After AI started doing math, a more dangerous idea might emerge: if machines can prove theorems, are human mathematicians no longer that important?

This essay, Automation Without Understanding, discusses exactly that issue.

The author’s core argument is straightforward: mathematical ability is not just the production of theorems and proofs — it is an infrastructure for understanding, verifying, and auditing complex reasoning.

The paper gives an example: in May 2026, AI helped overturn Erdős’s long-standing conjecture on the planar unit distance problem. But for this result to truly enter the mathematical community, it still relied on human mathematicians to verify, simplify, and transform it into knowledge that everyone could understand and build upon.

This is what the author really worries about:

AI’s mathematical capabilities are growing stronger, but the pipeline for training mathematicians in the United States is weakening. The paper mentions issues such as NSF budget turmoil, withdrawn funding for math programs, reduced doctoral admissions, and university hiring freezes.

What makes this piece interesting is that it doesn’t focus on “Will AI replace mathematicians?” Instead, it reminds us: the more machines are able to reason, the more humans need to preserve the ability to understand machine reasoning.

This is also important for AI agents and research agents.

Future systems may generate more proofs, papers, and scientific hypotheses. The truly scarce capability may become the ability to audit these outputs: to judge what assumptions they rely on, whether the chain of evidence is reliable, and whether the formalization process conceals problems.

The stronger automation becomes, the more important the people who understand automation become.

If we only pursue machine output but abandon training the people who can audit that output, what we end up with may not be a stronger knowledge system, but a more fragile dependency.

arXiv: https://arxiv.org/abs/2607.06377

References

Source: https://arxiv.org/html/2607.06377
Automation Without Understanding: Why the United States must preserve mathematical capacity in the age of AI – Jun–Yong Park

No one thinks about clean water until it turns brown. No one thinks about functioning courts until a verdict is corrupt. The defining feature of infrastructure is its invisibility: when it works, nobody notices; when it fails, everything downstream fails with it. Mathematical understanding is infrastructure of this kind. It is invisible and integral. And it is being dismantled, not by malice but by neglect, at the precise moment when the AI systems built on top of it are becoming more consequential and more opaque than any technology in history.

Two developments are unfolding at once. Artificial intelligence systems have begun to do genuine mathematics: not calculation but discovery, of a kind that until recently only trained human mathematicians could produce. And the United States, through budget chaos and institutional drift, is degrading the pipeline that produces humans capable of understanding what the machines are doing. Either development alone would deserve attention. The second, unfolding alongside the first, amounts to a strategic error.

THE MACHINES CAN DO MATHEMATICS

On May 20 of this year, OpenAI announced a breakthrough on the planar unit distance problem in discrete geometry[2 (https://arxiv.org/html/2607.06377#bib.bib2)], first posed by the Hungarian mathematician Paul Erdős in 1946[1 (https://arxiv.org/html/2607.06377#bib.bib1)]. The question sounds simple: scatter n points on a plane and ask how many pairs of them can sit exactly one unit apart. Erdős conjectured a ceiling on that count, and for eighty years the best minds in combinatorics believed him. The model proved him wrong. It constructed families of point arrangements that break the conjectured ceiling, and it did so by importing machinery from algebraic number theory, including class field towers, a corner of mathematics built for entirely different purposes.

The claim did not rest on the company’s word. The same day, nine mathematicians published a companion paper[3 (https://arxiv.org/html/2607.06377#bib.bib3),4 (https://arxiv.org/html/2607.06377#bib.bib4)] verifying the argument and translating it into human mathematical language. Among them was the Fields Medalist Timothy Gowers, who wrote that had a human submitted the result to the Annals of Mathematics, the field’s most prestigious journal, he would have recommended acceptance without hesitation.

Nor was the result a stunt. The system in question was not designed for mathematics; it was a general-purpose reasoning model powerful enough to do what no human had done. And the trajectory has been steep. In 2024, Google DeepMind’s specialized systems reached the silver-medal standard on International Mathematical Olympiad problems[5 (https://arxiv.org/html/2607.06377#bib.bib5)]. A year later, general-purpose models from both OpenAI and Google DeepMind cleared the gold-medal standard[6 (https://arxiv.org/html/2607.06377#bib.bib6),7 (https://arxiv.org/html/2607.06377#bib.bib7)]. Today, frontier models produce non-obvious constructions, identify connections across fields, and contribute to research-level work.

The machines are doing mathematics. That sentence was controversial three years ago. It is simply true today. The question is no longer whether they can. The question is whether the United States will retain the human capacity to understand, verify, direct, and, when necessary, refuse what the machines produce.

THE QUIET DISMANTLING

At exactly this moment, that capacity is being run down. Last year, the administration proposed cutting the National Science Foundation’s budget by more than half; within that request, funding for the mathematical and physical sciences would have fallen by roughly two-thirds, and for the mathematical sciences specifically, analysts put the reduction above 70 percent. Congress ultimately rejected most of the cuts: the appropriation signed in January keeps the NSF roughly flat, at $8.75 billion against roughly $9 billion the year before.

But the damage did not wait for the final number. During the months of uncertainty, more than $14 million in grants already promised to mathematics programs was clawed back, according to an analysis by Scientific American[8 (https://arxiv.org/html/2607.06377#bib.bib8)]. The NSF canceled funding for a national research symposium for women in mathematics four business days before it was scheduled to begin. The American Mathematical Society, a professional society rather than a funding agency, had to stand up $1 million in emergency “backstop” grants to keep stranded programs alive. And universities, facing the same pressure from every federal direction at once, pulled back: Harvard, Princeton, MIT, and the entire University of California system imposed hiring freezes.

Flat funding after a year of institutional chaos is not recovery. It is stabilization at a lower level of confidence, and even the stabilization has proven illusory. By June, according to reporting in Science, the NSF was quietly cutting hundreds of its basic research programs, in what agency insiders suspected was a bid to free more than a billion dollars for a new commercialization initiative that Congress had never funded[9 (https://arxiv.org/html/2607.06377#bib.bib9),10 (https://arxiv.org/html/2607.06377#bib.bib10)]. Some fields were cut far beyond the 5 percent limit that lawmakers had written into the appropriation itself, and some programs did not learn their actual budgets until five months after the money was approved. And the destination of the money completes the irony: the new initiative exists to commercialize quantum systems and AI-driven scientific instrumentation. The government is funding the machines with money taken from the people who understand them.

The universities, meanwhile, have already restructured around the expectation of decline. Retirements go unreplaced. Doctoral cohorts shrink: George Washington University admitted zero new funded mathematics doctoral students this year[11 (https://arxiv.org/html/2607.06377#bib.bib11),12 (https://arxiv.org/html/2607.06377#bib.bib12)], its funding sufficient only for the students it already had, while Harvard cut its doctoral intake by more than half and Chicago announced a 30 percent reduction[13 (https://arxiv.org/html/2607.06377#bib.bib13)]. The pipeline that produces the next generation of mathematical minds is narrowing just as the machines arrive.

None of this reflects a deliberate policy aimed at mathematics. It reflects a convergence of budget cuts, institutional restructuring, and a reasonable-sounding inference: if the machines can do mathematics, why pay humans to do it?

The answer is the same one that keeps engineers who understand water chemistry inside an automated treatment plant, and human jurists on the bench even if machines could draft flawless legal opinions. Automation works until it does not. Every sophisticated system eventually fails in a way its designers did not foresee, and when it does, the country needs people who understand the system from the inside: people who can trace the failure down to first principles. In domains of judgment, the need is not only technical but civic. A verdict that no human understands is a verdict no institution can stand behind. A system that nobody understands is a system that nobody can govern.

A PROOF IS A PRODUCT. UNDERSTANDING IS A CAPACITY.

Mathematical capacity is not a stockpile of theorems. It is a community of people transformed, by years of rigorous training, into minds able to see abstract structure inside complexity, detect errors that formal systems miss, and reason with unusual precision. No one learns to think as a mathematician by reading a book. One learns it by spending years confronting theoretical frameworks initially beyond one’s reach, under the supervision of people who learned the same way, inside institutions that believe such training is worth sustaining.

Remove the funding and weaken the institutions, and the training pipeline breaks. When it breaks, the capacity is not preserved elsewhere; it becomes thinner, more uneven, and far harder to rebuild. A country cannot conjure a mathematical workforce on demand any more than it can conjure an officer corps or a cadre of nuclear engineers on demand. The training takes years. The intellectual tradition takes generations.

The usual reply is that future students can simply learn from recorded lectures, archived textbooks, or AI tutors. But mathematical capacity is not a body of information; it is a practiced form of judgment. Every surgical procedure ever performed could be captured on video. No one would attempt open-heart surgery on that basis.

A proof is a product. Understanding is a capacity. Machines may increasingly manufacture the product. The capacity to interpret, test, extend, and challenge it must still be cultivated in human minds and sustained by institutions that train those minds. When the people who carry that judgment are gone, the books remain on the shelf. The capacity to read them does not.

The Erdős episode itself makes the point. The companion paper’s authors were explicit that human experts had to discuss, simplify, and improve the machine’s argument before the field could absorb it. The proof became knowledge, something other mathematicians can now build on, only because a community existed with the training to receive it. That community is the asset, and every additional machine-generated theorem raises its value.

The training produces habits that matter far beyond mathematics. A mathematician learns to ask questions that are often unwelcome in fast-moving institutions: What, exactly, is the object under discussion? Which assumptions are doing the work? What would count as a counterexample? Does the conclusion actually follow from the premises? Is the apparent pattern structural, or merely accidental? These are precisely the habits required to audit a model, expose a hidden assumption, distinguish a robust result from an artifact of the data, and recognize when a formal system is being asked a question it was not built to answer.

Pure mathematics also preserves a culture of unusually serious criticism. In a good seminar, authority is not enough. A famous speaker and a first-year graduate student are answerable to the same demands: define the terms, justify the inference, make the argument clear. That culture cannot be recreated by buying more chips or allocating more capital. It requires time, institutional continuity, and a community large enough to sustain standards that no individual can maintain alone.

That is what is at risk.

EMERGENCE WITHOUT UNDERSTANDING

The training objective behind today’s large language models is deceptively simple: predict the next token in a sequence. Yet systems trained at sufficient scale, then refined through post-training and extended inference, display forms of abstraction, planning, and multistep problem-solving that their creators do not fully understand. They never stop predicting tokens, but the behavior that emerges is increasingly difficult to describe in the vocabulary that built it. A system trained to continue text can now overturn an eighty-year-old conjecture. The gap between the simplicity of the rule and the complexity of the result is one of the central scientific facts of the present moment.

Serious work is under way on interpretability, formal verification, and the theory of learned representations. But these efforts remain immature relative to the systems they are trying to explain. There is not yet a systematic mathematical theory of how large artificial neural networks form internal concepts, why certain capabilities appear abruptly, or how to predict which forms of reasoning will emerge at a given scale.

There are glimpses of what such a theory would have to explain. A few years ago, researchers trained a small artificial neural network on modular arithmetic, a staple of undergraduate number theory, and then reverse-engineered what it had learned[14 (https://arxiv.org/html/2607.06377#bib.bib14)]. The network had not memorized its answers. It had independently converged on an algorithm built from discrete Fourier transforms and trigonometric identities, converting addition into rotation about a circle: mathematical structure that no one had taught it and that its designers did not anticipate. The episode is celebrated in the young field of mechanistic interpretability, but its most important lesson is usually missed. Decoding the network was possible only because the researchers could recognize Fourier structure when they saw it. The machine’s inner workings became legible only to humans who had the mathematics to read them. And that was a toy model; frontier systems are larger by many orders of magnitude. We cannot map what a frontier model has learned with the mathematical equivalent of a magnifying glass.

When mathematicians encounter an unfamiliar object, they do not ask only whether it performs well. They ask what kind of object it is: What are its invariants? Its symmetries? Its limiting behaviors? Which examples are canonical, and which apparent phenomena are artifacts of a particular presentation? Nothing like that framework yet exists for the internal representation spaces of frontier AI systems. Existing tools reveal fragments: circuits, features, scaling curves. But fragments are not a theory. We can inspect portions of the machine without knowing what sort of mathematical object the whole machine has become.

The institutional picture compounds the scientific one. Frontier models were trained on a vast public inheritance: papers, textbooks, lecture notes, and solved problems produced largely in universities and public research institutions. Yet the work of building and studying the most capable systems now occurs inside a small number of private firms, with limited outside access to the models, data, and experiments that independent scrutiny requires. The public intellectual infrastructure capable of understanding these systems has not kept pace with the private infrastructure capable of building them. Societies are coming to rely on systems whose capabilities are advancing faster than the science that explains them.

Pure mathematics cannot close this gap alone; neither can machine learning, computer science, statistics, formal methods, security engineering, or governance. But mathematics belongs at the center of the coalition, because the task is not only to make the systems more powerful. It is to develop the conceptual frameworks by which their structure can be understood, their claims audited, and their limits known.

THE FIRST-CONTACT PROBLEM

Artificial superintelligence may arrive soon, late, or never. No one knows, and a responsible argument does not pretend otherwise. The more immediate problem is already here: institutions are beginning to delegate judgment to systems whose reasoning they cannot fully examine. A model may advise on military logistics, approve loan applications, recommend medical treatments, or screen job candidates, with consequences that outrun the human oversight remaining in place. The problem is not that the systems are sometimes wrong. It is that when they are wrong, the error often resists easy characterization. It may stem from a flaw in training data, a misspecified objective, an unintended correlation, or an emergent behavior that no one anticipated. Without a human community trained to trace the reasoning, diagnose the failure, and decide what counts as a fix, the errors become authority by default.

The mathematics community does not yet have a clear answer to this problem. But it has the closest thing to a relevant culture: a tradition of demanding that every claim be made explicit, every step justified, and every assumption laid bare. That tradition, applied to AI systems, is the beginning of accountability. Without it, the systems will remain as opaque as they are powerful. With it, societies have a chance to govern the machines rather than being governed by them.

What is needed is not just more mathematicians, but a broader distribution of mathematical habits of mind across the technical and policy professions that interact with frontier AI. That is a long-term project of education and institutional design. It cannot be accomplished in a crisis. It can only be neglected until a crisis arrives.