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Mathematical practice is the study of mathematics as a human activity—what mathematicians actually do, why they do it, and how the resulting knowledge comes to be accepted, used, and revised. Where the philosophy of mathematics has traditionally focused on the nature of mathematical objects and the justification of mathematical truth, the philosophy of mathematical practice shifts attention to the processes, norms, and contexts in which mathematics is produced and maintained. It asks not only "What is a proof?" but "What counts as a proof, for whom, and under what circumstances?" It treats mathematics not as a static body of propositions but as a living, historically situated enterprise.
The subfield is organized around a cluster of interrelated questions. One concerns the nature of proof and rigor: What makes an argument convincing to mathematicians? How do standards of rigor change over time, and what role do informal reasoning, diagrams, and computational experiments play alongside formal derivation? A second concerns the role of representation: How do notations, diagrams, and visualizations shape mathematical thinking? Does a proof that relies on a diagram carry the same epistemic weight as a purely symbolic one? A third concerns the social dimension of mathematics: How do communities of mathematicians reach consensus? What role do refereeing, reputation, and institutional authority play in deciding what becomes accepted knowledge? A fourth concerns the relationship between mathematics and its applications: How do problems from physics, engineering, or other sciences enter mathematics, and how does mathematical knowledge travel back into those domains?
The stakes are philosophical but also practical. If mathematical knowledge is produced through social processes, then understanding those processes matters for evaluating claims about mathematical certainty, objectivity, and progress. The subfield also bears on education: understanding how mathematicians actually work can inform how mathematics is taught. And it bears on the philosophy of science more broadly, since mathematics is often taken as the gold standard of certainty; if that certainty is more complex and conditional than it appears, the implications ripple outward.
The philosophy of mathematical practice emerged as a self-conscious movement in the late twentieth century, but its concerns have deep roots. Earlier philosophers of mathematics—from Immanuel Kant to Henri Poincaré—paid close attention to how mathematics is actually done, even when their primary focus was on foundations. Kant's account of mathematical reasoning as grounded in the structure of intuition, and Poincaré's insistence on the role of intuition and convention in mathematics, both engage with practice in ways that later foundationalist philosophy tended to set aside.
The immediate background to the modern subfield was the foundational crisis of the early twentieth century. The competing programmes of logicism, intuitionism, and formalism each offered a different account of what mathematics is and how it should be justified. These debates were conducted largely at the level of ideal systems: logicism sought to reduce mathematics to logic, intuitionism rebuilt it on constructive grounds, and formalism treated it as the manipulation of formal symbols according to rules. By mid-century, the dominant view among philosophers—often associated with logical empiricism—treated mathematics as a formal, axiomatic science whose justification lay in its internal consistency and its applicability to empirical science. The actual practices of mathematicians, their heuristics, their diagrams, their informal talk, were regarded as psychological or sociological noise, not proper objects of philosophical analysis.
This began to change in the 1970s and 1980s. A number of philosophers and mathematicians grew dissatisfied with the gap between the idealized picture of mathematics in philosophy and the messy reality of mathematical work. Imre Lakatos's Proofs and Refutations (published in 1976 but circulating earlier) was a pivotal influence. Lakatos presented the development of mathematical knowledge as a process of conjecture, proof, counterexample, and concept refinement—not as the accumulation of indubitable truths. His work drew on the history of mathematics, particularly the history of Euler's polyhedron formula, to argue that mathematical concepts and proofs evolve together through a dialectical process. Lakatos did not use the phrase "mathematical practice," but his insistence on the historical and fallible character of mathematical knowledge opened the door for a practice-oriented approach.
In the 1990s and 2000s, the subfield coalesced as a distinct research programme. Philosophers such as Paolo Mancosu, Michael Detlefsen, and others began to argue explicitly that the philosophy of mathematics needed to take practice seriously. The founding of the Association for the Philosophy of Mathematical Practice in 2009 marked the institutional consolidation of the movement. Since then, the field has grown rapidly, drawing on methods and insights from history of mathematics, sociology of science, cognitive science, and mathematics education.
The field is not unified by a single method or doctrine. Instead, it comprises several overlapping approaches that share a commitment to studying mathematics as it is actually practiced, but differ in what they emphasize and how they go about their analyses.
One major approach treats the history of mathematics as the primary data for philosophical analysis. The guiding assumption is that mathematical concepts, methods, and standards of proof are not fixed but develop over time, and that understanding this development is essential to understanding mathematics itself. This approach draws on the work of historians of mathematics but uses historical material for philosophical ends: to show how concepts change, how proofs come to be accepted or rejected, and how the boundaries of mathematics shift.
Lakatos's work is the paradigmatic example, but the approach has been developed further by scholars who examine specific episodes in the history of mathematics—the development of calculus, the rise of set theory, the discovery of non-Euclidean geometry—to draw philosophical lessons. A key insight from this approach is that mathematical concepts are often refined in response to problems and counterexamples, rather than being fully formed from the start. The concept of a function, for instance, underwent dramatic changes over two centuries, from an algebraic expression to an arbitrary correspondence between sets, and these changes were driven by mathematical needs rather than by a priori reflection.
A limitation of the historical approach is that it can be difficult to draw general conclusions from particular episodes. What holds for the development of analysis in the eighteenth century may not hold for the development of category theory in the twentieth. Practitioners of this approach are aware of this difficulty and often resist grand generalizations, preferring careful case studies that illuminate specific aspects of mathematical practice.
A second approach looks to cognitive science and psychology to understand the mental processes underlying mathematical activity. This approach asks how human beings are able to do mathematics at all: What cognitive capacities support mathematical reasoning? How do visual and spatial thinking contribute to mathematical insight? What role do symbols and notations play in extending our cognitive abilities?
This approach has been influenced by work in cognitive psychology on number sense, spatial reasoning, and the psychology of problem solving. It also draws on the philosophy of mind and on research in embodied cognition, which argues that abstract thought is grounded in bodily experience. For example, the concept of a mathematical function may be understood through the metaphor of a machine that takes inputs and produces outputs, and this metaphorical grounding may shape how mathematicians think about functions.
A central question for this approach is whether mathematical cognition is continuous with ordinary cognition or whether it involves something special. Some researchers argue that mathematics is built from basic cognitive capacities that all humans share, such as the ability to perceive small quantities and to reason about spatial relations. Others emphasize the role of external representations—written symbols, diagrams, computer programs—in creating mathematical concepts that would be impossible to hold in the mind alone. This approach has important implications for mathematics education, suggesting that learning mathematics involves not just absorbing abstract rules but developing new ways of thinking that build on existing cognitive resources.
The main limitation of the cognitive approach is that it can be difficult to connect laboratory findings about cognition to the sophisticated practices of professional mathematicians. A study of how people solve simple arithmetic problems may have little to say about how mathematicians develop a new proof technique. Practitioners of this approach often acknowledge this gap and work to bridge it by studying mathematical cognition in more naturalistic settings.
A third approach examines the social structures and processes that shape mathematical knowledge. This approach draws on the sociology of science, particularly the work of Bruno Latour, Karin Knorr Cetina, and others who studied scientific laboratories and communities. It also draws on social epistemology, which studies how knowledge is produced and justified through social interactions.
From this perspective, mathematics is not the work of isolated geniuses but a collective enterprise. Mathematicians work in communities, share results at conferences and in journals, referee each other's papers, and build on each other's work. The social-epistemological approach asks how these processes contribute to the reliability of mathematical knowledge. Why should we trust a theorem that has been checked by only a handful of people? How does the mathematical community decide which results are important and which are marginal? What role do power, prestige, and institutional resources play in shaping the direction of mathematical research?
A key concept in this approach is that of "trust" or "testimony." Most mathematicians accept most results on the basis of the testimony of others, not by checking the proofs themselves. This reliance on testimony is not a weakness but a necessary feature of a large-scale collaborative enterprise. The social-epistemological approach examines how this trust is managed: through peer review, through the reputation of authors and institutions, through the replication of results in different contexts, and through the development of shared standards of rigor.
This approach has been criticized for relativism—for suggesting that mathematical truth is merely a social construction. Practitioners typically respond that they are not denying the existence of mathematical truth but rather studying how communities come to know and agree on it. The social processes they study are not alternatives to rational justification but the means by which rational justification is achieved in practice.
A fourth approach retains a connection to traditional philosophy of mathematics but redirects its attention to the norms that govern mathematical practice. Rather than asking what mathematical objects are, this approach asks what mathematicians ought to do: What makes a proof a good proof? When is it legitimate to introduce a new axiom? What role should computers play in mathematical verification?
This approach is sometimes called "normative" because it aims to evaluate mathematical practice, not just describe it. It draws on the tradition of philosophy of science that examines scientific method, but applies it to mathematics. For example, a normative approach might ask whether the use of computer-assisted proofs is epistemically legitimate, and under what conditions. It might examine the standards of rigor that mathematicians actually employ and ask whether those standards are justified.
This approach often engages with specific cases. The proof of the four-color theorem, which relied on extensive computer calculation, raised questions about whether a proof that cannot be checked by hand is really a proof. The classification of finite simple groups, which was completed only through a massive collaborative effort spanning decades, raised questions about the nature of mathematical knowledge when no single person can verify the entire proof. The recent development of formal proof verification systems, which allow proofs to be checked by computer, has raised new questions about the relationship between human understanding and mechanical verification.
The formal and normative approach is distinguished from the others by its willingness to make evaluative judgments. Where the historical approach describes how mathematics has developed and the sociological approach describes how it is practiced, the normative approach asks whether those developments and practices are good ones. This can put it in tension with the more descriptive approaches, but it also provides a bridge to traditional philosophy of mathematics.
These approaches are not mutually exclusive, and many scholars combine them. A historical study of a mathematical episode might draw on cognitive psychology to explain why certain conceptual changes were difficult. A sociological study of a mathematical community might use normative analysis to evaluate the community's standards. The boundaries between approaches are porous, and the field is characterized more by shared questions than by shared methods.
The main fault line runs between descriptive and normative approaches. The historical, cognitive, and sociological approaches are primarily descriptive: they aim to understand mathematics as it is, without judging whether it should be different. The formal and normative approach aims to evaluate and improve mathematical practice. This distinction is not absolute—descriptive studies often carry implicit normative implications, and normative studies must be grounded in accurate descriptions—but it marks a real difference in emphasis.
Another fault line concerns the unit of analysis. The cognitive approach focuses on individual minds, the sociological approach on communities, and the historical approach on the development of concepts and methods over time. These different units of analysis can lead to different questions and different kinds of explanations. A cognitive explanation of why a mathematician made a particular discovery is very different from a sociological explanation of why the discovery was accepted, and both are different from a historical explanation of why the discovery was possible at that particular time.
The philosophy of mathematical practice has become a well-established subfield with its own journals, conferences, and research networks. It has also influenced neighboring areas: historians of mathematics increasingly engage with philosophical questions, and some mathematicians have taken an interest in philosophical reflection on their own work.
Several topics are currently active. The role of computers in mathematics continues to generate discussion, particularly as machine learning and automated theorem proving become more sophisticated. The question of what constitutes understanding in mathematics—as opposed to mere proof—has received renewed attention. The relationship between mathematics and other disciplines, particularly physics and computer science, is another active area. And the field has begun to engage with questions of diversity and inclusion, asking how the demographics of the mathematical community affect the kinds of mathematics that are pursued and valued.
The subfield's greatest contribution has been to complicate the picture of mathematics inherited from the foundationalist tradition. Mathematics is neither a purely formal system nor a purely social construction. It is a human activity with its own standards, its own history, and its own forms of life. The philosophy of mathematical practice seeks to understand that activity in all its richness, without reducing it to something simpler.