Training Mathematicians in the Age of AI: Intellectual Agency, Cognitive Offloading, and the PhD Thesis

Abstract: Powerful artificial intelligence is weakening the traditional relationship between mathematical output and evidence of mathematical expertise. In particular, the production of an original theorem or a polished dissertation can no longer, by itself, certify the intellectual formation of its nominal author. I argue that graduate mathematical education should therefore be organized around the formation of intellectual agency: internal technical competence, mathematical judgment, understanding, and responsible participation in a shared intellectual culture. I distinguish productive from premature cognitive offloading, propose complementary independent and AI-augmented modes of training, and suggest a corresponding reformulation of the role of the PhD thesis and dissertation defense. More broadly, I argue that academic mathematics should understand itself increasingly as an institution for the reproduction and stewardship of human mathematical expertise rather than primarily as a mechanism for producing theorems.

• The paper argues that traditional measures of mathematical expertise are corrupted by AI-generated proof, urging a focus on 'intellectual agency'—a set of skills centers on research direction, ethics integration, problem-solving evaluations, and interdisciplinary connections—that captures more typical achievements of graduate-level training.
• Cognitive offloading should guide AI integration, distinguishing between 'productive offloading' (which enhances learning) and 'premature offloading' (which hinders skill development), recognizing a need for empirical research to determine this.

The paper argues that artificial intelligence has altered the epistemic relationship between mathematical output and mathematical expertise. A theorem, proof, or polished dissertation previously provided substantial evidence that its named author possessed technical competence, persistence, independence, and mathematical judgment. As AI systems increasingly generate sophisticated mathematical arguments, that inference becomes unreliable. The central proposal is therefore not to preserve a pre-AI conception of mathematical independence, but to reorganize graduate education around intellectual agency: the capacity to formulate worthwhile questions, evaluate arguments, direct research, integrate external assistance, explain mathematical significance, and participate responsibly in a shared intellectual culture.

The argument is normative rather than empirical. The paper reports no experiments, learning outcomes, or comparative assessment data, and it does not claim to provide a finalized institutional policy. Instead, it develops a conceptual framework for graduate training, cognitive offloading, dissertation assessment, and the social reproduction of mathematical expertise.

The epistemic problem created by AI

The paper begins from six axiomatic claims. Human expertise is desirable because it provides epistemic redundancy and permits machine outputs to be interrogated, criticized, and overruled. Tool access does not guarantee competent tool use: generating an apparently sophisticated argument may require little expertise, while diagnosing a subtle error may require years of mathematical training. Mathematical culture is not reducible to a set of formally valid propositions, because it includes interpretation, exposition, historical continuity, communal criticism, aesthetic judgment, and intergenerational transmission. Expertise must consequently be deliberately reproduced through apprenticeship and intellectual culture.

The fifth axiom is the decisive one: AI has permanently changed the evidentiary meaning of mathematical artifacts. This conclusion does not depend on continued capability improvements. Once systems capable of substantial mathematical assistance exist, a reader can no longer infer from a finished proof how much of the underlying mathematical cognition was performed by the nominal author. Disclosure norms may improve provenance, but they cannot restore the former presumption that sophisticated output ordinarily constituted evidence of sophisticated human activity. This is a strong claim about academic epistemology: the problem is not merely undisclosed AI use, but the loss of reliable information that was previously inferred from the artifact itself.

The implication is that theorem production can no longer function as a sufficient basis for judging mathematical formation. Theorems remain mathematically valuable, but their existence increasingly tells us less about the mathematical capacities of the person presenting them. The paper therefore distinguishes the value of a mathematical object from the evidentiary value of that object as a credential.

The discussion of formal verification sharpens this distinction. A machine-checked proof establishes that a formal statement follows from specified assumptions within a formal system; it does not by itself establish why the argument works, which ideas are essential, how the result relates to existing mathematics, or what mathematical meaning it has. Formal correctness and human understanding are thus treated as non-equivalent dimensions of mathematical achievement.

From independent researcher to intellectual agent

The conventional objective of doctoral training is often described as producing an independent researcher. The paper argues that this formulation is both historically inaccurate and conceptually inadequate. Mathematical research has always depended on advisors, collaborators, seminars, referees, literature, institutional resources, and inherited formal systems. Absolute independence is therefore a misleading ideal even before AI.

The proposed replacement is the intellectual agent. Intellectual agency is mathematical judgment exercised from within mathematics. It includes selecting important questions, assessing the feasibility of approaches, recognizing misunderstanding, identifying invalid arguments, rejecting poor suggestions from humans or machines, synthesizing disparate ideas, seeking outside expertise, and situating results within a broader research program. It also includes post-proof activity: reformulating results, identifying conceptual structure, generating consequences, and determining what other mathematicians should learn from the work.

Technical proficiency is a precondition for this agency, but agency is not equivalent to technical virtuosity. A mathematician need not execute every difficult argument unaided. They must instead possess enough internal mathematical structure to recognize the nature of a problem, distinguish conceptual from technical obstacles, assess whether a generalization is natural, and decide whether an external argument merits trust. The paper accordingly rejects the view that AI-supported work is necessarily non-independent. Heavy reliance on AI can coexist with substantial agency, while solitary work and technically strong output can coexist with weak agency over question selection, interpretation, and synthesis.

This conception overlaps with mathematical judgment as formulated in the related graduate-training proposal by Glickenstein, which emphasizes proof validation, definition assessment, monitoring understanding, transfer, and tool governance (Glickenstein, 17 Sep 2026). The present paper adopts a broader frame: mathematical judgment is one component of intellectual agency, which additionally encompasses research direction, synthesis, recognition of when expertise is needed, and stewardship of disciplinary culture.

Cognitive offloading and the formation of competence

The paper’s treatment of cognitive offloading is its principal account of how AI should be integrated into graduate formation. It rejects both unrestricted delegation and blanket technological abstinence. The relevant distinction is between productive offloading and premature offloading.

Productive offloading removes unnecessary drudgery, increases the scale of exploration, or permits attention to be redirected without undermining the competence required for judgment. Premature offloading delegates a task whose performance is itself part of the process by which the relevant competence would have formed. The distinction is developmental rather than moral: the same task may be productively delegated by an established expert but prematurely delegated by a student who has not yet internalized the underlying skill.

This framework yields two complementary modes of training. In independent mode, students work without AI assistance when the pedagogical objective is to develop or assess capacities that AI could otherwise perform, including sustained reasoning, reconstruction of details, management of failed approaches, and detection of invalid arguments. In augmented mode, students use AI extensively and deliberately in research, exploration, verification, exposition, and synthesis.

The independent mode is explicitly described as diagnostic and developmental, not ethically superior. AI is not treated as a contaminant of mathematical practice. Rather, an assessment cannot provide evidence that a student has internalized a capacity if the assessment permits an external system to perform that capacity. This point aligns with the distinction between AI-independent proving and AI-assisted proving in mathematics education, but the paper extends the distinction from individual assignments to the full process of doctoral formation (2609.28140).

The paper also identifies an institutional tension. Graduate education and academic employment continue to reward papers containing original theorems, while AI increasingly makes theorem production a poor proxy for human mathematical formation. Tao’s discussion of a transition from “proof scarcity” to “proof abundance” is used to characterize this change (Tao, 17 Aug 2026). The argument is not that theorems have become unimportant. Rather, an incentive system that treats theorem production as the principal currency of credit and prestige is no longer adequate if institutions intend to cultivate human mathematical expertise.

The paper leaves the operational boundary between productive and premature offloading unresolved. It concedes that this boundary will vary by individual, mathematical field, career stage, and task, and that empirical evidence will be needed. This concession is important: the conceptual distinction is clear, but the paper does not offer a validated protocol for determining which tasks students should perform unaided at particular stages of training.

Mathematics as an intellectual network

The paper situates individual mathematical achievement within a large network of dependencies. Definitions, proof techniques, terminology, foundational results, pedagogical practices, and research questions are inherited from prior generations. Advisors, collaborators, seminar participants, referees, students, and institutions all contribute to the production and validation of mathematical work. Authorship has traditionally compressed this distributed infrastructure into a small number of names.

AI makes this networked character more visible because the individual artifact is no longer reliable evidence of the quantity or kind of human cognition behind it. The appropriate professional question is consequently not only what a mathematician proved, but what their intellectual agency helped the community understand. This reorientation elevates teaching, exposition, criticism, synthesis, organization, mentoring, and institutional maintenance from peripheral activities to central components of mathematical stewardship.

The paper is careful not to equate stewardship with conservation. A steward may criticize inherited ideas, reorganize a theory, discard established approaches, or redirect collective attention. Nor does networked production weaken the importance of attribution. It instead supports more accurate attribution by recognizing forms of intellectual and institutional labor that theorem-centered credit systems often obscure.

A significant risk follows from this proposal. Judgment, taste, agency, and stewardship are less legible than theorem counts and therefore more vulnerable to patronage, prestige effects, and arbitrary authority. Replacing crude quantitative metrics with opaque elite judgment would create a different but equally serious failure. The paper therefore calls for distributed evaluation, multiple forms of evidence, transparent criteria where possible, careful attribution, and resistance to any single advisor or eminent mathematician becoming the arbiter of mathematical value.

Reconceiving the dissertation and defense

The dissertation should no longer function principally as evidence that a candidate has produced an original theorem. It should instead provide evidence of mathematical formation and intellectual agency. Original mathematical propositions remain an important component, but they are placed within a broader evidentiary structure.

The proposed dissertation has several components:

• A substantive mathematical contribution: The thesis should ordinarily contain significant mathematical work, whether or not AI participated in its production.
• An intellectual narrative: The candidate should explain how the problem arose, which background is relevant, where the main ideas came from, and which approaches failed.
• Synthesis: The thesis should locate the contribution within a wider mathematical landscape and explain what other mathematicians should understand or be able to do as a result.
• Machine-assisted methodology: Where AI was used, the candidate should document which tasks were delegated, how outputs were checked or rejected, and where human judgment entered.
• Independent reconstruction and defense: The candidate should be able to evaluate, reconstruct, modify, and defend relevant mathematics without delegating those capacities during the assessment.

The proposal assigns substantially greater evidentiary weight to the dissertation defense. Examiners might alter hypotheses, ask what breaks under modified assumptions, request reconstruction of a proof component, compare alternative approaches, identify conceptual landmarks, or ask the candidate to articulate plausible extensions. Such questioning tests understanding and agency rather than merely familiarity with a polished document. The paper’s proposal is closely related to “mathematical ownership” in Glickenstein’s account of AI-aware dissertation defenses (Glickenstein, 17 Sep 2026).

The defense is not presented as an infallible authenticity test. Oral performance is sensitive to language, disability, personality, and pressure, and live examination may reward speed or charisma rather than depth. It should therefore form one part of a broader evidentiary record. The central institutional principle is not that oral examination solves the provenance problem, but that assessment must include settings in which students demonstrate capacities that polished AI-assisted artifacts cannot reliably reveal.

Graduate education as apprenticeship and stewardship

The paper ultimately redefines doctoral training as an apprenticeship toward stewardship of mathematics and the formation of intellectual agency. Independent problem solving remains essential because it develops technical proficiency, but it is placed alongside activities that are often treated as secondary: research seminars, exposition, teaching, collaborative research, refereeing, synthetic writing, AI-assisted exploration, and mentoring.

These activities matter because they increase the community’s capacity to understand, criticize, use, and extend mathematics. In a high-productivity environment, conceptual synthesis, exposition, literature organization, formalization, and the construction of shared mathematical infrastructure may become more consequential than the production of another isolated theorem. The paper does not claim that these contributions should replace original research in every doctorate; it claims that they should receive substantially greater recognition in the training and evaluation of mathematicians.

The emphasis on collaboration follows from the same analysis. As AI expands the volume of potentially relevant mathematical output, human attention and coordination become scarce resources. Graduate students therefore need to learn how to combine human and machine-generated knowledge, communicate partial understanding, attribute credit, disagree productively, and recognize better judgment in collaborators.

The paper also addresses recruitment and professional identity. Students should not be given a nostalgic account in which mathematics consists primarily of writing papers containing proofs of finished results. At the same time, they should not be told that mathematical training has become meaningless. The paper’s claim is more specific: the content of mathematical work may change, but graduate education can remain valuable if it forms people capable of understanding, directing, evaluating, and stewarding mathematical activity under conditions of proof abundance.

Limitations and open questions

The paper’s principal limitation is its reliance on normative axioms rather than empirical validation. It assumes that preserving human mathematical expertise, epistemic self-government, and a living human mathematical culture are desirable. These assumptions are defended through arguments about robustness, intergenerational transmission, and the value of mathematical experience, but alternative institutional priorities are not systematically analyzed.

The proposal also leaves unresolved how intellectual agency should be measured. The paper correctly observes that agency and stewardship are difficult to operationalize, but it provides no rubric, reliability analysis, or evidence that defenses, portfolios, or distributed evaluation can distinguish genuine understanding from rehearsed performance. Its warning about charisma, patronage, and prestige effects applies directly to the proposed evaluative framework.

A further open question concerns the developmental schedule for independent and augmented modes. The paper does not specify which mathematical tasks should be AI-free, for how long, or according to what criteria a student may transition from restricted to extensive assistance. Nor does it determine whether competence acquired through AI-augmented pathways is equivalent to competence acquired through traditional independent work. These are empirical questions about learning, transfer, retention, and judgment.

Finally, the essay assumes that academia remains sufficiently adaptable to serve as the principal institution for reproducing mathematical expertise. It acknowledges institutional weaknesses and the possibility of new forms of organization, but it does not examine how existing incentives, labor conditions, unequal access to AI, or disciplinary hierarchies might obstruct the proposed reforms. The claim that AI should broaden recognized mathematical contribution therefore remains dependent on substantial changes in evaluation and resource allocation.

Conclusion

The paper’s central thesis is that AI has weakened the dissertation and theorem as proxies for mathematical expertise. Graduate education should consequently certify not only mathematical output but also internal competence, judgment, understanding, intellectual agency, and stewardship of mathematical culture. Independent and AI-augmented modes of training should be used for complementary purposes, while dissertations and defenses should provide evidence of both mathematical contribution and responsible cognitive participation. The resulting ideal is not the isolated independent researcher, but the technically competent intellectual agent who can direct, evaluate, explain, and sustain mathematical work in a networked and AI-assisted discipline.