Mathematics in the age of AI

1. A historical prologue

• For centuries, mathematicians worked successfully with naive foundations, largely leaving precise definitions of mathematical objects and proof to philosophers.They proved theorems about sets, numbers, and infinities without explicitly specifying what those entities were.Source: §1 · p. 1
• Russell’s paradox and Gödel’s incompleteness theorems [8] forced mathematicians to reconsider previously implicit foundational assumptions.Naive set theory contained contradictions; a formal system could not simultaneously be consistent, sufficiently expressive, and capable of proving its own consistency.Source: §1 · p. 1
• The foundational crisis produced an explicit, rigorous, standardized framework whose objects and reasoning rules can be inspected, taught, and mechanized.Although foundational research continues, the framework has survived a century of strenuous testing and provides a trusted environment for mathematics.Source: §1 · p. 1
• The author argues that mathematics now faces turbulence over values and practices, requiring explicit goals concerning contribution, reward, understanding, and attribution.The author expects examining and codifying these goals to leave the mathematical community stronger and more resilient.Source: §1 · pp. 1–2

• The central question is how the mathematical community should respond to modern AI technologies and their real or claimed capabilities to perform mathematical tasks.The author names this the Community Response Question.Source: §2 · p. 2
• The question belongs to the entire community; nobody has all the answers, and the author offers guidance on how to approach it.Source: §2 · p. 2
• The response question is metamathematical, political, ethical, sociological, and cultural, rather than mathematical, and cannot be settled by pure logical argument.The author nevertheless borrows mathematical language, including conjectures and hypotheses, to clarify the question’s structure.Source: §2 · p. 2

3. The first subquestion: AI capability

• The AI Capability Conjecture describes near-future research-level mathematical capabilities through parameters covering tools, expense, supervision, tasks, fields, success rates, correctness, and quality.These placeholders define many distinct conjectures; the article makes only a coarse distinction between weak and strong forms.Source: §3 · p. 2
• Failure of even weak conjectures would favor business as usual; the strongest forms would challenge unchanged mathematical practices, especially prioritizing solutions to unsolved problems.The dismissal in the weak-failure scenario concerns the current generation of AI tools and its long-term significance to mathematical research.Source: §3 · p. 2
• Disagreement over AI capabilities obstructs constructive discussion of mathematics’ response, but this article deliberately sets that dispute aside rather than adjudicating it.Most public debate about AI and mathematics has instead focused on which capability conjectures are true.Source: §3 · pp. 2–3
• Public evidence often lacks scientific controls and suffers reporting bias, non-scientific incentives, undisclosed variables, and conflation of capability claims with their desirability.Frequently undisclosed variables include attempt counts, human scaffolding, compute expenditure, and contamination from prior literature; successes are publicized while failures are not.Source: §3 · p. 3
• In First Proof’s second batch [6], seven of ten research-level mathematics problems received a passing grade from at least one of four AI systems under controlled conditions.First Proof uses problems whose solutions are known to contributors but have never been posted anywhere online; models were publicly accessible as of May 28, 2026, and experts reviewed correctness and exposition. Passing solutions were judged essentially flawless or required only minor revisions, with compute costs on the order of tens to hundreds of dollars per problem.Source: §3 · p. 3

4. The complement to the capability conjecture

• The article isolates what it calls the “orthogonal complement” of the AI Capability Conjecture within Question 2.1.Adopting the Working Hypothesis separates this component from the question of AI capability.Source: §4 · p. 3
• The Working Hypothesis assumes AI will reasonably soon perform a reasonable fraction of research-level mathematical tasks with reasonable success, quality, supervision, and cost.The precise meaning of “reasonable” is not critical to the analysis.Source: §4 · p. 3
• The analysis assumes the Working Hypothesis without asking readers to want, believe, or accept it as true.Evidence for or against the hypothesis is orthogonal to the ensuing discussion.Source: §4 · p. 3

• The community must identify not only its publicly stated objectives but also the implicit goals it actually optimizes in practice.The question concerns the goals, objectives, and values of both the mathematical community and mathematical research.Source: §5 · p. 3
• Mathematicians can no longer delegate reflection on their goals to the humanities under the Working Hypothesis, but such reflection remains valuable regardless of that hypothesis.Historically, mathematicians largely left these questions to historians, philosophers, and sociologists while concentrating on technical work.Source: §5 · p. 4
• Mathematical goals include solving problems, developing theories, structures, and techniques, understanding the world, sustaining community, training successors, accumulating shared knowledge, and creating enduring aesthetic works.The author presents this as a partial list, inviting readers to extend it.Source: §5 · p. 4
• AI threatens proxy metrics because generative systems optimize apparent satisfaction rather than underlying properties, while industry incentives reward demonstrable, quotable, benchmarkable achievements.This exemplifies Goodhart’s law: targeting a measure undermines its reliability, in Strathern’s formulation, following Goodhart’s monetary-policy observation.Source: §5 · pp. 4–5
• Excessive optimization for one or two mathematical goals may cause previously aligned goals to diverge.Historically, progress toward one goal typically advanced others, allowing selected goals to serve as convenient proxies.Source: §5 · pp. 4–5

• Maximizing solved problems requires correctness checks; AI and autoformalization into proof-assistant languages accelerate both proof generation and verification.Counting solutions alone encourages incorrect submissions. Formal verification makes correctness independent of the author’s reputation or diligence, and the Working Hypothesis predicts continued acceleration.Source: §6 · pp. 5–6
• Correct proofs must also be understandable: current AI exposition can obscure novel steps and omit literature context or high-level overviews despite near-flawless grammar and formatting.A verified proof could lack anyone able to explain it. Existing AI-generated submissions to the Erdős problems database already include cases without expert verification or endorsement.Source: §6 · p. 6
• Over-polishing exposition can erase informative difficulty cues, making proofs easy to read but hard to learn from and weakening transmission of tacit knowledge.Careful lemmas, notation changes, and visible revisions can signal where readers should slow down. Removing these cues differs from correcting typos, awkward phrasing, or disorganization.Source: §6 · p. 7
• Community acceptance requires slow human digestion and evaluation; AI screening may conserve expert attention but cannot substitute for acceptance or human referees.Authors can help by sharing insights and false starts, whereas current AI problem-solving processes remain opaque. Undercredited editorial and refereeing work converts individual achievements into collective progress.Source: §6 · p. 8
• Canonicalization incorporates results into definitive references through broad consensus; the author considers it the slowest, least AI-optimizable, and most valuable stage.Results acquire their natural generality, appropriate proofs, and connections before entering the standard toolkit. This digestion enables many applications, while canonical theories supply AI training data.Source: §6 · pp. 8–9
• The proposed final goal combines solving problems, verifying correctness, communicating clearly, securing community acceptance, and incorporating results into the field’s definitive theory.Deconstructing problem solving reveals implicit subgoals. This case study does not replace separate analyses of theory building, teaching, mentoring, or community service.Source: §6 · pp. 5, 9

7. Proof scarcity and proof abundance

• Proof abundance will strain mathematical institutions designed around scarcity.These include journals, priority conventions, hiring and promotion criteria, prizes, and the very notion of a research program.Source: §7 · p. 9
• Without suitable policy and cultural changes, AI-generated proofs will outpace verification, readable write-ups, volunteer expert peer review, and community efforts to produce definitive forms.Even correct, well-written proofs will overwhelm traditional peer review, while publication alone will not ensure their incorporation into definitive accounts.Source: §7 · p. 9
• AI will markedly exacerbate pre-existing stresses: growing literature, increasingly long and specialized major proofs, and an already strained refereeing system.These pressures predate modern AI rather than originating with it.Source: §7 · p. 10

8. From goals to recommendations

• Authors should transparently disclose automated tool use rather than conceal it to avoid peer criticism.Papers should include a tool and computational resource disclosure section; reviewers should follow publisher guidelines, disclose permitted AI use, and take responsibility for significant recommendations.Source: §8 · p. 10
• Authors should ease the additional reviewing demands AI can introduce by disclosing tools, supplying precise and complete references, and providing formal proofs where feasible and appropriate.Source: §8 · p. 10
• Mathematical culture should place less emphasis on generating proofs and being first, and more on proof digestion: exposition, refereeing, publication, and canonicalization.Bessis similarly argues for moving away from a theorem economy based primarily on proof generation.Source: §8 · p. 10
• Credit and responsibility should remain with humans, not automated systems, because AI does not replace the collective human labor behind mathematical results.Source: §8 · p. 11
• Automated tools’ attribution limitations oblige authors to proactively find and credit sources, explicitly acknowledging when satisfactory attribution is impossible.Source: §8 · p. 11
• Results should not be published unless their authors can convincingly demonstrate the ability to explain them correctly, expertly, and with proper attribution.A proof that no human can properly explain should count as incomplete, even if formally verified.Source: §8 · p. 11

9. Closing thoughts

• The goals-and-values analysis of problem solving should extend to teaching, mentoring, hiring, grant applications, refereeing, and public outreach.The Working Hypothesis potentially affects each of these aspects of mathematical work.Source: §9 · p. 11
• Education and the training of young mathematicians should emphasize irreducibly human work and tightly restrict AI use.Producing correct homework does not by itself achieve the goal of training a mathematician.Source: §9 · p. 11
• In other areas, mathematicians should define AI best practices on their own terms rather than vendors’ terms, while developing complementary workflows and infrastructure.Examples include collaborative formalization projects, structured problem databases, and new venues for exposition and negative or partial results.Source: §9 · p. 11
• The mathematical community should openly discuss both AI capabilities and its own goals and values.The essay connects this recommendation to the Leiden declaration.Source: §9 · p. 11
• Mathematicians should support serious science journalism and publicly contextualize AI-assisted results, especially where assessing depth, difficulty, and significance requires subfield expertise.They should also seek opportunities to cooperate with and support other researchers and creative professionals facing similar challenges.Source: §9 · p. 11

• Mathlib [12] provides a unified formalized library of mathematics.Source: Appendix A · p. 12
• The listed databases include the Erdős problems database and the optimization constants database.Source: Appendix A · p. 12
• Palomar is listed as a formalized proof registry.Source: Appendix A · p. 12
• Mathematical Discourse [4] is a peer-reviewed video journal for research talks.Source: Appendix A · p. 12
• The SAIR Foundation mathematics competitions are another listed example of new infrastructure.Source: Appendix A · p. 12
• The First Proof project [7] is also included among the infrastructure examples.Source: Appendix A · p. 12