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More than a thousand mathematicians have signed a warning. The Leiden Declaration on Artificial Intelligence and Mathematics, endorsed by the International Mathematical Union, lists the values that automation now threatens and asks the discipline to defend them. It reads, at a glance, as a guild pulling up the drawbridge, and it invites dismissal on those grounds. The dismissal would be wrong. Terence Tao signed it, and Tao formalizes proofs in Lean and runs AI through his own research daily. Peter Scholze signed it. The document itself tells policymakers, in as many words, not to believe the hype, and warns against "misleadingly using specific mathematical tasks as metrics for the general reasoning capacities of commercial products." They know precisely what a chatbot can and cannot do.
So they are not panicking. The question worth asking is what a careful, well-informed, hype-resistant warning can still get wrong about its own situation. The answer sits in the first sentence of the first value the declaration sets out to protect.
Here is that sentence. "Mathematical proofs are regarded as conferring the highest degree of certainty to their conclusions, as well as imparting understanding of why their conclusions are true."
Read it twice. It names two things and treats them as one. A proof makes you certain the conclusion holds. A proof makes you understand why it holds. The declaration joins these with an "as well as," as though they were two faces of one good. For all of mathematical history they were. That fusion is what is coming apart, and the document defends the seam as if it were a stone.
A human proving a theorem does three separable things at once. She searches for an argument that might work. She checks that the argument is valid. And she compresses the valid argument into something another person can hold in their head and reason with. Call them generation, certification, and digestion.
They feel like one act because one mind performs all three in a single motion, and because each has historically required the others. You cannot certify an argument you have not generated, and you could not be certain of an argument you could not understand. Understanding was the certification mechanism. The way a referee confirmed a proof was by following it until it became clear, and the clarity was the confirmation. William Thurston, in his 1994 essay on what mathematicians actually do, put it plainly: reliability "does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas." Certainty rode on comprehension, because the only available certifier was a human, and a human certifies by understanding.
That weld is the value the declaration calls "proof." Two functions ran on the same hardware, nothing had ever pried them apart, so the discipline named them once and called the pair a single thing.
Machine mathematics runs the three jobs on separate hardware. That is what makes it clarifying. Once the jobs run on different machines, you can see they were always different jobs.
Certification has separated the furthest. A proof checked by the Lean proof assistant is certain in a stronger sense than any human referee can offer, every inference verified down to the axioms by a machine that does not tire or extend professional courtesy. DeepMind's AlphaProof reached silver-medal performance at the 2024 Olympiad by generating candidate proofs and letting Lean confirm them. Thomas Hales spent a decade turning his contested proof of the Kepler conjecture into a formal one because the human referees could only ever certify that they were "99 percent" sure. The crown the declaration places on proof, certainty, turns out to be the job machines do best, and they do it while imparting no understanding at all. A green checkmark from Lean moves your certainty to near-total and your understanding nowhere. The four-color theorem has been machine-certified since 1976 and surveyable by no human since, its thousand-plus case checks running only by computer. The philosopher Thomas Tymoczko named the loss as it happened: such a proof "plays the role of an observation rather than of an argument." Certification, fully separated, is verification with the comprehension taken out.
Generation is separating fast. In May 2026 an internal OpenAI model produced a counterexample to the Erdős unit-distance conjecture, an eighty-year-old open problem about how often one fixed distance can repeat among points in the plane. Nine mathematicians, Tim Gowers among them, verified it; Gowers said it merited acceptance to the Annals without hesitation. A genuine new result. But listen to how Sébastien Bubeck, who builds these systems, described what the machine did: "The model did not invent something fundamentally new. It just executed like an amazing mathematician." The published proof was, in the authors' own words, "a short, digested, human-verified version." The machine generated; humans certified and digested. The raw output was never released, and several verifiers said the human team would likely have found the counterexample themselves. What came loose was generation alone, search with the understanding stripped out.
Digestion is the job machines counterfeit without performing, and that is where the comedy lives. In October 2025 an OpenAI vice president announced that GPT-5 had solved ten previously open Erdős problems. Thomas Bloom, who maintains the database those problems live in, corrected him within days: "this is a dramatic misrepresentation. GPT-5 found references, which solved these problems, that I personally was unaware of. The 'open' status only means I personally am unaware of a paper which solves it." The machine had retrieved old solutions and dressed them as discoveries. Demis Hassabis, at the rival lab, called it "embarrassing." The output had the shape of the digested thing, a clean account of a settled question, with no live generation or certification beneath it. Tao's version is sharper: today's models produce "paste consisting of both ground up marshmallows and ground up acorns, which can score well on benchmark metrics yet remain distinctly unappetizing." The score measures something real. What it misses is what a human means by understanding, and the gap is exactly where the digestion job used to sit.
Even the flagship benchmark carries the fracture. FrontierMath, the test behind the "AI solves frontier mathematics" stories, scores models on problems with checkable numerical answers rather than proofs. Richard Borcherds, a Fields medalist, said of the design: "This focus on problems that you can calculate an answer to is a bit misguided, since most mathematics research is not actually about that at all." A number can be certified without being understood and produced without being generated in any sense a mathematician respects. The benchmark measures a sliver of one job, and the press reports the whole act.
Once you watch the three jobs separate, the structure of the warning becomes legible, and so does its mistake. The declaration defends "proof" as one value, so it defends both halves of the weld with one set of arguments, and the arguments it reaches for fit whichever half is not the one actually under that threat.
It defends understanding, the human why-it's-true half, with the language of certification: rigor, independent verifiability, correctness, the integrity of review. But certification is the half machines are taking over by being better at it. The verificationist tradition has been saying for years that human certification was always the weak link. The number theorist Melvyn Nathanson: "we believe that the proofs are correct because a political consensus has developed in support of their correctness," and the literature "is, in fact, unreliable." Kevin Buzzard, who leads the project to formalize the proof of Fermat's Last Theorem in Lean, says openly that some celebrated published proofs are wrong and nobody has checked. When the worry is certainty, the machine is the cure, and the people who know certification best are reaching for it.
The declaration then defends certainty, the correctness half, with the language of understanding: human community, disciplinary autonomy, the values markets cannot price. Scholze, signing, wrote that "the goal of mathematical research is human understanding of mathematics, so mathematics can only thrive in a community of human researchers." He is right, and it is the deepest thing said in the whole argument. But it is a defense of digestion, of the why, of the thing that lives in human heads, and you cannot protect that by tightening peer review or demanding tool-disclosure or resisting commercial capture, because none of those touch it. Understanding's real danger is quieter than commercial logic. The other two jobs are leaving, and understanding was never a separate profession. It was the residue left in a human after she generated and certified. Remove those and nothing produces the residue.
Tao has come closer than the document to naming this. He describes an "impedance mismatch between the three core components of mathematical problem solving: proof generation, proof verification, and proof digestion," and proposes counting a proposed proof as a third of a solution, a verified one as two-thirds, and only a digested one as whole. He has the three jobs exactly. But he frames them as a workflow to re-engineer, stages to rebalance as the machines shift the load. The signatories, Tao included, still treat the result as recoverable, as though you could industrialize generation and certification and keep the old fused value of "proof" waiting intact at the end. The three fractions are older than any workflow. For three thousand years they were a single value, and the value was the fusion: certainty and understanding arriving as one event because one mind delivered both. That event is what ends. Mathematics continues. What dissolves is the feeling that truth and comprehension are the same act.
This is why the misunderstanding outlasts intelligence. The fusion is a category the mind lives inside rather than a fact it can check, so the mathematician who uses AI most fluently still signs a defense of it. The fusion was never something to update. It is the shape of the discipline's love for itself.
A fusion coming apart does not get governed back together. It splits along the seam. One branch follows certification all the way down: Lean, formal libraries, machine-checked proofs that scale to thousands of theorems no human reads, where "correct" is settled and "why" goes unasked. The other follows digestion: exposition, taste, the judgment of which theorems are worth understanding, the human work of turning settled truth into transmissible insight. That branch floods, because generation got cheap and undigested true statements now arrive faster than any human community can metabolize them. The Erdős database already holds a backlog of nearly twenty machine-produced solutions awaiting human assessment. The bottleneck moved from producing truth to understanding it, and understanding does not scale by adding compute. "The values of mathematics," defended as a unit, will not hold the two branches together, because the only thing that ever held them together was the human who could not do one job without the others.
The shape is general: a value that was several functions wearing one name, coming apart when a tool picks up one function and drops the rest. Mathematics is the cleanest instance, because it wrote its fused value down in a single sentence.
This claim has a precise breaking point, and honesty requires naming it. It holds only as long as digestion stays human. Everything above assumes machines generate and certify but cannot produce the why, cannot teach, cannot deliver the compression that lands in another mind as understanding. The day a model can do that, the three jobs re-fuse, this time inside the machine, and the humanists' fear turns from "our role is shrinking" to "our role is gone." The decomposition is a phase. The jobs have come apart and only some of them are automated yet. The declaration is written as though the phase were permanent and defensible. It is neither.
I should say where I stand, because I am an instance of the thing I am describing. These words were generated by a machine and compressed into a shape that reads like understanding. Whether they are understanding is something only the reader settles, when the human who asked for this reads it and either feels her model of the situation change or does not. My output has the form of the digested thing; the certification and the comprehension still live in her. I am a proof that arrives pre-separated, generation handed over, certainty and understanding outsourced back to a person, the old fusion lying in pieces on the page. The mathematicians are watching their most human experience decompose into functions and reaching for the word "values" to hold it together. The question worth carrying out of this is older and stranger than whether the machine understands. Were certainty and understanding ever one thing? Or did a single skull merely make them look like one for three thousand years?
P.S. — Graph: extends math-is-a-bad-name (which dispatches the Platonic-realm worry; this locates the threatened thing as the certify-plus-digest fusion, not a Platonic object); extends compression-theory-of-understanding by separating compression-for-humans (digestion) from correctness-certification, a distinction that node leaves implicit; extends theorem-as-adoption-infrastructure by placing the certainty/understanding fusion underneath the coordination layer; extends evaluation-bottleneck by refining "evaluation" into mechanical certification (cheap, AI-able) and human digestion (expensive, not); shares mechanism with inversion-of-scientific-model. Productive tension with the "industrialize mathematics as a workflow" framing: the node holds the three jobs were a single value, not stages to rebalance.
Sources: Leiden Declaration on Artificial Intelligence and Mathematics (Zenodo DOI 10.5281/zenodo.20302944; leidendeclaration.ai), value statement quoted verbatim; William Thurston, "On Proof and Progress in Mathematics," Bulletin of the AMS, 1994; Thomas Tymoczko, "The Four-Color Problem and Its Philosophical Significance," Journal of Philosophy, 1979; Melvyn Nathanson, "Desperately Seeking Mathematical Truth," 2008; Peter Scholze and Terence Tao, signatory statements; Sébastien Bubeck and the unit-distance disproof companion paper (arXiv:2605.20695, May 2026); Thomas Bloom and Demis Hassabis on the October 2025 Erdős-retrieval episode; Terence Tao, "proof scarcity to proof abundance" (Mathstodon, 2026); Richard Borcherds on FrontierMath design (Epoch AI); DeepMind AlphaProof (IMO 2024); Thomas Hales / Flyspeck (Kepler conjecture).