to study and to explain himself far more carefully from then on. It is a fitting note to end on, given where the conversation began. Asked, finally, whether the Collatz conjecture, a famously simple but unsolved rule about repeatedly transforming numbers, is true, Tao says he believes it is true but that mathematics currently has no tool that can rule out one freakishly lucky exception hiding somewhere in the infinite numbers still untested (42:39). Even for the field's most fluent translator of hard ideas into plain ones, some problems remain, for now, honestly unexplained.