What is mathematics now, and what should it be?

Abstract: Advances in neural theorem provers have been impressive, but the successes obscure a broader vision of what AI can do for mathematics and how mathematicians can engage with AI. This essay advances a more expansive and optimistic point of view.

• The paper argues that evaluating AI in mathematics requires considering versatile applications beyond neural theorem proving, including formalization, digitization, proof assistants, and machine learning, among other integrative roles.
• The central issue highlighted is the potential impact on the experience of mathematical work, particularly the motivational structure sustaining mathematical inquiry, and how automation may alter meaningful participation in mathematics
• The paper recommends adapting institutional and educational frameworks as readers must become both proficient users and developers of these AI-related tools.

The essay argues that the central question posed by contemporary AI is not whether machines can prove mathematical theorems, but what mathematicians want mathematics to become under conditions in which substantial parts of formal and informal reasoning can be automated. Its principal claim is that the current public and professional focus on neural theorem proving is too narrow. AI for mathematics should instead include formalization, digitization, proof assistants, automated reasoning, symbolic computation, machine learning, mathematical discovery, and new forms of collaboration. The paper therefore combines a diagnosis of disciplinary anxiety with a normative argument about mathematical values, institutional responsibility, and training.

The source of mathematical anxiety

The paper begins from the rapid improvement of LLM-based systems and neural theorem provers. It notes that these systems can increasingly produce formal and informal proofs, assist mathematicians with theorem proving, and address difficult mathematical problems. Such developments have generated concern not only about employment but also about the experience of mathematical work. The more important anxiety, in the author’s account, concerns the possible erosion of the activities through which mathematicians derive intellectual satisfaction: sustained struggle with a problem, informal exploration, spontaneous insight, and extended conversations with colleagues.

This distinction between occupational displacement and phenomenological displacement is important. The paper does not treat mathematics merely as a collection of problems whose solutions can be outsourced. Mathematical practice is also characterized by the process of searching, failing, reformulating, and eventually understanding. If an AI system supplies a plausible solution immediately, the motivational structure that sustains mathematical inquiry may change even when mathematicians remain employed. The concern is therefore not reducible to the standard claim that automation will eliminate jobs. It concerns the possibility that automation could alter what counts as meaningful participation in the subject.

The essay presents this anxiety as a legitimate response to technological change rather than as irrational resistance. Public descriptions of AI as “coming for mathematics” or replacing mathematicians reinforce the perception that a central human activity is being transferred to machines. The author also emphasizes that the emotional difficulty of this transition is particularly acute for mathematicians whose attachment to the subject is grounded in the experience of thinking itself. This framing establishes the paper’s broader thesis: evaluating AI for mathematics requires an account of the values that mathematical practice is intended to preserve.

Dedekind and the historical transformation of mathematics

The historical section uses Richard Dedekind’s 1888 essay “Was sind und was sollen die Zahlen?” as a structural analogue for the contemporary question. Dedekind characterized the natural numbers axiomatically as a system generated by an initial element and a successor operation, constructed such a system set-theoretically, and proved that any two systems satisfying the relevant axioms are isomorphic. The resulting notion of categoricity supported the conclusion that the essential nature of numbers lies not in a particular representation but in the structural relations determined by the axioms.

For the paper, Dedekind’s significance is methodological as much as foundational. His work represented a transition from explicit calculation and concrete representation toward axiomatic characterization and structural abstraction. That transition was historically contested. The essay recalls opposition from mathematicians who regarded the new abstraction as detached from mathematical simplicity and intelligibility, culminating in Carl Ludwig Siegel’s denunciation of modern algebraic styles as a threat to the identity of mathematics.

The historical analogy is not that AI is simply another instance of abstraction. Rather, it shows that mathematical communities have previously experienced methodological change as a threat to the subject’s character. The resistance to axiomatization and abstraction was partly intellectual but also personal: established mathematicians saw familiar mathematical objects and standards transformed into something they no longer recognized. This perspective allows the present response to AI to be interpreted as a recurring disciplinary conflict over methods, standards, and identity.

At the same time, the analogy has limits. The axiomatic transformation expanded the conceptual resources available to mathematicians while remaining a human intellectual practice. AI introduces systems that can generate, search, and verify mathematical artifacts with a degree of speed and scale that may directly displace parts of the activity of reasoning. The paper uses history to encourage institutional adaptability, not to claim that contemporary automation is merely a repetition of the foundational debates of the nineteenth and early twentieth centuries.

A broader conception of AI for mathematics

The essay’s most direct intervention is its rejection of the equation

AI for mathematics = neural theorem proving.

Neural theorem proving is treated as a significant development, but not as the defining application of AI to mathematics. The broader category includes the formalization and digitization of existing mathematics, the construction of proof-assistant libraries, automated reasoning, symbolic computation, and computational systems for interaction and collaboration. It also includes SAT, SMT, first-order and higher-order theorem proving, constraint solving, and symbolic optimization.

The paper further distinguishes LLMs from the wider set of machine-learning methods. Reinforcement learning and neural networks can be used to detect patterns in mathematical data, identify interesting objects, discover empirical regularities, approximate solutions to PDEs, and locate parameter regimes in which complex behavior occurs. These tasks differ fundamentally from reproducing a human-specified proof. They concern the generation and organization of mathematical data, the exploration of large search spaces, and the identification of structures that may subsequently require conceptual explanation.

This distinction has a significant methodological implication. A proof-producing system operates within a relatively well-defined objective: construct an object that satisfies a formal specification. Mathematical discovery often lacks such a specification in advance. The system may need to determine which objects, patterns, conjectures, or parameter values are worth investigating. Consequently, the most consequential AI systems for mathematics may not be those that solve the largest number of existing benchmark problems, but those that alter the processes by which mathematicians formulate questions and recognize structure.

The paper acknowledges that these broader applications have not yet achieved results comparable to neural theorem proving. Its explanation is partly sociological: large technology companies and startups have directed vastly greater resources toward LLM-based theorem proving than toward less standardized forms of mathematical exploration. The comparison between research areas is therefore confounded by investment, infrastructure, and publicity. The author’s optimism about other applications is explicitly not presented as an empirical conclusion that they have already delivered equivalent advances. It is a claim about underexplored potential and about the consequences of allocating research attention too narrowly.

Mathematics as a source of methods and values

The paper rejects the framing of mathematicians as competitors to AI. AI is characterized as technology whose significance depends on how it is designed and used. The relevant question is not whether a machine can perform a task faster than a human, but whether the task should be performed in that way and what purposes the resulting system serves.

This position leads to a substantive account of mathematical value. The legitimacy of mathematics has historically derived partly from its ability to provide conceptual and problem-solving resources for science, engineering, and other activities. If practitioners in these fields begin to rely on AI systems without mathematical mediation, mathematics could lose an important source of relevance. The paper states the concern in particularly strong terms: mathematics may be replaced not because mathematicians lose their jobs, but because other disciplines cease to seek mathematical guidance and turn directly to AI.

The argument does not imply that mathematics must justify itself solely through applications. The essay recognizes the autonomy of abstract mathematical inquiry. It nevertheless rejects the idea that the practical significance of mathematics is incidental. Mathematical abstraction, axiomatization, and symbolic representation have repeatedly enabled efficient reasoning about concrete problems. Their value lies not only in the truths they establish but also in the methods they provide for organizing complexity and reducing computational effort.

This is why the paper is skeptical of treating LLMs as the natural endpoint of mathematical mechanization. An LLM may be able to describe a procedure for multiplying large integers, but that does not make linguistic imitation an efficient implementation of arithmetic. Mathematical conceptualization and symbolization produce compact, reliable, and computationally effective procedures. The implication is that mathematicians should contribute to the design of reasoning systems rather than merely evaluate their outputs. Mathematical structure can improve AI systems at the level of representations, search procedures, verification, and algorithmic efficiency.

The paper also attributes value to dispositions associated with mathematical practice: asking precise questions, developing abstractions, building theory, pursuing problems over long periods, and resisting short-term incentives. These are not presented as uniquely human essences, but as disciplinary competencies that can guide the development and deployment of AI. The author’s normative claim is that mathematicians should help determine what AI systems are optimized to do and how their outputs are integrated into intellectual institutions.

Institutional and educational challenges

The essay identifies a substantial mismatch between the skills required for AI-enabled mathematical research and the evaluation criteria of mathematics departments. A growing group of researchers has acquired expertise in formal libraries, APIs, automated reasoning, neural-network training, reinforcement learning, SAT encodings, cluster-based experimentation, and solver diagnostics. Yet work involving these tools often receives little recognition in mathematical hiring and promotion.

This institutional asymmetry has already produced a talent-flow problem. Researchers interested in AI and formal methods leave mathematics departments for technology companies and startups, while mathematics undergraduates redirect themselves toward computer science. The paper presents this not simply as an employment trend but as a failure of disciplinary valuation. Researchers who combine mathematical expertise with computational and formal-methods competence are producing work that may be central to the future of mathematical practice, but they are frequently required to publish in computer science venues under standards established for a different research culture.

The tension is complicated by a genuine strength of mathematics: methodological conservatism. Stable standards protect the discipline from transient fashions and make it possible to distinguish durable contributions from technological novelty. The paper does not recommend abandoning this conservatism. Instead, it argues that conservatism must coexist with the capacity to adapt when the objects, methods, and social conditions of mathematical work change. The relevant institutional task is not to accept every new tool, but to develop criteria capable of recognizing technically sophisticated contributions whose mathematical content is expressed through computation, formalization, or machine learning.

The educational proposal is correspondingly moderate. The author does not argue that every mathematician must become an expert in machine learning or formal methods. Rather, future mathematicians should possess basic competence in these areas, analogous to the broad foundational competence expected in algebra, analysis, geometry, and topology. Specialization can remain diverse: some mathematicians may work extensively on computational reasoning systems, whereas others may use them only occasionally. The discipline should remain a “big tent” in which different combinations of theory, computation, experimentation, and formal verification are legitimate.

The paper’s program for mathematical participation

The paper’s program has two levels. First, mathematicians should become competent users of AI-related tools. This includes understanding what proof assistants, symbolic solvers, machine-learning systems, and formal libraries can and cannot do. Such competence is necessary to evaluate outputs, identify failure modes, and formulate productive research problems.

Second, mathematicians should become developers and conceptual architects of AI for mathematics. They should design scalable reasoning procedures, identify mathematical applications of AI beyond theorem proving, and exploit domain-specific understanding to improve the technology. The paper insists that passive consumption is inadequate. If mathematicians merely use commercial systems, control over mathematical infrastructure will remain concentrated in organizations whose incentives and understanding of the subject may not align with those of the mathematical community.

This claim also carries an infrastructural implication. Formal libraries, APIs, datasets, verification systems, and computational environments are not neutral accessories. They shape what can be represented, searched, reproduced, and evaluated. Mathematical participation in their design is therefore part of preserving the epistemic standards of the discipline. The essay does not offer a detailed governance model or implementation plan, but it identifies ownership and institutional involvement as central issues.

The paper places conceptual agreement before technical prescription. Before deciding how mathematics should incorporate AI, the community must clarify what it values and what forms of activity it wishes to sustain. A fixed commitment to the current division between “mathematics” and “computation” would make the discipline react defensively to each technological advance. A broader conception of mathematics, by contrast, would permit the integration of new methods without reducing the subject to automated theorem production.

Limitations and open questions

The essay is programmatic rather than empirical. It does not provide benchmarks, controlled studies of mathematicians’ workflows, analyses of employment outcomes, or demonstrations that the proposed applications of machine learning will produce mathematically significant discoveries. Its claims about underinvestment and unrealized potential are plausible within the account presented, but they are not quantified.

The paper also leaves unresolved how mathematical value should be operationalized when AI systems generate conjectures, proofs, or computational evidence. It argues that understanding, conceptualization, and human judgment remain important, but it does not specify when a machine-generated result should count as mathematical knowledge, nor how standards of explanation should change when exhaustive computation replaces a short conceptual proof. Similarly, the essay does not settle how departments should evaluate work combining mathematics with formal methods and machine learning, despite identifying this as a central institutional problem.

A further open question concerns the relationship between democratization and expertise. The availability of AI systems allows non-specialists to perform mathematical experiments, but the paper does not provide criteria for distinguishing productive amateur exploration from unsupported or misleading output. Its insistence on mathematical leadership implies that expertise remains necessary, yet the mechanisms by which expertise should be exercised in increasingly automated environments are left unspecified.

Finally, the historical comparison with the rise of axiomatic mathematics is illuminating but incomplete. The earlier transformation primarily changed the languages and standards through which humans conducted mathematics; contemporary AI may also change the allocation of agency between human researchers and computational systems. Whether the resulting practice preserves the forms of understanding that mathematicians value is an empirical and philosophical question the essay identifies but does not resolve.

Conclusion

The essay presents AI as a challenge to the definition, institutional organization, and values of mathematics rather than merely as a tool for accelerating proof production. Its central demand is that mathematicians broaden the category of AI for mathematics to include formalization, symbolic reasoning, machine learning, mathematical discovery, and computational collaboration. The discipline should retain its standards of depth, rigor, and sustained inquiry while expanding its methodological competence and recognizing researchers who work across mathematics, computer science, and formal methods. The unresolved question is not whether AI will enter mathematics, but which forms of mathematical understanding and activity the community will choose to cultivate as it does.