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A Response to
“A Severe Misalignment of AI in Mathematics”

Krzysztof Burdzy

September 2026

Abstract

The statement signed initially by twenty-five Fields Medalists correctly warns that solving famous problems is not the same as creating mathematical understanding. Its diagnosis is nevertheless too narrow. Landmark problems are only one source of mathematics; applications, models, and concepts create entire fields. Moreover, policy should anticipate not merely today’s powerful AI companies but a world in which superhuman mathematical tools are widely available. Human apprenticeship remains valuable, but even its intellectual products may eventually be supplied by AI. The proper response is therefore not professional protection. It is the construction of AI-inclusive institutions that serve society while preserving verification, interpretation, education, and open mathematical knowledge.

1. A serious warning, but an incomplete diagnosis

The declaration “A Severe Misalignment of AI in Mathematics,” whose initial signatories were twenty-five Fields Medalists, argues that the objectives of AI companies and those of the mathematical community have become seriously misaligned [1]. Its central insight is correct. A great open problem is not merely an unanswered yes-or-no question. Attempts to solve it generate definitions, examples, methods, connections, and people capable of seeing further. A final proof may be only the most visible product of a much larger process. A corporate race to announce solutions can damage this process, especially when exposition, attribution, and verification are treated as afterthoughts.

This warning does not determine what AI companies may properly do, nor does it tell us how mathematics should respond if machines become much better than humans at proving theorems. Four different things are easily conflated: mathematics as a body of truths and methods, mathematics as a human activity, the profession of mathematics, and the mathematical competence needed by society. AI might enlarge the first, transform the second, shrink the third, and increase the importance of the fourth. There is no reason to assume that all four must flourish or decline together.

The declaration speaks mainly from within the values and institutions of research mathematics. A broader analysis must ask whether its picture of research is representative, whether the interests of mathematicians coincide with those of society, what policies would survive the rapid diffusion of AI, and which human functions would remain valuable if AI produced concepts and questions as well as proofs.

2. Landmark problems are only one source of mathematics

The declaration places famous open problems near the center of mathematical research. Although it does not explicitly claim that most important mathematics is created around such problems, an outsider could easily receive that impression. The picture is badly distorted. Landmark problems are one engine of progress, but they are not the only engine and perhaps not the dominant one.

Applications are a major independent source of mathematics. Physics, engineering, statistics, economics, biology, and computation continually generate new concepts, methods, and theories. More often than not, the mathematics is developed because an application demands it, not because a celebrated conjecture is waiting to be solved. Once created, a theory develops internal questions of its own and may eventually become pure mathematics in everything but its history.

Modern AI is a good example. Large language models depend on linear algebra, probability and statistics, optimization, information theory, numerical analysis, and high-dimensional computation. Much of this mathematics predates LLMs and was not developed in pursuit of famous conjectures. It was assembled, adapted, and extended to meet practical needs. The technology now portrayed as a threat to mathematical culture is itself a product of the kind of mathematics that is developed independent of landmark problems.

Research may begin with a concept or model whose importance is recognized before it is attached to any famous open problem. Probability theory supplies many examples. Martingale theory grew from fair games, conditional expectation, and the evolution of information. Exchangeability begins with symmetry principles for random sequences and arrays. Random-matrix theory was driven by models from statistics and physics before forming connections with number theory, combinatorics, and data analysis. Superprocesses arose as continuum limits of branching particle systems and now link probability with nonlinear partial differential equations. None of these subjects owes its existence to a single landmark problem. Each created a mathematical world and, with it, a continuing supply of questions.

The present corporate interest in famous problems is partly advertising. A solution of a Millennium Prize Problem is instantly recognized as an extraordinary achievement; the hundredth solution of a difficult but obscure problem is not. Companies may eventually exhaust the small collection of universally recognized targets and turn to products with clearer commercial returns. On the other hand, mathematical reasoning may remain valuable for general AI training, formal verification, science, software, finance, and engineering, and a powerful system may solve thousands of lesser-known problems at low marginal cost. No confident prediction is possible.

Even the sudden solution of every currently celebrated open problem would not kill mathematics. It would disrupt research programs, eliminate familiar training grounds, and force a rapid reallocation of attention. But new applications would generate new models, new concepts would reveal new structures, and every sufficiently rich theory would create questions not visible before the theory existed. Mathematics is not a warehouse containing a finite stock of great problems that AI might empty. Solving the present landmarks would close a chapter, not end the subject.

3. A profession is not the public interest

The Association for Human Mathematics opens its statement with the sentence: “The Association for Human Mathematics (AHM) protects mathematics as a human endeavor against the threat of artificial intelligence” [2]. The sentiment is understandable, and a professional association should advocate for its members. But the interests of mathematicians, mathematics, and society are not identical.

Technological replacement is a normal feature of civilization. Automobiles displaced coachmen and much of the horse-based transportation economy. Word processors eliminated most professional typing. Software transformed accounting and routine calculation. These changes destroyed jobs, identities, and training systems. That pain did not create a social obligation to preserve the old division of labor. Society adopted the new technologies because they produced wanted goods and services more quickly, cheaply, or reliably.

Mathematics has no special exemption. If AI can produce correct, illuminating, and independently checkable arguments faster and more cheaply than people, society will have strong reasons to use it. The public primarily wants what mathematics helps provide: reliable knowledge, safe bridges, airplanes that fly, useful medicines, accurate predictions, secure communication, and intellectual discoveries worth knowing. If these goods can be obtained with far fewer professional mathematicians, many citizens may accept a drastic contraction of the profession.

Human mathematics may retain cultural value after losing much of its economic function. Horses survived the end of horse-based transportation, and painting survived photography, but in changed roles. Human mathematics might similarly continue as an art, a sport, an educational practice, or a source of conceptual interpretation even if machines perform most frontier theorem proving. Human chess remains valuable although computers are stronger than its champions.

Society will continue to need mathematical competence. Ordinary people need quantitative judgment to evaluate prices, interest, risk, and automated advice. Scientists and professionals must formulate objectives, inspect assumptions, connect formal results to reality, and accept responsibility for decisions. Teachers will be needed for at least as long as people learn mathematics from people. These needs may sustain a substantial profession, but that conclusion must be demonstrated rather than assumed.

The profession should therefore stop presenting its anxiety as a sufficient public argument. It must show which human capacities produce public value: independent oversight, adversarial checking, selection of significant questions, scientific interpretation, teaching, conceptual explanation, resilience when systems fail, and preservation of knowledge outside corporate control. A profession cannot establish a right to survive merely by describing its work as creative or culturally important.

4. Policy for widely distributed superhuman mathematics

Much of the present anger appears directed at large technology companies—or, in practice, at one especially visible company—that use famous problems to demonstrate their systems. Concentrated computing power, proprietary models, and control over publicity deserve scrutiny. But a policy designed only for this transitional configuration may be obsolete before it is implemented.

Within a few years, an ordinary mathematician might conceivably be able to ask a widely available system to solve a problem of what we now regard as Millennium Prize difficulty and receive an answer in minutes. This is not a prediction on which policy should depend. It is a scenario that policy must be able to survive. The relevant future may contain thousands or millions of individuals, universities, small companies, governments, and autonomous systems with mathematical abilities beyond the present human frontier.

The transitional challenge and the mature challenge are different. The former concerns concentrated corporate power, unequal access, proprietary infrastructure, confidential data, attribution, and publicity. The latter concerns the organization of mathematics after superhuman capacity becomes widely distributed: the status of machine-produced proofs, authorship and priority, journals and archives, education, employment, certification, and responsibility for errors.

Rules aimed only at today’s leading firms would fail both practically and ethically. Capabilities will move among companies, countries, universities, open projects, and personal devices. Restricting one visible company while thousands of equally capable users remain free would not preserve mathematical culture; it might merely transfer activity to less accountable actors. Comparable acts should also be judged by comparable standards. A proof of a public conjecture should not be legitimate when produced by a university and illegitimate when produced by a company merely because the company is large. Conversely, confidentiality, attribution, verification, and honest public description should bind universities and individuals as well as corporations. Differences in power, conduct, access, and harm may justify different obligations. Corporate identity alone does not.

A public mathematical problem is not owned by the person who formulated it, by those who have worked on it for years, or by the profession. Another mathematician may attack it without permission; the same presumption should apply to a company or an AI agent. Solving a public problem for publicity may be ethically unattractive, but publicity has long supported science and is not, by itself, misconduct.

Public problems must be distinguished from private material. Manuscripts, unpublished ideas, confidential communications, user data, and contractual promises create genuine obligations. A result that depends on prior human work requires proper attribution. A press release cannot substitute for an inspectable proof. These principles are consistent with the Leiden Declaration’s emphasis on verification, peer scrutiny, public infrastructure, and unequal bargaining power [3]. They should be capability-neutral and actor-neutral: the rules should govern what was done, with what information, and with what consequences, rather than who happened to possess the technology first.

5. The apprenticeship argument—and its possible expiration

The strongest part of the Fields Medalists’ case concerns apprenticeship. Graduate students struggle with problems whose answers are already known because the purpose is not the answer. It is to develop habits of proof, examples, skepticism, persistence, taste, and the ability to recognize a promising idea. Research continues this education at a higher level.

If young mathematicians delegate every hard step to AI, they may produce papers without acquiring these abilities. A generation could become skilled at requesting and editing machine output but unable to construct a proof when the machine lacks the right concept. Mathematical output might rise while human capacity quietly decays. New concepts are also often born from prolonged frustration: repeated failures expose an obstruction and motivate a landscape in which it becomes tractable. An instant proof may terminate an inquiry without creating a fertile theory.

This is a powerful argument today, but it may have a short half-life. Future AI may diagnose failed approaches, invent definitions, isolate reusable lemmas, generate examples and counterexamples, compare theories, pose better questions, and organize the results conceptually. It may provide, at greater speed and scale, the very intellectual by-products that the declaration values more than the final answer. If so, preserving major problems for human struggle would no longer be necessary for the progress of mathematics. It would instead preserve a human activity. That may be a legitimate cultural or educational objective, but it is a different argument.

Nor should we simply declare that humans will always be needed to choose questions, judge significance, explain results, or connect mathematics to science. These are plausible roles, not permanent truths. AI may encroach on each of them. Policy should distinguish three aims:

  1. producing the best mathematics, whether the decisive work is human or artificial;

  2. maintaining enough human competence to audit, interpret, govern, and recover from failures of AI; and

  3. preserving human mathematical activity where society values it as education, culture, intellectual development, or recreation.

The first concerns mathematics, the second social resilience, and the third the kind of human life we wish to sustain. They justify different policies.

Education should accordingly distinguish training from production. There should be protected periods of unaided proof construction, oral examinations and live discussion, assignments emphasizing examples and failed approaches, and explicit training in diagnosing AI-generated arguments. Advanced research can use AI fully while requiring the researcher to understand, interpret, and defend the result. The aim is not to preserve inconvenience. It is to preserve whatever human competence remains necessary.

The future role of mathematicians is genuinely unclear. They may become selectors of problems, interpreters of machine discoveries, teachers, auditors, formal certifiers, translators between mathematics and science, or governors of shared infrastructure. Some roles may support a large profession, others a small one, and still others may themselves be automated. The community should test new roles now rather than defend its present structure with assumptions that technological progress may soon invalidate.

6. Construction, not resistance

A comprehensive boycott of AI is implausible. Students, researchers, universities, companies, and governments will use tools that save time or open previously inaccessible problems. Competitive pressure makes abstention unstable. Human-only journals or competitions may have a legitimate place, but they should be voluntary cultural institutions, not the default rules of mathematical publication.

The Luddite label is historically crude: the original movement had serious grievances about wages, power, and the organization of work. Yet resistance to effective labor-saving technology has repeatedly failed as a strategy for preserving an occupational structure. Mathematics should not expect a different result merely because its practitioners consider their work intellectually elevated.

The useful response is institutional construction. A workable mathematical environment should include:

  1. inspectable arguments and independent verification, including formal proof where practical;

  2. open proof libraries and publicly governed alternatives to proprietary infrastructure;

  3. affordable access for researchers outside wealthy institutions;

  4. publication systems in which AI assists with routine checking while human judgment concentrates on significance, interpretation, and exposition;

  5. educational settings that separate the cultivation of human skill from the efficient production of new knowledge; and

  6. uniform standards for confidentiality, attribution, publicity, and responsibility across companies, universities, and individuals.

Professional evaluation should reward judgment, synthesis, explanation, and public value rather than count papers whose technical production can be automated. Journals may publish fewer isolated results and more verified syntheses, conceptual reorganizations, and connections to science.

AI companies currently have reasons to cooperate. They need expert judgment to choose meaningful benchmarks, distinguish a genuine advance from a reformulation, locate prior work, improve exposition, and earn scientific legitimacy. Universities and public funders therefore have room to negotiate access, verification, and responsible publication, and to support independent mathematical AI laboratories as proposed in recent discussions of public infrastructure [4, 3]. This creates leverage, not veto power.

What should be preserved is not every feature of the existing profession. Publication counts, priority for a bare theorem, and large numbers of professional theorem provers may lose their importance. The durable objectives are a trustworthy public record, independent verification, attribution, open knowledge, mathematical education, conceptual understanding where it performs a real function, and institutions capable of adapting again as AI changes.

7. Conclusion

The Fields Medalists are right that mathematical understanding is more valuable than the mass production of answers. They overstate, however, the centrality of landmark problems and leave the survival of the present profession too close to the center of the discussion. Mathematics also grows from applications, models, and concepts. AI may solve today’s famous problems without exhausting the sources from which new mathematics arises.

Policy cannot be organized around hostility toward whichever company currently leads the technology. It must remain defensible when superhuman tools are available to millions of people, and it must apply the same basic standards to companies, universities, governments, and individual mathematicians. It also cannot assume that AI will remain a proof engine while humans retain a monopoly on methods, concepts, and questions. The apprenticeship argument is powerful now, but its technological premise may disappear.

Mathematics will change, probably revolutionarily. Mathematicians should stop asking society to freeze that revolution at a convenient point. We should build institutions that use the strongest available AI, serve the public, produce mathematics of the highest quality, and preserve human understanding where it performs a genuine function. The strongest defense of human mathematics is not nostalgia, professional solidarity, or ownership of open problems. It is a successful demonstration that human mathematical judgment delivers what society needs.

References

  1. Artur Avila et al., “A Severe Misalignment of AI in Mathematics,” September 11, 2026. https://mathandai.org/
  2. Association for Human Mathematics, “The Association for Human Mathematics (AHM),” accessed September 23, 2026. https://www.ahmath.org/
  3. Leiden Declaration Working Group, “Leiden Declaration on Artificial Intelligence and Mathematics,” June 2, 2026. https://leidendeclaration.ai/ DOI: https://doi.org/10.5281/zenodo.20302944.
  4. Johan Commelin, Mateja Jamnik, Rodrigo Ochigame, Lenny Taelman, and Akshay Venkatesh, “Shaping the Future of Mathematics in the Age of AI,” Notices of the American Mathematical Society 73 (2026), no. 6, 480–483. https://doi.org/10.1090/noti3347