Mathematics in the age of AI

Abstract: An essay, based on a public lecture delivered at the 2026 International Congress of Mathematicians, on how the mathematical community might respond to the arrival of artificial intelligence tools that are capable of performing research-level mathematical tasks. Rather than debating the capabilities of such tools, we condition on the hypothesis that these capabilities will arrive, and examine instead a question that is orthogonal to it: what the goals and values of mathematical research actually are. The problem-solving component of mathematics is used as a case study.

• The paper argues that AI may produce research-level proofs faster than mathematicians can verify, explain, review, and integrate them into lasting theory.
• The paper presents a five-stage mathematical workflow—proof generation, verification, exposition, community acceptance, and canonicalization—and argues that formal correctness alone cannot ensure understanding or value.
• The paper recommends transparent AI-use disclosure, human authorship and accountability, stronger attribution practices, and greater institutional recognition for reviewing, exposition, and canonicalization.

The paper is a conditional analysis of how mathematical research should respond to advanced AI. Its central methodological decision is to set aside the empirical dispute over AI capability and assume that a sufficiently strong form of that capability will hold. Under this assumption, the principal problem is not whether AI can produce research-level mathematics, but whether the mathematical community has adequately articulated the goals and values that its institutions are intended to serve. The paper’s main claim is that AI will stress-test mathematics’ implicit social and evaluative framework in a manner analogous, though not identical, to the foundational crisis of the early twentieth century (2608.16753).

From foundational assumptions to institutional values

The historical prologue establishes the paper’s organizing analogy. Before the twentieth century, mathematicians worked within largely implicit conceptions of sets, infinity, and proof. Russell’s paradox and Gödel’s incompleteness theorems exposed limitations in this informal foundational regime, leading to the development of explicit formal systems. The resulting clarification did not eliminate foundational questions, but it produced a standardized environment in which mathematical objects and permissible inferences could be examined and mechanized.

The paper argues that contemporary mathematics is approaching a different kind of crisis. The object under examination is not mathematical truth or formal validity, but the community’s tacit criteria for evaluating mathematical work: what counts as a contribution, what is rewarded, what is considered understood, and how authorship and responsibility are assigned. The anticipated transformation is therefore sociological and epistemic rather than foundational in the narrow sense. The argument is not that existing mathematical values are defective in themselves, but that they have remained largely implicit because they historically operated in mutually reinforcing alignment.

This framing also limits the scope of the paper. It does not attempt to resolve the technical question of whether AI systems will achieve particular levels of mathematical competence. Instead, it treats capability as a parameter and studies the consequences for mathematical practice conditional on a sufficiently strong capability hypothesis.

Capability is treated as a conditional premise

The paper formulates an “AI Capability Conjecture” as a family of claims indexed by variables such as cost, human supervision, correctness, quality, field, and success rate. This formulation is intentionally non-specific: “some” AI systems will perform “some” research-level tasks in “some” mathematical fields with “some” nontrivial success rate. The paper distinguishes weak and strong versions but does not attempt to assign a probability to either.

A significant methodological point is that public evidence about AI mathematics is currently difficult to interpret. Reported successes are subject to selection and reporting bias, while important experimental variables—including the number of attempts, human scaffolding, compute expenditure, prior contamination, and evaluation procedures—are often omitted. The paper also distinguishes capability from desirability: evidence that a system can perform a task does not establish that the task should be delegated to it or that the resulting workflow is beneficial.

As a concrete data point, the paper cites the second batch of the First Proof project, which evaluated four publicly accessible AI systems on ten novel research-level problems under controlled conditions. At least one system received a passing grade on seven of the ten problems, with compute costs on the order of tens to hundreds of dollars per problem (Abouzaid et al., 16 Jun 2026). This result does not establish general research autonomy, nor does it resolve questions about scalability, reproducibility, or field dependence. Its significance within the paper is narrower: it illustrates why discussions of mathematical AI can no longer be restricted to elementary problem solving or retrospective demonstrations.

The paper then introduces its Working Hypothesis: AI tools will reasonably soon perform a reasonable fraction of research-level mathematical tasks with reasonable success, quality, supervision, and cost. The deliberately vague quantifiers are not intended as a predictive model. They isolate the normative question from the capability question. The subsequent analysis remains valid across a broad range of capability profiles, provided AI becomes sufficiently effective to interact materially with research workflows.

Mathematical goals have historically functioned as proxies

The paper presents a partial inventory of mathematical goals: solving unsolved problems, developing theories and techniques, understanding the world, sustaining a research community, training mathematicians, contributing to cumulative knowledge, and producing work of aesthetic value. These objectives have historically been positively correlated. A successful proof might simultaneously solve a problem, introduce a technique, generate a research community, train students, support applications, and become part of the subject’s canonical literature.

This correlation allowed the community to optimize one or two visible outputs as proxies for a more complex bundle of objectives. Publication counts, successful problem solutions, priority, and technical novelty could function tolerably because they often tracked broader forms of mathematical progress. The paper argues that AI threatens this alignment through a combination of technical and economic pressures.

Technically, generative systems optimize for outputs that appear satisfactory rather than directly optimizing every underlying property that mathematicians value. Economically, AI development is organized around measurable and benchmarkable achievements. These pressures make AI especially effective at exploiting gaps between a metric and the substantive value that the metric was intended to represent. The resulting danger is a form of Goodhart effect: once a proxy becomes an explicit optimization target, it ceases to reliably represent the broader goal.

The paper’s boldest institutional claim is therefore that the arrival of capable AI may cause the previously aligned objectives of mathematics to diverge. Maximizing the number of solved problems, for example, could increase the supply of formally correct results while reducing collective understanding, educational value, attribution quality, or the rate at which results are incorporated into theory.

Problem solving as a multi-stage epistemic process

The central case study decomposes mathematical problem solving into stages. The paper begins with the apparently natural goal of solving as many unsolved problems as possible. This objective is immediately shown to be inadequate because unconstrained optimization produces large numbers of false proofs, including the familiar stream of purported solutions to major problems such as the Riemann hypothesis.

The first refinement adds verification: mathematical research should solve problems and establish that the solutions are correct. Formal proof assistants such as Lean, Rocq, and HOL make this refinement technically plausible, and advances in autoformalization may substantially increase the throughput of both proof generation and proof checking. Formal verification removes dependence on the author’s reputation or diligence for the narrow question of whether a proof is accepted by a specified formal system.

However, correctness is not sufficient. A formally verified proof may be so long, opaque, or conceptually unstructured that no mathematician understands it. The paper treats this possibility as an emerging practical issue rather than a merely philosophical concern. AI-generated submissions to problem databases already include cases in which the submitter cannot independently verify or explain the proposed argument. Thus, the paper adds a third requirement: solutions must be communicated clearly enough to be understood by the mathematical community.

This refinement introduces a distinction between formal correctness and mathematical intelligibility. Current AI systems perform relatively well on surface-level exposition—grammar, spelling, and formatting—but often allocate disproportionate detail to routine steps while compressing or obscuring the novel part of an argument. They also frequently fail to situate a result within the relevant literature or provide the high-level conceptual overview needed for expert triage.

The paper makes a further, more subtle argument about exposition. Readability is not identical to pedagogical value. Human mathematical writing often contains traces of difficulty: carefully developed lemmas, changes of notation, qualifications, and uneven treatment of different parts of a proof. These features can identify where the actual intellectual obstacles lie. An excessively polished AI-generated proof may remove both accidental defects and informative signals of conceptual difficulty, producing text that is stylistically smooth but epistemically unhelpful.

Figure 1: An annotated page from Bourgain’s 1991 paper, illustrating how the friction of difficult mathematical exposition can support understanding and transmit tacit expertise.

The paper uses an annotated page from Bourgain’s work to exemplify this point. The annotations record the reader’s struggle with the text, but that struggle became part of the process of learning the author’s methods and mathematical perspective. The implication is not that poor exposition is intrinsically desirable. Rather, mathematical exposition should preserve the structure of difficulty instead of presenting every step as equally transparent.

The next refinement is community acceptance. A result must not merely be correct and readable; it must be digested and valued by other mathematicians. This process includes expert interpretation, comparison with prior work, identification of reusable ideas, and incorporation into ongoing research. It cannot be reduced to author-controlled optimization because its defining property is collective uptake.

The paper consequently arrives at a five-stage pipeline:

1. Proof generation from an open problem.
2. Proof verification establishing formal or expert-level correctness.
3. Proof exposition producing an intelligible account.
4. Publication and acceptance through editorial and refereeing processes.
5. Canonicalization into the definitive theory, textbooks, and reference literature.

The final stage is especially important. Canonicalization restates results in natural generality, replaces an initial proof with a more illuminating one when appropriate, connects the result to neighboring theory, and converts an isolated achievement into a reusable component of mathematical knowledge. The paper argues that this is the process through which mathematics accumulates durable understanding. It also observes that AI systems depend on canonicalized human mathematics as training data, creating a structural asymmetry: AI may generate new proofs, but its effectiveness is enabled by the prior human construction and organization of mathematical theory.

Proof abundance and institutional bottlenecks

Conditioned on the Working Hypothesis, the paper predicts a transition from proof scarcity to proof abundance. This does not mean that all mathematical knowledge will become abundant. It means that the production of candidate proofs may outpace the downstream human processes required to validate, explain, review, accept, and canonicalize them.

The paper identifies several corresponding bottlenecks:

• candidate proofs may accumulate faster than they can be verified;
• verified proofs may accumulate faster than they can be rewritten into useful expositions;
• correct and readable submissions may exceed the capacity of volunteer referees;
• published results may exceed the community’s capacity to integrate them into definitive theory.

The implication is that existing institutions will be evaluated against conditions for which they were not designed. Journals, priority conventions, hiring criteria, prizes, and research programs developed under proof scarcity, where generating a correct result was a relatively scarce achievement. Under abundance, the scarce resources may instead be expert attention, interpretation, attribution, and canonicalization.

The paper is careful to note that these pressures predate advanced AI. Literature growth, increasingly specialized mathematics, long proofs, and strain on peer review already challenge the existing system. AI is presented as an amplifier of these structural problems, not as their sole cause.

Recommendations for mathematical practice

The paper does not propose a complete institutional program. Instead, it endorses the Leiden Declaration on Artificial Intelligence and Mathematics as a starting point and interprets several of its recommendations through the problem-solving pipeline.

Tool-use disclosure is treated as a requirement for preserving accountability. Authors should disclose the use of LLMs, ML systems, proof assistants, and other automated tools, including the relevant computational resources. The purpose is not merely procedural transparency. Concealed AI use could create incentives for authors to hide assistance in order to avoid criticism, making it more difficult for referees and readers to assess provenance, reliability, and responsibility.

The paper also recommends supporting reviewing by providing precise references, formal proofs where appropriate, and sufficient information about tool use. AI-assisted production may increase the burden on reviewers even when it improves the probability that a submission contains a valid proof. Automatic systems could assist with triage by identifying missing citations, inadequate formal verification, or incoherent exposition. The paper nevertheless rejects the removal of human referees: automatic filtering can conserve expert attention, but it cannot supply community acceptance.

A central normative recommendation is to reduce the cultural emphasis on proof generation and priority while increasing recognition for proof digestion. Exposition, refereeing, publication, attribution, and canonicalization are not secondary administrative activities. They are the mechanisms that convert individual derivations into collective mathematical knowledge. In an environment of abundant candidate proofs, their relative importance should increase.

The paper affirms human authorship and responsibility. Automated systems should not receive credit as authors, because mathematical credit concerns not only the production of an output but also the ability to explain, defend, contextualize, and take responsibility for it. This position is connected to a strict attribution requirement: because automated tools have known limitations in tracing intellectual sources, authors have an enhanced obligation to search for and acknowledge antecedent ideas. If attribution cannot be established, that uncertainty should be stated explicitly.

The paper’s practical rule is demanding: if authors cannot give a clear, expert-level, correct, and properly attributed talk about their result, the result should not be published. A formally verified proof that no human can explain is treated as incomplete. This claim prioritizes mathematical understanding over formal certification alone, while leaving open the question of how the community should evaluate results whose significance is apparent but whose conceptual explanation has not yet been found.

Limitations and open questions

The analysis depends explicitly on the Working Hypothesis and therefore does not establish that AI will attain the assumed level of competence. The First Proof result is a useful empirical datapoint, but seven passing systems on ten problems cannot support broad claims about general mathematical research, long-horizon autonomy, or performance across fields. The paper also does not specify quantitative thresholds for “reasonable” cost, success rate, supervision, or quality.

The problem-solving pipeline is acknowledged to be schematic. Mathematical research does not always proceed linearly from generation to verification to canonicalization, and theory building may alter the formulation of the original problem. The paper focuses on problem solving and states that teaching, mentoring, hiring, grant writing, refereeing, and outreach require separate analyses. It therefore leaves unresolved how the proposed values should be operationalized in evaluation systems, how credit should be assigned in AI-assisted collaborations, and how human understanding should be measured without reducing it to another vulnerable proxy.

A further open question concerns the relationship between formal verification and comprehension. The paper argues that formal correctness is insufficient, but it does not provide a general criterion for when a proof is “understood” or for how much explanation is necessary. It also leaves open whether AI-generated proofs might eventually contribute useful conceptual abstractions rather than merely verified derivations. More broadly, the essay identifies canonicalization as the most valuable and least easily optimized stage without specifying how institutions should fund, recognize, or coordinate that work.

Conclusion

“Mathematics in the age of AI” (2608.16753) argues that capable mathematical AI should be analyzed primarily as a challenge to the implicit value structure of mathematical practice. Its case study shows that problem solving cannot be adequately defined as the generation of correct proofs. Mathematical progress also requires verification, intelligible exposition, community acceptance, and canonicalization into durable theory. If AI produces proofs faster than these downstream processes can absorb them, the central institutional scarcity will shift from derivation to expert attention and collective understanding. The paper’s conclusion is consequently normative but specific: mathematical institutions should make their goals explicit, disclose AI use, preserve human responsibility and attribution, strengthen rather than eliminate expert review, and assign substantially greater value to the digestion and canonicalization of mathematical knowledge.