Structuralism in the Philosophy of Mathematics

First published Mon Nov 18, 2019; substantive revision Mon Mar 31, 2025

The core idea of structuralism concerning mathematics is that modern mathematical theories, always or in most cases, are meant to characterize abstract structures, instead of referring to more concrete objects, including sets, or providing mere calculating techniques for applications. Thus, arithmetic characterizes the natural number structure, analysis the real number structure, and traditional geometry the structure of Euclidean space. As such, structuralism is a position about the subject matter and content of mathematics. But it also includes, or is closely connected with, views about its methodology, since studying such structures involves distinctive tools and strategies. Hence there are two related kinds of structuralism: metaphysical and methodological.

In English-speaking philosophy of mathematics, the introduction of structuralist views is often taken to have happened in the 1960s, in works by Paul Benacerraf and Hilary Putnam. Our survey will start there as well. Debates about structuralist views picked up steam in the 1980s–90s, when Michael Resnik, Stewart Shapiro, Geoffrey Hellman, Charles Chihara, and Charles Parsons entered the fray. And they have been reshaped again during the last 25 years, in terms of philosophical challenges to and novel variants of structuralism. Most of those challenges and variants involve metaphysical issues, especially questions about how to think about mathematical objects and about mathematical truth along structuralist lines; but a few also concern epistemological questions.

However, mathematical structuralism played a role already before the 1960s, especially if it is understood in the methodological sense, where questions about how to organize, innovate, and articulate mathematical practice are at stake. Structuralist positions in that sense go back to the nineteenth and early twentieth centuries, in writings by Dedekind, Klein, Hilbert, Poincaré, Noether, Bourbaki, etc. Today the two most influential versions of methodological structuralism are set-theoretic and category-theoretic structuralism, both of which lead back to metaphysical issues, mainly by being related to foundational ones. Relevant debates will be covered in the later parts of this survey. It will end with some brief remarks about structuralism beyond mathematics.

1. Eliminative vs. Non-Eliminative Structuralism

1.1 Beginnings of the Structuralism Debate in the 1960s

The discussion of structuralism, as a major position in English-speaking philosophy of mathematics, is usually taken to have started in the 1960s. A central article in this connection was Paul Benacerraf’s “What Numbers Could Not Be” (1965; cf. Benacerraf 1996). The background and foil for this article was the position, dominant at the time, that axiomatic set theory provides the foundation for modern mathematics, including identifying all mathematical objects with sets. For example, the natural numbers 0, 1, 2, … can be identified with the finite von Neumann ordinals (starting with \(\emptyset\) for 0 and using the successor function \(f: x \rightarrow x\cup \{x\})\); similarly, the real numbers can be identified with Dedekind cuts (constructed out of the set of rational numbers). Arithmetic truths are then truths about these set-theoretic objects. And this generalizes to other mathematical theories, whose objects can be identified with sets as well.

As Benacerraf argued in his article, such a set-theoretic foundational position misrepresents the structuralist character of arithmetic in particular and of mathematics more generally. Thus, instead of working with the finite von Neumann ordinals, we can work equally well with the finite Zermelo ordinals (starting again with \(\emptyset\) for 0 but using the alternative successor function \(f: x \rightarrow \{x\})\); and infinitely many other choices are possible too. Similarly, instead of identifying the real numbers with Dedekind cuts, we can work with equivalence classes of Cauchy sequences (again constructed set-theoretically out of the rational numbers), as suggested by Cantor and others. This insight about a multiplicity of options is hard to deny and even set-theorist foundationalists can agree with it. (More on set-theoretic responses to this challenge below.) But Benacerraf drew further, more controversial conclusions from his basic point.

Benacerraf suggested, in particular, that the natural numbers should not be identified with any set-theoretic objects. In fact, they should not be taken to be objects at all. Instead, numbers should be treated as “positions in structures”. In contrast to the set-theoretic representatives mentioned above, all that matters about such positions are their structural properties, i.e., those “stem[ming] from the relations they bear to one another in virtue of being arranged in a progression” (1965: 70). What we talk about in modern mathematics are, then, the corresponding abstract structures, e.g., “the natural number structure” and “the real number structure”. It is in this sense that Benacerraf suggests a structuralist position. The further details of that position are left open, however, including how to think about abstract structures, except that they are not to be identified with set-theoretic relational systems, i.e., models of standard model theory.

A second article from the 1960s that was influential in the rise of structuralism is Hilary Putnam’s “Mathematics without Foundations” (1967). Like in Benacerraf’s case, for Putnam the foil was a set-theoretic foundational position, where all mathematical objects are identified with sets. This position is sometimes, although not always, understood in a realist sense (most famously by Gödel), i.e., as the description of an independent realm of abstract objects, namely the universe of sets characterized by the Zermelo-Fraenkel axioms. In opposition to such set-theoretic realism, Putnam suggested a form of if-then-ism (a suggestion that can be traced back to Bertrand Russell’s works). This alternative can again be illustrated in terms of arithmetic, now taken to be based on the Dedekind-Peano axioms. How should an arithmetic theorem, say “\(2+3=5\)”, be understood now? It should be analyzed as having the following form:

For all relational systems M, if M is a model of the Dedekind-Peano axioms, then \(2_M +3_M =5_M\).

(Here \(2_M\), \(3_M\), and \(5_M\) are those objects that “play the roles” of 2, 3, and 5 in the model M.) Similarly for the real numbers, based on the axioms for a real closed field originally formulated by Dedekind and Hilbert.

Instead of if-then-ism, one can characterize Putnam’s position also as a kind of universalist structuralism, since it involves universal quantification over relevant systems and those systems are a kind of structures (cf. Reck & Price 2000 for this terminology). Often an objection to such a position is the non-vacuity problem. It is based on the observation that if-then statements of the given form are vacuously true if there is nothing that satisfies the antecedent, e.g., if there is no model of the Dedekind-Peano axioms. (A sentence like “\(2+3=6\)” would then also be true etc.) In response, axiomatic set theory can be invoked as providing the needed models. But from Putnam’s point of view this is unsatisfactory, for two reasons: It relies on a foundational view about set theory, thus undermining the anti-realist thrust of if-then-ism; and set theory is then treated differently from other mathematical theories, on pain of circularity. As a way out, Putnam suggested to work with modal logic instead; but the exact details were again left open.

1.2 Consolidation and Further Discussions in the 1980s

One way to understand Benacerraf’s 1965 article is that it proposes to treat the natural number structure as a new kind of abstract entity, different from all set-theoretic objects. Mathematics then concerns what holds true in such abstract structures. Along such lines, all depends on what exactly this amounts to, especially whether one should treat such entities as objects, thus reifying them in a substantive way (still to be worked out). Benacerraf himself was reluctant to do so, in line with his overall hesitancy to talk about mathematical objects.

A subsequent writer who picked up on Benacerraf’s ideas in the early 1980s is Michael Resnik (cf. Resnik 1981, 1982, 1988, and, most systematically, 1997). For him too, modern mathematics involves a “structuralist perspective”. Following Benacerraf, mathematical objects are viewed as positions in corresponding “patterns”; and as Resnik adds, this is meant to allow for taking mathematical statements at face value, in the sense of seeing ‘0’, ‘1’, ‘2’, etc. as singular terms referring to such positions. At the same time, doing so is still not supposed to require reifying the underlying structures, which for Resnik would mean specifying precise criteria of identity for them, something he avoids intentionally. (He presents himself as a Quinean on this point, by adopting Quine’s slogan: “No entity without identity!”) Instead, Resnik’s main focus is on the epistemological side of structuralism (more on which later).

Stewart Shapiro is a second philosopher of mathematics who attempted to build on Benacerraf’s paper (see Shapiro 1983, 1989, and most systematically, Shapiro 1997). By focusing on metaphysical questions and by leaving behind Benacerraf’s and Resnik’s hesitations about structures as objects, Shapiro’s goal is to defend a more thoroughly realist version of mathematical structuralism. Such realism includes the semantic aspect just mentioned (taking mathematical statements at face value), but Shapiro also clarifies the talk about “positions in structures” further, by distinguishing two perspectives on them. According to the first, the positions at issue are treated as “offices”, i.e., as slots that can be filled or occupied by various objects (e.g., the position “0” in the natural number structure is occupied by \(\emptyset\) in the series of finite von Neumann ordinals). According to the second perspective, the positions are treated as “objects” themselves; and so are the abstract structures overall.

For Shapiro, the structures at issue thus have a dual nature: they are “universals”, in the sense that the natural number structure, say, can be instantiated by various relational systems (the system of finite von Neumann ordinals, the system of Zermelo numbers, etc.); but they are also “particulars”, to be named by singular terms and treated as objects. To defend the latter further, Shapiro develops a general structure theory, i.e., an axiomatic theory that specifies which structures exist and how to identify them. While clearly modeled on set theory, this theory is justified independently (more on how below). As such, it is meant to underwrite “ante rem structuralism”, a terminology that refers to Medieval discussions of universals. The crucial point is that the structures specified in the theory are meant to be ontologically independent of, indeed prior to, any instantiations of them. In other words, the structures do not just exist in their instantiations, but separate to and before them.

While Resnik’s and Shapiro’s structuralist positions are sometimes identified, this is misleading given the differences already mentioned. Nevertheless, there is some overlap. Both recognize mathematical structures as patterns with positions in them (whether these patterns are treated as full-fledged objects or not). Also for both, the notion of isomorphism is crucial (or perhaps some related, more general notion of equivalence; cf. Resnik 1997 and Shapiro 1997). That is to say, for Resnik and Shapiro a structure/pattern can be instantiated by any of a relevant class of isomorphic relational systems. This corresponds to the fact that the axiomatic systems at issue—those for the natural numbers, the real numbers, set theory, etc.—are categorical (or quasi-categorical in the case of set theory). Not every mathematical theory has that feature; e.g., the axiom systems for group theory or ring theory allow for non-isomorphic models. According to both Resnik and Shapiro, such “algebraic” theories are to be treated in a different, more derivative way. Their structuralist suggestions are meant to apply primarily to “non-algebraic” theories, paradigmatically arithmetic and analysis.

There is another structuralist position first introduced in the 1980s that is quite different and explicitly anti-realist, namely that promoted by Geoffrey Hellman (cf. Hellman 1989, 1996, and later articles). While for Resnik and Shapiro the inspiration was Benacerraf’s 1965 article, the starting point for Hellman is Putnam’s 1967 article. In fact, Hellman’s modal structuralism is meant to be a systematic development of Putnam’s if-then-ism. The modal aspect, only hinted at by Putnam, is now spelled out in detail, also including the case of set theory (building on work by Zermelo etc.). For Hellman, an arithmetic sentence such as “\(2+3 = 5\)” is to be analyzed as follows:

Necessarily, for all relational systems M, if M is a model of the Dedekind-Peano axioms, then \(2_M +3_M =5_M\).

And to address the non-vacuity problem, Hellman adds the following assumption:

Possibly, there exists an M such that M is a model of the Dedekind-Peano axioms.

(We will come back to its justification below, which is surprisingly close to Shapiro’s in his structure theory.) Similarly for the real numbers and for set theory.

As Hellman makes clear, his goal is to develop a “structuralism without structures” (Hellman 1996). In it, the existence of abstract structures, postulated by Shapiro, is replaced by the modal aspects of his position, i.e., the assumptions about necessity and possibility just mentioned. In fact, Hellman’s position is meant to be a form of nominalism, by eliminating the appeal to any kind of abstract entities (not only abstract structures but also sets etc.). On the other hand, it is not meant to rely on possibilia, i.e., possible objects existing in some shadowy sense. This leads Hellman to take the modalities at issue to be basic, i.e., the relevant possibilities and necessities are not reducible to anything further. They are specified directly in terms of laws of modal logic (those of the system S5). According to Hellman, such possibilities and necessities are an irreducible feature of mathematics, as his approach is meant to make explicit.

1.3 A First Taxonomy of Structuralist Positions

From the late 1980s on, Shapiro’s and Hellman’s positions have often been treated as the two main structuralist options. (This is reflected in Hellman & Shapiro 2019, among others.) As they are quite different from each other, this already indicates that it is wrong to see “structuralism in the philosophy of mathematics” as a unique position or a unified perspective, even though Shapiro’s and Hellman’s positions share some structuralist features. And from the early 1990s on, other variants of structuralism have started to play a role as well. As a result, the discussions about structuralism in the philosophy of mathematics became richer and more complex (although it was still mostly restricted to metaphysical issues, as opposed to adding methodological ones).

To clarify the situation, Charles Parsons made several crucial contributions. In particular, he suggested a first taxonomy, based on the distinction between two main kinds of structuralist positions (Parsons 1990). Namely, there are eliminative forms of structuralism, like Hellman’s; and there are non-eliminative forms, like Shapiro’s. (The elimination at issue concerns the postulation, or avoidance thereof, of structures as separate abstract objects.) Or put in Hellman’s slightly later terminology, there is structuralism without structures, on the one hand, and structuralism with structures, on the other hand. Besides Shapiro (and Resnik, with the qualifications above), another proponent of the non-eliminative form became Parsons himself (see Parsons 1990, 2004, and, most systematically, 2008). And on the other side, another proponent of the eliminative form of structuralism became Charles Chihara (cf. Chihara 2004).

Nevertheless it remains widespread in the literature, even today (with exceptions), to identify non-eliminative structuralism with Shapiro’s position and eliminative structuralism with Hellman’s. Moreover, many discussions of structuralism then focus on Shapiro’s ante rem side, explicitly classified by him as a realist position as we saw. Given the latter focus, some critics go on to dismiss “philosophical structuralism” as a misguided form of metaphysics, one that is seen as irrelevant for mathematical practice (cf. Awodey 1996 and Carter 2008, more recently Carter 2024). But such a general dismissal is too quick and inadequate in the end, as we want to emphasize. Even if one just considers non-eliminative forms of structuralism, Shapiro’s position isn’t the only option, as a further consideration of Parsons’ position already makes clear.

Unlike Shapiro, Parsons does not offer a novel, metaphysically motivated structure theory. According to him, we can and should instead remain closer to mathematical practice, as it developed from the late nineteenth century on. In fact, for him structuralist views should be seen as growing out of that practice, rather than being imposed on it from outside by philosophers. For Parsons this means, among others, to see abstract structures as introduced directly by categorical axiom systems, a practice he spells out and justifies further in a “meta-linguistic” way (again inspired by Quine, see Parsons 2008). It also means for him that we should refrain from cross-structural identifications, such as identifying the natural number 1 and the real number 1 (as it can be found in Shapiro’s early works, although he later disavowed it). Such putative identities are to be left indeterminate, as Parsons assumes is done in mathematical practice.

As this makes evident, Parsons’ structuralist position (like Resnik’s), is less realist than Shapiro’s. In fact, Parsons is explicit that adopting a “structuralist view of mathematical objects” should be seen as separable from, and orthogonal to, the realism/nominalism dichotomy. Consequently, for him one can be a non-eliminative structuralist without being a realist in any strong sense; his own position is meant to be a case in point. Then again, Parsons’ structuralism still takes mathematical statements at face value, so that it remains realist in this minimal semantic sense. In that respect too, Parsons takes himself to follow mathematical practice, e.g., with respect to talking about “the natural numbers” in arithmetic.

2. Later Developments and a Broader Taxonomy

2.1 Metaphysical and Epistemological Challenges

So far we have traced discussions of structuralism in the philosophy of mathematics from Benacerraf and Putnam, in the 1960s, to Resnik, Shapiro, Hellman, Chihara, and Parsons, in the 1980s–90s. During the last 25 years, a number of further philosophers have started to address this topic. We now turn to the ensuing debates, beginning with some metaphysical and epistemological challenges to structuralism. Most of them concern non-eliminative structuralism, with Shapiro’s position as the main target. (This reflects again its prominence.) But some considerations are broader, including the comparison of basic commitments underlying eliminative and non-eliminative forms of structuralism. In what follows, we will not try to be comprehensive about the relevant debates, but just provide six illustrative examples. (Later we will also move beyond such metaphysical and epistemological issues; but as these examples illustrate, they have dominated debates about structuralism to a good degree.)

Non-eliminative structuralism, in Shapiro’s and other versions, involves the thesis that all that matters about mathematical objects are their structural properties (as opposed to further intrinsic properties, e.g., set-theoretic ones). In fact, such properties are taken to determine the objects’ identities. But then, objects that are indistinguishable in this respect—“structural indiscernibles”—should be identified, shouldn’t they? As several critics argued around the year 2000, this leads to the identity problem for structuralism (cf. Keränen 2001, earlier Burgess 1999). It arises for relational systems or structures that are non-rigid, i.e., allow for non-trivial automorphisms. In such cases there are supposedly distinct objects that are indiscernible in the relevant sense. A widely known example is the system of complex numbers, with the conjugate numbers i and −i; but geometry and graph theory provide further illustrations. The simplest example is probably an unlabeled 2-element graph with no edges, whose two vertices are structurally indiscernible.

How can and should such cases be handled, if they can be handled at all? Is structuralism (of the non-eliminative kind) simply incoherent, as some critics charge? Or is it at least not applicable to non-rigid cases, which would limit its reach significantly? Several structuralist responses to the identity problem have been offered in the literature. One is to rigidify such structures, by enlarging the vocabulary that is used, e.g., by adding the constant symbol ‘i’ for the complex numbers (either into the original language or into the language for the “setting” used in the background, as suggested in Halimi 2019). However, this still seems problematic in certain cases, especially when there are uncountably many indiscernibles. Another suggestion is to treat identity as primitive, as is arguably done in mathematical practice. Yet along such lines too, a number of follow-up questions arise (see Ladyman 2005, Button 2006, Leitgeb & Ladyman 2008, Shapiro 2008, Menzel 2018, and Leng 2020, among others).

A second and more basic challenge for structuralism starts again with the assumption that all that matters about mathematical objects are their structural properties. With respect to non-eliminative structuralism, this is sometimes understood or formulated in the sense that the positions in mathematical structures, as well as the abstract structures themselves, only have structural properties. But that again leads to counterexamples quickly. For instance, does the natural number structure as a whole not have the property that it is the most widely used example in debates about structuralism? Similarly, isn’t it a property of the number 8, as a position in that structure, to be the number of planets in the Solar System? Both seem clearly non-structural properties. Once again, structuralism appears incoherent, or at the very least, in need of further clarification. (One advantage of eliminative forms of structuralism, like Hellman’s, is that they avoid both problems mentioned so far; but they lead to others instead.)

A natural response to this challenge is to refine the structuralist thesis just mentioned, e.g., by saying that abstract structures only have structural properties essentially, or that only such properties are constitutive for them, while admitting that they do have other properties (cf. Reck 2003, Schiemer & Wigglesworth 2019, also Assadian 2022). This leads to the question of how exactly to make that distinction. Several structuralist responses to it have again been proposed. (We will come back to some of them in the next subsection.) But even if it we assume to have a satisfactory response to this question, another one arises, thus a third basic problem, namely: How can we distinguish structural from non-structural properties in the first place? Several answers to that question have been suggested in turn, e.g., that structural properties are those definable in a certain way, or that they are those preserved under relevant morphisms. Yet in that respect as well, the debates continue (see Korbmacher & Schiemer 2018, also for more references).

A fourth challenge directed again especially, or even exclusively, at non-eliminative structuralism concerns the following: From a structuralist point of view, positions are always “positions in a structure”; i.e., the structure is primary and the positions are secondary. Hence, a particular mathematical object, such as the natural number 2, seems “ontologically dependent” on the background structure, here the structure of the natural numbers. (The fact that for a structuralist it is misguided to consider the number 2 as existing in itself reflects this aspect. It also illustrates a main difference between structuralism and, say, set-theoretic foundationalism or neo-logicism.) Yet how should this ontological dependence be understood? Are there perhaps connections to grounding or related notions in current analytic metaphysics? In this context too, a lively debate has started and continues until today (cf. Linnebo 2008, also, e.g., MacBride 2005, Wigglesworth 2018).

A fifth challenge for structuralism in its non-eliminative form is how we can have access to structures taken to be abstract objects (see, e.g., Hale 1996). To some degree, this revives an older, more general debate about the possibility of knowledge about such objects (cf. Benacerraf 1973). An initial response, shared by Resnik, Shapiro, and Parsons (all following Quine in this respect), is to talk about the positing of structures, which is meant to undercut the access problem from the start. Then again, under what conditions is such positing legitimate (given the threat of paradox familiar from naïve set theory)? A plausible answer to that question points to the coherence of the relevant axiomatic theories, which, after Gödel’s incompleteness results, is seen as taking the place of provable consistency. (Compare Hilbert’s earlier suggestion that mathematical existence simply amounts to the consistency of the theories at issue.) But this leads to our sixth basic question: What exactly does such coherence amount to and how can we establish it?

With this sixth challenge, we have moved from metaphysics over to epistemology. An interesting outcome of the resulting debates is that it also brings us back to eliminative structuralism. Thus, Shapiro and Hellman, coming from very different directions, have actually arrived at suggestions that are very close to each other in this connection (cf. Hellman 2005, Hellman & Shapiro 2019). Namely, with respect to basic commitments—ultimate conditions for existence in Shapiro’s case and for possibility in Hellman’s case—their approaches converge, since they both appeal to coherence understood in a similar, non-reducible sense. That convergence may be seen as providing support for both sides, since they have arguably identified the same core feature of mathematics. But it might also be taken to undermine the realism/nominalism dichotomy, since it seems to imply that the existence of abstract structures, properly understood, amounts to the same thing as the possibility that underwrites modal structuralism.

There are more challenges to structuralism in the philosophy of mathematics that one can find in the literature of the last 25 years. While usually connected to the six problems just mentioned, sometimes they go further. For example, additional questions about the semantics involved in structuralism have been raised, typically again for the non-eliminative variant but now including questions about “relational positions” in structures (cf. Button & Walsh 2016, Assadian 2018, Button & Walsh 2022, and Zalta 2024). As a different kind of example, there is the question of how to generalize eliminative structuralism, along Hellman’s modal lines and beyond, so as to cover all of of mathematics systematically (cf. Zalta 2024). But instead of detailing such questions further, we hope that our six examples above provide enough illustration of the kinds of debates that have occurred in the recent literature. As should have become evident, many of those debates concern metaphysical issues, to a smaller degree also epistemological ones. (Later we will come back to parallel debates about categorical structuralism.) Moreover, they have led to additional variants of structuralism, as we will see next.

2.2 Several Additional Variants of Structuralism

As already noted, in many discussions of structuralism, from the 1980s to the early 2000s and beyond, a few positions have taken center stage: often Shapiro’s and Hellman’s, sometimes Parsons’, occasionally also Resnik’s. But several other variants of structuralism exist, as we want to make clear now. Those positions deserve, and have started to get, more attention in recent years. Moreover, some of them go back to before the 1960s, sometimes all the way to the mid- or late nineteenth century. As this shows, the still widespread assumption that debates about mathematical structuralism started during the 60s needs to be revised (even more so if we bring in methodological structuralism, as will become evident later). In this subsection, we will go over several especially noteworthy examples. A major one, in several respects, is set-theoretic structuralism (cf. Reck & Price 2000, Burgess 2015, and for its history, Reck & Schiemer 2020b). To introduce it, let us return to the illustration central to Benacerraf’s 1965 paper: the natural numbers.

As Benacerraf argued, there is something wrong with identifying “the natural numbers” with a particular set-theoretic system; or more precisely, it is wrong to do so in an absolute sense. As we saw, he concluded that numbers are not sets, or in fact, not objects of any kind. Now, one can agree with almost everything Benacerraf writes and still want to identify the natural numbers with a set-theoretic system in a less absolute sense. Doing so involves admitting that any other model of the Dedekind-Peano axioms “would do as well”, i.e., could have been chosen instead, except perhaps for pragmatic reasons (suggested by the context). Yet we can still pick one and let it play the role of the natural numbers. For most mathematical purposes such a provisional and pragmatic identification is sufficient. In fact, this is exactly what is done in standard ZFC set theory (where the system of finite von Neumann ordinals is usually picked, among others because it can be generalized systematically to the infinite). Moreover, the resulting position, which goes back to Zermelo in the early twentieth century, deserves to be counted as a form of structuralism. What makes it structuralist is, among others, the “indifference to identify” the natural numbers in any more absolute sense (cf. Burgess 2015).

The core of set-theoretic structuralism is to choose one of several isomorphic systems as the provisional, pragmatic referent for “the natural numbers” (similarly for “the real numbers” etc.). As one may put it, our talk of “the natural numbers”, then also of “the number 0”, “the number 1”, etc., is relative to this initial, somewhat arbitrary choice. This is taken to be unproblematic since, no matter how we choose, we will get the same arithmetic theorems (because of the categoricity of the axiom system, which implies its semantic completeness). And as background for this approach, it is usual to employ ZFC. But we generalize it by also allowing for “atoms” or “urelements” (objects that are not sets). We can even include Julius Caesar or some beer mug in our ___domain, with the consequence that (pace Frege) either of them can “be” the number 2, say, by taking up the “2-position” in the model for arithmetic we choose to work with. This generalized approach deserves the name relativist structuralism (cf. Reck & Price 2000). Moreover, such an approach, particularly in its set-theoretic version, is arguably accepted by a large number of mathematicians today, explicitly or implicitly. (Committed category theorists will be exceptions, as we will see later.)

In the case of relativist structuralism, the only mathematical objects at play are those that axiomatic set theory, possibly with urelements, provides. We do not need to postulate separate abstract structures, along Shapiro’s or similar lines. For that reason, the position is another variant of eliminative structuralism, although it does not eliminative all abstract objects (since it countenances sets). In fact, set-theoretic relational systems, in the sense of the standard models of theories, are often taken to be the relevant structures. (It is exactly such systems that are called “structures” in many mathematics textbooks.) Yet there are at least two further options at this point, which lead to two additional variants of structuralism. One of them is to identify the structure of the natural numbers with the (higher-order) concept defined by the Dedekind-Peano axioms; similarly again for other categorical axiom systems. And this bring us to yet another form of eliminative structuralism: concept structuralism (cf. Isaacson 2010, Feferman 2014, also Ferreirós 2021, where this is called “conceptual structuralism”). Furthermore, that position can be traced back even to the mid-nineteenth century, e.g., to the works of Hermann Grassman (cf. Cantu 2020).

According to concept structuralism, what matters in mathematics aren’t really objects, especially not problematic abstract objects. Rather, crucial are mathematical concepts, e.g., the concept “natural number system” (“model of the Dedekind-Peano axioms”, or in Russell terminology, “progression”); similarly for the concept “complete ordered field” etc. More precisely, what matters is what follows from those concepts, in the sense of what can be derived from the relevant axioms. It is true, as concept structuralists can admit, that in mathematical practice we often reason by talking about objects falling under those concepts. (If set theory is the framework used, then we use sets for this purpose.) But as they would want to add, such talk can be explained away in the end (perhaps by taking up a formalist position). In this or similar forms, concept structuralism is again a fairly widespread position among mathematicians and logicians, explicitly or implicitly, even though it has not been very prominent in the philosophical debates about structuralism so far. An immediate challenge for the view is, however, whether talking about concepts is less problematic than talking about mathematical objects in the end (cf. Parsons 2018).

We just introduced relativist structuralism, set-theoretic structuralism as a special case, and concept structuralism. For the sake of a comprehensive survey and to show that there really is a large variety of structuralist positions, we need to go further. As the next step, there are two forms of structuralism that are closely related to relativist and concept structuralism but not identical with either. Both are forms of abstractionist structuralism, and in both cases it is convenient to work again within a set-theoretic framework (although this too can be varied, e.g., by using second-order logic instead). We can start once more with the concept “natural number system”, together with the set-theoretic relational systems falling under it. But instead of identifying the relevant structure either with that concept or with a pragmatically chosen system falling under it, let us now consider the equivalence class of relational systems determined by the concept. Finally, given that equivalence class we can go down one of two “abstractionist” paths.

On the first path, we now identify the structure at issue with that equivalence class, i.e., we identify it with the “concept in extension” as it is sometimes put, thereby highlighting the proximity to concept structuralism. In our basic example, connected with the concept “natural number system” (defined by the Dedekind-Peano axioms), as well as with each set-theoretic models corresponding to it (e.g., the system of finite von Neumann ordinals, also the system of Zermelo natural numbers, etc.), there is the whole equivalence class of all such models as a third entity; and it is this class that we now call “the natural number structure”. (Here the primary focus is again on categorical axiom systems; but the approach can be generalized, e.g., by considering broader notions of equivalence.) This entity is not a set, of course, but a proper class. Nevertheless, it can be studied logico-mathematically, e.g., within the framework of Von Neumann-Bernays-Gödel set theory, NBG. (By using what is known as “Scott’s trick”, as is often done in such contexts, we could also avoid working with the whole class and restrict ourselves to a large subset of it.)

Compared to relativist structuralism, this new approach can also be characterized as follows: Its core is to go from a particular, arbitrarily chosen system falling under a higher-order concept, such as the concept “natural number system”, to the whole equivalence class of such systems, i.e., the class of all models isomorphic to it. Moreover, we can think of this move as a kind of abstraction, specifically in the sense of the “principle of abstraction” introduced and made prominent by Bertrand Russell (cf. Russell 1903). This is the sense in which the resulting position is a form of abstractionist structuralism. In terms of its history, this approach can be traced back far as well. In an explicit and systematic form, it was studied by Rudolf Carnap in the 1930s-40s, who saw it as a development of Russellian ideas from the early twentieth century (cf. Schiemer 2020, and see the discussion of Carnap’s structuralism in the entry on Carnap). But there is another abstractionist path we can take, involving a different form of “abstraction”; and it too can be traced back far in time, in this case to Dedekind’s works from the 1870s-80s.

For that second abstractionist option, let us start again from a relevant higher-order concept, or from the axiom system defining it, together with an arbitrarily chosen system falling under it (a set-theoretic relational system, possibly with urelements). The alternative suggestion is then the following: Given the arbitrarily chosen model, we “abstract away from the particular nature of its elements” so as to arrive at a “pure” system, in the sense of a separate, distinguished abstract structure; and we call that structure “the natural numbers” (cf. Dedekind 1888). The intention along such lines is that the objects introduced by such abstraction only have structural properties, or better, they only have them essentially, since we have abstracted away from all “foreign properties” (non-arithmetic ones in the case of the natural numbers). And in contrast to the previous approach, where the outcome of the abstraction is a whole equivalence class, the resulting system is supposed to be isomorphic to the one from which we started (cf. Reck 2003).

Using this second form of abstraction—“Dedekind abstraction”, as it is sometimes called—leads to a position that, at least in terms of its outcome, is close to Shapiro’s ante rem structuralism (who occasionally appeals to “abstraction” himself, e.g., in Shapiro 1997). Similarly it is close to Parsons’ non-eliminative position in terms of its outcome. Yet the current approach involves neither an appeal to Shapiro’s structure theory nor to Parsons’ meta-linguistic procedure. Instead, the relevant structures are introduced “by abstraction” from set-theoretic relational systems (possibly with urelements). What that amounts to can be explicated further, also more formally, in terms of an abstraction operator and corresponding abstraction principles. Another comparison suggests itself, then, namely with the use of abstraction principles in neo-logicism (cf. Linnebo and Pettigrew 2014, Reck 2018a). Having said that, informal versions of such abstraction can already be found in mathematical practice (often within a set-theoretic framework), namely when mathematicians think of “the natural numbers”, “the real numbers”“the cyclic group with five elements”, etc. as separate from the usual set-theoretic models.

The five additional forms of structuralism just considered—relativist, set-theoretic, conceptual, and abstractionist in two variants—all have historical roots in works from before the 1960s. (They grew out of forms of methodological structuralism, as we will see more below.) Each of them also has proponents and implicit adherents later, including in the twentieth century among mathematicians. And most of them rely on ZFC set theory in the background (which thus needs to be conceptualized differently in itself). Yet this is still not all in terms of available structuralist options. If we look forward into the twenty-first century now, there are several further variants of structuralism introduced by philosophers and that should be covered in our survey as well. We will mention three (the most interesting ones, in our view), without going into much details in each case. One aspect these three final variants share is the appeal to novel background theories, different from set theory. Moreover, each arose in defense of non-eliminative structuralism against the metaphysical and epistemological challenges mentioned in the previous subsection.

As a first illustration of such new approaches, Edward Zalta and Uri Nodelman have introduced a form of structuralism that relies on an axiomatic objects theory, inspired by Alexius Meinong, to account for both the existence and the nature of abstract structures (Nodelman & Zalta 2014, Zalta 2024). A particular strength of this approach is its ability to distinguish systematically between “accidental” and “essential” properties of objects in such structures, in terms of the difference between “encoding” and “exemplifying” properties provided in their object theory. As such, it provides novel answers to several of the philosophical challenges above, especially the one in which that distinction is challenged. More generally, the approach offers an explicit and detailed metaphysical background theory for non-eliminative structuralism. A remaining question is then whether or to what degree that metaphysics is compatible with those implicit in other non-eliminative positions, e.g., in the second form of abstractionist structuralism mentioned above.

Second, by building on Kit Fine’s theory of arbitrary objects, as another new background theory, Leon Horsten has introduced generic structuralism (Horsten 2018, 2019). This approach too allows for responses to some of the philosophical challenges mentioned earlier. However, the unusual kind of objects appealed to by Horsten (which share some properties with variables, given how they are introduced in Fine’s theory of arbitrary objects) raises new questions in turn. As a third case, Hannes Leitgeb has recently proposed to start from graph theory for similar purposes, i.e., to develop an alternative framework for non-eliminative structuralism (Leitgeb 2020a, 2020b). Leitgeb’s approach is well suited for addressing the identity problem for structuralism from a mathematical point of view. And in other ways as well, it stays closer to mathematical practice than either Zalta and Nodelman’s or Horsten’s approach. Nevertheless, Leitgeb’s main goal, like theirs, is to clarify metaphysical aspects of non-eliminative structuralism.

As a final option, there is a whole family of category-theoretic forms of structuralism. However, we will postpone their discussion until later (Section 4), for two reasons: First, this approach is especially important mathematically, thus deserving treatment in a separate section. Second, it is harder to compare to the forms of structuralism mentioned so far, since it is motivated more by methodological considerations. Before turning to it, we will do two things: We round off Section 2 by offering further taxonomic distinctions for structuralist positions, thus refining the taxonomic framework for our survey. After that (in Section 3), we discuss several forms of methodological structuralism, both to enrich our survey further and to provide a better background for the subsequent treatment of categorical structuralism.

2.3 Finer Taxonomic Distinctions for Structuralism

All the variants of structuralism discussed up to this point are forms of metaphysical structuralism, or as it is sometimes also put, of philosophical structuralism. What that means is that they are primarily intended to provide answers to philosophical questions about what mathematical structures are (even when this involves an eliminative stance), in the sense of theses about their existence, abstractness, identity, dependence, etc. Now, one can distinguish this whole variety of positions from methodological structuralism, or in alternative terminology (less helpful, we think), from mathematical structuralism (cf. Reck & Price 2000, earlier Awodey 1996). In fact, we suggest that recognizing this basic dichotomy—metaphysical versus methodological—is crucial for further advancing both the systematic and the historical discussions of structuralism in the philosophy of mathematics. (Among others, it will allow to include and compare categorical forms of structuralism more naturally; cf. Corry 2004 and Marquis 2009.)

As its name suggests, and as already mentioned briefly, methodological structuralism concerns the methodology of mathematics, thus mathematical practice. Or as one might also say, it concerns a certain “style” of doing mathematics (cf. Reck 2009, Ferreirós and Reck 2020). Basically, that style consists of studying whole systems of objects in terms of their global, relational, or structural properties, while neglecting the intrinsic nature of the objects involved. This can be done in two ways that are often intertwined in practice: by proceeding axiomatically, i.e., by deriving theorems from basic axioms for the systems at issue; and by considering morphisms between those systems (homomorphisms, isomorphisms, etc.), together with invariants under those morphisms (more on both side later). As this kind of approach typically involves infinite sets, non-decidable properties, and classical logic, it tends to be opposed to more computational or constructivist ways of doing mathematics (cf. Stein 1988, Reck and Schiemer 2020a).

A structuralist methodology in this sense can be elaborated further in several ways, beyond the general characterization just given, with various forms of methodological structuralism as the result (see Section 3). But at bottom, the methodology is tied to a general assumption about the subject matter of mathematics, namely: mathematics is the study “structures” (the natural numbers, the real numbers, also various groups, rings, geometric spaces, topological spaces, function spaces, etc.). At the same time, accepting that assumption does not, by itself, involve views about the nature of those structures, at least not in any detailed metaphysical or otherwise philosophically loaded way. (Practicing mathematicians who are guided by that assumption, often implicitly, typically keep their distance from such metaphysical issues.) In contrast, all the forms of metaphysical structuralism considered earlier are meant to provide the latter. This is exactly how they go beyond methodological structuralism (while presumably building on it).

So far we have clarified what we mean by “methodological structuralism” (or again, what is sometimes meant by “mathematical structuralism” in the literature, although that label has also been used more broadly). With respect to metaphysical positions, Parsons’ eliminative vs. non-eliminative distinction remains helpful (while a more positive label for the latter kind of positions, along the lines of “structuralism with structures”, seems more adequate for some purposes). Yet the significant differences between metaphysical forms of structuralisms are not exhausted by it. In fact, only working with Parsons’ distinction can obscure philosophically important points. In particular, Shapiro’s ante rem structuralism is far from the only version of non-eliminative structuralism; nor is Hellman’s modal structuralism the only form of eliminative structuralism. As tools for clarifying the corresponding debates further, we will now suggest several more fine-grained distinctions (many already implicit in what we wrote above).

Let us first look again at non-eliminative structuralism. Several positions that fall under that label introduce abstract structures by means of corresponding basic theories. This includes Shapiro’s structure theory, but also, e.g., Nodelman and Zalta’s object theory, Leitgeb’s adaptation of graph theory, and Horsten’s theory of arbitrary objects. All of them lead to forms of ante rem structuralism, but significantly different ones. In addition, there are forms of structuralism, including one variant of non-eliminative structuralism, that are based on abstraction principles; we therefore called them “abstractionist forms of structuralism”. We discussed two versions of them, which involved different kinds of structures as the outcome of the abstraction process. (If we explicated the respective abstractions as mathematical operations or functions, their arguments would be the same but their values different.) One thing this makes evident is that there are abstractionist and non-abstractionist forms of non-eliminative structuralism.

If we reflect further on such issues, the roles of some other distinctions and subdivisions become apparent, namely: between ante rem positions and either in re or post rem positions. Shapiro’s is explicitly an example of ante rem structuralism. In contrast, our first version of abstractionist structuralism can be seen as a form of post rem structuralism, because the equivalence classes used as the structures in it are “built out of” their elements to which they are thus posterior. In our second version of abstractionist structuralism, there is a form of posteriority involved as well. Here too, we start with a more concrete relational system, usually a system of sets perhaps containing urelements, and we introduce an abstract structure on that basis. The prior versus posterior relation in that case is not based on the element-class relation, but on a more fundamental argument-function-value relation. And we end up with abstract structures, as opposed to classes, as the result of the abstraction process.

On the side of eliminative structuralism, further distinctions can be introduced as well. In particular, there are fully eliminative positions, which avoid commitment to any kind of abstract objects. Hellman’s modal structuralism is of this sort. But there are also semi-eliminative positions, which avoid commitment to sui generis abstract structures, but accept other, more concrete mathematical objects. Set-theoretic structuralism is a good illustration; universalist structuralism is another, at least when backed up by set theory. Beyond that, should relativist structuralism, and set-theoretic structuralism more particularly, be seen as a case of in re structuralism, in the sense that here abstract structure exist “in” their more concrete instantiations? Perhaps; but this thesis does not seem forced on us (cf. Leitgeb 2020a/b for more). The same question arises for in re forms of structuralism more generally; the details of the position will matter. (For more on the latter positions, sometimes labeled “Aristotelian” as opposed to “Platonist”, cf. Pettigrew 2008 and Franklin 2014.)

Yet another version of eliminative structuralism is concept structuralism. If it is meant to work without any appeal to abstract objects (e.g., by basing itself on formalism), it amounts to a fully eliminative position. Questions remain, however, about the appeal to concepts, including their existence, nature, and identity (cf. Parsons 2018). Depending on the answers, a strict nominalist may find this position unacceptable, namely if the concepts are seen as another problematic kind of abstract entities. On the other hand, if a concept structuralist allows for sets to play a secondary role, it becomes a semi-eliminative position; i.e., structures conceived of as separate abstract objects are eliminated (by reconceiving them as concepts), but set-theoretic systems remain. Finally, another option might be to consider concept structuralism merely as a form of methodological structuralism, in which case additional metaphysical questions are brushed aside as irrelevant. Here again, the details will matter. But let us now turn to methodological structuralism in itself.

3. Epistemological and Methodological Aspects

3.1 Patterns, Postulation, Construction, Abstraction

While discussions of structuralism in the philosophy of mathematics often focus on metaphysical questions, occasionally these are tied to epistemological questions. In the English-speaking literature, an early illustration is provided by Michael Resnik’s works. As we already noted, his approach is distinctive in two ways: by emphasizing the role of structures, but not fully objectifying them; and by addressing, more than most, questions about how we can know anything about such structures (cf. Resnik 1982, 1997). The main phenomenon Resnik points to in connection with the latter is our ability to recognize patterns. Two simple examples would be intuitive patters in geometry (involving points, lines, etc.) and numerical patterns in arithmetic (concerning numerical sequences represented by means of numerals etc.). The basic suggestion for the latter is, then, that knowledge about the natural number structure is grounded in recognizing the beginnings of such patters, usually involving finitely many numbers, and then generalizing to the infinite. If one considers education in mathematics, there is clearly something to this suggestion. Yet questions about the process of generalization remain.

A second suggestion for how to account for our knowledge of structural facts in mathematics, including our knowledge of structures as abstract objects, has come up already as well. Namely, Resnik, Shapiro, as well as Parsons all argue that modern mathematics involves the postulation of various structures based on corresponding axiom systems. Along such lines no access to some “Platonic heaven“, as rejected by nominalists, is required (and more positively, one can combine postulating with pattern recognition). In this case, what remains are questions about when such postulations are legitimate. Other variants of metaphysical structuralism, including eliminative ones, involve related views. Thus in Hellman’s nominalist structuralism any claim about knowledge of abstract objects is avoided, while mathematical knowledge is tied to establishing certain if-then claims by mathematical proof, thereby assuring their necessity. Yet an additional kind of modal knowledge is required as well, involving mathematical possibility. And as noted earlier, Hellman points to the coherence of basic axiom systems here, parallel to Shapiro’s justification for postulating abstract structures. Here one can ask: What kind of coherence exactly; and how can it be established?

Finally, in our two versions of abstractionist structuralism and in related positions we begin with set theory. A basic kind of mathematical knowledge consists, then, in learning how to perform set-theoretic constructions. Here too, questions about the consistency, or perhaps again the coherence, of the set-theoretic framework remain, related to which such constructions are legitimate. But an additional appeal to abstraction, which can take several different forms, is important. Some kind of knowledge by abstraction is thus involved as well, including questions about how it is to be justified. This is not unrelated to pattern recognition, postulation, and their legitimacy; but here the details need to be spelled out in terms of abstraction principles, in one way or another. Dedekind’s version of structuralism, in particular, stays close to the methodology of modern mathematics in this connection, which provides the basis for more advanced kinds of knowledge and understanding as well—the topic to which we turn next.

3.2 Levels of Methodological Structuralism

With the appeal to set-theoretic constructions, related forms of abstraction, etc., we have arrived at methodological questions about mathematics. Broadly speaking again, a structuralist methodology consists in studying whole systems of objects in terms of their global, relational, or structural properties, while neglecting the intrinsic nature of the objects involved. Such a methodology has roots in traditional mathematics, which can be traced all the way back to Ancient times (cf. Reck 2020). But it was further developed and became more prevalent in the second half of the nineteenth and the early twentieth centuries, especially as far as infinite structures are involved. Moreover, arguably the first systematic employment and discussion of this methodology can be found in the works of Dedekind (cf. Reck 2003, 2009, Ferreiros and Reck 2020), although numerous other mathematicians contributed to it as well (cf. Reck and Schiemer 2020a).

This description of a structuralist methodology is still very general, of course. In the recent literature one can find the beginnings of a more differentiated treatment of it. In this context, we can distinguish four levels or kinds of methodological structuralism. (Probably more can be added in the end.) The most basic level consists of studying an area of mathematics by first characterizing the main structure at issue (a task that relies on pattern recognition etc.), and based on that, proving theorems about it. This is typically done axiomatically, especially in the sense of the formal axiomatics introduced and made prominent by Hilbert. Along such lines, Dedekind, and in more familiar form Peano, characterized the natural number structure by means of the Dedekind-Peano axioms and proved theorems about it. Structuralism at this first level is the culmination of doing mathematics more conceptually than before (thus leading to concept structuralism, among others), as opposed to earlier computational methods (cf. Stein 1988, Ferreiros and Reck 2020).

A second level of proceeding structurally in mathematics, one that builds on the first, involves not just considering one structure but the relationships between several of them. This can be done by either invoking abstract structures or working with more concrete instantiations of such structures, like in standard model theory. What becomes crucial at this level is the consideration of various structure-preserving mappings between such structures, especially homomorphisms and isomorphisms. Using Dedekind as an early source again, one example is his theorem that all models of the Dedekind-Peano axioms (all “simple infinities”) are isomorphic (the basis of his abstractionist structuralism). Perhaps a more striking example is Dedekind’s proof that the natural numbers, as characterized by him, can be embedded homomorphically into any recursively constructed system (thus anticipating category-theory, cf. McLarty 1993). Later this approach was pushed further and made more abstract in works by Hilbert, Noether, Tarski, etc.

We reach a third level of structuralist mathematics when we don’t just consider one or a few closely related kinds of structure, but attempt to characterize all the main kinds of structures in mathematics, including their interrelations. Paradigm examples of such an approach can be found in the works of the Bourbaki group, including their distinction between three kinds of mother structure: algebraic, topological, and order-theoretic (cf. Heinzmann and Petitot 2020). In terms of historical shifts, Bourbaki used set theory as the relevant framework initially, but it was later replaced by category-theory (including the notion of functor, universal mapping properties, etc.). In fact, category theory can be seen as a systematic, even more abstract development of exactly this kind of methodological structuralism. Moreover, it is no exaggeration to say that such structuralism, either in a set-theoretic or a category-theoretic form, has dominated large parts of mathematics since the mid-twentieth century (cf. Corry 2004, especially for algebra).

A fourth level of methodological structuralism, or perhaps an especially interesting procedure on the second level instead (since it is more concrete again), has not found much attention in the philosophical literature yet, while some mathematicians have started to highlighted it. Here mathematicians investigate how some mathematical theorem, proven in terms of structuralist background assumptions in one part of mathematics, can be transferred to other parts in fruitful ways. The terminology typically used for successful cases is that of structure theorems. (A relatively recent illustration is Szemerédi’s Theorem; cf. Tao 2006.) In this context the structuralist methodology is not used in a way that is as global and abstract as on level three. Instead, the focus is on particular theorems, while general assumptions remain in the background. This strategy is again very significant in mathematical practice and, as such, deserves more philosophical attention (cf. Ryan 2023, more generally also Carter 2024, for some forays into this area).

3.3 Structuralist Understanding in Mathematics

When philosophers such as Resnik, Shapiro, and Parsons appeal to pattern recognition and the postulation of structures, their goal is usually to address very basic questions about knowledge in mathematics, questions that are highlighted in traditional philosophy. This still applies to the investigation of basic kinds of possibility in mathematics, e.g., involving the notion of coherence in Hellman and Shapiro. Usually the motivation for such appeals consists, at least in part, in a desire to respond to skeptical or nominalistic challenges to the very possibility of mathematical knowledge. Hence we are dealing again with philosophical structuralism, although its epistemological rather than its metaphysical dimension; and working mathematicians are seldom interested in such work, since it doesn’t affect their practice directly.

With the issues mentioned in the previous subsection this starts to change, however. We have moved closer to mathematical methodology, in a way that concerns mathematicians in their own pursuits. Nevertheless, we are dealing with broadly epistemological issues, in the sense of questions about how we know things in mathematics, including fruitful strategies for obtaining such knowledge. At the same time, what is at stake is not so much basic knowledge but more advanced mathematical understanding (also related notions such as mathematical explanation). Traditional epistemology and philosophy of mathematics have focused mostly on knowledge, usually in a fairly exclusive way. In contrast, recent “philosophy of mathematical practice” has become broader and reached further, including with respect to methodological issues (cf. Carter 2024). In the philosophy of science as well, there is renewed interest in the notion of understanding, as opposed to knowledge, in that case tied to the relevant practices in other sciences (see, e.g., De Regt, Leonelli & Eigner 2013). In our remarks, what has come into focus more specifically is structuralist understanding in mathematical practice, including its philosophical significance.

The step-by-step crystallization of such structuralist understanding, via the novel methodologies that made it possible, played a crucial role in the emergence of modern mathematics in the late nineteenth and early twentieth centuries. Arguably, it is what drove the rise of structuralism in mathematics (see again Reck and Schiemer 2020a). But even for current philosophers of mathematics who are skeptical about “mathematical structuralism“, perhaps because of their (mistaken) identification of it with purely metaphysical positions, this is an aspect they should not ignore. Actually, a charitable way to interpret recent metaphysical or philosophical structuralism, at least in some of its variants (perhaps most clearly Parsons’), is that it was motivated by the realization that methodological structuralism had become quite prevalent in mathematics, including raising new philosophical questions. In any case, our discussion of structuralist understanding has led to epistemological issues that are relevant for both philosophers and mathematicians.

Another advantage of including an explicit treatment of structuralist understanding in this survey (more than before) is that this allows us to incorporate category-theoretic structuralism better. As already noted, category theory grew out of reflections on the methodology introduced by Dedekind, Klein, Hilbert, Poincaré, Noether, Bourbaki, etc. As such, it is primarily a form of methodological structuralism, one meant to enhance structuralist understanding further. But it has also been proposed as a new foundation for mathematics, thus leading back to metaphysical questions, as we will see in the next section.

4. Category-Theoretic Structuralism

4.1 Category Theory as the Study of Mathematical Structures

Over the past two decades, a number of proposals have been made to formulate a theory of mathematical structuralism based on category theory, thus a theory, or theories, of category-theoretic structuralism, or in slightly different terminology, of categorical structuralism. We are now in a better position to consider these proposals, although we will still proceed indirectly, starting with more background. Category theory was first introduced as a branch of abstract algebra, in Samuel Eilenberg and Saunders Mac Lane’s famous article, “General Theory of Natural Equivalences” (1945). It subsequently developed into an autonomous mathematical discipline, in work by Mac Lane, Alexander Grothendieck, Daniel Kan, William Lawvere, and many others, with important, wide-ranging applications in algebraic topology and homological algebra, more recently also in logic and computer science (cf. Landry & Marquis 2005, as well as the entry on category theory in this encyclopedia).

Building on these developments, the philosophical discussion of categorical structuralism was initiated by Steve Awodey, Elaine Landry, Jean-Pierre Marquis, and Colin McLarty in the 1990s. To understand their contributions, our distinction between metaphysical and methodological structuralism, which is drawn explicitly by Awodey (1996), will again be helpful. Or rather, Awodey distinguishes between the use of category theory as a framework for “mathematical” and for “philosophical structuralism”. He describes mathematical structuralism as a general way of “pursuing a structural approach to the subject“, i.e., a style of practicing mathematics that employs structural concepts and methods; and he argues that category theory provides the best way of capturing structuralist mathematics in this sense. However, he also presents it as a framework for philosophical structuralism, i.e., “an approach to the ontology and epistemology of mathematics”. Let us first consider the former suggestion and argument.

Category theory, understood as a branch of mathematics, had been described as a “general theory of mathematical structures” already earlier, e.g., by Mac Lane (1986, 1996). But what exactly is meant by “structure” here? In connection with category theory, it will help to clarify this issue further. At least two relevant notions are mentioned in the literature concerning categorial structuralism. First, a structure can be understood in the set- and model-theoretic sense, which we can specify further as a tuple consisting of a ___domain and an ordered sequence of relations, functions, and distinguished elements used for the interpretation of a formal language. (This is the notion of “set-theoretic relational system” appealed to in earlier sections.) Such structures are also called “Bourbaki structures” in this context, because of Bourbaki’s early reliance on set theory as the framework. Their properties are usually defined axiomatically, e.g., by the Dedekind-Peano axioms for arithmetic or by the group axioms.

Second and alternatively, there is a categorical notion of structure based on the primitive concept of morphism between mathematical objects as used in category theory. Typically, a category consists of two types of entities, namely objects and morphisms between them, in the sense of mappings represented by arrows that preserve some of the internal structural composition of the objects at issue. An axiom system that defines the general concept of a category, along such lines, was first introduced by Eilenberg and Mac Lane in their 1945 article. It describes a suitable composition operation on arrows, its associativity, as well as the existence of an identity morphism for each object (cf. Awodey 2010 for a textbook presentation).

Now, why might category theory be considered a more adequate framework for a mathematical structuralism than other disciplines, in particular traditional set theory? According to Awodey, the “Bourbaki notion of structure” is a direct result of the modern axiomatic tradition, from Dedekind and Hilbert to the Bourbaki group. This tradition did introduce a structuralist perspective on mathematics. Yet set theory is not an ideal framework for capturing a structuralist understanding of mathematical objects, as Awodey goes on to argue. To begin with, it is closely tied to a model-theoretic conception of mathematical theories, including the view that such theories study their models only “up to isomorphism”. But according to Awodey, central to a structuralist point of view is the principle (more on which below) to identify isomorphic objects; and this principle is well motivated from a category-theoretic viewpoint, but less so if mathematical objects are constructed set-theoretically.

A second advantage of category theory over set theory, also mentioned by Awodey, is that the categorical notion of structure is syntax invariant. That is to say, unlike in standard model theory the categorical specification of objects in terms of their mapping properties is independent of the choice of a particular signature used for their description (a choice of basic relations, functions, and distinguished elements). Third and most importantly, characteristic for category theory is its focus on mappings between mathematical objects that preserve (some of) their internal structure. Besides the use of set-theory, it is the emphasis on structure-preserving mappings that is often viewed as a central feature of methodological structuralism, as we saw above already (see also again Reck and Schiemer 2020a).

Category theory was first worked out, against the background of these developments, as a unifying framework for the study of the relations between different mathematical structures (cf. Landry and Marquis 2005, Marquis 2009). Several types of mappings were introduced for that task. One type involves the morphisms between objects of the same category, e.g., group homomorphisms in the category of groups, or linear maps in the category of vector spaces. Another important type of mappings are functors between different categories. (Roughly, a functor between two categories is a mapping of objects to objects and arrows to arrows that preserves the categorical properties in question.) It is such functors that are the crucial tools in category theory for comparing the objects of different mathematical categories, and thus, to “relate structures of different kinds” (Awodey 1996). As such, they are central in categorical structuralism. In particular, they allow for structuralist understanding of an advanced kind.

4.2 Categorical Foundations and Debates About Them

As has been argued repeatedly along such lines, category theory provides a more natural framework than traditional set theory for mathematical or, as we prefer to call it, methodological structuralism in mathematics. But what about its prospects as a form of philosophical or metaphysical structuralism, i.e., an alternative to the recent theories of Resnik, Shapiro, and Hellman, or those of Nodelman and Zalta, Horsten, Leitgeb, etc.? We already indicated that Awodey also presents it as such (cf. Awodey 1996); but this has led to ongoing controversies. A well-known article by Hellmann (2003) contains a first critical discussion of philosophical claims such as Awodey’s. More specifically, Hellman’s piece raises several objections against the view that category theory provides an adequate framework for a structuralist account of mathematics in the philosophical sense, let along a better such framework. As we will see, these objections are closely related to the status of category theory as a foundational discipline.

In the last few years there has been much debate on which criteria a theory has to meet in order to serve as a proper “foundation” for mathematics. According to a helpful proposal by Tsementzis, a foundational system consists of three items: (i) a formal language; (ii) an axiomatic theory expressed in that language; and (iii) a rich universe of objects, described by the theory, in which all mathematical structures can be located, represented, or encoded (cf. Tsementzis 2017). Zermelo-Fraenkel set theory clearly represents a foundational system in this sense. The axioms of ZFC are usually formulated in a formal first-order language; and they describe a comprehensive universe, the cumulative hierarchy of sets, in which mathematical objects such as number systems, groups, rings, topological spaces, function spaces, etc. can be represented.

In research on category theory from the 1960s onward, after its establishment as a fruitful methodological framework, several axiomatizations of specific categories have been proposed as alternative foundations for mathematics. This includes the axiom systems describing the category of sets and functions, on the one hand, and the category of categories, on the other hand (cf. Lawvere 1964, 1966). Both were explicitly introduced as foundational systems, and thus, as alternatives to Zermelo-Fraenkel set theory. More recently, elementary topos theory has been developed as a form of categorical set theory, which can serve as a foundational system in the above sense as well (cf. Landry and Marquis 2005, Marquis 2013).

Returning to Hellman’s 2003 challenge, the question of whether category theory is usable to defend a version of philosophical structuralism can then be directly related to the presumed autonomy of these new approaches from traditional set theory. Building on work by Solomon Feferman (especially Feferman 1977), Hellman formulates two general objections. The first is the logical dependence objection (cf. Linnebo & Pettigrew 2011). Its core is the argument that category theory, general topos theory, etc. are not autonomous of set theory in the end. The reason given is that the axiomatic specification of categories and topoi presupposes the concepts of operation, collection, and function, and the latter need to be defined in a set theory such as ZFC. Categorical foundations are therefore dependent on non-structural set theory.

The second argument against the autonomy of categorical foundations is the mismatch objection. It concerns the general status of category theory or topos theory; and it is based on the distinction between two ways of understanding mathematical axioms: as “structural”, “algebraic”, “schematic”, or “Hilbertian”, on the one hand, and as “assertoric” or “Fregean”, on the other hand. As Hellman also argues, foundational systems need to be assertoric in character, in the sense that their axioms describe a comprehensive universe of objects used for the codification of other mathematical structures. Zermelo-Fraenkel set theory is an assertoric, “contentual” theory in that sense. Its axioms (e.g., the power set axiom or the axiom of choice) make general existence claims regarding the objects in the universe of sets.

In contrast, category theory represents a branch of abstract algebra, as its origin indicates. Thus it is, by its very nature, non-assertoric in character; it lacks existence axioms conceived as truths about an intended universe. For example, the Eilenberg-Mac Lane axioms of category theory are not “basic truths simpliciter”, but “schematic” or “structural”. They function as implicit definitions of algebraic structures, similar to the way in which the axioms of group theory or ring theory are “defining conditions on types of structures”. This point is related to another argument against the autonomy of category theory, the problem of the home address. As Hellman asks, “where do categories come from and where do they live?” Given the “algebraic-structuralist perspective” underlying category theory and general topos theory, its axioms make no assertions that particular categories or topoi actually exist. Classical set theories, such as ZFC with its strong existence axioms, have to step in again in order to secure the existence of such objects.

Hellman’s and Feferman’s arguments against the foundational character of category theory have been examined from various angles in the subsequent literature. One can distinguish between two main types of responses, namely: (i) by proponents of categorical foundations, who defend the autonomous character of category theory relative to classical set theory; and (ii) by non-foundationalists, who call into question that category theory should be viewed as a foundational discipline. A series of articles by McLarty represent the first line of response well (cf. McLarty 2004, 2011, 2012). Roughly speaking, his reply to Hellman is as follows: Whereas category theory and general topos theory did originate as algebraic theories and are, as such, not feasible as foundational systems, certain theories of particular categories and toposes have been introduced as alternative foundations. McLarty’s central examples are Lawvere’s axiomatizations of the category of categories and his “Elementary Theory of the Category of Sets” (ETCS).

According to McLarty, these theories should be understood as assertoric in Hellman’s sense. That is to say, their axioms are not merely implicit definitions, but general existential claims about categories, sets, and functions. ETCS, for instance, presents a function-based set theory, where sets and mappings between them form a topos. In contrast to ZFC, with its primitive membership relation, in ETCS a set is not specified via its internal composition, but in terms of its mapping properties with respect to other sets, formulated independently of ZFC. McLarty’s response to the objections mentioned above is, then, that categorical set theories such as ETCS do provide a foundation for mathematics, one that is logically autonomous from traditional set theories. Moreover, given that mathematical structures can only be encoded in ETCS as objects up to isomorphism, such categorical set theories provide a more adequate foundation for modern structural mathematics than ZFC. (We will return to this point below.)

The second, quite different line of response to Hellman is provided by Awodey (cf. Awodey 2004, earlier also Landry 1999). Awodey outlines a category-theoretic form of structuralism that is decidedly anti-foundationalist. He holds, in agreement with Hellman and McLarty, that both the Eilenberg-Mac Lane axioms for category theory and the axioms of general topos theory are schematic. But he then argues that category theory in general does not, and should not be taken to, provide a foundation for mathematics, either in a logical or ontological sense. Rather, it presents a general and unifying language for structural mathematics. He thus rejects Hellman’s assumption that the success of categorical structuralism is dependent on whether category theory is usable as a foundational enterprise or not.

In fact, according to Awodey the central motivation for a category-theoretic approach to mathematics is to sidestep foundational issues concerning the nature of mathematical objects or the study of a single comprehensive universe in which all structures can be represented. While topos theory, say, might well serve as a structural foundation for mathematics, for Awodey such a foundational approach runs against the structuralist perspective embodied in category-theory. In his own words, “the idea of ‘doing mathematics categorically’ involves a different point of view from the customary foundational one” (Awodey 2004: 55). In light of such fundamental, ongoing debates about categorical foundations for mathematics, what are the implications for categorical structuralism in the metaphysical sense, along Awodey’s lines and more generally? Or does it just amount to a form of methodological structuralism, after all? We will conclude our discussion of category-theoretic structuralism by addressing this question.

4.3 Philosophical Features of Category-Theoretic Structuralism

Beyond issues involving methodological structuralism (including broadening and strengthening structuralist understanding in mathematics), the literature on category-theoretic structuralism during the last 25 years centers around two questions that have already been mentioned. First, in what sense does category theory provide a framework for philosophical or metaphysical structuralism? Second, why is it supposed to be better suited for that task than other frameworks, such as set theory, Shapiro’s structure theory, Hellman’s modal structuralism, etc.? In recent work on these topics, one can find three related philosophical assumptions that characterize categorical structuralism and distinguish it further from the versions of structuralism surveyed earlier. We will treat each of them in turn, starting again with Awodey’s writings.

A first characteristic assumption is that all mathematical theorems are schematic statements that have a conditional form. (This point is explicit in Awodey 2004.) We already saw that according to Awodey a category-theoretic approach is non-foundational in character. This includes that mathematical axioms and theorems, as expressed in category theory, should be understood as schematic statements. As such, they do not express truths about the specific nature of mathematical objects. In addition, mathematical theorems are, at least in principle, all of hypothetical form, i.e., they can be reconstructed as if-then statements. Note that this view of the logical form of mathematical theorems is prima facie similar to the if-then-ism one can find in Putnam’s and Hellman’s works, building on Russell’s earlier work.

However, according to Awodey there is an important difference between standard if-then-ism and a category-theoretic approach in terms of the ontological commitments involved. According to standard if-then-ism, any mathematical statement can be translated into a universally quantified conditional statement, where the quantifiers are meta-theoretic in nature, ranging over all set-theoretic systems of the right type. (This is essentially what we called “universalist structuralism” above.) As such, the approach presupposes a rich ontology of sets. In contrast, along category-theoretic lines mathematical theorems do not involve such ontological commitments. There is no implicit generalization over the set-theoretic structures of a theory. Rather, a mathematical theorem is “a schematic statement about a structure […] which can have various instances” (Awodey 2004: 57). These instances remain undetermined on purpose, unless a further specification of them is needed for the proof of a theorem in question.

A second distinctive feature of categorical structuralism, both for Awodey and others, concerns a certain top-down conception of mathematical objects characteristic for category theory. According to standard set theory, mathematical objects are constructed from the bottom up, in successive steps starting from some ground level (the empty set or also a ___domain of urelements). Every object is thus determined, as a set, in terms of its members. In contrast, mathematical objects in category theory are characterized in a top-down fashion, starting with the Eilenberg-Mac Lane axioms and using the notion of morphism. Hence, the objects in a given category, such as that of rings or topological spaces, are not considered independently of the relevant morphisms. They are fully determined by their mapping properties, as expressible in the language of category theory. Nothing further is assumed about their inner constitution. In particular, questions about their set-theoretic nature are considered redundant (cf. Landry and Marquis 2005.)

Third, arguably the most important feature of categorical structuralism is that it verifies a version of the structuralist thesis (hinted at earlier). Recall here Benacerraf’s argument, in his 1965 paper, that numbers should not be identified with particular sets, but rather, with positions in an abstract structure. Benacerraf also emphasizes that only certain properties are relevant in arithmetic. For him, these are the number-theoretic properties, such as “being prime” or “being even”, that can be defined in terms of the primitive relations and functions of the theory at issue. The general structuralist thesis holds, then, that all (relevant) properties of the objects treated by a mathematical theory should be structural in a specified sense. (The question what “structural” means also came up earlier.)

Categorial structuralists typically argue that category theory presents the most adequate framework for a structuralist conception of mathematical objects, given that all the properties expressible in its language turn out to be structural (cf. McLarty 1993, Awodey 2004, Marquis 2013). This is so, presumably, because the category-theoretic study of mathematical objects, such as rings or topological spaces, allows us to express just the right kind of structural information, namely information about the structural properties of these objects. In this context, structural properties are usually characterized in terms of the notion of isomorphism invariance. (Given a category C, a morphism \(f: A \rightarrow B\) presents an isomorphism between objects A and B if and only if there is a morphism \(g: B \rightarrow A\) such that \(g \circ f = 1_A\) and \(f \circ g =1_B\). A property P of an object A in category C is then structural if it remains invariant under the isomorphisms in C, that is, if \(P(A) \leftrightarrow P(f(A))\), for all isomorphisms f; cf. Awodey 1996.)

We saw above that the representation of mathematical objects in traditional set theory (proceeding “bottom up”) brings with it the possibility of expressing all sorts of properties about their set-theoretical constitution that are not isomorphism invariant in this sense. The central advantage of category theory over traditional set theory (and similar approaches) is, according to this argument, that such non-structural properties are simply ruled out in the categorical framework. This point was first stressed in McLarty’s “Numbers Can Be Just What They Have To” (1993), which, as its title suggests, is a rejoinder to Benacerraf 1965 article. McLarty’s central thesis is that Benacerraf’s structuralist program is most successfully realized if one considers numbers to be represented in categorical set theory, such as in ETCS.

To elaborate this point further, number systems of basic arithmetic can be characterized in such categorical frameworks as “natural number objects”, as was first shown by Lawvere. In contrast to their ZFC-based representation, such objects are not just isomorphic but share “exactly the same properties”, namely those expressible in the language of the category of sets. In other words, any two natural number objects are “provably indiscernible” (McLarty 1993). Moreover, all of these properties are structural in the above sense. As a consequence, Benacerraf’s dilemma of isomorphic numbers systems with different set-theoretic properties does not arise in the context of categorical set theory. The conclusion is that numbers can, after all, be identified with sets, but with structural ones as defined in ETCS.

As McLarty and others went on to argue, this observation generalizes from numbers to all other mathematical objects studied in category theory. The claim is that any property of the objects in a given category that is expressible in the language of category theory is structural in the sense of being isomorphism invariant. The main consequence for a structuralist conception of mathematical objects is then summarized by Awodey as follows: “Since all categorical properties are thus structural, the only properties which a given object in a given category may have, qua object in that category, are structural ones” (Awodey 1996: 214). This would seem to illustrate a main advantage of categorical forms of structuralism over set-theoretic and similar ones (but not over all forms of structuralism considered earlier, as we can note).

A rejoinder should be added, however. McLarty’s and Awodey’s claim that all mathematical properties expressible in categorical set theory are isomorphism invariant has been contested (cf. Tsementzis 2017). In fact, it has become clear that neither ZFC nor ETCS provide fully structuralist foundations for mathematics, since their respective languages do not, after all, exclusively allow for the formulation of invariant properties. Then again, both Michael Makkai’s FOLDS system (Makkai 1995, Other Internet Resources, 1998) and the Univalent Foundations program developed in Homotopy Type Theory (Univalent Foundations Program 2013) seem to meet this condition. Here we have reached another ongoing debate about mathematical structuralism in the literature, one that partly depends on technical results.

Debates like the one just mentioned are clearly relevant for us, but we cannot explore them further here. (Cf. Awodey 2014 for more on the relation between structuralism and Univalent Foundations.) Some additional questions can be raised as well. For example, is Awodey’s position best understood as a deflationary form of metaphysical structuralism; or does his anti-foundationalism reduce category-theory to an advanced form of methodological structuralism? On the epistemological side, in which ways exactly does category theory involve new forms of structuralist understanding (e.g., with the notion of functor)? However, we have to reserve answering such questions again for future occasions. Instead, we will conclude this survey with a few general remarks about structuralism, also beyond the philosophy of mathematics.

5. Conclusion

5.1 Varieties of Mathematical Structuralism

In retrospect, we pursued two main goals in this survey. The first was to provide an introduction to the discussions of structuralism in contemporary philosophy of mathematics, usually traced back to the 1960s. The second was to broaden and deepen those discussions, by making clear that a much larger variety of structuralist positions has played a role than is often acknowledged. To be sure, some amount of variety was recognized before, as reflected in Parsons’ distinction between eliminative and non-eliminative positions, with Shapiro’s ante rem structuralism and Hellman’s modal structuralism as paradigms. Also, category-theoretic forms of structuralism have been acknowledged as a third main alternative, although their relationship to the others needed clarification. Yet the range of positions at play encompasses much more than that, including older variants, like set-theoretic structuralism, and several more recent ones. Overall, “mathematical structuralism” is not the name of a single position, but of a multifaceted family of them.

Three additional, more specific goals influenced the content and overall shape of this survey further. First, while numerous variants of structuralism for mathematics have been proposed, a number of of them have not received adequate attention yet, including clarifying their relationships to Shapiro’s, Hellman’s, set-theoretic, and category-theoretic positions. Besides including those variants explicitly, our discussion was meant to build bridges and to provide conceptual tools for that purpose. Second, several important forms of structuralism played a role already before the 1960s, i.e., before the publications by Benacerraf and Putnam with which debates about structuralism are often assumed to have started. We wanted to make that evident as well (and point the reader again to Reck & Schiemer 2020a for more). Third, this concerns both metaphysical and methodological structuralism, a distinction we highlighted repeatedly; it also leads to new questions about structuralist understanding, which deserve more philosophical attention in themselves.

5.2 Structuralism Beyond Mathematics

If one widens one’s perspective even further, it becomes clear that another theme relevant for our discussion in a general way has not been covered in this survey, although it should be mentioned at least briefly before closing. This concerns debates about “structuralism” outside the philosophy of mathematics. There are two main areas in which one can find such debates (each of which has had broader reverberations). The first is the philosophy of physics, where structuralist positions have played a significant role as well (including a position called “structural realism”, which comes in both an “ontic” and an “epistemic” form). The second includes various parts of the humanities and social sciences, primarily linguistics and anthropology, but also psychology, sociology, etc. There too, “structuralism” (and “post-structuralism”) has been a major topic, indeed already for a relatively long time. In both cases there are systematic and historical ties to mathematical structuralism, although sometimes those ties are only loose.

At a basic level, the debate about structuralism in the philosophy of physics concerns how to think about the “objects” of modern physics, given the revolutionary changes quantum mechanics and relativity theory have brought about. More specifically, it concerns the move to appeal to “structures” in this connection, in a way that is closely related to the philosophy of mathematics (cf. French 2014, Ladyman 2007 [2019], etc.). Second and in addition, this debate concerns the question how to characterize what, if anything, remains constant through theory changes in the ontology of physics (cf. Worrall 1989, see also the entry on structuralism in physics in this encyclopedia). Third, there are debates about scientific representation that are related to structuralism (cf. van Fraassen 2008). While one could pursue several resulting points of contact further (e.g., concerning “structural indiscernibles”, or versions of “set-theoretic vs. category-theoretic structuralism” in the context of physics), we refrain from doing so here.

The structuralism introduced into linguistics by Ferdinand de Saussure and Roman Jakobson, then adopted by various thinkers in the humanities and social sciences—most prominently by Claude Levi-Strauss in anthropology and by Jean Piaget in psychology (cf. Levi-Strauss 1958 [1963], Piaget 1968 [1970], also Caws 1988)—does have ties to mathematical structuralism as well. But they are looser than in the case of the philosophy of physics and there are more systematic differences. In particular, the kind of psychological determinism that is often associated, or even identified, with structuralism in the humanities and social sciences (and criticized in post-structuralism) has no analogue on the side of mathematics (or physics). Still, some striking historical ties exist, e.g., personal contacts between Levi-Strauss and members of the Bourbaki group (cf. Dosse 1991–92 [1997]). Those ties might be worth exploring further too, at least if one wants to understand the role of structuralist ideas in human thought more fully.

Bibliography

  • Assadian, Bahram, 2018, “The Semantic Plights of the Ante-Rem Structuralist”, Philosophical Studies, 175(12): 3195–3215. doi:10.1007/s11098-017-1001-7
  • –––, 2022, “The Insubstantiality of Mathematical Objects as Positions in Structures”, Inquiry: An Interdisciplinary Journal of Philosophy, 20: 1–22
  • Awodey, Steve, 1996, “Structure in Mathematics and Logic: A Categorical Perspective”, Philosophia Mathematica, 4(3): 209–237. doi:10.1093/philmat/4.3.209
  • –––, 2004, “An Answer to Hellman’s Question: ‘Does Category Theory Provide a Framework for Mathematical Structuralism?’”, Philosophia Mathematica, 12(1): 54–64. doi:10.1093/philmat/12.1.54
  • –––, 2010, Category Theory, second edition, Oxford: Oxford University Press.
  • –––, 2014, “Structuralism, Invariance, and Univalence”, Philosophia Mathematica, 22(1): 1–11. doi:10.1093/philmat/nkt030
  • Benacerraf, Paul, 1965 [1983], “What Numbers Could Not Be”, The Philosophical Review, 74(1): 47–73. Reprinted in Benacerraf and Putnam 1983: 272–294. doi:10.2307/2183530 doi:10.1017/CBO9781139171519.015
  • –––, 1973, “Mathematical Truth”, The Journal of Philosophy 70(19): 661–679.
  • –––, 1996, “Recantation or Any Old ω-Sequence Would Do after All”, Philosophia Mathematica, 4(2): 184–189. doi:10.1093/philmat/4.2.184
  • Benacerraf, Paul and Hilary Putnam (eds.), 1983, Philosophy of Mathematics: Selected Readings, second edition, Cambridge: Cambridge University Press. doi:10.1017/CBO9781139171519
  • Burgess, John P., 1999, “Book Review: Stewart Shapiro. Philosophy of Mathematics: Structure and Ontology (1997)”, Notre Dame Journal of Formal Logic, 40(2): 283–291. doi:10.1305/ndjfl/1038949543
  • –––, 2015, Rigor and Structure, Oxford: Oxford University Press. doi:10.1093/acprof:oso/9780198722229.001.0001
  • Button, Tim, 2006, “Realistic Structuralism’s Identity Crisis: A Hybrid Solution”, Analysis, 66(3): 216–222. doi:10.1093/analys/66.3.216
  • Button, Tim and Sean Walsh, 2016, “Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics”, Philosophia Mathematica, 24(3): 283–307. doi:10.1093/philmat/nkw007
  • –––, 2018, Philosophy and Model Theory, Oxford: Oxford University Press
  • Cantù, Paolo, 2020, “Grassmann’s Concept Structuralism”, in Reck and Schiemer 2020a: 21–58.
  • Carter, Jessica, 2008, “Structuralism as a Philosophy of Mathematical Practice”, Synthese, 163(2): 119–31. doi:10.1007/s11229-007-9169-6
  • –––, 2024, Introducing the Philosophy of Mathematical Practice (Elements in the Philosophy of Mathematics), Cambridge: Cambridge University Press.
  • Caws, Peter, 1988, Structuralism. A Philosophy for the Human Sciences, Atlantic Highlands, NJ: Humanities Press.
  • Chihara, Charles S., 2004, A Structural Account of Mathematics, Oxford: Oxford University Press
  • Cole, Julian C., 2010, “Mathematical Structuralism Today”, Philosophy Compass, 5(8): 689–699. doi:10.1111/j.1747-9991.2010.00308.x
  • Corry, Leo, 2004, Modern Algebra and the Rise of Mathematical Structures, second edition, Basel: Birkhäuser Basel. doi:10.1007/978-3-0348-7917-0
  • De Regt, Henk, Sabina Leonelli, and Kai Eigner, eds., 2013, Scientific Understanding: Philosophical Perspectives, Pittsburgh: Pittsburgh University Press
  • Dedekind, Richard, 1888, Was sind und was sollen die Zahlen?, Braunschweig: Vieweg. Translated as “The Nature and Meaning of Numbers”, in his Essays on the Theory of Numbers, Wooster Woodruff Beman (trans.), Chicago: Open Court, 1901, pp. 29–115.
  • Dosse, François, 1991–92 [1997], Histoire du structuralisme, 2 volumes, Paris: Éditions La Découverte. Translated as History of Structuralism, 2 volumes, Deborah Glassman (trans.), Minneapolis, MN: University of Minnesota Press, 1997.
  • Eilenberg, Samuel and Saunders MacLane, 1945, “General Theory of Natural Equivalences”, Transactions of the American Mathematical Society, 58(2): 231–294. doi:10.2307/1990284
  • Feferman, Solomon, 1977, “Categorical Foundations and Foundations of Category Theory”, in Logic, Foundations of Mathematics, and Computability Theory, Robert E. Butts and Jaakko Hintikka (eds.), Dordrecht: Springer Netherlands, 149–169. doi:10.1007/978-94-010-1138-9_9
  • –––, 2014, “Logic, Mathematics, and Conceptual Structuralism”, in The Metaphysics of Logic, Penelope Rush (ed.), Cambridge: Cambridge University Press, 72–92. doi:10.1017/CBO9781139626279.006
  • Ferreirós, José and Erich H. Reck, 2020, “Dedekind’s Mathematical Structuralism: From Galois Theory to Numbers, Sets, and Functions”, in Reck and Schiemer 2020a: 59–87.
  • Franklin, James, 2014, An Aristotelian Realist Philosophy of Mathematics, London: Palgrave Macmillan UK. doi:10.1057/9781137400734
  • French, Steven, 2014, The Structure of the World: Metaphysics and Representation, Oxford: Oxford University Press. doi:10.1093/acprof:oso/9780199684847.001.0001
  • Hale, Bob, 1996, “Structuralism’s Unpaid Epistemological Debts”, Philosophia Mathematica, 4(2): 124–147. doi:10.1093/philmat/4.2.124
  • Halimi, Brice, 2019, “Settings and Misunderstandings in Mathematics”, Synthese, 196(11): 4623–4656. doi:10.1007/s11229-017-1671-x
  • Heinzmann, Gerhard and Jean Petitot, 2020, “The Functional Role of Structures in Bourbaki”, in Reck and Schiemer 2020a: 187–214.
  • Hellman, Geoffrey, 1989, Mathematics without Numbers: Towards a Modal-Structural Interpretation, Oxford: Oxford University Press. doi:10.1093/0198240341.001.0001
  • –––, 1996, “Structuralism Without Structures”, Philosophia Mathematica, 4(2): 100–123. doi:10.1093/philmat/4.2.100
  • –––, 2001, “Three Varieties of Mathematical Structuralism†”, Philosophia Mathematica, 9(2): 184–211. doi:10.1093/philmat/9.2.184
  • –––, 2003, “Does Category Theory Provide a Framework for Mathematical Structuralism?”, Philosophia Mathematica, 11(2): 129–157. doi:10.1093/philmat/11.2.129
  • –––, 2005, “Structuralism”, in Shapiro 2005: 536–562.
  • Hellman, Geoffrey and Stewart Shapiro, 2019, Mathematical Structuralism (Elements in The Philosophy of Mathematics), Cambridge: Cambridge University Press. doi:10.1017/9781108582933
  • Horsten, Leon, 2018, “Generic Structure”, Philosophia Mathematica (III), 27(3), 362–380.
  • –––, 2019, The Metaphysics and Mathematics of Arbitrary Objects, Cambridge: Cambridge University Press
  • Isaacson, Daniel, 2011, “The Reality of Mathematics and the Case of Set Theory”, in Truth, Reference, and Realism, Zsolt Novák and András Simonyi (eds), Budapest: CEU Press, pp. 1–75.
  • Keränen, Jukka, 2001, “The Identity Problem for Realist Structuralism”, Philosophia Mathematica, 9(3): 308–330. doi:10.1093/philmat/9.3.308
  • Ketland, Jeffrey, 2006, “Structuralism and the Identity of Indiscernibles”, Analysis, 66(4): 303–315. doi:10.1093/analys/66.4.303
  • Korbmacher, Johannes and Georg Schiemer, 2018, “What Are Structural Properties?”, Philosophia Mathematica, 26(3): 295–323. doi:10.1093/philmat/nkx011
  • Ladyman, James, 2005, “Mathematical Structuralism and the Identity of Indiscernibles”, Analysis, 65(3): 218–221. doi:10.1093/analys/65.3.218
  • –––, 2007 [2023], “Structural Realism”, Stanford Encyclopedia of Philosophy (Fall 2023 edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/fall2023/entries/structural-realism/>
  • Landry, Elaine, 1999, “Category Theory as a Framework for Mathematical Structuralism”, The 1998 Annual Proceedings of the Canadian Society for the History and Philosophy of Mathematics, pp. 133–142
  • –––, 2006, “Category Theory as a Framework for an in re Interpretation of Mathematical Structuralism”, in The Age of Alternative Logics: Assessing Philosophy of Logic and Mathematics Today, Johan van Benthem, Gerhard Heinzmann, Manuel Rebuschi, and Henk Visser (eds.), Dordrecht: Springer Netherlands, 163–179. doi:10.1007/978-1-4020-5012-7_12
  • –––, 2011, “How to Be a Structuralist All the Way Down”, Synthese, 179(3): 435–454. doi:10.1007/s11229-009-9691-9
  • ––– (ed.), 2017, Categories for the Working Philosopher, Oxford: Oxford University Press. doi:10.1093/oso/9780198748991.001.0001
  • Landry, Elaine and Jean-Pierre Marquis, 2005, “Categories in Context: Historical, Foundational, and Philosophical”, Philosophia Mathematica, 13(1): 1–43. doi:10.1093/philmat/nki005
  • Lawvere, F. William, 1964, “An Elementary Theory of the Category of Sets”, Proceedings of the National Academy of Sciences of the United States of America, 52(6): 1506–1511.
  • –––, 1966, “The Category of Categories as a Foundation for Mathematics”, in Proceedings of the Conference on Categorical Algebra, La Jolla 1965, S. Eilenberg, D. K. Harrison, S. MacLane, and H. Röhrl (eds.), Berlin, Heidelberg: Springer Berlin Heidelberg, 1–20. doi:10.1007/978-3-642-99902-4_1.
  • Leitgeb, Hannes, 2020a, “On Non-Eliminative Structuralism: Unlabelled Graphs as a Case Study, Part A”, Philosophia Mathematica (III), 28(3), 317–346.
  • –––, 2020b, “On Non-Eliminative Structuralism: Unlabelled Graphs as a Case Study, Part B”, Philosophia Mathematica (III), 29(1), 64–87.
  • Leitgeb, Hannes and James Ladyman, 2008, “Criteria of Identity and Structuralist Ontology”, Philosophia Mathematica, 16(3): 388–396. doi:10.1093/philmat/nkm039
  • Leng, Mary, 2020, “An ‘i’ for an i, a Truth for a Truth”, Philosophical Mathematica, 28(3): 347–359.
  • Lévi-Strauss, Claude, 1958 [1963], Anthropologie structurale, Paris, Plon. Translated as Structural Anthropology, Claire Jacobson and Brooke Grundfest Schoepf (trans.), Basic Books, 1963.
  • Linnebo, Øystein, 2008, “Structuralism and the Notion of Dependence”, The Philosophical Quarterly, 58(230): 59–79. doi:10.1111/j.1467-9213.2007.529.x
  • Linnebo, Øystein and Richard Pettigrew, 2011, “Category Theory as an Autonomous Foundation”, Philosophia Mathematica, 19(3): 227–254. doi:10.1093/philmat/nkr024
  • –––, 2014, “Two Types of Abstraction for Structuralism”, The Philosophical Quarterly, 64(255): 267–283. doi:10.1093/pq/pqt044
  • Makkai, Michael, 1998, “Towards a Categorical Foundation of Mathematics”, in Logic Colloquium ‘95: Proceedings of the Annual European Summer Meeting of the Association of Symbolic Logic, held in Haifa, Israel, August 9–18, 1995, Johann A. Makowsky and Elena V. Ravve (eds), (Lecture Notes in Logic, 11), Berlin: Springer-Verlag Berlin Heidelberg, pp. 153–190. [Makkai 1998 available online]
  • Mac Lane, Saunders, 1986, Mathematics Form and Function, New York, NY: Springer New York. doi:10.1007/978-1-4612-4872-9
  • –––, 1996, “Structure in Mathematics”, Philosophia Mathematica, 4(2): 174–183. doi:10.1093/philmat/4.2.174
  • MacBride, Frazer, 2005, “Structuralism Reconsidered”, in Shapiro 2005: 563–589.
  • Marquis, Jean-Pierre, 1995, “Category Theory and the Foundations of Mathematics: Philosophical Excavations”, Synthese, 103(3): 421–447. doi:10.1007/BF01089735
  • –––, 1997 [2019], “Category Theory”, The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.), URL: <https://plato.stanford.edu/archives/fall2019/entries/category-theory/>
  • –––, 2009, From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory, (Logic, Epistemology, and the Unity of Science 14), Dordrecht: Springer Netherlands. doi:10.1007/978-1-4020-9384-5
  • –––, 2013, “Categorical Foundations of Mathematics: Or How to Provide Foundations for Abstract Mathematics”, The Review of Symbolic Logic, 6(1): 51–75. doi:10.1017/S1755020312000147
  • McLarty, Colin, 1993, “Numbers Can Be Just What They Have To”, Noûs, 27(4): 487–498. doi:10.2307/2215789
  • –––, 2004, “Exploring Categorical Structuralism”, Philosophia Mathematica, 12(1): 37–53. doi:10.1093/philmat/12.1.37
  • –––, 2008, “What Structuralism Achieves”, in The Philosophy of Mathematical Practice, Paolo Mancosu (ed.), Oxford: Oxford University Press, 354–369. doi:10.1093/acprof:oso/9780199296453.003.0014
  • –––, 2011, “Recent Debate over Categorical Foundations”, in Foundational Theories of Classical and Constructive Mathematics, Giovanni Sommaruga (ed.), (Western Ontario Series in Philosophy of Science 76), Dordrecht: Springer Netherlands, 145–154. doi:10.1007/978-94-007-0431-2_7
  • –––, 2012, “Categorical Foundations and Mathematical Practice”, Philosophia Mathematica, 20(1): 111–113. doi:10.1093/philmat/nkr041
  • –––, 2020, “Sauders Mac Lane: From Principia Mathematica through Göttingen to the Working Theory of Structures”, in Reck and Schiemer 2020a: 215–237.
  • –––, 2024, “Structuralism in Differential Equations”, Synthese 203(83), available online February 28, 2024.
  • Menzel, Christopher, 2018, “Haecceities and Mathematical Structuralism”, Philosophia Mathematica, 26(1): 84–111. doi:10.1093/philmat/nkw030
  • Nodelman, Uri and Edward N. Zalta, 2014, “Foundations for Mathematical Structuralism”, Mind, 123(489): 39–78. doi:10.1093/mind/fzu003
  • Parsons, Charles, 1990, “The Structuralist View of Mathematical Objects”, Synthese, 84(3): 303–346. doi:10.1007/BF00485186
  • –––, 2004, “Structuralism and Metaphysics”, The Philosophical Quarterly, 54(214): 56–77. doi:10.1111/j.0031-8094.2004.00342.x
  • –––, 2008, Mathematical Thought and Its Objects, Cambridge: Cambridge University Press. doi:10.1017/CBO9780511498534
  • –––, 2018, “Concepts versus Objects”, in Reck 2018b: 91–112.
  • Pettigrew, Richard, 2008, “Platonism and Aristotelianism in Mathematics”, Philosophia Mathematica, 16(3): 310–332. doi:10.1093/philmat/nkm035
  • Piaget, Jean, 1968 [1970], Le structuralisme, Paris: Presses Universitaires de France. Translated as Structuralism, Chaninah Maschler (trans.), New York: Basic Books, 1970.
  • Putnam, Hilary, 1967, “Mathematics without Foundations”, The Journal of Philosophy, 64(1): 5–22. Reprinted in Benacerraf and Putnam 1983: 295–312. doi:10.2307/2024603 doi:10.1017/CBO9781139171519.016
  • Reck, Erich H., 2003, “Dedekind’s Structuralism: An Interpretation and Partial Defense”, Synthese, 137(3): 369–419. doi:10.1023/B:SYNT.0000004903.11236.91
  • –––, 2009, “Dedekind, Structural Reasoning, and Mathematical Understanding”, in New Perspectives on Mathematical Practices, Bart van Kerkhove, ed., Singapore: WSPC Press: 150–173.
  • –––, 2018a, “On Reconstructing Dedekind Abstraction Logically”, in Reck 2018b: 113–138.
  • ––– (ed.), 2018b, Logic, Philosophy of Mathematics, and their History: Essays in Honor of W.W. Tait, London: College Publications.
  • –––, 2020, “Cassirer’s Reception of Dedekind and the Structuralist Transformation of Mathematics”, in Reck and Schiemer 2020a: 329–351.
  • Reck, Erich H. and Michael P. Price, 2000, “Structures And Structuralism In Contemporary Philosophy Of Mathematics”, Synthese, 125(3): 341–383. doi:10.1023/A:1005203923553
  • Reck, Erich H. and Georg Schiemer (eds.), 2020a, Prehistory of Mathematical Structuralism, Oxford: Oxford University Press.
  • –––, 2020b, “The Prehistory of Mathematical Structuralism: Introduction and Overview”, in Reck and Schiemer 2020a: 1–18.
  • Resnik, Michael D., 1981, “Mathematics as a Science of Patterns: Ontology and Reference”, Noûs, 15(4): 529–550. doi:10.2307/2214851
  • –––, 1982, “Mathematics as a Science of Patterns: Epistemology”, Noûs, 16(1): 95–105. doi:10.2307/2215419
  • –––, 1988, “Mathematics from the Structural Point of View”, Revue Internationale de Philosophie, 42(167): 400–424.
  • –––, 1996, “Structural Relativity”, Philosophia Mathematica, 4(2): 83–99. doi:10.1093/philmat/4.2.83
  • –––, 1997, Mathematics as a Science of Patterns, Oxford: Oxford University Press. doi:10.1093/0198250142.001.0001
  • Russell, Bertrand, 1903, The Principles of Mathematics, Cambridge: Cambridge University Press. Republished by Norton & Company, New York, 1996. [Russell 1903 available online].
  • Ryan, Patrick, 2023, “Szemerédi’s Theorem: An Exploration of Impurity, Explanation, and Content”, The Review of Symbolic Logic 16(3), 700–739.
  • Richart, Charles E., 1995, Structuralism and Structures: A Mathematical Perspective, London: World Scientific.
  • Shapiro, Stewart, 1983, “Mathematics and Reality”, Philosophy of Science, 50(4): 523–548. doi:10.1086/289138
  • –––, 1989, “Structure and Ontology”, Philosophical Topics, 17(2): 145–171.
  • –––, 1996, “Space, Number and Structure: A Tale of Two Debates”, Philosophia Mathematica, 4(2): 148–173. doi:10.1093/philmat/4.2.148
  • –––, 1997, Philosophy of Mathematics: Structure and Ontology, Oxford: Oxford University Press. doi:10.1093/0195139305.001.0001
  • ––– (ed.), 2005, The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford: Oxford University Press. doi:10.1093/oxfordhb/9780195325928.001.0001
  • –––, 2008, “Identity, Indiscernibility, and ante rem Structuralism: The Tale of i and −i”, Philosophia Mathematica, 16(3): 285–309. doi:10.1093/philmat/nkm042
  • Schiemer, Georg, 2020: “Carnap’s Structuralist Thesis”, in Reck and Schiemer 2020a: 383–420.
  • Schiemer, Georg and Wigglesworth, John, 2019: “The Structuralist Thesis Reconsidered”, British Journal for the Philosophy of Science 70(4): 1201–1226, doi:10.1093/bjps/axy004
  • –––, 2023, “Structuralism and Informal Provability”, Synthese, 202:55, https://doi.org/10.1007/s11229-023-0427-4
  • Stein, Howard, 1988: “Logos, Logic, Logistiké: Some Philosophical Remarks on the 19th-Century Transformation of Mathematics”, in History and Philosophy of Modern Mathematics, W. Aspray and P. Kitcher, eds., Minneapolis: University of Minnesota Press: 238–259.
  • Tao, Terence, 2006, “The Ergodic and Combinatorial Approaches to Szemerédi’s Theorem”, in Procceedings of the International Congress of Mathematicians, Madrid, Spain: 581–608.
  • Tsementzis, Dimitris, 2017, “Univalent Foundations as Structuralist Foundations”, Synthese, 194(9): 3583–3617. doi:10.1007/s11229-016-1109-x
  • Univalent Foundations Program, 2013, Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program. [Homotopy Type Theory available online]
  • Van Fraassen, Bas C., 2008, Scientific Representation: Paradoxes of Perspective, Oxford: Oxford University Press. doi:10.1093/acprof:oso/9780199278220.001.0001
  • Wigglesworth, John, 2018, “Grounding in Mathematical Structuralism”, in Reality and Its Structure: Essays in Fundamentality, Ricki Bliss and Graham Priest (eds.), Oxford: Oxford University Press, 217–236. doi:10.1093/oso/9780198755630.003.0012
  • Worrall, John, 1989, “Structural Realism: The Best of Both Worlds?”, Dialectica, 43(1–2): 99–124. doi:10.1111/j.1746-8361.1989.tb00933.
  • Zalta, Edward, 2024, “Mathematical Pluralism”, Noûs, 58(2): 306–332. doi:10.1111/nous.12451

Other Internet Resources

Acknowledgments

The authors wish to thank Steve Awodey, Francesca Biagioli, Norbert Gratzl, Henning Heller, Johannes Korbmacher, Hans-Christoph Kotzsch, Hannes Leitgeb, Øystein Linnebo, Jean-Pierre Marquis, Michael Price, Patrick Ryan, Andrea Sereni, Matthew Warren, John Wigglesworth, and Edward Zalta for helpful comments and discussions. In addition, we would like to thank two anonymous reviewer for constructive suggestions and Dilek Kadıoğlu for noting some infelicitous typographical errors. Finally, we are grateful to the editors for The Stanford Encyclopedia of Philosophy both for their patience and their general support.

Research by Georg Schiemer on this project received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 715222).

Copyright © 2025 by
Erich Reck <erich.reck@ucr.edu>
Georg Schiemer <georg.schiemer@univie.ac.at>

Open access to the SEP is made possible by a world-wide funding initiative.
The Encyclopedia Now Needs Your Support
Please Read How You Can Help Keep the Encyclopedia Free