Applied mathematics has met the machine before

Denys Dutykh, Associate Professor at Khalifa University of Science and Technology

Most voices on AI and mathematics, on this blog too, come from pure mathematics; the applied mathematician who lives between models, equations and code has stayed silent, and I would like to speak from the adjacent side. I work on fluid mechanics, water waves, numerical methods and, recently, black holes; the views below are my own, and by “AI” I mean the large language models and the tools built on them. My point is simple: applied mathematics welcomes these tools, and I expect a major boost for our discipline if we embrace them with our usual standards of verification and validation, because we have been here before.

Applied mathematics has met the machine before

Mathematics began as a service: Herodotus tells us that geometry was born when the Egyptian surveyors measured the fields again after each flood of the Nile.¹ Courant, Friedrichs and Lewy found their famous condition while using finite differences to prove existence theorems, and Courant’s piecewise linear functions on triangles became the finite element method.² Pure motives produced applied tools, and applied needs produced theorems. The electronic computer arrived for military reasons, ENIAC was built for artillery tables, and it computed the first numerical weather forecast in 1950;³ in 1955 Fermi, Pasta, Ulam and Mary Tsingou performed at Los Alamos the first numerical experiment, to which I return below, and a whole branch of pure mathematics was born from that computer run.

Until the 1950s, “computer” was a job title. Computations were done by chains of humans, mostly women, and before his orbital flight of 1962 John Glenn asked that Katherine Johnson recompute by hand the trajectory produced by the new IBM machine.⁴ The digital computer destroyed these jobs, but the same people took the higher jobs which it created: the first programmers of ENIAC were recruited among the human computers, and Dorothy Vaughan taught her group FORTRAN before the machine arrived at Langley. 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 I believe that AI will do the same, destroying some jobs and creating higher and more creative ones. Each time the machine became better than us at something we did by hand, applied mathematics absorbed it and grew. The crisis was, in fact, a promotion.

Why the mood is different on the applied side

I read with much interest Terence Tao‘s essay “Mathematics in the age of AI” and Hugo Duminil-Copin‘s post on this blog, and I understand them. Pure mathematics has its lighthouses, to use Hugo’s beautiful word: the great conjectures which give direction and identity to whole fields and careers. When he writes that the current use of AI “nukes the mathematical landscape”, I sympathise with him, and Tasmin Chu‘s defence of the dissenter viewpoint deserves the same respect.

Events gave Hugo’s post a cruel confirmation while I was writing mine. His main example was the $\theta(p_c) = 0$ conjecture of percolation theory, which says that there is no infinite cluster at the critical probability, and he wrote that it was only a matter of time before it also fell. Three days later a document produced by Claude, with a formal verification in Lean, claimed a proof through the inequality conjectured by Gady Kozma and Shahaf Nitzan, and Gil Kalai reported it with the words “if verified, this is a remarkable breakthrough”.⁵ I am sincerely sorry for Hugo and for his community, whose landscape was nuked again three days after he asked for a little more time. Alonso Castillo-Ramirez offered under his post the only consolation I know: we can still take great joy in the beauty of mathematics, whoever created it.

But applied mathematics has never been driven by big conjectures. It is driven by the challenges of the times, from the Manhattan project to the climate; our identity is attached to a service, not to a conjecture. This makes us a hybrid: internally we function like pure mathematicians, proving that a scheme converges, while externally the problems which come to us decide our agenda. An AI which solves a problem we care about does not take a lighthouse away; it brings the shore closer. Most applied mathematicians do not even carry the title: they are called engineers, actuaries, meteorologists or data scientists, so the discipline is much larger than its departments, and we are used to sharing the credit with a team, with another profession, with a machine. The question of who receives the credit for a theorem, which Ruodu Wang analyses as a prisoner’s dilemma, matters less to us.

Nilima Nigam objects that calculators automated tasks of no value to do by hand, while AI automates the parts of our work which we value most. The human computers of Langley would not have agreed that their work had no value; it was a profession, and it still moved up. Where I agree with her entirely is that the transition must be managed deliberately, and that this is our responsibility, as is the energy cost of these systems raised by Alexis Marchand and Vadim Lebovici, which applied mathematicians, who work on the climate, should help to measure and to reduce.

Better players, better mathematicians

Chess and Go show what happens when the machine takes from humans the very thing they valued most, the search for the best move. Deep Blue beat the world champion in 1997, engines have been out of human reach since the 2000s, and AlphaGo beat Lee Sedol in 2016;⁶ humans did not stop playing. The largest chess website passed a quarter of a billion accounts this year, every serious player prepares with engines, and Magnus Carlsen credits AlphaZero for making him “a very different player in terms of style”.⁷ A study of 5.8 million moves of professional Go players found that the quality of human decisions improved significantly after 2016, through novel moves which nobody had played before.⁸ The machine did not petrify the players; it made them measurably better and more creative, and their games, because they are human, remain the ones everybody watches.

Applied mathematics had its own AlphaGo moment in 1955. 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.⁹ A surprise produced by a machine and digested by humans became one of the most beautiful chapters of twentieth-century mathematics, and it did not extinguish a lighthouse; it lit a new one. This is why I am quite sure that thanks to AI human mathematicians will become better mathematicians. A machine which proves a conjecture by an unexpected route, as in the proof of Feige’s conjecture described by Guanyang Wang, teaches the humans who study it a move they had never considered, and the young mathematician whom Hugo hoped for, the one who will show us what we all missed, will still come, having studied the proofs of the machine as Carlsen studied the games of AlphaZero.

A new field will also appear, call it AI mathematics, whose subject is not the theorems but the machines which produce them, with conferences on the merits of the different models, the prompting strategies, the types of agents and the ways in which they interact; I would be surprised if this did not happen within a few years. Chess programs have had their own world championship since 1974,¹⁰ the first conferences and workshops on AI and theorem proving exist already,¹¹ and Guanyang Wang’s controlled experiment on this blog with two prompts which differ by a single sentence is one of the first papers of the field. The question is whether mathematicians will shape it or leave it to others.

Three trades in one person

An applied mathematician must master three trades at once, mathematics, the field of application and scientific programming, and AI changes all three. First, the code. Some of my fellow PhD students spent six months debugging their programs; today AI removes most of this friction, and three hours often replace three months. We do not trust these tools blindly: checking that we solve the equations correctly and that we solve the correct equations matters more, not less, when the code is produced faster. We also program in the language best suited to the problem rather than the one we know, because the cost of learning has almost disappeared, and the agents generate and test the code while the humans decide, in the words of Shaowu Zhang, where to point the telescope. Second, the field of application. When friendship led me from fluid mechanics to the quasi-normal modes of black holes, a transition which would have taken five years before, it happened without any gap in my publications, with an expert of the new field at the tips of my fingers; AI will improve the intellectual mobility of applied mathematicians, and the borders between application fields will matter less. Third, the mathematics itself. We do not prove enough in applied mathematics: we present numerical evidence and leave the theorem for later. If AI helps us to increase the number of statements actually proved in our field, it will be the most positive development of all. Martin Hairer‘s rule that an argument produced by an AI deserves more scrutiny than a colleague’s and must never be pasted into our papers applies to us without change, and formal verification will play a growing role. We have lived in the regime where proofs were scarce and numerical evidence abundant; a little abundance of proofs would do us good, and it would bring applied mathematics closer to pure mathematics, not further from it.

One mathematics

The differences explain the mood; let me end with what unites us. We love the same mathematics and serve humanity in complementary ways, pure mathematics by extending what humans can understand, applied mathematics by putting this understanding to work, and we carry the same responsibility towards the next generation. I agree with Álvaro Lozano-Robledo that we must invest in human mathematicians: the virus of mathematics is transmitted from human to human, by a teacher who shows by example that this life is possible, and no tool will replace the moment when a person you admire believes that you can do it. Terence Tao distinguishes 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 the checkers verify, the digestion will remain ours, and here pure mathematics leads, with digestions of recent AI-found results which set the standard we should follow instead of inventing lower ones. Together we should develop the workflows for these tools, verification, disclosure, attribution and reproducibility; the Leiden Declaration and Hairer’s recommendations are good starting points, and we bring decades of verification and validation.

On one point in particular I would like to join forces with my pure colleagues. Some AI companies announce presumed progress on famous conjectures in the language of business communication, and a new kind of mathematics has appeared, where a result exists from the moment it is posted on a social network; Tian Lan has documented how misleading such announcements can be. We should condemn this practice together. A counterexample can be checked, as the Jacobian counterexample was, before being digested. A proof is a different object, and we should admit no proof until it has received independent verification and validation by the community. The percolation claim of this week is closer to what we ask, and I want to be fair to it: no press release, but a public repository with a Lean formalisation and the statement that nobody independent had yet refereed the work. In the language of my field, the machine has done the verification: the argument is correct with respect to the definitions it was given. The validation remains, and it is human work: do these definitions say exactly that there is no infinite cluster at the critical probability, and nothing weaker? Until the experts have done this reading, the result is a claim, an impressive one, and not yet a theorem. The journals, with their imperfect peer review, should remain the channel through which a result enters mathematics; a post is not a replacement. In applied mathematics we have had a name for this discipline for a long time: verification and validation.

To conclude: applied mathematics welcomes and embraces the new technology, and we anticipate a rapid development of our discipline, an acceleration of discovery and an improvement in the quality of our work. These views may appear overly optimistic, and I admit it: I am an optimist. 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, and to propose a few supporting historical examples, which I then verified. Following the guidelines of this blog, I state this here.

1. Herodotus, *The Histories*, Book II, 109 (Perseus Digital Library). ↩︎
2. R. Courant, K. Friedrichs and H. Lewy, Mathematische Annalen 100 (1928), 32–74, doi:10.1007/BF01448839; R. Courant, Bulletin of the American Mathematical Society 49 (1943), 1–23, doi:10.1090/S0002-9904-1943-07818-4. ↩︎
3. T. Haigh, M. Priestley and C. Rope, ENIAC in Action, MIT Press, 2016; J. G. Charney, R. Fjørtoft and J. von Neumann, Tellus 2 (1950), 237–254, doi:10.3402/tellusa.v2i4.8607. ↩︎
4. NASA, biographies of Katherine Johnson and Dorothy Vaughan; Columbia University Computing History, The ENIAC programmers. ↩︎
5. G. Kalai, “Amazing: there is no percolation at the critical probability in all dimensions”, Combinatorics and more, 3 September 2026; G. Kozma and S. Nitzan, arXiv:2401.12397 (2024), a preprint; the AI documents are described in Anthropic’s public repository. ↩︎
6. IBM, Deep Blue; D. Silver et al., Nature 529 (2016), 484–489, doi:10.1038/nature16961. ↩︎
7. “Chess.com reaches 250 million members”, 27 February 2026; P. Doggers, “Carlsen wins 2019 Norway Chess with round to spare”, Chess.com, 14 June 2019. ↩︎
8. M. Shin, J. Kim, B. van Opheusden and T. L. Griffiths, Proceedings of the National Academy of Sciences 120 (2023), e2214840120, doi:10.1073/pnas.2214840120. ↩︎
9. E. Fermi, J. Pasta and S. Ulam, Los Alamos report LA-1940 (1955); T. Dauxois, Physics Today 61 (2008), 55–57, doi:10.1063/1.2835154; N. J. Zabusky and M. D. Kruskal, Physical Review Letters 15 (1965), 240–243, doi:10.1103/PhysRevLett.15.240; C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Physical Review Letters 19 (1967), 1095–1097, doi:10.1103/PhysRevLett.19.1095. ↩︎
10. Chessprogramming wiki, WCCC 1974. ↩︎
11. Conference on Artificial Intelligence and Theorem Proving, held every year since 2016; The 6th Workshop on Mathematical Reasoning and AI, NeurIPS 2026. ↩︎

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Received 5 September 2026.