Applied mathematics has met the machine before

This is the long version of my essay Applied mathematics has met the machine before, published on 28 September 2026 on Proofs and Prompts, to which I contributed it as a guest. The discussion takes place under the published version.

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Dr. Denys Dutykh, Associate Professor at the Mathematics Department of Khalifa University of Science and Technology, Abu Dhabi, UAE, and Associate Dean of Graduate Studies at the College of Computing and Mathematical Sciences

Abstract. Why does AI worry applied mathematicians less? We have met the machine before, from human computers to the soliton. Driven by problems rather than great conjectures, we expect AI to make us better mathematicians. When it reaches our own equations, as with Navier–Stokes, the answer remains verification and validation.

The voices we hear most often about artificial intelligence, mathematics, and the crisis which AI is supposed to bring to mathematics come from the pure mathematics community, and the same is true of most of the essays published on Proofs and Prompts so far. There are exceptions here, and I read them with particular attention: the reflections of Nilima Nigam, the dilemma described by Ruodu Wang, and the responses of two data scientists, Anatoly Vitold Stankyavichyus and Simon Hayward, who look at our community from downstream. But the applied mathematician who spends the day between models, equations and code has remained rather silent. I would like to say something from the other side, or rather from the adjacent side, because applied mathematics has never been far from pure mathematics.

Let me say first what this post is and what it is not. I am an applied mathematician. My work is in fluid mechanics and water waves, in the numerical methods which allow us to compute them and, more recently, in the perturbation theory of black holes. The views below are my own. I do not pretend to represent a community which is huge and very diverse, and if any applied mathematician reading these lines disagrees, or has something to add, I invite them to complement or correct me in the comments below. A small clarification on words: by "AI" I mean, roughly speaking, the large language models and the generative tools built on them, nothing broader; I use the short word only for simplicity.

My main point is simple. Applied mathematics welcomes the current developments in AI, and I expect a major boost in our discipline if our community embraces these tools with the standards of verification and validation which we have always applied to our software and to our results. To explain why the mood on the applied side is so different from the mood of many of my pure colleagues, I have to begin with some history, because we have been here before.

Applied mathematics has met the machine before

The history of applied mathematics is as old as the history of mathematics itself. Herodotus tells us that geometry was born in Egypt, where the king sent his surveyors to measure the fields again after each flood of the Nile, so that an owner whose land had been taken by the river would pay a smaller tax; the word geometry itself means measuring the earth.¹ Long before anyone distinguished pure from applied mathematics, mathematics was a service. More than two thousand years later, when the first asteroid, Ceres, was lost in the light of the Sun a few weeks after its discovery in 1801, the young Gauss computed its orbit from these few weeks of observations and told the astronomers where to look; they found it there at the end of the year.² Computation was the craft of the mathematician, and nobody thought that it diminished him.

The modern story begins with Lewis Fry Richardson. In 1910 he used finite differences, computed by hand, to find the stresses in a masonry dam.³ Then he tried to compute the weather. In his book of 1922 he described a six-hour forecast for 20 May 1910, carried out with pencil and paper during the First World War, while he served as an ambulance driver in France, and the result was famously absurd: a change of the surface pressure of 145 hPa in six hours, while the real pressure had hardly moved. I used to believe, as many of us do, that his scheme was unstable. The truth is more interesting. Peter Lynch showed, decades later, that the method was sound and that the failure came from the initial data, which contained unbalanced winds and pressures; once these data are filtered, the same computation gives a sensible answer.⁴ Richardson dreamed of a "forecast factory" in which sixty-four thousand human computers would compute the weather of the globe in real time. Machines realised his dream within thirty years.

The mathematics needed to understand such computations came from a pure motive. In 1928 Courant, Friedrichs and Lewy used finite differences as a tool to prove the existence of solutions of partial differential equations, and on the way they found the condition on the time step which every student of numerical analysis learns today under their names.⁵ Fifteen years later Courant proposed piecewise linear functions on triangles to solve a variational problem, an idea which engineers rediscovered independently and which we now call the finite element method.⁶ Pure motives produced applied tools, and applied needs produced theorems. The two communities have always fed each other, and this is the first thing I want to keep in mind when we speak about them as if they were separate.

Then the electronic computer arrived, and it arrived for military reasons. ENIAC was built to compute firing tables for the artillery, and its first large computation, in December 1945, was a feasibility study for the hydrogen bomb.⁷ Five years later Charney, Fjørtoft and von Neumann produced on the same machine the first weather forecast ever computed by a computer: twenty-four hours of computing for a twenty-four-hour forecast.⁸ In 1955 Fermi, Pasta, Ulam and Mary Tsingou, who wrote the program, performed at Los Alamos what is now considered the first numerical experiment. The surprising recurrence which they observed in a chain of nonlinear springs led Zabusky and Kruskal, ten years later, to the discovery of solitons, and from there to the inverse scattering transform and to the theory of integrable systems.⁹ A whole branch of pure mathematics, and a large part of my own field, was born from a computer run.

I want to stop for a moment on the people who did the computing before the machines. Until the 1950s, "computer" was a job title.¹⁰ Computations were performed by chains of humans sharing the same office; each person carried out one mechanical and often boring step and passed the sheet to the next, with a desk calculating machine and sometimes an analogue device to help. These offices were mostly staffed by women, for reasons which had little to do with talent: educated women were available, they were paid less, and most other scientific careers were closed to them. The first NASA missions were computed in this way. In early 1962, before his orbital flight, John Glenn asked that Katherine Johnson recompute by hand the trajectory which the new IBM 7090 had produced; she worked for a day and a half on her desk calculator, and the numbers agreed.¹¹ The digital computer destroyed these jobs. There is no doubt about it. But the same people took the new and higher jobs which it created: the six programmers of ENIAC were recruited among the human computers of the Moore School, and Dorothy Vaughan, who supervised the West Area computers at Langley, taught herself and her group FORTRAN before the IBM machine arrived.¹² Nobody wrote that applied mathematics was finished. The discipline moved one level up, from executing arithmetic to designing algorithms and proving that they converge, and the people who had spent their days on repetitive mechanical steps could finally spend them on more interesting and more creative tasks. I believe that the same will happen with AI. It will destroy some jobs, without any doubt. It will also create new ones, which will allow us to do higher-level and more creative work, and to flourish more in our professional lives.

The rest of the twentieth century confirmed the pattern. The finite element method, first published by aeronautical engineers in 1956 and named by Ray Clough in 1960, and the finite volume method, whose modern form goes back to Godunov's scheme of 1959 for gas dynamics, completely transformed engineering practice, from solid mechanics and viscous flows to compressible aerodynamics.¹³ Digital computers allowed humanity to replace nuclear tests in nature by numerical simulation: in the 1990s, after the last tests and the test-ban treaty, both the American Stockpile Stewardship programme and the French Simulation programme of the CEA took this road, and, whatever one thinks of the weapons themselves, it is remarkable that a physical experiment was replaced by a validated calculation.¹⁴ Those who order such a test want, most of the time, one number above all: the energy released by the event. Applied mathematics can now estimate this number while preserving the environment, and I find this fantastic. And since the real purpose of these weapons is deterrence, not use, a credible calculation serves this purpose as well as an explosion did. Topology optimisation designed the droop-nose ribs of the wing of the Airbus A380 by removing material wherever it carried no load, and saved several hundred kilograms per aircraft.¹⁵ Since the middle of the last century computers have performed computations better than any human, and nobody is screaming about an existential crisis of applied mathematics. Each time the machine became better than us at something we used to do by hand, applied mathematics absorbed it and grew. The crisis was, in fact, a promotion.

I am glad to see that this historical argument is now being made from the pure side as well, and made without my professional bias. Orr Shalit opens his post on Proofs and Prompts with Kohelet, "there is nothing new under the sun", and works through the anxieties one by one to show that most of them are older than the machines: being scooped, standing on results one has not verified down to the bone, drowning in more papers than anyone can read, and students who were copying homework long before there was anything to copy it from.¹⁶ Jeremy Avigad reaches back further, to the moment when axiomatic and set-theoretic methods displaced explicit calculation, and reminds us how personal that quarrel was: Carl Ludwig Siegel, writing to Mordell in 1964 about the new style, said that he saw "a pig broken into a beautiful garden and rooting up all flowers and trees", and feared that mathematics would perish before the end of the century.¹⁷ It did not perish. It absorbed the new methods, argued about them for thirty years, and came out larger. I am not offering this as a proof that everything will be fine, because history proves nothing of the sort. I offer it as a reason to distrust the feeling that this time the ground is being taken away for good, a feeling which our discipline has had before and survived.

Why the mood is different on the applied side

I read with a lot of interest Terence Tao's essay "Mathematics in the age of AI" and the post of Hugo Duminil-Copin on Proofs and Prompts, and I understand their points. Pure mathematics has its lighthouses, to use Hugo's beautiful word: the great conjectures which give direction and identity to whole fields and to whole careers. When he writes that the current use of AI "nukes the mathematical landscape" and makes it more difficult to inhabit after each blast, I understand him and I sympathise with him. Tasmin Chu's defence of the dissenter viewpoint deserves the same respect: she says with honesty what many think in silence.

Events have given Hugo's post a cruel confirmation while I was writing mine. He used as his main example the θ(p_c) = 0 conjecture, the most famous open problem of percolation theory: at the critical probability, with probability one, there is no infinite cluster. This was known in the plane and in high dimensions, and open in every dimension in between, starting with dimension three. He wrote that it now seemed only a matter of time before this conjecture also fell to the bulldozers, and he asked us to let the lighthouses shine a little longer. Three days later, a document produced by Claude, with a formal verification in Lean, claimed a proof of the inequality which Gady Kozma and Shahaf Nitzan conjectured in 2024 and from which they had derived the conjecture in every dimension. Gil Kalai reported the news on his blog with the words "if verified, this is a remarkable breakthrough".¹⁸ I am sincerely sorry for Hugo and for the whole percolation community. His mathematical landscape was nuked again, to use his own word, three days after he asked for a little more time, and the conjecture which once inhabited his dreams is now, if the claim holds, a theorem found by a machine. Nothing in what follows should be read as a lack of respect for this loss. If there is a consolation, it was offered under his post by Alonso Castillo-Ramirez, who wrote that we can still take great joy in the beauty of mathematics on its own, whether it was created by a human or by an AI. I do not know whether this is enough for those who spent years on the problem. But I know that Hugo's lighthouses were also ours, because the mathematics is the same, and I come back to this at the end.

But applied mathematics has never been driven by big unsolved conjectures. It is driven by applications and by the challenges of the times in which we live. At one moment it was the Manhattan project and then the race to the Moon; today we may think about the climate, the energy transition, water, health and the safety of the structures we build. Our identity is attached to a service, not to a conjecture. This makes our discipline a hybrid. Internally we function like pure mathematicians: we want to understand our methods, we prove that a scheme converges and that a model is well posed, and we admire an elegant algorithm as much as anyone admires an elegant proof. Externally we are pulled by the problems which come to us from outside, and when a big challenge appears, it decides our agenda. An AI which solves a problem we care about does not take a lighthouse away from us; it brings the shore closer. Mark Wildon asks how impressed we would be by a candidate who ends a job talk by saying that all of it was done deliberately without computer algebra. In applied mathematics we passed that point long ago.

There is a second reason. Most applied mathematicians do not carry the title on their business cards. In industry and in government they are called engineers, actuaries, quantitative analysts, meteorologists, data scientists or research scientists. Anatoly Stankyavichyus calls the data scientists "the truffle pigs of mathematics", who search through our papers for ideas to bring to life, and he is right that this downstream community cares deeply about the health of the whole discipline. The title on the card hides their contribution to humanity, and it means that the discipline is much larger than the departments which carry its name. But it also means that we are used to sharing the credit: with a team, with another profession, with a machine. The question of who will receive the credit for a theorem, which Ruodu Wang analyses as a prisoner's dilemma, matters less to us. In our world the result was always the problem which got solved, and the name written on it was often not ours.

Nilima Nigam objects to the parallel with calculators and computers: those tools, she writes, automated tasks which humans had decided were of no value to do by hand, while AI automates the parts of our work which we value most. I do not think that the human computers of Langley would have agreed that their work had no value; it was a profession, with pride in it, and the profession still moved on, and moved up. The value was never in the execution of the calculation but in what the calculation told us. Where I agree with her entirely is that the transition must be managed deliberately, and that this is our responsibility, not the market's. I also do not ignore the energy cost of these systems, which Alexis Marchand and Vadim Lebovici put at the centre of their posts; applied mathematicians work on the climate and on the energy transition, and we should be among those who measure this cost honestly and help to reduce it.

Nobody stopped playing chess

There is one domain where the machine has already taken from humans the very thing which they valued most, and it deserves a closer look, because it answers both Hugo's fear and Nilima Nigam's objection. In May 1997 Deep Blue defeated Garry Kasparov, the reigning world champion, in a match played under tournament conditions.¹⁹ Nine years later, in Bonn, the world champion Vladimir Kramnik lost to Deep Fritz, a commercial program running on a personal computer, and no world champion has played a match against a machine since, because there would be no point: a chess program on a telephone now plays far better than any human.²⁰ The machine did not automate a chore here. It automated the search for the best move, which is the whole of chess, the thing for which the grandmasters had been admired for centuries. Did humans stop playing chess? They did not. The largest chess website passed a quarter of a billion accounts in February 2026,²¹ the world championship is followed by millions, and in December 2024 an eighteen-year-old, Gukesh Dommaraju, became the youngest world champion in history, four years younger than Kasparov had been.²²

What the players did with the machine is more interesting than the defeat. Kasparov himself, one year after his loss, invented what he called advanced chess, in which each player consults a program during the game, and he later described a tournament of 2005 in which two amateurs with three ordinary computers beat grandmasters equipped with the same engines: "weak human + machine + better process was superior to a strong computer alone and, more remarkably, superior to a strong human + machine + inferior process".²³ Today every serious player prepares with engines, and the engines have changed the way the best humans play. When AlphaZero, a program which learned chess in a matter of hours by playing against itself, was presented in 2017 and 2018, two strong players wrote a whole book on what it had discovered,²⁴ and Magnus Carlsen, the strongest player of our time, said in 2019 that he had become "a very different player in terms of style", influenced by "my heroes recently, which is AlphaZero".²⁵ The machine did not make the humans weaker or duller. It made them stronger and more original, and the games of the human champions, precisely because they are human, with nerves, mistakes and courage in them, are more interesting to watch than the games of the engines. Only the specialists watch the engines play against each other; everybody watches Carlsen.

Go is an even better test, because the game is older, deeper and more revered than chess, and because here we have measurements. AlphaGo beat the European champion Fan Hui in October 2015,²⁶ Lee Sedol, one of the greatest players of the last decades, in March 2016, and the world number one Ke Jie in May 2017, after which the program was retired.²⁷ Lee Sedol left professional Go in November 2019, saying that even if he became the number one, there was an entity that could not be defeated,²⁸ and when I read this sentence I understand Hugo better than through any argument. But the game did not stop, and something remarkable happened to those who stayed. A study published in 2023 analysed 5.8 million move decisions of professional players between 1950 and 2021, evaluating each of them against a superhuman program, and found that the quality of human decisions improved significantly after 2016, and that the improvement came from novel moves, never played by humans before, which the authors attribute to the players breaking away from traditional strategies after the arrival of the machine.²⁹ Ke Jie, who lost the last match, said afterwards that he had fundamentally reconsidered the game, and that this reflection had helped him greatly.³⁰ The superhuman machine did not petrify the human players. It set them free from the traditional strategies, and it made them measurably better and more creative than they had been in seventy years of records.

I am quite sure that the same will happen to mathematics, and that thanks to AI human mathematicians will become better mathematicians. The mechanism is already visible on Proofs and Prompts. A machine which proves a conjecture by an unexpected route, as in the proof of Feige's conjecture described by Guanyang Wang, where the key was a connection between a paper of a few weeks earlier and a classical result of Grünbaum, teaches the humans who study it a move which they had never considered, exactly as AlphaZero taught the grandmasters to push the rook's pawn and to give up material for activity. Applied mathematics has, in fact, its own AlphaGo moment, and it took place in 1955. I have told the story above: Fermi, Pasta, Ulam and Tsingou expected the computer to show the energy of their chain of nonlinear springs spreading over all the modes, and the machine showed instead that the energy came back, almost entirely, to the mode in which it had started. Nobody had predicted this, and nobody understood it. The humans who took the surprise of the machine seriously, Zabusky and Kruskal, found the soliton, again in a computer run, and the humans who digested the soliton, Gardner, Greene, Kruskal and Miura, found the inverse scattering transform with pencil and paper.³¹ A surprise produced by a machine and digested by humans became one of the most beautiful chapters of the mathematics of the twentieth century, and it did not extinguish a lighthouse; it lit a new one. A machine which knows the literature of every field ends the isolation of the subfields, and a machine which never tires is the sparring partner of which every player dreams. The young mathematician whom Hugo hoped for, the one who will show us what we all missed, will still come, and will have studied the proofs of the machine as Carlsen studied the games of AlphaZero. The theorems will be more abundant, and the humans who understand them, explain them and find the next question will be more valuable, not less.

A second thing will happen, and I would be surprised if it did not happen within a few years: a new field will appear, which we may call AI mathematics for want of a better name, and whose subject is not the theorems but the machines which produce them. A typical conference in this field will discuss the merits of the different models, the prompting strategies, the types of agents and the ways in which they interact, the harnesses and the verification pipelines, the benchmarks and their failures. Chess went through exactly this. The first world championship for programs was held in Stockholm in 1974, twenty-three years before Deep Blue, and was won by the Soviet program Kaissa; the programmers founded their own association in 1977, with its own journal and its own conferences; and since 2010 the strongest engines have played in a championship of their own, which enthusiasts follow as a spectator sport.³² Nobody confuses that championship with the human one, and both flourish. In mathematics the seeds are visible. A conference on artificial intelligence and theorem proving has met every year since 2016, and its eleventh edition took place in Aussois in the first days of September 2026;³³ the MATH-AI workshop of the machine-learning community, born in 2021, will hold its sixth edition in December 2026 with a call for papers on building reliable mathematical agents, on evaluating them, and on automated mathematical research;³⁴ the AI for Math workshop of the ICML conference attaches prize competitions to its meetings;³⁵ and there has even been a journal called Annals of Mathematics and Artificial Intelligence since 1990, although its subject is the mathematics used in AI rather than the AI used in mathematics.³⁶ On Proofs and Prompts itself, Guanyang Wang ran a controlled experiment with two prompts which differ by a single sentence, one of which produced the proof three times out of four while the other failed four times out of four; Qi Guo proposes to account for the architecture of the prompts as a unit of human contribution separate from the proof; and Segev Gonen Cohen asks whether prompting and harnessing should be seen as a skill comparable to a talent. These are the first papers of the new field. What is missing is the mathematicians. Today this field lives in the workshops of the machine-learning conferences and is run mostly by computer scientists, as numerical analysis once lived in the computing centres before it took its place in the mathematics departments. The question is not whether AI mathematics will exist; it exists. The question is whether mathematicians will shape it or leave it to others, and I hope that we will shape it, from both sides of our discipline.

While I was writing this, Jeremy Avigad made the same argument from the pure side, and made it better than I have. He warns that the sprint of the companies to settle as many famous conjectures as possible is "sucking the air out of the room", crowding out everything else these technologies offer mathematics: proof assistants and the curated libraries built around them, automated reasoning, satisfiability and constraint solvers, and machine learning used to detect patterns in mathematical data, to search for objects of interest, to compute solutions of partial differential equations and to find the parameters at which interesting behaviour begins. That list is a research programme, and most of it is nearer to my side of the discipline than to his. He adds a point I had not seen, and it is the uncomfortable one. Our discipline has punished the people who went into this field: their papers appear in computer-science venues and count for nothing on a mathematics job application, so they have left for the companies and the startups, where they are doing interesting work and earning good money. "The irony", he writes, "is that as much as we may not want to admit it, we need them more than they need us." If a new field is going to be built, it will be built by exactly those people, and we have spent a decade making it rational for them to build it somewhere else.

Let me now come back to my own side of the discipline, and look at what the machine changes in the daily work of an applied mathematician.

Three trades in one person

Applied mathematics is a complicated discipline in the sense that an applied mathematician should be proficient in three trades at once: mathematics, the field of application, and scientific programming. Let me take them one by one, because AI changes all three.

First, the code. When I did my PhD, some of my fellow students spent six months, and sometimes more, debugging their programs to get their computations right and convergent. This was accepted as part of the training, but most of it was friction. Today the assistance which we get from AI in writing, testing and reading code removes a huge part of this friction. It is not rare now to do in three hours, with the right tools, what required three months of work before. Moreover, the code is the part of our work which has real commercial value, so this is not a marginal gain. Of course, we cannot trust these tools blindly, and we do not: the verification and validation of scientific software, checking that we solve the equations correctly and that we solve the correct equations, with manufactured solutions, convergence studies and benchmarks, remain the standard, and I would say that they matter more, not less, when the code is produced faster. One more change is easy to miss. We no longer program in the language which we know best; we program in the language which is the most appropriate and the most efficient for the problem at hand, because the cost of learning it has almost disappeared. This was not possible before the AI era. The account by Shaowu Zhang of the last open sporadic case of the inverse Galois problem describes exactly the workflow which I recognise: the AI agents generated and tested the code, organised large computations and recovered from failures, while the humans decided what to compute, when to stop, and where to point the telescope. Mark Wildon lists code generation among the good uses of these tools, and I agree. Simon Hayward writes that generative AI has failed to keep its promise in programming; my daily experience is the opposite, and I think that the difference lies in the discipline of verification which the applied mathematician brings with them.

Second, the field of application. Here I welcome AI because of its general knowledge. I appreciate having an expert of the application area on which I am working at the tips of my fingers, and this is fantastic. Recently, thanks to friendship and encounters, I had to move from fluid mechanics to general relativity, to the quasi-normal modes of black holes. Before, such a transition would have required at least five years before I became operational in the new field. In the AI era, and with the support of my friends and collaborators from relativity, the transition happened without any gap in my list of publications. Anatoly Stankyavichyus makes the same point from industry: tools which help a researcher understand arguments outside their own domain allow small teams to achieve more. Mark Wildon recommends using the model as a Socratic partner to learn an adjacent field, and Amruth Srivathsan describes it as a middleware between a fixed text and a particular reader. For these reasons I predict that AI will improve the intellectual mobility of applied mathematicians. The borders between application fields will matter less, and a mathematician will be able to follow the problems wherever they lead.

Third, the mathematics itself. To tell the truth, I have always believed that we do not prove enough in applied mathematics. We are content to observe, to present numerical evidence, to show that a method works on a set of test cases and to leave the theorem for later; some schemes have been used for decades with a convergence proof which nobody ever wrote. If AI can help us to increase the number of statements which are actually and rigorously proved in applied mathematics, I believe that it will be a very positive development, perhaps the most positive of all. Here the warnings of the pure mathematicians apply to us without any change. Martin Hairer asks that an argument produced by an AI receive more scrutiny than an argument produced by a trusted colleague, and that we never paste its output into our papers. Shmuel Weinberger reminds us that these systems are extremely convincing, and that the idea of a proof, refined over millennia, is precisely our defence against a convincing but wrong argument. Formal verification, as in the Lean formalisation which accompanied the AI-found proof of Feige's conjecture described by Guanyang Wang, will play a growing role here, and I expect applied mathematicians to adopt it faster than one might think, because, as Anatoly Stankyavichyus writes, in the applied sciences sometimes a proof is just a proof. Alonso Castillo-Ramirez writes that when proofs become abundant, understanding becomes scarce, and Andrew Ng recalls Thurston's view that what we want is not a collection of answers but understanding. In applied mathematics we have lived in the opposite regime, where proofs were scarce and numerical evidence abundant, and a little abundance of proofs would do us good. If the proof culture spreads among our practitioners, applied mathematics will become closer to pure mathematics, not further from it.

Now it is my own equations

I wrote the sections above over the first days of September. Since then the machine has come for the equations on which I have spent my working life. On 8 September OpenAI announced that an internal model, driving a system of cooperating agents of the order of ten thousand of them, had produced in about eighty-eight hours a proof that the three-dimensional Navier–Stokes equations can develop a singularity in finite time, with a further seventeen hours spent on a formalisation in Lean.³⁷ Let me give credit where it is due before I say anything else. There was a press release, but there were also a manuscript of one hundred and sixty-six pages, a companion manuscript, and a public repository containing the formal code. That is a great deal closer to what our profession should require of such an announcement than a claim posted on a social network, and I want to acknowledge it plainly.

Now the fine print, which is where an applied mathematician earns his keep. When the Clay Mathematics Institute posed the problem in 2000, Charles Fefferman wrote the official statement and offered four alternatives, a proof of any one of which counts as a solution.³⁸ In (A) and (B), which assert that a smooth flow stays smooth for ever, the external force is required to vanish identically. In (C) and (D), which assert instead that the equations break down, a smooth external force is permitted. The main theorem of the new manuscript constructs, for every positive viscosity, a flow which starts from rest, is driven by a smooth force compactly supported in space and in time, keeps its kinetic energy bounded throughout, and whose velocity becomes unbounded in finite time; and the authors state in their introduction, without any ambiguity, that this establishes alternative (C), and with it (D). So the Millennium Problem has been resolved as it was posed, and the question which my whole community means by the Navier–Stokes problem, whether a fluid left to itself can break down, stands this morning exactly where it stood last week. Fefferman had written that he was giving "reasonable leeway to solvers while retaining the heart of the problem". The leeway turned out to be wider than the heart. Nobody concealed any of this: the asymmetry is written into the official statement, the authors declare which alternative they prove, and the company adds that it does not intend to claim the prize. The fine print simply had to be read, and reading the fine print of a problem statement is not a pedantic activity. It is the whole of validation.

There is a second result, and I find it the more interesting of the two. In a footnote of the same announcement, and in a separate manuscript of fifty-six pages, the same system claims finite-time blowup for the three-dimensional Euler equations with no external force at all, from smooth, compactly supported, divergence-free initial data. Euler is not on the Clay list, and the claim received a small fraction of the attention. Yet the smooth unforced case was genuinely open: Tarek Elgindi's theorem gives blowup for axisymmetric velocities without swirl in the Hölder class C^{1,α}, not for smooth data.³⁹ About a hundred agents worked on it for some fifty hours. If it survives examination, most specialists in fluid mechanics would tell you that it is the deeper of the two events of that week, and it went into a footnote while the Millennium Problem went into the headline. What is placed in the headline and what is placed in the footnote is a decision, and it is not always the mathematics that decides.

The machine did not begin from nothing, and this is easy to check, because the bibliography is printed at the back of the manuscript like anybody else's. The construction rests on Elgindi; on Diego Córdoba and Luis Martínez-Zoroa, who built finite-time singularities for the forced three-dimensional Euler equations by amplifying increasingly concentrated vortex layers; on the same two authors with Fan Zheng for the hypodissipative Navier–Stokes equations; on Tristan Buckmaster and Vlad Vicol's convex integration; on Dallas Albritton, Elia Brué and Maria Colombo; and on Terence Tao's blowup for an averaged Navier–Stokes equation.⁴⁰ In the same fortnight Levent Alpöge and Tristan Buckmaster, building directly on Córdoba and Martínez-Zoroa, proved finite-time blowup with a smooth forcing term for the incompressible porous medium equation, the two-dimensional Boussinesq system and the three-dimensional Euler equations, and formalised the results in Lean; the company acknowledges their priority on the forced Euler problem. Tao, who digested that work the day before the announcement, records that the arguments are heavily AI-assisted, and reports that the authors themselves described their first write-up as the worst they had ever seen in the history of mathematics, before human hands turned it into something of professional quality.⁴¹ There is the division of labour I described above, visible in the field rather than in a diagram: the machine generates, and the humans verify, digest and explain. That is not a consolation prize. It is the part which turns a proof into something anybody else can use.

I have to record the uncomfortable part as well, because it now belongs to our professional landscape. Within a day the announcement was followed by a public dispute about priority, and about whether unpublished work developed in private sessions with a commercial assistant had reached a competitor. I will not name the people involved and I will not adjudicate between them: the company denies it while conceding, in the same paragraph, that it cannot rule out that de-identified data derived from that usage improved its models, and I have no means of settling such a question from the outside. What I will record is the professional conclusion, because it applies to every reader of Proofs and Prompts and because it is genuinely new. Our drafts are now typed into somebody else's machine. An accusation of this kind is close to unfalsifiable in both directions, which is exactly what makes it corrosive, and the remedy is not indignation but the dull apparatus of dated records, early preprints and explicit agreements. That is a governance problem for our community, and I would rather we solved it before it happens to one of our students.

As for the mathematics, apply the two words of my trade. Verification asks whether the argument is correct with respect to the definitions it was given, and here a proof assistant has answered, which is a real thing and not nothing. It is also less than it sounds, and Seewoo Lee has just explained on Proofs and Prompts exactly how much less: a formal certificate establishes that the formalised statement follows from the axioms the proof uses, and it does not establish that the formalised statement means what the English sentence means, nor that the formal proof follows the argument you believed you had. His remedy is three words long, and every one of us should adopt it: read the code.⁴² Validation asks the other question, the one my discipline was built to ask: does the theorem answer what we wanted to know? Here the answer is documented, and, to the credit of its authors, documented honestly. It establishes (C) and (D). It does not establish (A) or (B). Until my colleagues in fluid mechanics have read one hundred and sixty-six pages and told the rest of us what they think, this is a claim, an impressive one, and not yet a theorem of the mathematical community.

And how do I feel about it, now that it is my own equations? Less bereaved than I expected. The shore came closer, which is precisely what I said applied mathematics wants from these tools. The lighthouse my community actually steers by, whether a fluid left entirely to itself can break down, is still lit. If it goes out next month I will write this paragraph again with a different ending, and I suspect I will still conclude that a machine capable of constructing a flow of that kind is an instrument I would rather have than not.

One mathematics

I have insisted on the differences between the pure and the applied sides, because they explain the difference in mood. Let me end with what unites us, because I believe that it matters more. We have different goals and we work in different ways, but we love the same mathematics, and we serve humanity in different ways: pure mathematics by extending what human beings are able to understand, applied mathematics by putting this understanding to work. And we carry exactly the same responsibility towards the future generations. The students who enter our departments today will practise their profession in a world which none of us can describe, and they will practise it with the values which we manage to transmit to them now.

I agree with Álvaro Lozano-Robledo that we must invest in human mathematicians, and I would put it in my own way: the virus of mathematics is transmitted from human to human, and a machine cannot transmit it. Every one of us does mathematics because at some stage of our life we met a teacher, a supervisor or a colleague who inspired us and who showed us, by their example, that this life was possible. This is as true in applied mathematics as in pure mathematics, and no tool, however good it is at explaining, will replace the moment when a person you admire believes that you can do it.

There is also a ground on which we already stand together, and on which, I believe, we will stand together forever. Terence Tao distinguishes three components of mathematical work: the generation of a proof, its verification, and its digestion, the slow work by which a correct argument becomes human understanding. Whatever the machines generate and whatever the formal checkers verify, the digestion will remain ours, in pure and in applied mathematics alike, and the digestions which Tao has published of recent AI-found results show what this work looks like when it is done well. Here we acknowledge without hesitation the leadership of pure mathematics: proofs and their digestion have been its business for centuries, and the standards of this work were set there. I hope that the proof culture will become more and more present in applied mathematics, and that we will follow the best standards of proof digestion set by our pure colleagues, instead of inventing lower ones of our own.

This is why I think that the two communities should unite their forces instead of arguing about whose landscape is more damaged. Together we should develop the new and responsible workflows and standards for working with these tools: verification of every argument, disclosure of every significant use, attribution of every idea to its source, reproducibility of every computation. The Leiden Declaration and Martin Hairer's concrete recommendations for writing are good starting points, and the applied community has decades of experience in verification and validation to add to them. Together we should define the professional, intellectual and scientific ethics which go with the new tools, and, above all, we should transmit these values to the future generations of scientists, whether they end up proving theorems or designing wings.

On one point in particular I would like to join forces with my pure colleagues, and several posts on Proofs and Prompts have already prepared the ground. Some AI companies announce presumed spectacular progress on famous conjectures in the language of business communication, and a new kind of mathematics has appeared, announced on social networks, where a result exists from the moment it is posted. Tian Lan has documented how misleading such announcements can be, when a conjecture is declared resolved while its most famous case remains open; Qi Guo reminds us that these announcements are written to attract attention, customers and investors; Mark Wildon warns about the few spectacular hits selected from a mass of unreported misses; and Hugo Duminil-Copin asks what a young researcher should do when the proof of a major conjecture may appear from one day to the next on some social network. We should condemn this practice together, from both sides of our discipline. Let me be fair about one distinction, because it matters. A counterexample can be checked: the Jacobian counterexample in dimensions three and higher was verified by a direct calculation, and the valuable part of that episode came after the announcement, in Terence Tao's digestion of the counterexample and in Zihan Zhang's note on its direct consequences for other conjectures. That work of human understanding is what makes the edifice of mathematics solid and attractive, and I think that we all agree on this. A proof is a different object, and it must be treated differently: we should hold the line which mathematics has always held, and admit no proof as a proof until it has received many independent verifications and validations by the mathematical community. The percolation claim I described above shows what this means when the proof comes with a formal certificate, and I want to be fair to it, because it is closer to what we ask than most announcements. There was no press release. There was a public repository with a Lean formalisation, which checks every logical step down to the standard axioms, a guide of fifteen pages written for human readers, and an explicit statement that nobody independent of the author had yet refereed the work and that independent examination was sought. The two words of my trade apply to it exactly as they applied to the fluid equations above. The machine has done the verification. The validation is the human question, and here it is a sharp one: do the definitions in that repository say exactly that there is no infinite cluster at the critical probability, for bond percolation on the lattice in every dimension, and nothing weaker? The document asks its own readers to check, which is the right instinct, and until the experts of percolation have read and digested it the result is a claim rather than a theorem of the community. A social network is not an acceptable venue for scientific communication. The traditional scientific journals, with their imperfect peer review, should remain the main channel through which a result enters mathematics. Yes, the present system is imperfect, and Anatoly Stankyavichyus and Andrew Ng describe its defects on Proofs and Prompts with justified impatience; but we do not have a new one yet, and a press release or a post is not a replacement. In applied mathematics we have had a name for this discipline for a long time, and it is the same one: verification and validation. After the two episodes of these two weeks I think we can state concretely what a responsible announcement looks like, and it is not a high bar. The manuscript and the formal code are published with the claim and not after it. The authors say which precise statement they have established, including, and above all, the neighbouring statement they have not. The prior work on which the construction rests is cited, and the priority of others is acknowledged. And nobody calls the thing a theorem of the community until the community has read it. Both of the announcements I have discussed clear most of that bar, which is real progress over a year ago, and neither of them is the end of the process.

To conclude: in applied mathematics we welcome and we embrace the new technology, and we anticipate a rapid development of our discipline and an improvement in the quality of our scientific work. The pace of discovery will accelerate. I know that these views may appear overly optimistic to some readers, and I admit it: I am an optimist. Nobody is perfect. If you are an applied mathematician and you see things differently, the comments section below is yours.

Disclosure. The ideas, the structure and the opinions in this post are mine and were written from my own notes. I used an AI assistant (Claude) to improve the English, to check the historical dates and references against the sources, to verify the September 2026 results in fluid mechanics against the original manuscripts and against the official statement of the Millennium problem rather than against press coverage, and to propose a few supporting historical examples, which I then verified. Following the guidelines of Proofs and Prompts, I state this here.

Notes

1. Herodotus, The Histories, Book II, 109; see for instance the text in the Perseus Digital Library. ↩

2. D. Teets and K. Whitehead, "The discovery of Ceres: how Gauss became famous", Mathematics Magazine 72 (1999), 83–93, doi:10.1080/0025570X.1999.11996710. ↩

3. L. F. Richardson, "The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam", Philosophical Transactions of the Royal Society A 210 (1911), 307–357, doi:10.1098/rsta.1911.0009. ↩

4. L. F. Richardson, Weather Prediction by Numerical Process, Cambridge University Press, 1922 (Internet Archive); P. Lynch, The Emergence of Numerical Weather Prediction: Richardson's Dream, Cambridge University Press, 2006; P. Lynch, "Richardson's forecast: the dream and the fantasy", arXiv:2210.01674 (2022). ↩

5. R. Courant, K. Friedrichs and H. Lewy, "Über die partiellen Differenzengleichungen der mathematischen Physik", Mathematische Annalen 100 (1928), 32–74, doi:10.1007/BF01448839. ↩

6. R. Courant, "Variational methods for the solution of problems of equilibrium and vibrations", Bulletin of the American Mathematical Society 49 (1943), 1–23, doi:10.1090/S0002-9904-1943-07818-4. ↩

7. T. Haigh, M. Priestley and C. Rope, ENIAC in Action: Making and Remaking the Modern Computer, MIT Press, 2016; see also "The ENIAC computer runs its first, top-secret program", APS News, November 2022. ↩

8. J. G. Charney, R. Fjørtoft and J. von Neumann, "Numerical integration of the barotropic vorticity equation", Tellus 2 (1950), 237–254, doi:10.3402/tellusa.v2i4.8607. ↩

9. E. Fermi, J. Pasta and S. Ulam, "Studies of nonlinear problems. I", Los Alamos report LA-1940 (1955); on the part of Mary Tsingou, T. Dauxois, "Fermi, Pasta, Ulam, and a mysterious lady", Physics Today 61 (2008), 55–57, doi:10.1063/1.2835154; N. J. Zabusky and M. D. Kruskal, "Interaction of 'solitons' in a collisionless plasma and the recurrence of initial states", Physical Review Letters 15 (1965), 240–243, doi:10.1103/PhysRevLett.15.240. ↩

10. D. A. Grier, When Computers Were Human, Princeton University Press, 2005, publisher's page. ↩

11. NASA, Katherine Johnson biography; M. L. Shetterly, Hidden Figures, William Morrow, 2016. ↩

12. Columbia University Computing History, The ENIAC programmers; NASA, Dorothy Vaughan biography. ↩

13. M. J. Turner, R. W. Clough, H. C. Martin and L. J. Topp, "Stiffness and deflection analysis of complex structures", Journal of the Aeronautical Sciences 23 (1956), 805–823, doi:10.2514/8.3664; R. W. Clough, "The finite element method in plane stress analysis", Proceedings of the 2nd ASCE Conference on Electronic Computation, Pittsburgh, 1960; S. K. Godunov, "A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics", Matematicheskii Sbornik 47 (1959), 271–306, mathnet.ru. ↩

14. National Research Council, The Comprehensive Nuclear Test Ban Treaty: Technical Issues for the United States, National Academies Press, 2012, nap.nationalacademies.org/catalog/12849; CEA, Defence and security: the Simulation programme, annual report 2020. ↩

15. L. Krog, A. Tucker, M. Kemp and R. Boyd, "Topology optimisation of aircraft wing box ribs", 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, 2004, AIAA 2004-4481, doi:10.2514/6.2004-4481; L. Krog, A. Tucker and G. Rollema, "Application of topology, sizing and shape optimization methods to optimal design of aircraft components", Airbus UK and Altair Engineering, 2002. ↩

16. O. Shalit, "What's not new", Proofs and Prompts, 8 September 2026. ↩

17. J. Avigad, "What is mathematics now, and what should it be?", Proofs and Prompts, 7 September 2026. ↩

18. G. Kalai, "Amazing: there is no percolation at the critical probability in all dimensions", Combinatorics and more, 3 September 2026. The reduction is G. Kozma and S. Nitzan, "A reduction of the θ(p_c) = 0 problem to a conjectured inequality", arXiv:2401.12397 (2024), a preprint at the time of writing. The AI documents are the description of the formalisation and the guide to the proof in Anthropic's public repository of formal mathematics. ↩

19. IBM, Deep Blue, IBM History. ↩

20. "Kramnik vs Deep Fritz: computer wins match by 4:2", ChessBase, 5 December 2006. ↩

21. "Chess.com reaches 250 million members", Chess.com, 27 February 2026. ↩

22. P. Doggers, "18-year-old Gukesh becomes youngest-ever undisputed chess world champion", Chess.com, 12 December 2024. ↩

23. G. Kasparov, "The chess master and the computer", The New York Review of Books, 11 February 2010; see also G. Kasparov with M. Greengard, Deep Thinking: Where Machine Intelligence Ends and Human Creativity Begins, PublicAffairs, 2017. ↩

24. D. Silver et al., "A general reinforcement learning algorithm that masters chess, shogi, and Go through self-play", Science 362 (2018), 1140–1144, doi:10.1126/science.aar6404; M. Sadler and N. Regan, Game Changer: AlphaZero's Groundbreaking Chess Strategies and the Promise of AI, New in Chess, 2019. ↩

25. P. Doggers, "Carlsen wins 2019 Norway Chess with round to spare", Chess.com, 14 June 2019. ↩

26. D. Silver et al., "Mastering the game of Go with deep neural networks and tree search", Nature 529 (2016), 484–489, doi:10.1038/nature16961. ↩

27. DeepMind, "AlphaGo's next move", 27 May 2017. ↩

28. "Go grandmaster Lee Se-Dol retires saying artificial intelligence cannot be defeated", ABC News, 29 November 2019. ↩

29. M. Shin, J. Kim, B. van Opheusden and T. L. Griffiths, "Superhuman artificial intelligence can improve human decision-making by increasing novelty", Proceedings of the National Academy of Sciences 120 (2023), e2214840120, doi:10.1073/pnas.2214840120. ↩

30. DeepMind, "2017: DeepMind's year in review", 21 December 2017. ↩

31. C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, "Method for solving the Korteweg–de Vries equation", Physical Review Letters 19 (1967), 1095–1097, doi:10.1103/PhysRevLett.19.1095. ↩

32. Chessprogramming wiki, WCCC 1974 and ICGA; Top Chess Engine Championship. ↩

33. Conference on Artificial Intelligence and Theorem Proving. ↩

34. The 6th Workshop on Mathematical Reasoning and AI, NeurIPS 2026. ↩

35. 3rd AI for Math Workshop, ICML 2026. ↩

36. M. C. Golumbic, "35 years of math and AI", Annals of Mathematics and Artificial Intelligence 93 (2025), 1–3, doi:10.1007/s10472-025-09969-7. ↩

37. OpenAI, "On the Navier–Stokes Millennium Prize Problem", 8 September 2026, together with the Navier–Stokes manuscript, the Euler manuscript and the Lean repository. ↩

38. C. L. Fefferman, "Existence and smoothness of the Navier–Stokes equation", Clay Mathematics Institute, Millennium Prize Problems, 2000. The force is required to vanish in (A) and (B); in (C) and (D) it need only be smooth and suitably decaying, which a compactly supported force is. ↩

39. T. M. Elgindi, "Finite-time singularity formation for C^{1,α} solutions to the incompressible Euler equations on R^3", Annals of Mathematics 194 (2021), 647–727, doi:10.4007/annals.2021.194.3.2. ↩

40. The references are those of the Navier–Stokes manuscript itself. See in particular D. Córdoba and L. Martínez-Zoroa on blow-up for the incompressible three-dimensional Euler equations, D. Córdoba, L. Martínez-Zoroa and F. Zheng on the hypodissipative case, T. Buckmaster and V. Vicol, "Nonuniqueness of weak solutions to the Navier–Stokes equation", Annals of Mathematics 189 (2019), 101–144, and T. Tao, "Finite time blowup for an averaged three-dimensional Navier–Stokes equation", Journal of the American Mathematical Society 29 (2016), 601–674. ↩

41. T. Tao, "Finite time blowup with smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations", What's new, 7 September 2026. ↩

42. S. Lee, "(auto)formalization, but why", Proofs and Prompts, 8 September 2026. ↩