261031 – AI Machine Proofs and The Future of Mathematics

by | Oct 31, 2026 | Physics | 0 comments

The Proof and the Why:

Mathematics After the Machine Proofs

· @Thomas Lee Abshier, ND

A season of machine proofs

In about five weeks, machines produced proofs of problems that mathematicians expected to take whole careers. In early September 2026, Anthropic’s models produced a complete, machine-checked formalization of Fermat’s Last Theorem, some 13 million lines of the Lean proof language built up from the axioms. On September 8, OpenAI announced that an unreleased model had settled one formulation of the Navier–Stokes Millennium Prize problem. On October 6, OpenAI released 722 manuscripts grouped into 372 families of results, including claimed proofs of the quasi-Riemann hypothesis and the Unique Games Conjecture.

Shortly afterward, Lex Fridman sat down with three mathematicians to make sense of it: Steven Brunton, a fluid dynamicist at the University of Washington; Ankur Moitra, a theoretical computer scientist at MIT, visiting the Institute for Advanced Study this year; and Alex Kontorovich, a number theorist at Rutgers who works closely with Lean. Two of them learned of the October release on their phones, drinks in hand, at a National Museum of Mathematics gala in New York. This essay draws on that conversation.

My thesis is simple. The machines have taken over the most visible part of mathematics: producing proofs of problems already stated. They have not taken over its purpose: understanding. A proof that no human understands adds a true sentence to the world, but it adds no knowledge to anyone. The question now facing mathematics, and physics with it, is not whether machines can prove theorems. It is what human beings are for once they can.

What fell, in plain terms

Four results anchor the discussion. Each is worth understanding on its own terms, because the details decide how much each one means.

Result Field What was shown Who
Quasi-Riemann hypothesis Number theory No Dirichlet L-function, the Riemann zeta function included, has a zero with real part above 7/8 OpenAI, Oct 6
Unique Games Conjecture Theoretical computer science Khot’s 2002 conjecture is true OpenAI, Oct 6
Navier–Stokes, forced case Fluid dynamics / analysis A smooth external force can drive a 3D viscous flow to infinite vorticity in finite time OpenAI, Sep 8
Fermat’s Last Theorem Formal verification The full proof, checked by Lean from the axioms Anthropic, early Sep

The quasi-Riemann hypothesis. The prime numbers are the atoms of multiplication: every whole number is a product of them. Riemann discovered that a function of complex numbers, the zeta function, encodes how the primes are spread out. Kontorovich compares it to signal processing. The zeta function decomposes the primes into waves, and its zeros set the frequencies. The Riemann hypothesis says every nontrivial zero lies on the line where the real part equals one-half, so every wave has the same amplitude.

The zeros are known to lie in a strip between real part 0 and 1. For about seventy years the best result was a zero-free region that hugged the line at 1 and grew thinner the higher one looked. The new proof clears an entire strip of fixed width: nothing above 7/8, at any height. That is not the Riemann hypothesis. Kontorovich says the method fundamentally cannot reach one-half, so a new idea is still required. But he did not expect to see this much in his lifetime.

The Unique Games Conjecture. Many important problems are too hard to solve exactly, so we settle for approximate answers. The conjecture says a particular kind of constraint puzzle is hard even to approximate. If true, it tells us that for a large family of problems, the simple approximation algorithms we already have are as good as possible. Moitra describes a decade in which the field’s opinion swung back and forth. Human work, notably the 2-to-1 games theorem of Dor Minzer, Irit Dinur, and colleagues, had opened a plausible path. The machine finished the road.

Navier–Stokes. These equations state momentum conservation for anything that flows. The open question is whether a smooth flow can blow up, reaching infinite velocity gradients in finite time. Brunton gives the simplest model: a quantity that grows in proportion to its own square, x′ = x², runs to infinity at a finite moment, whereas ordinary exponential growth never does. The fluid equations contain a term of the same quadratic kind, and viscosity is the friction that might damp it. The machine constructed a smooth, carefully orchestrated forcing that stretches a vortex into an ever-thinner filament, like a skater pulling in his arms, until its spin becomes infinite while the total energy stays finite. This settles one of the four formulations posed by the Clay Institute. The result also came with a public dispute over credit with mathematician Tristan Buckmaster.

Fermat’s Last Theorem in Lean. Lean is software that checks every logical step of a proof down to the axioms. Kevin Buzzard had a five-year grant to formalize Wiles’s proof, and even he did not plan to reach the deepest foundations. Two days before the drop, Kontorovich told a Princeton audience that AI could formalize material only one or two steps beyond the existing library. Then it formalized the whole thing. The code is, in his words, jerry-rigged rather than elegant, and turning it into library-quality mathematics is real work still ahead.

Is it new mathematics?

Many mathematicians’ first instinct was to ask whether the machine invented anything, or only recombined what humans already knew. Kontorovich’s honest answer for the quasi-Riemann proof is the second. Every move was known to analytic number theorists, and a 2022 breakthrough on cubic Gauss sums by Alexander Dunn and Maksym Radziwiłł supplied the path. What astonished him was the length and order of the chain, a long cascade in which each step was standard and the sequence was not.

Skeptics call this staying inside the convex hull: picture a balloon whose skin is the boundary of human ideas, and the machine only filling in the interior. Kontorovich prefers an image he credits to Tim Gowers. Known ideas are tiles, and you may lay a new tile beside any tile already down. Such tiling can reach far farther than the balloon picture suggests. Many famous problems, he notes, were eventually solved by short, elementary arguments that could have been found decades earlier. What looked like a wall was a missing combination.

Brunton adds that even pure interpolation would be valuable. The balloon’s interior is not transparent. We do not know what lies inside it or what sits next to what, and a tool that lifts that fog is a kind of radar. Moitra makes the decisive point: whatever the method’s novelty, the results move the boundary in practice because they change which problems are worth pursuing. He cites an Anthropic result on the 3SUM problem, written up by Virginia Vassilevska Williams and Josh Alman. A whole theory had been built on the assumption that 3SUM was hard. Learning it is easier than believed sends the field back to studying its problems separately.

I think the convex-hull question matters less than it first appears. Human beings also mostly combine known ideas; genuinely new definitions are rare in any generation. The better test is the one Kontorovich gives later in the conversation: has the result added to anyone’s understanding? A new definition passes that test almost by its nature. A long, correct, unexplained chain may not.

The doorman fallacy

Moitra brought an idea from the social sciences that I found the most useful single frame in the conversation. The doorman fallacy is the mistake of reducing a job to its most visible task and concluding that automating the task replaces the job. A hotel sees a man who opens doors and installs an automatic door. It loses the person who greeted guests, recognized returning customers, handled luggage, kept out trouble, and made the building feel cared for.

The visible task of a mathematician is proving theorems, especially famous ones. That is the part we have popularized, and it is the part the machines now do. But Moitra describes a mathematician as someone who thinks deeply, and proof as one byproduct of deep thinking. Others include new theories, new ways of seeing old things, and connections between fields that turn out to be unreasonably powerful. When one activity becomes cheap, he argues, the activities orthogonal to it become more valuable, not less.

All three named the same orthogonal activity first: asking the right question. Brunton says good problem solving has always been measured by the quality of the sub-questions. He would be impressed by a machine that told him which questions to ask; so far, humans are still the ones posing the problems. Kontorovich pushes back honestly. He has tried several times to name something he does that a language model could never do, and failed; he can always ask the machine what questions to ask. He therefore takes seriously what he calls the solved axiom: that anything a human working with AI could solve, AI alone could solve.

Then he says he does not believe it. Nor does Moitra, who thinks it may be half true, and that finding out which half will be one of the interesting projects of the coming years. Neither is a statement of faith. It is a working judgment from people who use these systems every day.

“But how do you know?”

Moitra retold a story he heard from Camillo De Lellis at the Institute for Advanced Study. A play written by a mathematician in the 1950s imagines John von Neumann arriving in heaven and asking God whether the Riemann hypothesis is true. God says yes. Everyone exhales. Then von Neumann asks how He knows, and God answers that He simply knows where all the zeros are, and they are all on the one-half line. Von Neumann is not satisfied, and in the play he ends up in the other place.

The joke carries the whole argument. Kontorovich put it plainly: mathematics is not the fact but the why, and a fact one is told without understanding adds nothing to one’s knowledge. Moitra warned against reading the machine’s success through human eyes. Deep networks can recognize cats and dogs as well as people do, yet a faint pattern invisible to us can fool them, because they find statistical trapdoors to the answer rather than the shapes we see. He reports that the OpenAI proofs simply do not look as if a human wrote them. A correct proof reached by an alien route may verify the theorem while explaining very little.

The practical consequence is a change in what a paper is for. Kontorovich distinguishes two jobs that papers have always tried to do at once: engineering the proof so every step is valid, and communicating the idea to another mind. These pull against each other. When he teaches real analysis, he no longer spends the first two weeks constructing the real numbers from sequences of rationals. Students already have the number line. He starts with calculus and waits until a theorem fails over the rationals, so that the question “what are the reals?” arrives when it is needed.

With Lean checking the engineering, he expects future papers to carry the story: where the obvious path stalls, why one goes left instead of right, what the key special case reveals. The formal file guarantees the details; the prose carries the understanding. That division strikes me as the most hopeful practical idea in the conversation, because it lets each partner do what it does best.

The physicist’s angle: equations are not phenomena

For readers of a physics site, Brunton’s part of the conversation deserves the most attention. He is blunt: the Navier–Stokes result is an important theorem, and it changes almost nothing in practical fluid dynamics. Turbulence is as hard to simulate and predict today as it was two months ago. His summary is that we solved the million-dollar mathematics problem while the trillion-dollar engineering problem remains: predicting flows quickly and designing wings, turbines, and artificial hearts from the outcome backward.

Kontorovich asked him the right question. When Schwarzschild found a singularity in Einstein’s equations, physicists suspected an error, until black holes turned out to be real. Does a blow-up in Navier–Stokes predict something new about water? Brunton’s answer is no. The equations average over many molecules; long before a vortex filament becomes infinitely thin, the fluid description fails, and molecular physics takes over. The singularity marks the edge of the model, not a property of water.

That distinction is at the heart of physics. Brunton recalls Feynman’s observation that we can write down the general equations, Maxwell’s, Navier–Stokes, Schrödinger’s, yet we cannot read off from them the phenomena they contain. No one who had never seen Earth could deduce sunsets, tides, and hurricanes from a partial differential equation. Whether a solution exists at all is, in Brunton’s words, a baby step toward knowing what an equation actually holds.

He adds a historical pattern I think is right. Our great physical breakthroughs were preceded by a change of coordinates. Newton’s laws came after the sun replaced the Earth at the center. A gas of 10²³ molecules is described by three numbers, pressure, volume, and temperature, because someone found the right variables. Turbulence has no such description yet; every scale matters. Yet insects fly through turbulent gusts using a few dozen strain-sensing neurons on their wings, correcting faster than a signal could reach the brain. That is proof by existence that a small set of the right variables suffices for control. Finding them is a problem of understanding, and AI may help with it. But it is not the same problem as proving theorems about the equations we already have.

The lesson I draw is that a proof about a model is not yet knowledge of nature. Physics has always lived in the gap between the equation and the world, deciding which mathematics describes reality and where it stops. That judgment is not made inside the formal system, and no Lean file can make it.

The cost to students and institutions

The human cost is real, and the conversation did not hide it. Fridman read a statement from Terence Tao, who said he was horrified that years of patient, deliberately slow work by graduate students and postdocs had been casually disrupted. Tao compared it to a movie spoiler: once you know a solution exists, you cannot explore the problem as richly, and he mourned the paths that will now not be taken. Moitra reported that when he asked Subhash Khot, who proposed the Unique Games Conjecture, whether he would read a machine proof of it, Khot said no.

Moitra named the institutional loss precisely. For generations, a young researcher had a simple blueprint: pick a problem people care about, improve it, and doors open. That blueprint is gone, and nothing has replaced it yet. He sees students thinking in three-month horizons, afraid the next model will take whatever they choose, and he worries researchers will start hiding their key ideas, since one published missing ingredient may be all a machine needs. Journals were already overloaded before any of this. Kontorovich defends their slowness anyway, recalling how Perelman’s proof of the Poincaré conjecture was confirmed only after years of lectures and line-by-line questioning. Truth in mathematics has always been a community verdict, not a file.

The prizes are in question too. Kontorovich says mathematicians at the Fields Medal ceremony this summer were already wondering whether it would be the last. A prize celebrates what a community values. If solving famous problems becomes common, the prize must learn to honor something else.

The advice the three gave was sound and old-fashioned:

  • Learn the hardest, most abstract mathematics your mind can hold. Kontorovich sees undergraduate grades splitting in two: students who use AI to test their understanding are rising, and students who use it to produce homework are falling.
  • Master the basics anyway. Calculators arrived decades ago, and students told to skip the times tables suffered for it.
  • Be curious, and protect the curiosity. Brunton thinks boredom is underrated; focus often grows out of it.
  • Work for reasons of your own. Moitra’s advice, passed down from Sanjeev Arora, is to act as if you already have tenure and do what you would do then.
  • Remember who matters. Kontorovich said that after the Fermat formalization he took his children to the beach for Labor Day, and that is the foundation of his life; the mathematics is the cherry on top.

Where I stand

I stand with those who say mathematics is not ending but shifting its center of gravity. Its center has been producing proofs. It is moving toward what proofs were always meant to serve: understanding them, choosing the questions worth asking, building the concepts that make whole fields of problems fall at once, and deciding which mathematics describes the physical world.

There are honest areas of uncertainty, and I want to name them without letting them blur the claim:

  • Capability. No one knows where these systems level off. Brunton sees two possible worlds within five or ten years: one in which we are dwarfed by another intelligence, and one in which AI becomes an extraordinary instrument for human curiosity. Either could happen.
  • The record. The companies’ accounts of how much human guidance went into these results are not independently known. Moitra notes that only about one in ten attempted problems succeeded, and we see only the successes. The Navier–Stokes result carries an unresolved credit dispute.
  • Words. “Solve,” “new idea,” and “understand” are used loosely, including by me. The convex-hull debate shows how much depends on their definitions.

The claim holds for reasons that do not depend on how those uncertainties resolve. Understanding cannot be delegated, by definition: to understand is to have the thing whole in a mind, and a result that lives only in a data center is understood by no one. Mathematics has been “killed” before; the nineteenth-century crisis over infinity, when Cauchy’s and Fourier’s theorems contradicted each other, broke it and forced its rebuilding, and most of today’s graduate curriculum is the result. Strong incentives pull the other way. Companies gain from presenting a single prompt as the whole story, and institutions that reward paper counts will reward flooding the archives with machine output. Brunton added a warning worth repeating: intelligence is not morality, and greater intelligence often brings a greater power to rationalize. A society that equates the two will defer to whatever system argues best.

The practical stance follows. Use these systems as Kontorovich uses them: tireless tutors who find a person’s next step and checkers of detail. Do not let them replace the struggle that produces understanding. When Andrew Wiles finally saw how to repair his proof, he later said he would never again have a moment like it. A helicopter that sets a climber on the summit gives him the view but not the climb, and in mathematics the climb is how one learns to see.

Coda: understanding is an experience

Kontorovich said something early in the conversation that I want to end on. The value of pure mathematics, he said, is not that theorems exist as ones and zeros in data centers; it is that mathematics lives in human heads. Brunton added that the universe’s willingness to be described by simple mathematics is a miracle we have no a priori right to.

I agree with both, and I would say why from my own conviction. I hold that God inhabits and experiences every person’s life from the inside, so that our seeing is His experience of His own creation through us. If that is so, understanding is not a byproduct of mathematics. It is the point of it. When a person finally sees why the primes fall as they do, or why a vortex stretches and spins up, something happens that a stored proof cannot do: the order built into the world is known from within by a mind that can delight in it. A proof no one understands is like a letter no one reads. It may be true in every word, and still no one has been addressed.

The machines can now write the letters. It remains for us to read them, and to keep asking the question von Neumann asked: not only whether it is true, but how we know, and why.

Sources

This essay was drafted with Claude, an AI model made by Anthropic, one of the companies whose results it discusses.

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