Thinking with Notations: Epistemic Actions and Epistemic Activities in Mathematical Practice

Abstract

In a text that has become an instant favorite among physicists and mathematicians alike, Eugene P. Wigner expressed his sense of wonder at “the enormous usefulness of mathematics in the natural sciences,” which he likened to “something bordering on the mysterious.”

The Applicability ‘Problem’

In a text that has become an instant favorite among physicists and mathematicians alike, Eugene P. Wigner expressed his sense of wonder at “the enormous usefulness of mathematics in the natural sciences,” which he likened to “something bordering on the mysterious,” for which “there is no rational explanation.”¹ This phenomenon, for which Wigner coined the phrase “the unreasonable effectiveness of mathematics in the natural sciences” (which also serves as the title of his original lecture in 1960), brings into sharp focus one of the core issues at the intersection of philosophy of mathematics and philosophy of science: the applicability of mathematics to the world at large. For as long as the ‘Book of Nature’ was thought—at least within the Western intellectual tradition—to have been authored in the language of mathematics (as Pythagoreans, in their various guises, would have it), and Nature was seen as bearing the hallmarks of a divine Being’s rationality, the natural world’s amenability to mathematical description did not constitute a salient problem, but was simply assumed. In turn, no special ‘applicability problem’ arose for mathematics—which, after all, was thought to be omnipresent in Nature ‘by design.’ Yet, once “numbers and geometrical forms [were] no longer assumed to be inherent as such in Nature,” as Michael Polanyi has argued, “‘pure’ mathematics, formerly the key to nature’s mysteries, became strictly separated from the application of mathematics to the formulation of empirical laws.”² Marveling at the divinely mandated concordance between mathematics and Nature gradually gave way to a sense of puzzlement at the very possibility of applying mathematics to the empirical world.

Mathematics, of course, underwent significant transformations itself, and it may well be argued that it was precisely by outstripping the domain of what was needed in order to explain natural phenomena, that the proliferation of mathematical theories and subfields contributed to this sense of puzzlement. Indeed, this seems to have been Wigner’s main point: If certain mathematical concepts are useful for describing natural phenomena, who is to say that other—hitherto undiscovered—mathematical frameworks could not be even more ‘unreasonably effective’? As Wigner argues, “because we do not understand the reasons for their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate.”³ Wigner’s version of the applicability problem, then, must be seen against the backdrop of the vast proliferation of mathematical frameworks since the nineteenth century and their subsequent uptake in early twentieth-century physics, notably of Riemannian geometry in general, relativity theory, group theory in particle physics, etc. Yet throughout some of its most formative periods, mathematicians and philosophers hardly reflected on the applicability problem and related foundational issues. Around 1800, it was quite common to speak of mathematics, as Leonhard Euler did, as “nothing more than a science of magnitudes.”⁴ The term ‘magnitudes’ here refers to the act of measuring physical quantities: perhaps the amount of water used to irrigate a field, or the total weight of the harvest it generated. The thought was not that mathematics should study the qualitative physical features of the world, but rather that it deals with the relations in which real-world magnitudes stand to one another. At the same time, its applicability to the physical world was, again, simply presupposed.

In the late nineteenth/early-twentieth century, the tables initially seemed to turn. Foundational issues had slowly moved to the center of mathematics, not least due to attempts to formalize mathematics and give it a programmatic dimension. Thus, David Hilbert had this to say about the relation between mathematics and the physical world:

We are confronted with the peculiar fact that matter seems to comply well and truly to the formalism of mathematics. There arises an unforeseen unison of being and thinking, which for the present we have to accept like a miracle.⁵

We can see here a nascent—if unresolved—concern with ‘bridging the gap’ between mathematical knowledge and knowledge of the physical world, formulated from within the field of mathematics. At the same time, the rise of logical empiricism in philosophy largely denied the existence of any special problem of the applicability of mathematics. As Torsten Wilholt has convincingly argued, the rise of Carnap-style logical empiricism largely neutralized the philosophical significance of the applicability problem—“not because [the applicability problem] was considered to have received an answer, but because all matters of applicability of frameworks were considered practical questions.”⁶

Where the logicist tradition within mathematics had at least kept alive the hope of a resolution by grounding applicability in a general theory of how any concept can apply to the world, for logical empiricism, mathematics was simply to be equated with whatever conceptual framework was empirically found to be most “serviceable”⁷ for the time being. Arguably, such a philosophical view left little room for ‘problematizing’ the fact that mathematics was, indeed, found to be a useful resource in describing the world. Yet once again, developments in mathematics pulled in a different direction. As José Ferreirós observes, in mid-twentieth century mathematics, “it was usual to emphasize that ‘all mathematical theories can be considered as extensions of the general theory of sets’”⁸ (“as everyone knows”⁹). This recreated, or at least widened, the gulf between mathematics—now understood as a set-theoretic enterprise whose ontology followed from the existential assumptions implicit in the axioms of set theory—and the empirical world, which seemed to conform to it. If the applicability problem has in recent years regained some of its prominence,¹⁰ then this is as much a reflection of the ‘fading’ of (certain commitments of) logical empiricism, as it is the long-term effect of intra-mathematical developments in the second half of the twentieth century.

Whatever the merits and varied fortunes of the applicability problem in the philosophy of mathematics, one aspect of the debate is immediately obvious: it invariably construes applicability in ‘global’ terms, as the challenge of relating a domain of abstract objects to the empirical world-at-large. The applicability problem, thus understood, refers to the difficulty of specifying the global conditions of possibility for how abstract entities can manifest themselves in a world of concrete events and facts. Such a perspective on the ‘applicability’ of mathematics, however, is far removed from any actual—concrete—actions of applying mathematics. Applying mathematics—proving theorems, deriving corollaries, performing calculations, etc.—takes work and requires significant cognitive effort. Which acts of inferring, deriving, representing, manipulating, or rearranging (e.g. of formulas and equations), are appropriate, and when? Such questions cannot be answered with reference to the global relationship between the realm of mathematical entities and the empirical realm, but instead require attention to mathematical actions and activities.

Mathematical actions and activities, I wish to argue in this paper, are typically mediated by symbol systems, notations, and formalisms, in a way that not only allows for the seamless expression of mathematical concepts, but also actively shapes mathematical practice, both at the level of individual derivations and at the collective level of mathematics as a discipline. That is, notations and formalisms do not play a merely auxiliary role, in that they give neutral expression to the underlying foundational concepts, but they are themselves constituents of mathematical practice, without which certain connections and developments within mathematics (and certain ways of applying mathematics to the natural world) would not be possible. The realization that mathematical notations and formalisms are more than just passive ‘vessels’ for the transmission of mathematical concepts has gradually been catching on in contemporary philosophy of mathematics. Thus, James Robert Brown argues that “the source of the attraction of formalism [as a general position in the philosophy of mathematics] stems from the evident power of notations themselves,” and less from its “nominalistic hostility to abstract entitites” (which he deems “silly”);¹¹ Mark Colyvan notes “that the notation can help reveal hitherto unknown mathematical facts;”¹² finally, Helen De Cruz and Jan De Smedt (more about whom below) have gone so far as to claim that “symbols are not merely used to express mathematical concepts,” but are “constitutive of the concepts themselves.”¹³

While the coalescence of such sentiments into a self-proclaimed philosophy of mathematical practice is a fairly recent development (see next section), there are multiple historical precursors. I wish to conclude this introductory survey of the debate about the applicability problem with an especially illustrative example from Edmund Husserl’s Philosophie der Arithmetik (1891, transl. Philosophy of Arithmetic, 2003). Among other things, Husserl chides nominalists such as Hermann von Helmholtz and Leopold Kronecker for putting the cart before the horse when they take ordinal numbers to be the most natural starting point for our number concept. On this view, the number ‘five’ would be “nothing other than a sign for the totality of the signs ‘first,’ ‘second,’ ‘third,’ ‘fourth,’ ‘fifth’”¹⁴—a view Husserl rejects as fanciful. What is telling is Husserl’s analysis of what has gone wrong in this hasty identification of the “number of objects” with the “entirety of the designations” that went before (as Kronecker puts it). What “these great mathematicians” have misunderstood is the autonomy of “the process of symbolic enumeration, which we carry out as a blind routine,”¹⁵ and which precisely allows us to set apart any considerations of content and treat symbol sequences as mnemonic devices—which need to be given a conceptual interpretation only at the final step of the process. We do not ‘grasp,’ in the final step of an enumeration, the totality of conceptual designations that went before; instead we generate, with certainty, a symbolic output which, thanks to the rule-governed operation we have been following, is assured to have a definite mathematical content. As Husserl puts it later in his treatise, not every derivation “is an essentially conceptual operation;” rather, much mathematical activity takes the form of “essentially sense perceptible” operations “which, utilizing the system of number signs, deriv[e] sign from sign according to fixed rules.”¹⁶ For Husserl, the latter—sense-perceptible—approach has considerable advantages over the ‘method of concepts’:

The method of concepts is highly abstract, limited, and, even with the most extensive practice, laborious. That of signs is concrete, sense perceptible, all-inclusive, and it is, already with a modest degree of practice, convenient to work with. […] Thus, it makes the conceptual method entirely superfluous, its use being no longer suited to the scientific state of mind, but only to a childish backward one instead.¹⁷

One need not endorse Husserl’s broadside against conceptual approaches in order to agree with his characterization of the immense usefulness of notations, formalisms, and sign systems that allow us to ‘externalize’ reasoning processes—especially in contexts where, due to the complexity of the issues at hand, we cannot trust our own intuitions about whether or not we have, indeed, grasped the “conceptual substrata”¹⁸ correctly and have managed to track our conceptual operations successfully over time. Indeed, it seems only a small step to extend Husserl’s insight concerning “essentially sense-perceptible operations” also to the use of other representational formats and media, such as the three-dimensional material models that were common in late 19th and 20th-century mathematics—especially in instructional settings—and which, through their concrete realizations, facilitated the haptic and visual examination of mathematical concepts and relationships. As we shall see, it is through paying attention to the interplay of models, notational systems, actions, and operations on the part of its practitioners, that much of mathematical activity can be reconstructed as meaningful, in ways that a purely foundationally oriented approach in the philosophy of mathematics would be likely to miss.

Philosophies of Mathematical Practice

It has become customary to distinguish, among the various conceivable approaches to the philosophy of mathematics, between ‘mainstream’ and so-called ‘maverick’ approaches.¹⁹ The mainstream has its origin in well-known foundational approaches—such as Platonism, intuitionism, and formalism—and subsequently gave rise to analytic approaches, such as the ones associated with logical positivism, which mainly focused on ontological and epistemological issues. In opposition to these discussions, which basically align well with core questions of philosophy in general—pertaining to realism, foundationalism, ontology—‘maverick’ approaches are marked by anti-foundationalism, anti-logicism, and generally a greater attention to mathematical practice.²⁰ What these maverick commitments entail is a view that considers mathematical knowledge not a matter of being acquainted with mathematical objects or structures that exist independently, but rather as the fallible product of practices—such as conjecturing, proving, refuting, etc.—that cannot be adequately captured by the inferential rules of logic. Imre Lakatos, in the introduction to his Proofs and Refutations (1976), gives a vivid characterization of the motivation behind such early practice-oriented (‘maverick’) approaches:

The history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, has become empty.²¹

However, these early ‘maverick’ approaches, it has been argued, have “not managed to substantially redirect the course of philosophy of mathematics,”²² and the balance of philosophical work continued to be in the ontological and epistemological tradition outlined earlier.

At the same time, the growth of interest in the role of mathematical practice over time led to a proliferation of methodological approaches, just as the so-called ‘practice turn’ did in the philosophy of science more generally. At one end of the spectrum are those who are keen to highlight the continuity of mathematics with all sorts of practical concerns—such as measuring, estimating, and counting, which have historically developed across various occupations, such as agriculture, crafts, and trades. This, for example, is the preferred approach in the historical parts of the Oxford Handbook of the History of Mathematics (2008). By almost exclusively looking for commonalities between mathematics and other practical projects, it is perhaps not surprising that such an approach risks losing track of anything specific that might characterize mathematical practice. At the other end of the spectrum, we find approaches such as that of Philip Kitcher, who conceives of a mathematical practice as a quintuple $<L,M,S,R,Q>$, consisting of a language L in which statements, S, are derived by reasoning processes R as answers to questions Q, all informed by a metamathematical framework M.²³ Specific practices—such as the production of proofs, the conjecturing of mathematical relationships, the development of new notations in mathematics, etc.—are all lumped together in a single, overarching, paradigm-like meta-mathematical framework M. As Ferreirós rightly criticizes, this renders Kitcher’s account “rather abstract”—resulting in a notion of “practice without practitioners.”²⁴

In recent years, a small, but growing number of scholars have begun to develop what one might call a ‘meso-level’ approach to the philosophy of mathematical practice: that is, a perspective that is not content with merely positing overarching paradigms (while holding on to a largely abstract view of the content of what mathematics is about), nor with assimilating mathematical practice to other—non-mathematical—practical projects. Instead, this intermediate perspective acknowledges, amongst others, that mathematics is both a tool and offers de facto foundations for many scientific pursuits, while also maintaining that it differs from a mere “science of magnitudes” by its “concentration on conceptual and theoretical issues independently of their potential role as models for physical phenomena.”²⁵ In other words, mathematics—and mathematical knowledge—cannot be reduced to formal and symbolic systems, but must be understood, in Ferreirós’s apt phrase, as a “network of practices”²⁶: more accurately, of practices that are specific to mathematics—such as conjecturing, deriving proofs (as well as explicating them), demonstrating, deriving, formalizing, and so forth. Similar to the rapprochement between practice-oriented approaches and foundational methodologies in more recent philosophy of science, philosophy of mathematics, too, stands to flourish “under the combined influence of both general methodology and classical metaphysical questions […] interacting with detailed case studies.”²⁷ Other sources of inspiration for this recent emergence of an intermediate-level philosophy of mathematical practice include the growing literature in cognitive science, especially in relation to the range of inferential abilities that underpin all mathematical reasoning, and recent work on cultural techniques such as written representations and their “notational iconicity.”²⁸ It is the professed hope, at least of some of those engaged in this enterprise, that such a practice-based approach “ceases to pose the problem of the ‘applicability’ of mathematics as external to mathematical knowledge itself,” instead recognizing it as “internal to its analysis.”²⁹

Given the richness of mathematics as a discipline, where can one hope to gain a theoretical foothold in order to make sense of mathematical practice? One place to look—not the only one, to be sure, but the one that will guide my inquiry in this paper—is the assemblage of tools (symbol systems such as notations and formalisms as well as other, non-symbolic tools, including models) that mathematicians avail themselves of on a daily basis. Such notational systems, it is argued, mediate—in the broadest sense—mathematical action and activities, thereby shaping mathematical practice, both at the individual and collective level. Any approach that aims “to address epistemological issues having to do with fruitfulness, evidence, visualization, diagrammatic reasoning, understanding, explanation, and other aspects of mathematical epistemology which are orthogonal to the problem of access to ‘abstract objects,’”³⁰ will be well-advised to also pay significant attention to the notations and formalisms that constitute the ‘formats’ and tools through which mathematical work unfolds.

Notations, Formalisms, Models

Anyone who begins to learn mathematics is immediately struck by its pervasive reliance on symbolic notations, starting from the simple ‘+’-sign designating addition—an operation that can still be expressed reasonably well in natural language—via the symbols for differentiation and integration (${f}^{^{\prime}},\frac{df}{dx},\int f\left(x\right)dx$), all the way to diagrammatic notations in, say, knot theory. The same pervasiveness of symbols, it could be argued, can be found in physics and chemistry, yet the fact that mathematical notations have proven useful also across the empirical sciences suggests that, by analyzing the more basic function of mathematical notations, we may be able to learn something fundamental about the role of symbolic notations in our epistemic practices.

At the most basic level, for the purposes of this paper, notations can be understood as symbol systems of sorts. While this is not the place for developing a full definition of what constitutes a symbol system, typically such a system would require that any well-formed arrangement of tokens, e.g. of physical marks on a piece of a paper, can be registered semantically as instances of a character, and that certain other features (such as rules for substituting, rearranging, and interpreting symbols) enable a competent user to manipulate and interpret any such expressions within that system. Yet, notations are more than just syntactically articulate symbol systems that allow for the precise specification of what any well-formed expression is designating: notations also function as inferential resources. In making this claim, we are no longer confining ourselves to features immanent to the notational system; instead, the claim is about the function of notations to human users. That is, there is no underlying assumption in what follows that a physical system that manipulates symbols, such as a digital computer, “has the necessary and sufficient means for intelligent action,”³¹ nor do we need to take a stand on this (or any other similarly controversial) issue. Mathematical notations, and similar symbolic tools in the physical sciences (such as the operator formalism in quantum mechanics, Feynman diagrams in high-energy physics, structural formulas in chemistry, etc.) are, for present purposes, conceived of as thoroughly human constructs: they are what we teach students when they are being inducted into a discipline, what scientists and mathematicians use on a daily basis in their own work and in communication with their peers, and they are subject to changes and modifications over time.

In science, mathematical notations—often referred to as ‘mathematical formalisms’ by scientific practitioners—have, of course, long been recognized as eminently useful. What makes mathematical formalisms such a valuable resource for those working in the empirical sciences, is nicely captured by Mary Hesse’s remark that

Mathematical formalisms, when used as hypotheses in the description of physical phenomena, may function like the mechanical model of an earlier stage in physics, without having in themselves any mechanical or other physical interpretations.³²

Hesse is here referring to the ‘mechanical analogies’ of nineteenth-century physics, which accounted for physical phenomena “in terms of mechanical models whose behavior is known apart from the experimental facts to be explained.” She further notes two desiderata of physical hypotheses that are specified in a certain way: first, that “it must be possible to deduce the data from the hypothesis when the symbols in the latter are suitably interpreted;” second, that “it is necessary that the hypothesis itself should be capable of being thought about, modified and generalized, without necessary reference to the experiments,”³³ so that derivations—of numerical predictions, or more theoretical results—can take place with a certain degree of autonomy from the empirical data. Mathematics, then, is not merely a convenient shorthand that allows for the expression of physical relationships, but it furnishes us with inferential resources that can guide subsequent inquiry. As Hesse puts it, it is in “the nature of the mathematical formalism itself” that

any particular piece of mathematics has its own ways of suggesting modification and generalisation; it is not an isolated collection of equations having no relation to anything else, but is a recognisable part of the whole structure of abstract mathematics, and this is true whether the symbols employed have any concrete physical interpretation or not.³⁴

Mathematical formulations of physical hypotheses do not merely record physical facts, but express them in a way that has an ‘internal dynamic’ built into them—in virtue of being so formulated. This “internal dynamic,” as Richard I. G. Hughes notes, “is supplied, at least in part, by the deductive resources of the mathematics,” which is “one reason why mathematical models are the norm in physics.”³⁵ Hughes here relies on current scientific usage of the term ‘mathematical model’ which, for the most part, refers to a set of mathematical equations that are to be interpreted as (partial) representations of, and in this sense ‘stand in for,’ certain aspects of empirical reality. Yet, a similar case could be made for material models in mathematics itself—that is, for models of mathematical concepts and relationships (as briefly mentioned above in the first section). Where symbolic derivations are made compelling by the logical force of deductive reasoning, material manipulations of ‘hands-on’ models derive their power from the vivacity and immediacy of direct sensory experience; both are constrained by the rules and affordances of their respective representational media. Ideally—this, at least, was the goal of many of those trafficking in material models of mathematics—logical derivation (of equations and theorems) and haptic manipulation (of material models) would coincide.³⁶

When enriched with rules for the manipulation and physical interpretation of well-formed characters and formulas, symbolic notations that originate in mathematics may constitute what I have elsewhere called “mature mathematical formalisms” viz.

a system of rules and conventions that deploys (and often adds to) the symbolic language of mathematics; it typically encompasses locally applicable rules for the manipulation of its notation, where these rules are derived from, or other systematically connected to, certain theoretical or methodological commitments.³⁷

What makes a mathematical formalism ‘mature’ in this sense, is partly a matter of how entrenched it is in a given scientific (sub)discipline; partly, it depends on its fruitfulness—again, judged by the standards of the corresponding scientific field—in facilitating problem-solving and generating novel questions that may then guide future inquiry. As Hesse also hints at, not just any “particular piece of mathematics”³⁸ will automatically qualify as a (mature) mathematical formalism, since the versatility of mathematics as an inferential resource will typically underdetermine the specific rules and conventions to be adhered to—whose function, after all, it is to delimit and guide our scientific inferences. Mature mathematical formalisms in physics thus occupy an interesting middle ground between physical theory and mathematics-at-large. Typically, the rules and conventions that govern their application—that is, the criteria that determine whether a given output of the formalism (a series of creation and annihilation operators in quantum many-body physics, say) is formally correct by the lights of the locally applicable rules and conventions (and thus can be interpreted as, for example, a series of changes in a many-body system)—ensures that certain theoretical constraints (say, preservation of particle number) are automatically satisfied. This may only work for a relevant subclass of theoretically permissible scenarios, beyond which the formalism would lose its legitimacy, but within its domain of applicability, the formalism may be said to both enforce and constrain the underlying general physical theory for the cases in question.³⁹ In this regard, mature mathematical formalisms may be considered as forming the basis for a specific form of “operative writing,” which functions both as “a medium for representing a realm of cognitive phenomena” and as “a tool for operating hands-on with these phenomena in order to solve problems or to prove theories”⁴⁰ pertaining to the corresponding realm of phenomena.

Mature mathematical formalisms in the sciences have an important role to play in generating scientific representations.⁴¹ This orientation towards representing empirically observable phenomena in the world might seem to detract from the relevance of what has been said so far to the realm of mathematics itself. After all, it is one thing to recognize the enabling and constraining role of mathematical formalisms in generating scientific representations, with which to make inferences about empirical phenomena; it is quite another to argue that similar considerations also extend to the role of notations in mathematics in general. Yet, the two cases may be less dissimilar than it might seem. Just as, in science, we are faced with decisions about which aspects of the world to represent—which is often not a free choice, but is heavily constrained by what is feasible, given our background knowledge—so, in mathematics, there is “theoretical reason for thinking that different notations would capture different truths,” since any particular choice of a notation consisting of countably many basic elements entails that “many [mathematical] facts would have to go unrepresented in that notation”⁴²—which is, of course, why mathematics as a whole operates with a great diversity of coexisting notations. While mathematics does not need to grapple with the vagaries of an imperfect physical reality, which has been known to harbor many empirical surprises and which resists attempts to fully understand it, this does not mean that mathematical relationships are readily transparent. The resistance we experience when working through mathematical problems is not due to hidden empirical factors, but is a form of cognitive resistance—working through conceptual relationships in mathematics, as Husserl puts it in the passage quoted above, is “even with the most extensive practice, laborious” (“selbst bei grösster Uebung mühsam”). A well-chosen notation can overcome, or at least mitigate, this resistance. Again, the notion of “operative writing,” or “operative systems of notation”⁴³ is helpful here. Just as the graphic-visual dimension of (ordinary) writing, with its various ways of structuring recorded speech (e.g., via punctuation), makes abstract relationships—in this case, grammatical structure—“accessible to the perceptual register of the ‘aisthetic’” (ibid.), so mature mathematical formalisms, too, may well become entrenched as ways of depicting abstract structures and relations.

Practices, Agents, Actions

A good mathematical notation must be apt in two ways: at the individual level, it must be easily accessible by cognitively limited beings like us, for example by utilizing distinct and visually salient signs, along with straightforward rules for how to manipulate them. At the collective level, notations should not only be easily communicable, so as to facilitate the exchange of mathematical results and approaches, but should—ideally—aid the advancement of mathematics as a discipline. Both aspects are frequently acknowledged by those writing on the philosophical significance of mathematical notation, as in Husserl’s emphasis on “sense-perceptible” signs that are “convenient to work with” (and which help to make “the conceptual method entirely superfluous,” at least locally) and in Alfred North Whitehead’s remark that “[b]y relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems,” enabling us to “make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain.”⁴⁴ Mathematical notation would hardly compel much interest, were it not for the fact that, as one mathematical practitioner puts it,

notation can have a decisive influence on the development of mathematics: Differences of notation hindered or promoted mathematical advances, and seemingly small changes in notation or concepts led to new and profound mathematical results.⁴⁵

Similar considerations also apply to diagrams, even where these may not meet the formal requirements for notations—which is why it has been argued that, more generally, “the introduction and development of systems of representation that are indispensable for the practice, such as, symbols, notations and diagrams” is a “crucial target for the study of the practice of mathematics.”⁴⁶

Important though it is to recognize that mathematical practice merits at least as much philosophical attention as, say, the question of applicability, the notion of ‘a practice’—in the singular—is of limited use when analyzing specific instances or historical episodes of mathematics. For one, mathematical practice—just like scientific practice—is really a plurality of interconnected practices (of conjecturing, demonstrating, and so forth). While such recognition of the plurality of practices goes some way toward correcting the traditional—overly abstract—view of mathematics as a single, unitary intellectual endeavor, in an important sense it is still too global, since it continues to maintain the illusion of a ‘practice without practitioners.’ Conjecturing, demonstrating, proving a theorem (and re-proving it, perhaps more elegantly!), refuting a conjecture: these are not impersonal, ‘disembodied’ steps in the advancement of mathematics as a discipline; instead, they are moves that are pursued and attempted—sometimes successfully, sometimes not—by individual agents (or, increasingly, groups of agents).

The need for an agential perspective in the philosophy of mathematics is perhaps most evident in discussions of mathematics education, where learning mathematics—as well as instructing others in various mathematical techniques—is, sometimes casually, equated with ‘doing mathematics.’ Traditional philosophy of mathematics, with its emphasis on foundational issues and cutting-edge mathematical research, has been largely silent on the murky territory of mathematics instruction. Yet, it only takes a moment’s reflection to realize that, without successful induction of every new cohort into the mathematical techniques of its time, mathematics as a discipline would disintegrate. This is not to say that mathematics is constituted by a fixed set of established techniques; rather—like any other scientific discipline and collective human endeavor—it needs to be underwritten by a set of ‘live’ practices. The basic idea is familiar from Kuhnian accounts of science: On the one hand, we have the global notion of a scientific ‘paradigm’ (or ‘disciplinary matrix’) that supposedly shapes a given discipline; on the other hand, the continued existence and reproduction of anything resembling a unified discipline would remain an utter mystery, were it not for the recurrence of specific instances of shared techniques and applications—Kuhn’s exemplars—which correspondingly are used both in scientific training and as heuristic tools for tackling new problems. In mathematics, such exemplars will crucially “have to do with the manipulations and rules that an agent will apply while confronting a certain configuration of [symbols].”⁴⁷ Mathematicians themselves speak of “common tricks that you might see in proofs across a variety of [branches of mathematics],”⁴⁸ they exhort their students “to look for what is linked to what”⁴⁹ and, if one’s initial search for a problem solution is unsuccessful, “to find consolation with some easier success, [and to] try to solve first some related problems.”⁵⁰ It is easy to see in these (and similar) pronouncements an “explicit recognition of the importance of exemplars in the practice of mathematics”⁵¹ and, by extension, also a recognition of the centrality of agents who master a practice via acquiring the requisite know-how on the basis of working with exemplars.

Yet stable practices do not just come down to mere regularities in whatever it is that agents do in a certain context. For something to be recognizable as a distinct practice, we need standards of what is a correct or valid move within that practice. And for that, we need to look at the level of individual actions. This is why I wish to argue that we must move not only from a wholesale notion of ‘practice’ (in the singular) to a multiplicity of interconnected practices (in the plural), and from practices to those who engage with it—that is, human agents—but even further down, as it were, to the analysis of individual actions. The guiding assumption, of course, is that actions themselves display certain similarities, such that different agents can be said to perform, if not the same action, then at least the same type of action. Identifying distinctive patterns of action, thus, is a necessary complement to an agent-centered philosophy of mathematical practice.

Epistemic Actions and Their Limits

Before turning in the next section to distinctively mathematical actions—that is, actions which can reasonably be considered as being constitutive of, or at least as contributing to, mathematical practice—the present section will lay the necessary groundwork by distinguishing between two broader classes of actions. Following the work of David Kirsh and Paul Maglio (1994), I shall distinguish between pragmatic actions, which are performed in order to “bring one physically closer to a goal,”⁵² and epistemic actions that “are performed to uncover information that is hidden or hard to compute mentally.”⁵³ Arguably, the philosophy of action has traditionally focused on pragmatic actions which, broadly speaking, adhere to the standards of instrumental rationality, whereas epistemic actions would previously have been assimilated to the (more or less pre-theoretical) domain of heuristic behavior.

Writing from the perspective of a computational approach to problem solving, Kirsh and Maglio give the following qualitative characterization of what epistemic actions aim for:

Epistemic actions – physical actions that make mental computation easier, faster, or more reliable – are external actions that an agent performs to change his or her own computational state.⁵⁴

Why, one might ask, is there a need for the concept of ‘epistemic action,’ and for distinguishing it from ‘pragmatic action,’ which is itself hardly a simplistic concept and may well accommodate multiple functions and interpretations? Why introduce a new, distinct category in the first place? The novel point of the concept of ‘epistemic action,’ according to Kirsh and Maglio, is its ability to resolve a puzzle that would result if we had to evaluate actions solely in virtue of their contribution to a proximate physical goal:

At times, an agent ignores a physically advantageous action and chooses instead an action that seems physically disadvantageous.⁵⁵

This appears to be at odds with the traditional focus—common, for example, in early theories of AI—on how an intelligent agent ‘chooses physically useful actions.’ Only by allowing for epistemic actions—that is, actions performed in order to change the agent’s own cognitive state—can one recognize such apparent physically disadvantageous moves as also being goal-oriented, not towards a proximate physical outcome, but to cognitive goals:

When viewed from a perspective which includes epistemic goals – for instance, simplifying mental computation – such actions once again appear to be a cost-effective allocation of the agent’s time and effort.⁵⁶

It is important to be clear about what the introduction of the novel concept of ‘epistemic action’ is, and is not, meant to challenge. What is not at issue is the general idea that actions are goal-oriented at some structural level; rather, the existence of epistemic actions undermines the “assumption that the point of action is always pragmatic,”⁵⁷ along with what one might call a linear model of the relationship between cognition and action, according to which agents first identify goals, then plan accordingly, and finally carry out actions designed to achieve those goals. What epistemic actions demonstrate is that, on occasion, action can also “be undertaken in order to alter the way that cognition proceeds.”⁵⁸

As an application and illustration of what they have in mind, Kirsh and Maglio use the well-known computer game Tetris. The reasoning behind their choice of example is as follows: Tetris “is a fast, repetitive game requiring split-second decisions of a perceptual and cognitive sort,” as variously shaped blocks move into view and need to be placed in such a way as to minimize any gaps between them. When a new block (or ‘zoid,’ as Kirsh and Maglio call it) appears, the ‘clock starts ticking;’ one could wait for the shape to become fully visible, but this might then not leave enough room for maneuvering the block to its optimal position. Rotating the block early can disclose additional information about its shape, by rotating into view features (e.g. ‘hooks’) that have not yet appeared as the block moves downward. More importantly, every action in the game inevitably contributes in measurable (positive or negative) ways to the desired final outcome, by bringing a given block closer to (or removing it further from) its final position in the playing field (also known as the ‘matrix’). This means that one can clearly distinguish between moves that contribute to the pragmatic goal—to minimize gaps and eliminate completed rows of blocks—and those that fail to do so:⁵⁹

Thus, if epistemic actions are found in the time-limited context of Tetris, they are likely to be found almost everywhere.⁶⁰

A further advantage of Kirsh and Maglio’s choice of example, solely for the purposes of experimental study design, is the fact that it is easy to recruit participants for this game, which many people are already familiar with—even if they do not know some of the regularities (e.g. that certain types of blocks always emerge from the same columns of the matrix). As a result, they were able to carry out empirical-psychological experiments identifying the various actions that Tetris players (of different proficiencies) routinely engage in.

In particular, Kirsh and Maglio identify a number of specific moves that, they argue, are best interpreted as epistemic actions. For example, their data show that players are more likely to rotate blocks which are not yet fully visible and which are ambiguous in both shape and position, as compared with blocks that are merely ambiguous in shape. (Blocks of different shapes move gradually into view from the top of the matrix, so until a block has fully emerged—and depending on whether identifying markers of the block are still hidden from view—there remains some uncertainty as to which specific shape one will be encountering next; see Fig. 1.) Yet, early rotation can only effectively help with disambiguating shape, whereas the desired physical outcome—creating gap-free rows of blocks—depends predominantly on moving the (correctly oriented) block in its final position, once its shape is known. This suggests that, from an outcome-oriented perspective, the two cases should not be treated significantly differently, yet subjects do invest time and effort into (superfluous) early rotations. Thus, Kirsh and Maglio argue, “[e]arly rotation is a clear example of an epistemic action.”⁶¹ They are, however, aware that

one might try arguing against this view by suggesting that there is pragmatic value in orienting the zoid early, and so its epistemic function is not decisive. Such an explanation, however, fails to explain why partial displays that are ambiguous in shape and position are rotated more often than those that are not ambiguous in shape and position.⁶²

Fig. 1

Copyright Clearance Center Inc., all rights reserved

‘Zoids’ moving into view; at the top, the visible portions are identical both in position and in shape; at the bottom, they are identical in shape alone. Republished with permission of Annual Reviews Inc. from: David Kirsh and Paul Maglio, “On Distinguishing Epistemic from Pragmatic Action,” Cognitive Science 18, no. 4 (1994): 528, Fig. 8; permission conveyed through

The reasoning here is that non-ambiguous shapes need to be rotated no less often (and ambiguous ones no more often) in order to place them properly—that is, in order to achieve a satisfactory realization of the explicit goal, as defined by the rules of the game. If ambiguous shapes are, in fact, rotated more often, then this additional expenditure of time and effort, so the argument goes, is best explained by the status of such rotations as epistemic actions that aim at, for example, preparing individual decision-paths for the application of some learned strategy. This is not to say that such actions are somehow irrational, but rather that their performance cannot be solely justified as contributing directly to the determinate goal as explicitly defined within the game. Rather, such actions promote distinctly epistemic goals, regardless of their immediate contribution to the outcome demanded by the game.

It would be easy to misunderstand the term ‘epistemic action’ as referring to any action that advances, or otherwise affects, our epistemic state—which, of course, would be most actions. Turning my upper body in order to scratch my heel inevitably leads me to turn my head, which entails that I may perceive objects in my environment—and gain perceptual knowledge about them—which I would otherwise have missed, had I continued to look straight ahead. Yet turning one’s body to scratch one’s heel is not an epistemic action. Neither are ‘focusing attention’ or ‘directing one’s gaze to a source of new information.’ Including these and similar sensor actions under the umbrella term ‘epistemic action,’ simply because they result in new information, would rob the term of its specific meaning—which, according to Kirsh and Maglio, pertains to the issue of “control of activity.”⁶³ Epistemic actions are ordinary actions that come at a cost, when judged in terms of their contribution to a (pre-defined) physical goal. Contrary to the view that actions are predominantly performed after intelligent planning—by searching for the most advantageous position among all the available physical states, and then attempting to reach that physical outcome—proponents of the idea of ‘epistemic action’ argue that informational states are an integral part of the state-space of options that a cognizer must consider. This point might seem trivial. Are there not many techniques—such a symbolic representation—that allow cognizers to bridge the gap between informational states and physical representations (which may then be manipulated through ordinary actions)? While there are indeed, Kirsh and Maglio insist that assimilating informational actions to the realm of symbolic representation risks obscuring the “less appreciated” fact “external actions can be [valuable] for simplifying the mental computation that takes place in tasks which are not clearly symbolic.”⁶⁴

It is this intended restriction of the term ‘epistemic action’ to non-symbolic actions (or at least to types of action that are not clearly symbolic) that poses a problem for any attempt to identify symbolic operations in mathematics with epistemic actions. Yet this is precisely how at least some recent philosophers of mathematics have viewed epistemic actions, and the use of symbolic notations, in mathematics. Thus, De Cruz and De Smedt (2013), argue:

Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare [unassisted] brains.⁶⁵

One natural way to understand this statement is to read it as elliptical: while symbols themselves cannot be actions, manipulations of such symbols may well be. The claim then becomes one about the uses to which mathematical symbols can be put—through individual acts of manipulating token instances, designating mathematical entities with new symbols, enforcing certain notational rules, etc.—in order to enable inferences that would otherwise exceed our cognitive capacity.

Yet, the identification of symbolic manipulations with epistemic actions in mathematics remains problematic. Consider one of De Cruz and De Smedt’s core examples: the introduction, by Leonhard Euler, of the imaginary unit $i$, which rendered calculations using complex numbers tractable—especially when combined with Jean Robert Argand’s innovation of the complex plane, in which complex numbers can be ‘located,’ and thereby uniquely represented, according to their real and imaginary parts:

Once [$i$ had been] introduced, mathematicians no longer needed to worry about square roots of negative numbers, because the symbolism effectively masks this cognitively intractable operation […]. Denoting a cognitively intractable operation with a symbol makes it more manipulable, which effectively enables mathematicians to overcome human cognitive limitations.⁶⁶

While this general characterization is eminently plausible, and also coheres with our earlier discussion (see section “Notations, formalisms, models” above), its designating the introduction of $i$ as an epistemic action is problematic, at least if one applies the criteria spelled out by Kirsh and Maglio. For one, it is not clear which underlying physical action poses a puzzle. Recall that the concept of epistemic action was intended to explain why certain effortful physical actions that seem to be incompatible with goal-oriented behavior could nonetheless be vindicated as serving cognitive ends. Two options naturally present themselves. First, one could try to reinterpret the example as one that, while falling short of the status of ‘epistemic action,’ nonetheless exhibits similar surface features. For example, introducing $i$ as a placeholder for $\sqrt{-1}$, thereby enabling us to perform calculations that we could not otherwise simultaneously survey in our mind, may be seen as just another form of cognitive offloading. This interpretation is supported by De Cruz and De Smedt’s endorsement of Whitehead’s remark that “a good notation sets [the brain] free to concentrate on more advanced problems, and in effect increases [its] mental power.” While cognitive offloading is also one of the functions of epistemic actions explicitly mentioned by Kirsh and Maglio,⁶⁷ the two are not identical, and so, on this first interpretation, the introduction of the imaginary unit is an example of the former, yet not of the latter.

However, there is a second option—which may be implicitly intended by De Cruz and De Smedt—according to which we need to widen the scope of the term ‘epistemic action’ so as to pick out actions whose primary value to the cognizer consists in enabling cognitive processes that would not otherwise be available. Perhaps, then, Kirsh and Maglio’s focus on the relationship between planning and intelligent behavior—though initially well-motivated for the purposes of critiquing certain types of AI approaches to the so-called “planning problem”⁶⁸—is simply too limiting and stands in need of widening. After all, once it is acknowledged that this epistemic activity needs to be properly accounted for, AI practitioners may decide to expand the relevant state space of possible configurations to also include informational states along with physical ones;⁶⁹ yet, even then it would be useful to distinguish those actions that primarily aim at the advancement of a cognizer’s epistemic state from those that predominantly serve pragmatic goals. On this more ‘liberal’ view, the term ‘epistemic action’ would simply refer to the first kind of actions, yet with no restriction on how these actions can manifest themselves (e.g., symbolic or non-symbolic) and without insisting that a ‘hard’ trade-off be involved between the epistemic and pragmatic contributions of a specific action. Such an approach would also need to acknowledge the contribution of other factors, e.g. specific qualities of the requisite representational media: the introduction of the imaginary unit $i$ as a mere placeholder for $\sqrt{-1}$ might not have been nearly as compelling, had it not been quickly supplemented by Argand’s more easily visualizable representation in the complex plane.⁷⁰

While I have broad sympathies for being more ‘liberal’ than Kirsh and Maglio about the extension of the term ‘epistemic action,’ their position is arguably both well motivated and possesses the virtue of specificity. Hence, in the remainder of this paper, I will be guided by the idea that, when it comes to giving a reconstruction of mathematical practice, it is worth adhering to Kirsh and Maglio’s (restrictive) notion of ‘epistemic action’—up to a point, that is. This, I believe, will shed light on what one might call different scales of action—some of which pertain to individual actions, others to broader disciplinary activities. As we shall see, there is no reason to restrict the term epistemic action to non-symbolic activities; however, the symbolic dimension of notations and formalisms is best introduced only once we have gained a better understanding of the richness—and the range—of relevant mathematical activities.

What ‘Epistemic Actions’ in Mathematics Might Be

Let us grant, for the purpose of argument, that mathematical notation and the introduction of new symbols are not epistemic actions in the narrow sense intended by Kirsh and Maglio. That is, they do not promote cognitive goals at the overt expense of proximate pragmatic goals. Or at least, they do not do so in any more than the unspecific sense, in which any pursuit of ‘pure inquiry’ may be said to be cognitively costly, inasmuch as it makes demands on our cognitive and attentional resources without offering any immediately identifiable pragmatic ‘pay-off’ in return. Yet, the cost of deploying a symbolic notation tends to be low and, since the symbolic realm offers less resistance to our manipulations than the physical world around us, it does not typically detract from our proximate physical goals. This diagnosis should not come as a surprise. For, the existence—and even the ‘bringing into existence,’ i.e. the coining—of a new term does not, in and of itself, do any epistemic work. If anything, it is specific acts of deploying a term or symbol that need to be assessed in terms of their contribution to our cognitive and/or pragmatic goals.

Does this mean that the notion of ‘epistemic action’ has no place in the analysis of mathematical practice? As already suggested in the previous section, the answer is no. Yet, in order to appreciate the contribution of epistemic actions, narrowly understood, to mathematical practice, one needs to adopt a wider perspective. Rather than offering a systematic account of the conditions under which epistemic actions can be expected to facilitate mathematical thinking (or to advance the discipline of mathematics in other ways), I shall proceed by way of example. In particular, I shall highlight four areas where one might fruitfully look for epistemic actions in the course of studying mathematical practice. The four examples in question are the use of gestures and other anticipatory bodily movements (in particular in instructional settings), the application of material models to mathematics, the initially puzzling disciplinary practice of re-proving theorems (which, after all, should be epistemically redundant), and—finally—the creation of collectively shared symbolic and notational resources. As I shall argue, it is the latter example—the emergence of mature notational formalisms—which leads to a partial vindication of symbol use as a type of epistemic action, provided we make a slight (though well-motivated) modification of the very notion of epistemic action in question.

The Use of Gestures and Symbolic Operations in Instructional Settings

Since the publication, in 1986, of Eric Livingston’s book The Ethnomethodological Foundations of Mathematics, it has been a commonplace in the science studies of mathematical practice that “the heart of mathematical work is at and around the blackboard.”⁷¹ Sometimes, this is put the other way around, for example by Michael Barany and Donald MacKenzie (2014) when they emphasize that mathematicians work with—and prefer working with—chalk.

Coming out of such ethnographic studies are close-up observations of how mathematical instruction occurs in concrete settings. As an example, consider this excerpt from Christian Greiffenhagen’s and Wes Sharrock’s ethnovideographic study of the use of “Gestures in the blackboard work of mathematics instruction” (2005), which records how an instructor works through a particular derivation with his students and, in the course of doing so, introduces a specific mathematical notation (see Fig. 2):

Going over this new notation prepares the way for commenting upon this new notation, i.e., commenting on a ‘nice property that this notation has’ (lines 5-6). This is spelled out by tracing the equivalence relation that has been written in the top right corner of the board (lines 6-8). Mathematically, the ‘nice property’ consists in the possibility of moving from the typically used symbols for propositions (namely ‘p’ or ‘not p’) to numbers (1 or 0) representing these two possibilities. In effect, it is putting in place a powerful mechanism to ‘talk’ about propositions of the formal language.⁷²

Fig. 2

Excerpt from the transcript of the instructor’s classroom demonstration. From: Christian Greiffenhagen and Wes Sharrock, “Gestures in the blackboard work of mathematics instruction,” in Interacting Bodies. Proceedings of 2nd Conference of the International Society for Gesture Studies (June 2005), 1-24, here 12. © Used with kind permission by Christian Greiffenhagen; all rights reserved

This passage is significant for three reasons. First, it illustrates the importance of bodily actions—such as hand gestures, orienting one’s body and directing one’s gaze—to the effective instruction of learners, even in the context of a seemingly abstract operation such as introducing a new mathematical notation. Physical actions and verbal assertions nicely complement each other, as indicated by the instructor’s verbal testimony, which highlights certain commendable features (‘nice properties’) of the notation in question. Second, the transcribed sequence documents actions that can properly be called ‘epistemic actions’ (in Kirsh and Maglio’s sense), such as the extensive giving of ‘meta-commentary’ on the part of the instructor—including, among other things, a number of evaluative statements and endorsements which, at least on one level, detract from the proximate goal of proving the theorem being derived. Engaging in ‘meta-talk,’ after all, delays the finalization of the proof—the proximate goal, even by the lights of the instructional aim of giving a sample demonstration—and so can only be understood as well-motivated if it serves other, specifically cognitive goals. Third, there is a clear anticipatory element in the way that gestures, verbalization, and the sequence of steps in the derivation link up:

We can note the anticipatory character of the gestures, of the use of hands, and of the lecturer positioning himself with respect to the board. […] these are coordinated with the oral exposition, often in such a way that the positioning, the gesture and the identification of the relevant point in the written out proof are designed to coincide.⁷³

This echoes the case of early rotation of zoids in Kirsh and Maglio’s Tetris example: some actions are being carried out simply in order to highlight relevance—again emphasizing the primarily epistemic character of such actions.

Of course, not all actions an instructor performs while demonstrating a proof on the blackboard, or while carrying out similar instructional work, are epistemic actions in the narrow sense. Apart from obviously irrelevant actions (e.g., interrupting the writing process in order to scratch an itch), there are also actions a presenter may carry out—including, again, gestures such as pointing—in order to direct his or her own attention, or that of the audience. Such sensor actions, for the reasons discussed earlier, would not be epistemic actions of the kind distinguished by Kirsh and Maglio. Other actions, though technically superfluous and even delaying the completion of the proximate task—such as completing a mathematical proof—aim at organizing or structuring a proof. Some may serve “to locate the license provided by an earlier entry for a current move”⁷⁴; that is, they contribute directly to, or make explicit, the justificatory relationships that underwrite a given sequence. Depending on which among the “network of practices”⁷⁵ we focus on—the instructional goal of conveying mathematical relationships to the audience, or the goal of establishing the truth of a theorem—such actions may contribute only indirectly to the explicitly stated practical goal. The ‘remainder,’ as it were,—that is, the extent to which these actions constitute detours from the explicit goal—may well be best interpreted as due to epistemic goals.

Applying Material Models to Mathematics

The use of three-dimensional models in mathematics—which were variously made of plaster, paper, wire, and other materials—was by no means an oddity in late 19th-century mathematics. Indeed, as a number of historical studies in recent years have shown, demand for such models was high enough to sustain a significant level of commercial activity. Companies would offer series of dozens of models, mainly for use in instructional settings, and there was vigorous trade and export (e.g., from Germany to the United Kingdom). Felix Klein, in a lecture held at a mathematical congress that was part of the Chicago Columbian Exposition in 1893, describes the rationale for the use of models in mathematics instruction as follows: “Collections of mathematical models and courses in drawing are calculated to disarm, in part at least, the hostility directed against the excessive abstractness of the university education.”⁷⁶ Others, such as Hermann Wiener,⁷⁷ emphasized that “the capacity to form concepts through abstraction” is trained “far more via visual aids [Anschauungsmittel] of all sorts” (and especially via models, because of their ‘immediacy’) than via calculating and even geometrical drawings.⁷⁸ Such pronouncements might make it seem that, just as in the earlier case of gestures and symbolic derivations using chalk on a blackboard, the use of material models served primarily pedagogical purposes. If the goal had been exclusively that of instilling mathematical knowledge in one’s audience, rather than creating new knowledge, then it might be possible to sidestep the question of whether the actual deployment of models in mathematics had any genuinely epistemic features. Models in mathematics would simply be effective teaching tools, and the decision to use them would be an instrumentally rational choice to make—there would hardly be any reason to consider the effect that such models had on advancing the instructor’s epistemic state.

Arguably, however, the function of material models in 19th-/early-twentieth century mathematics went well beyond that of serving as an effective teaching tool. Besides functioning as pedagogical tools, material models served as heuristic devices and interpretative tools for practicing mathematicians. Consider the development of non-Euclidean geometry in the second half of the nineteenth century. At a time when most mathematicians took the truth of Euclidean geometry to be self-evident and exclusive—even though indirect challenges and doubts had surfaced as early as in the eighteenth century (ironically, in the course of attempted vindications of Euclid)—the Italian mathematician Eugenio Beltrami (1835–1900) began to study mathematical objects that, while being represented as curved surfaces in three-dimensional Euclidean space, exhibited internal relations that could be considered as realizations of non-Euclidean geometries (of the type proposed by Nikolai Lobachevsky and János Bolyai). In particular, such structures included surfaces of constant negative curvature, which could be thought of as being generated by rotating a suitable curve around an axis in three-dimensional space. Material models of such surfaces—or, more precisely, material models that approximated portions of such surfaces—could be made from plaster or, in Beltrami’s case, even from paper (see Fig. 3 for one of the paper models of these surfaces, which Beltrami made in 1869). On the one hand, it is clear that any actual encounter, or action, involving material models is merely an empirical occurrence that does not, in and of itself, lend weight to any particular mathematical conclusion. On the other hand, as props for the theoretical imagination, Beltrami’s models opened up ways of conceiving of concrete components (e.g. line segments marked on a plaster model) as representing abstract elements that, in turn, realized a system of geometrical principles.⁷⁹ The very co-existence of such models eased the way for acknowledging that principles of geometry allowed for multiple ‘interpretations’—while, at the same time, demanding that a proper theoretical appreciation of the new geometries would require transcending, in thought, the limitations associated with the models’ concreteness and materiality. As Moritz Epple puts it, Beltrami’s models constituted “one of the early epistemic objects of mathematics beyond the sphere of ordinary spatial intuition that would become so characteristic of mathematical modernity.”⁸⁰ By extension, the concrete uses and manipulations of material models in mathematics typically do not serve proximal, pragmatic goals, but are instead best thought of as epistemic actions—even when, as in Beltrami’s case, they aim to challenge the accepted bounds of knowledge, as it were.

Fig. 3

© Used with permission from the Department of Mathematics of the University of Pavia, Fulvio Bisi and Maurizio Cornalbaa; all rights reserved

One of Eugenio Beltrami’s models of the pseudosphere, preserved at the Department of Mathematics of the University of Pavia.

Re-proving Theorems

One potential objection to the analysis of what one might call individual ‘micro-level’ actions—among others, anticipatory uses of gestures, pausing to highlight aspects of the derivation that do not themselves directly contribute to the outcome, and so forth—is that these are common to all contexts of instruction and, thus, are not specific to mathematics. They would, in other words, fail the specificity test for any prospective ‘meso-level’ philosophy of mathematical practice, where such an approach would not content itself with merely noting commonalities between mathematics and other practical endeavors, but would aim to pinpoint genuinely mathematical aspects of the various interlinked practices that give rise to mathematical knowledge.

As a more compelling case which is not limited to instructional contexts and which does appear to be specific to the practice of mathematics, consider the puzzling practice of re-proving of mathematical theorems. From an outcome-oriented perspective, which highlights the “discovery and proof of new theorems [as] the sine qua non of mathematical endeavor,”⁸¹ re-proving appears to be redundant since, by definition, it does not contribute new theorems to the corpus of mathematical knowledge. After all, we do not stand to gain new mathematical knowledge from proving an already known theorem again. This raises legitimate questions:

What reasons are there for re-proving known results? And how do mathematicians judge whether a proof is conceptually distinct from those that have been given before?⁸²

A practice-oriented philosophy of mathematics—one that is less fixated on the final outcomes and abstract claims of mathematics—is better able to make sense of this phenomenon. For, as John Dawson notes, new proofs are often “presented as such, because they are deemed to be particularly novel, ingenious, or elegant, or (like the free climber who disdains mechanical aids) because they attain the goal through restricted means.”⁸³ In other words, novel proofs are valued not because they generate new results, but because they promote values such as elegance or ingenuity. Sometimes a new proof may lead to the desired outcome more efficiently—by shortening the derivation—but even then it is technically redundant. Sometimes, a novel proof may not even shorten the derivation, but may lengthen it—just like the free climber who may take longer to reach the mountain top than his better-equipped competitors—yet may succeed in achieving the same, already known outcome under conditions of self-imposed constraints. Its value, then, consists not in contributing novel mathematical conclusions, but in furthering epistemic goals such as methodological knowledge or knowledge of how things ‘hang together.’ In much the same way as Beltrami’s material models were suggestive of new interpretations of geometrical principles, novel proofs can also point to new intra-mathematical relationships and properties and suggest new lines of inquiry. Far from being redundant, then, the act of re-proving a theorem may be seen to be an epistemic action that serves (perhaps tacit) epistemic goals.

Notations as ‘Institutionalized’ (Long-Term) Epistemic Actions?

At the beginning of this section, I promised that a proper consideration of the emergence of collectively shared notational resources may yet vindicate the thought that certain deployments of notations should count as epistemic actions. In order for this to be possible, the initial notion of ‘epistemic action’ needs to be adjusted, but only moderately so. This avoids losing all specificity of the term—as would inevitably be the case if one were to call any action (symbolic or non-symbolic) with epistemic consequences an ‘epistemic action.’ A full discussion would need to look at the specific affordances of a range of specific formalisms and notations as well as the specific ways in which they enhance, or extend, human cognitive abilities.

In the context of the present paper, I can only refer to a select few examples and hint at some general reasons why it may be legitimate to consider the introduction and deployment of notations a type of epistemic action. An important motivation for entertaining this possibility is the oft-noted role of notations in cognitive offloading. Whitehead, in a passage already quoted earlier, gives vivid expression to this sentiment:

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases [our] mental power […]. In mathematics, granted that we are giving any serious attention to mathematical ideas, the symbolism is invariably an immense simplification [...] [B]y the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain.⁸⁴

At a less global level, similar claims have been advanced for specific mathematical notations, formalisms, and diagrammatic methods.

One branch of mathematics that has received some attention in this regard is knot theory. Thus, Brown devotes half a chapter of his Philosophy of Mathematics (1999) to knot theory and the various properties and relationships that different notations—or, ‘different forms of representation’ (since, for Brown, mathematics seeks ways of representing an independently existing realm of mathematical entities and relations)—highlight. He is keen to point out—unsurprisingly, given his philosophical commitments—that different notations will typically capture different truths; that is, mathematical reality exceeds what any one notational system may be able to represent. Notations ‘create’ as much as ‘reveal’ the boundaries of what can be known about the properties of mathematical objects.⁸⁵ Taking their lead from Brown’s discussion, Silvia De Toffoli and Valeria Giardino (2014) have analyzed the cognitive features and epistemic roles of diagrams in knot theory in greater detail. In particular, they propose an “operational account for knot diagrams, based on: (i) the moves allowed on them and (ii) the space they define.”⁸⁶ Diagrammatic notations, such as those used in knot theory, are “effective in promoting inference” because they enable the cultivation of “a form of manipulative imagination” that is developed and enhanced through continued use of shared symbolic resources.⁸⁷ A properly cultivated manipulative imagination, according to De Toffoli and Giardino, prompts experts “to re-draw diagrams and calculate with them, performing epistemic actions.”⁸⁸ It would be easy for a casual reader to misread this as a definition of what makes something an epistemic action in the narrow sense; thanks to our earlier discussion, we know better. Yet, De Toffoli and Giardino are entirely right to highlight the close affinity that exists between (certain types of) epistemic actions and routinized inferential strategies. What remains to be shown is how this affinity arises and how it can guide mathematical practice.

Extending our cognitive reach and facilitating inferences about the world is arguably one of the key motivations for introducing a novel formalism or notation. It takes only a moment’s reflection, however, to realize that this desired effect cannot be a matter of a single act of deploying a particular notation. Instead, it requires a long-term commitment—both at the individual level and, if a formalism is to become a source of scientific progress, at the collective level—to deploy the same types of operations in order to denote the same types of inferential moves. The emergence—and often-deliberate invention—of notational formalisms aims at the creation of common symbolic and epistemic resources, which anyone who has been trained in their usage can then ‘tap into.’ As we have seen, the term ‘epistemic action’ has occasionally been co-opted—notably by De Cruz and De Smedt, as well as by De Toffoli and Giardino—in order to describe just such (epistemically oriented) activities of notational innovation. Yet, at the very least, what is required for such an interpretation to be defensible, is a modified type of epistemic action, namely long-term epistemic action, in contradistinction to short-term epistemic action.⁸⁹

Whereas short-term epistemic actions “tune interactions between agent(s) and settings in such a way that a strong mutual fit and fluent problem-solving for single and concrete tasks in specifiable local conditions”⁹⁰—as in specific instances of rotating a ‘zoid’ in a game of Tetris—long-term epistemic actions allow an agent “to plan, design and put [subsequent] short-term epistemic actions to use;”⁹¹ this may also require robust commitments to certain ways of prioritizing problems—which, in turn, determine the way in which an agent would respond if things were different. All of this rings true of mathematical notations and formalisms, which often serve the purpose of preparing problem situations for systematic analysis. Once a problem has been ‘formalized’—that is, has been rendered representable within the symbolic resources afforded by the notation—what we can do with it is heavily constrained: only certain symbolic manipulations will be deemed admissible by the lights of the formalism at hand. Formalization requires cognitive effort—and, as in the case of mathematics, often needs to be aided by physical actions (and, sometimes, material models) in order to ‘work things out.’ But, in the case of fruitful formalisms and notations, this effort—as well as the constraints that come with the choice of a given formalism—will be more than compensated for by the ease with which they allow us to take further steps of inquiry and, for example, derive meaningful and relevant results. We forego some leeway in the description of our problem situation, but in turn gain greater inferential depth—and, one hopes, greater insight into the problem as a whole. Whatever cognitive cost may be incurred in the form of first having to invent, and then having to master, a new notation or formalism, may simply be the price we have to pay for greater long-term pay-offs. It is this aspect of symbolic notations in mathematics—the fact that formalization does not always make specific derivation easier, because it requires considerable mastery, yet nonetheless promises greater epistemic returns ‘further down’—rather than the mere fact of cognitive offloading that renders their development and mastery a case of epistemic actions, at least in the long haul.