The core idea of structuralism concerning mathematics is that modernmathematical theories, always or in most cases, characterizeabstract structures, instead of referring to more traditionalkinds of objects, including sets, or of providing mere calculatingtechniques for applications. Thus, arithmetic characterizes thenatural number structure, analysis the real number structure, andtraditional geometry the structure of Euclidean space. As such,structuralism is a position about the subject matter and content ofmathematics. But it also includes, or is closely connected with, viewsabout its methodology, since studying such structures involvesdistinctive tools and strategies. Hence there are two related kinds ofstructuralism:metaphysical andmethodological.

In English-language philosophy of mathematics, the introduction ofstructuralist views is often taken to have happened in the 1960s, inworks by Paul Benacerraf and Hilary Putnam. Our survey will startthere as well. Debates about structuralist views picked up steam inthe 1980s–1990s, when Michael Resnik, Stewart Shapiro, GeoffreyHellman, Charles Chihara, and Charles Parsons entered the fray. Andthey have been reshaped again during the last 25 years, throughphilosophical challenges to and novel variants of structuralism. Mostof those challenges and variants involvemetaphysical issues,especially questions concerning how to think about mathematicalobjects and about mathematical truth along structuralist lines; but afew also concernepistemological questions.

However, mathematical structuralism played a role already before the1960s, especially if it is understood in the methodological sense,where questions about how to organize, articulate, and innovatemathematical practice are at stake. Structuralist positions in thatsense go back to the nineteenth and early twentieth centuries, inwritings by Dedekind, Klein, Hilbert, Poincaré, Noether,Bourbaki, etc. Today the two most influential versions ofmethodological structuralism areset-theoretic andcategory-theoretic structuralism, both of which lead back tometaphysical issues, mainly by being related to foundationalones. Relevant debates will be covered in the later parts ofthis survey. It will end with some brief remarks about structuralismbeyond mathematics.

• 1. Eliminative vs. Non-Eliminative Structuralism
  • 1.1 Beginnings of the Structuralism Debate in the 1960s
  • 1.2 Consolidation and Further Discussions in the 1980s
  • 1.3 An Initial Taxonomy of Structuralist Positions
• 2. Later Developments and a Broader Taxonomy
  • 2.1 Metaphysical and Epistemological Challenges
  • 2.2 Several Additional Variants of Structuralism
  • 2.3 Finer Taxonomic Distinctions for Structuralism
• 3. Epistemological and Methodological Aspects
  • 3.1 Patterns, Positings, Constructions, Abstractions
  • 3.2 Levels of Methodological Structuralism
  • 3.3 Structuralist Understanding in Mathematics
• 4. Category-Theoretic Structuralism
  • 4.1 Category Theory as the Study of Mathematical Structures
  • 4.2 Categorical Foundations and Debates About Them
  • 4.3 Philosophical Features of Category-Theoretic Structuralism
• 5. Conclusion
  • 5.1 Varieties of Mathematical Structuralism
  • 5.2 Structuralism Beyond Mathematics
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Eliminative vs. Non-Eliminative Structuralism

1.1 Beginnings of the Structuralism Debate in the 1960s

The discussion of structuralism as a major position inEnglish-speaking philosophy of mathematics is usually taken to havestarted in the 1960s. A central article in this connection was PaulBenacerraf’s “What Numbers Could Not Be” (1965; cf.Benacerraf 1996). The background and foil for this article was theposition, dominant at the time, that axiomatic set theory provides thefoundation for modern mathematics, including identifying allmathematical 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 beidentified with Dedekind cuts (constructed out of the set of rationalnumbers). Arithmetic truths are then truths about these set-theoreticobjects. And this generalizes to other mathematical theories, whoseobjects can be identified with sets as well.

As Benacerraf argued in his article, such a set-theoretic foundationalposition misrepresents the structuralist character of arithmetic inparticular and of mathematics more generally. Thus, instead of workingwith the finite von Neumann ordinals, we can work equally well withthe finite Zermelo ordinals (starting again with \(\emptyset\) for 0but 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 (againconstructed set-theoretically out of the rational numbers), assuggested by Cantor and others. This insight about a multiplicity ofoptions is hard to deny and even set-theorist foundationalists canagree with it. (More on set-theoretic responses to this challengebelow.) But Benacerraf drew further, more controversial conclusionsfrom his basic point.

Benacerraf suggested, in particular, that the natural numbers shouldnot be identified withany set-theoretic objects. In fact,they should not be taken to be objects at all. Instead, numbers shouldbe treated as “positions in structures”. In contrast tothe set-theoretic representatives mentioned above, all that mattersabout such positions are their structural properties, i.e., those“stem[ming] from the relations they bear to one another invirtue of being arranged in a progression” (1965: 70). What wetalk about in modern mathematics, then, are the correspondingabstract structures, e.g., “the natural numberstructure” and “the real number structure”. It is inthis sense that Benacerraf suggests a structuralist position. Thefurther details of that position are left open, however, including howto think about abstract structures, except that they are not to beidentified with set-theoretic relational systems, i.e., models ofstandard model theory.

A second article from the 1960s that was influential in the rise ofstructuralism is Hilary Putnam’s “Mathematics withoutFoundations” (1967). As in Benacerraf’s case, for Putnamthe foil was a set-theoretic foundational position, where allmathematical objects are identified with sets. This position issometimes, although not always, understood in arealist sense(most famously in Gödel 1964), i.e., as the description of anindependent realm of abstract objects, namely the universe of setscharacterized by the Zermelo-Fraenkel axioms. In opposition to suchset-theoretic realism, Putnam suggested a form ofif-then-ism(a suggestion that can be traced back to the discussion ofnon-Euclidean geometry in Russell 1903). This alternative can again beillustrated in terms of arithmetic, now taken to be based on theDedekind-Peano axioms. How should an arithmetic theorem, say“\(2+3=5\)”,be understood now? Itshould be analyzed as having the following form:

For all relational systemsM, ifM is a model of theDedekind-Peano axioms, then \(2_M +3_M =5_M\).

(Here \(2_M\), \(3_M\), and \(5_M\) are those objects that “playthe roles” of 2, 3, and 5 in the modelM.) Similarlyfor the real numbers, based on the axioms for a real closed fieldoriginally formulated by Dedekind and Hilbert (see Reck 2003).

Instead of if-then-ism, one can characterize Putnam’s positionalso as a kind ofuniversalist structuralism, since itinvolves universal quantification over relevant systems and thosesystems are kinds of structures (see Reck & Price 2000 for thisterminology). Often an objection to such a position is thenon-vacuity problem. It is based on the observation thatif-then statements of the given form are vacuously true if there isnothing that satisfies the antecedent, e.g., if there is no model ofthe Dedekind-Peano axioms. (A sentence like “\(2+3=6\)”would then also be true etc.) In response, axiomatic set theory can beinvoked as providing the needed models. But from Putnam’s pointof view this is unsatisfactory, for two reasons: it relies on afoundational view about set theory, thus undermining the anti-realistthrust of if-then-ism; and set theory is then treated differently fromother mathematical theories, on pain of circularity. As a way out,Putnam suggested working with modal logic instead; but the exactdetails were again left open.

1.2 Consolidation and Further Discussions in the 1980s

One way to understand Benacerraf’s 1965 article is that itproposes to treat the natural number structure as a new kind ofabstract 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, thusreifying them in a substantive way (still to be worked out).Benacerraf himself was reluctant to do so, in line with his overallhesitancy to talk about mathematical objects.

A subsequent writer who picked up on Benacerraf’s ideas in theearly 1980s is Michael Resnik (see Resnik 1981, 1982, 1988, and, mostsystematically, 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 fortaking mathematical statementsat face value, in the sense ofseeing ‘0’, ‘1’, ‘2’, etc. assingular terms referring to such positions. At the same time, doing sois still not supposed to require reifying the underlying structures,which for Resnik would mean specifying precise criteria of identityfor them, something he avoids intentionally. (He presents himself as aQuinean on this point, by adopting Quine’s slogan: “Noentity without identity!”) Instead, Resnik’s main focus ison the epistemological side of structuralism (more on whichlater).

Stewart Shapiro is a second philosopher of mathematics who attemptedto build on Benacerraf’s paper (see Shapiro 1983, 1989, and,most systematically, 1997). By focusing on metaphysical questions andby leaving behind Benacerraf’s and Resnik’s hesitationsabout structures as objects, Shapiro’s goal is to defend a morethoroughlyrealist version of mathematical structuralism.Such realism includes the semantic aspect just mentioned (takingmathematical statements at face value), but Shapiro also clarifies thetalk about “positions in structures” further, bydistinguishing two perspectives on them. According to the first, thepositions at issue are treated as “offices”, i.e., asslots that can be filled or occupied by various objects (e.g., theposition “0” in the natural number structure is occupiedby \(\emptyset\) 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 structuresoverall.

For Shapiro, the structures at issue thus have a dual nature: they are“universals”, in the sense that the natural numberstructure, say, can be instantiated by various relational systems (thesystem of finite von Neumann ordinals, the system of Zermelo numbers,etc.); but they are also “particulars”, to be named bysingular terms and treated as objects. To defend the latter further,Shapiro develops a generalstructure theory, i.e., anaxiomatic theory that specifies which structures exist and how toidentify them. While clearly modeled on set theory, this theory isjustified independently (more on how below). As such, it is meant tounderwrite “ante rem structuralism”, aterminology that refers to medieval discussions of universals. Thecrucial point is that the structures specified in the theory are meantto be ontologically independent of, indeed prior to, anyinstantiations of them. In other words, the structures do not justexist in their instantiations, but separate to and before them.

While Resnik’s and Shapiro’s structuralist positions aresometimes identified with each other, this is misleading given thedifferences already mentioned. Nevertheless, there is some overlap.Both recognize mathematical structures as patterns with positions inthem (whether these patterns are treated as full-fledged objects ornot). Also for both, the notion ofisomorphism is crucial (orperhaps some related, more general notion ofequivalence; seeResnik 1997 and Shapiro 1997). That is to say, for Resnik and Shapiroa structure/pattern can be instantiated by any of a relevant class ofisomorphic relational systems. This corresponds to the fact that theaxiomatic systems at issue—those for the natural numbers, thereal numbers, set theory, etc.—arecategorical (orquasi-categorical in the case of set theory). Not every mathematicaltheory has that feature; e.g., the axiom systems for group theory orring theory allow for non-isomorphic models. According to both Resnikand Shapiro, such “algebraic” theories are to be treatedin a different, more derivative way. Their structuralist suggestionsare meant to apply primarily to “non-algebraic” theories,paradigmatically arithmetic and analysis.

There is another structuralist position first introduced in the 1980sthat is quite different and explicitlyanti-realist, namelythat promoted by Geoffrey Hellman (see Hellman 1989, 1996, and laterarticles). While for Resnik and Shapiro the inspiration wasBenacerraf’s 1965 article, the starting point for Hellman isPutnam’s 1967 article. In fact, Hellman’smodalstructuralism is meant to be a systematic development ofPutnam’s if-then-ism. The modal aspect, only hinted at byPutnam, is now spelled out in detail, also including the case of settheory (building on work by Zermelo etc.). For Hellman, an arithmeticsentence such as“\(2+3 = 5\)”is to be analyzed as follows:

Necessarily, for all relational systemsM, ifM is amodel of the Dedekind-Peano axioms, then \(2_M +3_M =5_M\).

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

Possibly, there exists anM such thatM is a modelof the Dedekind-Peano axioms.

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

As Hellman makes clear, his goal is to develop a “structuralismwithout structures” (Hellman 1996). In it, the existence ofabstract structures, postulated by Shapiro, is replaced by the modalaspects of his position, i.e., the assumptions about necessity andpossibility just mentioned. In fact, Hellman’s position is meantto be a form ofnominalism, by eliminating the appeal to anykind of abstract entities (not only abstract structures but also setsetc.). On the other hand, it is not meant to rely on possibilia, i.e.,possible objects existing in some shadowy sense. This leads Hellman totake the modalities at issue to be basic, i.e., the relevantpossibilities and necessities are not reducible to anything further.They are specified directly in terms of laws of modal logic (those ofthe systemS5). According to Hellman, such possibilities andnecessities are an irreducible feature of mathematics, as his approachis meant to make explicit.

1.3 An Initial Taxonomy of Structuralist Positions

From the late 1980s on, Shapiro’s and Hellman’s positionshave often been treated as the two main structuralist options. (Thisis reflected in Hellman & Shapiro 2019, among others.) As they arequite different from each other, this already indicates that it iswrong to see “structuralism in the philosophy ofmathematics” as a unique position or a unified perspective, eventhough Shapiro’s and Hellman’s positions share somestructuralist features. And from the early 1990s on, other variants ofstructuralism have started to play a role as well. As a result,discussions of structuralism in the philosophy of mathematics becamericher and more complex (although it was still mostly restricted tometaphysical issues, as opposed to adding methodological ones).

To clarify the situation, Charles Parsons made several crucialcontributions. In particular, he suggested an initial taxonomy, basedon the distinction between two main kinds of structuralist positions(Parsons 1990). Namely, there areeliminative forms ofstructuralism, like Hellman’s; and there arenon-eliminative forms, like Shapiro’s. (The eliminationat issue concerns the postulation, or avoidance thereof, of structuresas separate abstract objects.) Or put in Hellman’s slightlylater terminology, there isstructuralism without structures,on the one hand, andstructuralism with structures, on theother hand. Besides Shapiro (and Resnik, with the qualificationsabove), Parsons himself became another proponent of thenon-eliminative form (see Parsons 1990, 2004, and, mostsystematically, 2008). And on the other side, Charles Chihara becameanother proponent of the eliminative form of structuralism (Chihara2004).

Nevertheless it remains widespread in the literature, even today (withexceptions), to identify non-eliminative structuralism withShapiro’s position and eliminative structuralism withHellman’s. Moreover, many discussions of structuralism thenfocus on Shapiro’sante rem side, explicitly classifiedby him as a realist position. Given the latter focus, some critics goon to dismiss “philosophical structuralism” as a misguidedform of metaphysics, one that is seen as irrelevant for mathematicalpractice (see 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-eliminativeforms of structuralism, Shapiro’s position isn’t the onlyoption, as a further consideration of Parsons’s position alreadymakes clear.

Unlike Shapiro, Parsons does not offer a novel, metaphysicallymotivated structure theory. According to him, we can and shouldinstead remain closer to mathematical practice, as it developed fromthe late nineteenth century on. In fact, for him structuralist viewsshould be seen as growing out of that practice, rather than beingimposed on it from outside by philosophers. This means, among otherthings, seeing abstract structures as introduced directly bycategorical axiom systems, a practice Parsons spells out andjustifies further in a “meta-linguistic” way (againinspired by Quine, see Parsons 2008). It also means for him that weshould refrain fromcross-structural identifications, such asidentifying the natural number 1 and the real number 1 (anidentification that can be found in Shapiro’s early works,although he later disavowed it). Such putative identities are to beleftindeterminate, as Parsons assumes is done inmathematical practice.

As this makes evident, Parsons’s structuralist position (likeResnik’s), is less realist than Shapiro’s. In fact,Parsons is explicit that adopting a “structuralist view ofmathematical objects” should be seen as separable from, andorthogonal to, the realism/nominalism dichotomy. Consequently, for himone can be a non-eliminative structuralist without being a realist inany strong sense; his own position is meant to be a case in point.Then again, Parsons’s structuralism still takes mathematicalstatements at face value, so that it remains realist in this minimalsemantic sense. In that respect too, Parsons takes himself to followmathematical practice, e.g., with respect to talking about “thenatural 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 philosophyof mathematics from Benacerraf and Putnam, in the 1960s, to Resnik,Shapiro, Hellman, Chihara, and Parsons, in the 1980s–1990s.Since then, a number of further philosophers have started to addressthis topic. We now turn to the ensuing debates, beginning with somemetaphysical and epistemological challenges to structuralism. Most ofthem concern non-eliminative structuralism, with Shapiro’sposition as the main target. (This reflects again its prominence.) Butsome considerations are broader, including the comparison of basiccommitments underlying eliminative and non-eliminative forms ofstructuralism. In what follows, we will not try to be comprehensiveabout the relevant debates, but just provide six illustrativeexamples. (Later we will also move beyond such metaphysical andepistemological issues; but as these examples illustrate, they havedominated 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 objectsare theirstructural properties (as opposed to furtherintrinsic properties, e.g., set-theoretic ones). In fact, suchproperties are taken to determine the objects’ identities. Butthen, objects that are indistinguishable in thisrespect—“structural indiscernibles”—should beidentified, shouldn’t they? As several critics argued around theyear 2000, this leads to theidentity problem forstructuralism (see Keränen 2001, earlier Burgess 1999). It arisesfor relational systems or structures that arenon-rigid,i.e., allow for non-trivial automorphisms. In such cases there aresupposedly distinct objects that are indiscernible in the relevantsense. A widely known example is the system of complex numbers, withthe conjugate numbersi and −i; but geometryand graph theory provide further illustrations. The simplest exampleis probably an unlabeled 2-element graph with no edges, whose twovertices are structurally indiscernible.

How can and should such cases be handled, if they can be handled atall? Is structuralism (of the non-eliminative kind) simply incoherent,as some critics charge? Or is it at least not applicable to non-rigidcases, which would limit its reach significantly? Severalstructuralist responses to the identity problem have been offered inthe literature. One is torigidify such structures, byenlarging the vocabulary that is used, e.g., by adding the constantsymbol ‘i’ for the complex numbers (either intothe original language or into the language for the“setting” used in the background, as suggested in Halimi2019). However, this still seems problematic in certain cases,especially when there are uncountably many indiscernibles. Anothersuggestion is to treat identity asprimitive, as is arguablydone in mathematical practice. Yet along such lines too, a number offollow-up questions arise (see Ladyman 2005, Button 2006, Leitgeb& Ladyman 2008, Shapiro 2008, Menzel 2018, and Leng 2020, amongothers).

A second and more basic challenge for structuralism starts again withthe assumption that all that matters about mathematical objects aretheir structural properties. With respect to non-eliminativestructuralism, this is sometimes understood or formulated in the sensethat the positions in mathematical structures, as well as the abstractstructures themselves,only have structural properties. Butthat again leads to counterexamples quickly. For instance, does thenatural number structure as a whole not have the property that it isthe most widely used example in debates about structuralism?Similarly, isn’t it a property of the number 8, as a position inthat structure, to be the number of planets in the Solar System? Bothseem clearly non-structural properties. Once again, structuralismappears incoherent, or at the very least in need of furtherclarification. (One advantage of eliminative forms of structuralism,like Hellman’s, is that they avoid both problems mentioned sofar; but they lead to others instead.)

A natural response to this challenge is to refine the structuralistthesis just mentioned, e.g., by saying that abstract structures onlyhave structural propertiesessentially, or that only suchproperties areconstitutive for them, while admitting thatthey do have other properties (see Reck 2003, Schiemer &Wigglesworth 2019, also Assadian 2022). This leads to the question ofhow exactly to make that distinction. Several structuralist responsesto it have again been proposed. (We will come back to some of them inthe next subsection.) But even if we are assumed to have asatisfactory response to this question, another one arises, thus athird basic problem, namely: how can we distinguishstructural fromnon-structural properties in thefirst place? Several answers to that question have been suggested inturn, e.g., that structural properties are those definable in acertain way, or that they are those preserved under relevantmorphisms. Yet in that respect as well, the debates continue (seeKorbmacher & Schiemer 2018, also for more references).

A fourth challenge directed again especially, or even exclusively, atnon-eliminative structuralism concerns the following: from astructuralist point of view, positions are always “positions ina structure”; i.e., the structure isprimary and thepositions aresecondary. Hence, a particular mathematicalobject, such as the natural number 2, seems “ontologicallydependent” on the background structure, here the structure ofthe natural numbers. (The fact that for a structuralist it ismisguided to consider the number 2 as existing in itself reflects thisaspect. It also illustrates a main difference between structuralismand, say, set-theoretic foundationalism or neo-logicism.) Yet howshould thisontological dependence be understood? Are thereperhaps connections togrounding or related notions incurrent analytic metaphysics? In this context too, a lively debate hasstarted and continues until today (see Linnebo 2008, also, e.g.,MacBride 2005, Wigglesworth 2018).

A fifth challenge for structuralism in its non-eliminative form is howwe can haveaccess to structures taken to be abstract objects(see, e.g., Hale 1996). To some degree, this revives an older, moregeneral debate about the possibility of knowledge about such objects(see Benacerraf 1973). An initial response, shared by Resnik, Shapiro,and Parsons (all following Quine in this respect), is to talk aboutthepositing of structures, which is meant to undercut theaccess problem from the start (see, e.g., Resnik 1997). Then again,under what conditions is such positinglegitimate (given thethreat of paradox familiar from naïve set theory)? A plausibleanswer to that question points to thecoherence of therelevant axiomatic theories, which, after Gödel’sincompleteness results, is seen as taking the place of provableconsistency. (Compare Hilbert’s earlier suggestion thatmathematical existence simply amounts to the consistency of thetheories at issue.) But this leads to our sixth basic question: whatexactly does such coherence amount to and how can we establish it?

With this sixth challenge, we have moved from metaphysics over toepistemology. An interesting outcome of the resulting debates is thatit also brings us back to eliminative structuralism. Thus, Shapiro andHellman, coming from very different directions, have actually arrivedat suggestions that are very close to each other in this connection(see Hellman 2005, Hellman & Shapiro 2019). Namely, with respecttobasic commitments—ultimate conditions forexistence in Shapiro’s case and forpossibility in Hellman’s case—their approachesconverge, since they both appeal to coherence understood in a similar,non-reducible sense. That convergence may be seen as providing supportfor both sides, since they have arguably identified the same corefeature of mathematics. But it might also be taken to undermine therealism/nominalism dichotomy, since it seems to imply that theexistence of abstract structures, properly understood, amounts to thesame thing as the possibility that underwrites modalstructuralism.

There are more challenges to structuralism in the philosophy ofmathematics that one can find in the literature of the 21st century.While usually connected to the six problems just mentioned, sometimesthey go further. For example, additional questions about the semanticsinvolved in structuralism have been raised, typically again for thenon-eliminative variant but now including questions about“relational positions” in structures (see Button &Walsh 2016, Assadian 2018, Button & Walsh 2018, and Zalta 2024).As a different kind of example, there is the question of how togeneralize eliminative structuralism, along Hellman’s modallines and beyond, so as to cover all of of mathematics systematically(see Zalta 2024). But instead of detailing such questions further, wehope that our six examples above provide enough illustration of thekinds of debates prevalent in the recent literature. As should havebecome evident, many of those debates concern metaphysical issues, toa smaller degree also epistemological ones. (Later we will come backto parallel debates about categorical structuralism.) Moreover, theyhave led to additional variants of structuralism, as we will seenext.

2.2 Several Additional Variants of Structuralism

As already noted, in many discussions of structuralism, from the 1980sto the early 2000s and beyond, a few positions have taken centerstage: often Shapiro’s and Hellman’s, sometimesParsons’s, occasionally also Resnik’s. But several othervariants of structuralism exist, as we want to make clear now. Thosepositions deserve, and have started to get, more attention inthemselves. Moreover, some of them go back to before the 1960s,sometimes all the way to the mid- or late nineteenth century. As thisshows, the still widespread assumption that debates about mathematicalstructuralism started during the 1960s needs to be revised (even moreso if we bring in methodological structuralism, as will become evidentlater). In this subsection, we will go over several especiallynoteworthy examples. A major one, in several respects, isset-theoretic structuralism (see Reck & Price 2000,Burgess 2015, and for its history, Reck & Schiemer 2020b). Tointroduce it, let us return to the illustration central toBenacerraf’s 1965 paper: the natural numbers.

As Benacerraf argued, there is something wrong with identifying“the natural numbers” with a particular set-theoreticsystem; or more precisely, it is wrong to do so in an absolute sense.He concluded that numbers are not sets, or in fact, not objects of anykind. Now, one can agree with almost everything Benacerraf writes andstill want to identify the natural numbers with a set-theoretic systemin a less absolute sense. Doing so involves admitting that any othermodel of the Dedekind-Peano axioms “would do as well”,i.e., could have been chosen instead, except perhaps for pragmaticreasons (suggested by the context). Yet we can still pick one and letitplay the role of the natural numbers. For mostmathematical purposes such a provisional and pragmatic identificationis sufficient. In fact, this is exactly what is done in standard ZFCset theory (where the system of finite von Neumann ordinals is usuallypicked, since it can be generalized systematically to the infinite).Moreover, the resulting position, which can be traced back to Zermeloin the early twentieth century (see Ebbinhaus 2015), deserves to becounted as a form of structuralism. What makes it structuralist is the“indifference to identify” the natural numbers in any moreabsolute sense (see Burgess 2015).

The core of set-theoretic structuralism is to choose one of severalisomorphic systems as the provisional, pragmatic referent for“the natural numbers” (similarly for “the realnumbers” etc.). As one may put it, our talk of “thenatural numbers”, then also of “the number 0”,“the number 1”, etc., isrelative to thisinitial, somewhat arbitrary choice. This is taken to be unproblematicsince, no matter how we choose, we will get the same arithmetictheorems (because of the categoricity of the axiom system, whichimplies its semantic completeness). And as background for thisapproach, it is usual to employ ZFC. But we generalize it by alsoallowing for “atoms” or “urelements” (objectsthat are not sets). We can even include Julius Caesar or some beer mugin our domain, with the consequence that (pace Frege) eitherof them can “be” the number 2, say, by taking up the“2-position” in the model for arithmetic we choose to workwith. This generalized approach deserves the namerelativiststructuralism (Reck & Price 2000). Moreover, such anapproach, particularly in its set-theoretic version, is arguablyaccepted by a large number of mathematicians today, explicitly orimplicitly. (Committed category theorists will be exceptions, as wewill see later.)

In the case of relativist structuralism, the only mathematical objectsat 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 positionis another variant of eliminative structuralism, although it does noteliminate all abstract objects (since it countenances sets). In fact,set-theoreticrelational systems, in the sense of thestandard models of mathematical theories, are often taken to be therelevant structures. (It is exactly such systems that are called“structures” in many mathematics textbooks.) Yet there arefurther options at this point, which lead to additional variants ofstructuralism. One of them is to identify the structure of the naturalnumbers with the (higher-order)concept defined by theDedekind-Peano axioms; similarly again for other axiom systems. Thisgives us another form of eliminative structuralism:conceptstructuralism (see Isaacson 2010, Feferman 2014, alsoFerreirós 2023, where this is called “conceptualstructuralism”). Furthermore, that position can be traced backeven to the mid-nineteenth century, e.g., to the works of HermannGrassman (see Cantu 2020).

According to concept structuralism, what matters in mathematicsisn’t really objects. Rather, it is mathematicalconcepts that are crucial, e.g., the concept “naturalnumber system” (“model of the Dedekind-Peanoaxioms”, or in Russell terminology, “progression”);similarly for the concept “complete ordered field”, etc.More precisely, what matters is what follows from those concepts, inthe sense of what can be derived from the relevant axioms. It is true,as concept structuralists can admit, that in mathematical practice weoften reason by talking about objects falling under those concepts.(If set theory is the framework used, then we use sets for thispurpose.) But as they would want to add, such talk can be explainedaway in the end (perhaps by taking up a formalist position). In thisor similar forms, concept structuralism is again a fairly widespreadposition among mathematicians and logicians, explicitly or implicitly,even though it has not been very prominent in the philosophicaldebates about structuralism so far. An immediate challenge for theview is, however, whether talking about concepts is less problematicthan talking about mathematical objects in the end (see Parsons2018).

We just introduced set-theoretic structuralism, relativiststructuralism as a generalization, and concept structuralism. For thesake of a comprehensive survey and to show that there really is alarge variety of structuralist positions, we need to go further. Asthe next step, there are two forms of structuralism that are closelyrelated to relativist and concept structuralism but not identical witheither. Both are forms of another main kind of structuralism:abstractionist structuralism. In both cases it is convenientto work again within a set-theoretic framework (although this too canbe varied, e.g., by using second-order logic instead). We can startonce 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 thatconcept or with a pragmatically chosen system falling under it, let usnow consider theequivalence class of relational systemsdetermined by the concept. Finally, given that equivalence class wecan go down one of two “abstractionist” paths.

On the first path, we nowidentify the structure at issuewith that equivalence class, i.e., we identify it with the“concept in extension” as it is sometimes put (therebyhighlighting the proximity to concept structuralism). In our basicexample, connected with the concept “natural numbersystem” (defined by the Dedekind-Peano axioms), as well as witheach set-theoretic models corresponding to it (e.g., the system offinite von Neumann ordinals, also the system of Zermelo naturalnumbers, etc.), there is the whole equivalence class of all suchmodels as a third entity; and it is this class that we now call“the natural number structure”. (Here the primary focus isagain on categorical axiom systems; but the approach can begeneralized, 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 ofVon Neumann-Bernays-Gödel set theory, NBG. (By using what isknown as “Scott’s trick”, as is often done in suchcontexts, we can also avoid working with the whole class and restrictourselves to a large subset of it.)

Compared to relativist structuralism, this new approach can also becharacterized as follows: its core is to go from a particular,arbitrarily chosen system falling under a higher-order concept, suchas the concept “natural number system”, to the wholeequivalence class of such systems, i.e., the class of all modelsisomorphic to it. Moreover, we can think of this move as a kind ofabstraction, specifically in the sense of the“principle of abstraction” introduced and made prominentby Bertrand Russell (Russell 1903). This is the sense in which theresulting position is a form ofabstractionist 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 inthe 1930s-1940s, who saw it as a development of Russellian ideas fromthe early twentieth century (see Schiemer 2020, and the discussion ofCarnap’s structuralism in the entry onCarnap). But there is another abstractionist path we can take, involving adifferent form of “abstraction”; and it too can be tracedback far in time, in this case to Dedekind’s works from the1870s-1880s.

For that second abstractionist option, let us start again from arelevant higher-order concept, or from the axiom system defining it,together with an arbitrarily chosen system falling under it (aset-theoretic relational system, possibly with urelements). Thealternative suggestion is then the following: given the arbitrarilychosen model, we “abstract away from the particular nature ofits elements” so as to arrive at a “pure” system, inthe sense of a separate, distinguishedabstract structure;and we call that structure “the natural numbers” (Dedekind1888). The intention along such lines is that the objects introducedby such abstraction only have structural properties, or better, theyonly have them essentially, since we have abstracted away from all“foreign properties” (non-arithmetical ones in the case ofthe natural numbers). And in contrast to the previous approach, wherethe outcome of the abstraction is a whole equivalence class, theresulting system is meant to beisomorphic to the one fromwhich we started (see Reck 2003).

Using this second form of abstraction—“Dedekindabstraction”, as it is sometimes called—leads to aposition that, at least in terms of its outcome, is close to theante rem structuralism of Shapiro (who occasionally appealsto “abstraction” himself, e.g., in Shapiro 1997).Similarly it is close to Parsons’s non-eliminative position interms of its outcome. Yet the current approach involves neither anappeal to Shapiro’s structure theory nor an appeal toParsons’s meta-linguistic procedure. Instead, the relevantstructures are introduced “by abstraction” fromset-theoretic relational systems (possibly with urelements). What thatamounts to can be explicated further, also more formally, in terms ofanabstraction operator and correspondingabstractionprinciples. Another comparison suggests itself, then, namely withthe use of abstraction principles in neo-logicism (see Linnebo andPettigrew 2014, Reck 2018a). Having said that, informal versions ofsuch abstraction can already be found in mathematical practice (oftenwithin a set-theoretic framework), namely when mathematicians think of“the natural numbers”, “the realnumbers”“the cyclic group with five elements”, etc.as separate from the usual set-theoretic models.

The five additional forms of structuralism justconsidered—set-theoretic, relativist, conceptual, andabstractionist in two variants—all have historical roots inworks from before the 1960s. (They grew out of forms of methodologicalstructuralism, as we will see more below.) Each of them also hasproponents and implicit adherents later, including in the twentiethcentury among mathematicians. And most of them rely on ZFC set theoryin the background (which thus needs to be conceptualized differentlyin itself). Yet this is still not all in terms of availablestructuralist options. If we look to the twenty-first century now,there are several further variants of structuralism introduced byphilosophers that should be covered in our survey as well. We willmention three (the most interesting ones, in our view), without goinginto much details in each case. One aspect these three final variantsshare is the appeal to novelbackground theories, differentfrom set theory. Moreover, each arose in defense ofnon-eliminative structuralism against the metaphysical andepistemological challenges mentioned in the previous subsection.

As a first illustration of such new approaches, let us take EdwardZalta and Uri Nodelman’s introduction of a form of structuralismthat relies on an axiomaticobject theory, inspired byAlexius Meinong, to account for both the existence and the nature ofabstract structures (Nodelman & Zalta 2014, Zalta 2024). Aparticular strength of this approach is its ability to distinguishsystematically between “accidental” and“essential” properties of objects in such structures, interms of the difference between “encoding” and“exemplifying” properties provided in their object theory.As such, it provides novel answers to several of the philosophicalchallenges above, especially the one in which that distinction ischallenged. More generally, the approach offers an explicit anddetailed metaphysical background theory for non-eliminativestructuralism. A remaining question is then whether or to what degreethat metaphysics is compatible with those implicit in othernon-eliminative positions.

Second, by building on Kit Fine’s theory ofarbitraryobjects, as another new background theory, Leon Horsten hasintroducedgeneric structuralism (Horsten 2018, 2019). Thisapproach too allows for responses to some of the philosophicalchallenges mentioned earlier. However, the unusual nature of theobjects appealed to by Horsten (which share some properties withvariables, given how they are introduced in Fine’s theory ofarbitrary objects) raises new questions in turn.

As a third case, Hannes Leitgeb has recently proposed to start fromgraph theory for similar purposes, i.e., to develop analternative framework for non-eliminative structuralism (Leitgeb2020a, 2020b). Leitgeb’s approach is well suited for addressingthe identity problem for structuralism from a mathematical point ofview. And in other ways as well, it stays closer to mathematicalpractice than either Zalta and Nodelman’s or Horsten’sapproach. Nevertheless, Leitgeb’s main goal, like theirs, is toclarify metaphysical aspects of non-eliminative structuralism.

As a final option, there is a whole family ofcategory-theoreticforms of structuralism. Yet we will postpone their discussionuntil later (Section 4), for two reasons. First, this approach isespecially important mathematically, thus deserving treatment in aseparate section. Second, it is harder to compare to the forms ofstructuralism mentioned so far, since it is motivated more bymethodological considerations. Before turning to it, we will do twothings. We round off Section 2 by offering further taxonomicdistinctions for structuralist positions, thus refining the taxonomicframework for our survey. After that (in Section 3), we discussseveral forms ofmethodological structuralism, both to enrichour survey further and to provide a better background for thesubsequent treatment of categorical structuralism.

2.3 Finer Taxonomic Distinctions for Structuralism

All the variants of structuralism discussed up to this point are formsofmetaphysical structuralism, or as it is sometimes alsoput, ofphilosophical structuralism. What that means is thatthey are primarily intended to provide answers to philosophicalquestions about what mathematical structures are (even when thisinvolves an eliminative stance), in the sense of theses about theirexistence, abstractness, identity, dependence, etc. Now, one candistinguish this whole variety of positions frommethodologicalstructuralism, or in alternative terminology (less helpful, wethink), frommathematical structuralism (see Reck & Price2000, earlier Awodey 1996). In fact, we suggest that recognizing thisbasic dichotomy—metaphysical versus methodological—iscrucial for further advancing both the systematic and the historicaldiscussions of structuralism in the philosophy of mathematics. (Forinstance, it will allow to include and compare categorical forms ofstructuralism more naturally; see Corry 2004 and Marquis 2009.)

As its name suggests, and as already mentioned briefly,methodological structuralism concerns the methodology ofmathematics, thus mathematical practice. Or as one might also say, itconcerns a certain “style” of doing mathematics (see Reck2009, Ferreirós and Reck 2020). Basically, that style consistsof studying whole systems of objects in terms of their global,relational, or structural properties, while neglecting the intrinsicnature of the objects involved. This can be done in two ways that areoften intertwined in practice: by proceedingaxiomatically,i.e., by deriving theorems from basic axioms for the systems at issue;and by consideringmorphisms between those systems(homomorphisms, isomorphisms, etc.), together with invariants underthose morphisms (more on both side later). As this kind of approachtypically involves infinite sets, non-decidable properties, andclassical logic, it tends to be opposed to more computational orconstructivist ways of doing mathematics (see Stein 1988, Reck andSchiemer 2020a).

A structuralist methodology in this sense can be elaborated further inseveral ways, beyond the general characterization just given, withvarious forms of methodological structuralism as the result (seeSection 3). But at bottom, the methodology is tied to a generalassumption about thesubject matter of mathematics, namely:mathematics is the study of “structures” (the naturalnumbers, the real numbers, also various groups, rings, geometricspaces, topological spaces, function spaces, etc.). At the same time,accepting that assumption does not, by itself, involve views about thenature of those structures, at least not in any detailedmetaphysical or otherwise philosophically loaded way. (Practicingmathematicians who are guided by that assumption, often implicitly,typically keep their distance from such metaphysical issues.) Incontrast, all the forms of metaphysical structuralism consideredearlier are meant to provide the latter. This is exactly how they gobeyond methodological structuralism, while presumably building onit.

So far we have clarified what we mean by “methodologicalstructuralism” (or again, what is sometimes meant by“mathematical structuralism” in the literature, althoughthat label has also been used more broadly). With respect tometaphysical positions, Parsons’seliminative vs.non-eliminative distinction remains helpful (while a morepositive label for the latter kind of positions, along the lines of“structuralism with structures”, seems more adequate forsome purposes). Yet the significant differences between metaphysicalforms of structuralisms are not exhausted by it. In fact, working onlywith Parsons’s distinction can obscure philosophically importantpoints. Moreover, Shapiro’sante rem structuralism isfar from the only version of non-eliminative structuralism; nor isHellman’s modal structuralism the only form of eliminativestructuralism (see section 2.2). As tools for clarifying thecorresponding debates further, we will now suggest severalmorefine-grained distinctions (many already implicit in what we wroteabove).

Let us first look again atnon-eliminative structuralism.Several positions that fall under that label introduce abstractstructures by means of correspondingbasic theories. Thisincludes Shapiro’s structure theory, but also, e.g., Nodelmanand Zalta’s object theory, Leitgeb’s adaptation of graphtheory, and Horsten’s theory of arbitrary objects (see section2.2). All of them lead to forms ofante rem structuralism,but significantly different ones. In addition, there are forms ofstructuralism, including one variant of non-eliminative structuralism,that are based onabstraction principles; we therefore calledthem “abstractionist forms of structuralism”. We discussedtwo versions of them, which involved different kinds of structures asthe outcome of the abstraction process. (If we explicated therespective abstractions as mathematical operations or functions, theirarguments would be the same but their values different.) One thingthis makes evident is that there areabstractionist andnon-abstractionist forms of non-eliminativestructuralism.

If we reflect further on such issues, the roles of anotherdistinctions and subdivisions become apparent, namely: betweenante rem positions and eitherin re orpostrem positions. Shapiro’s is explicitly an example ofante rem structuralism. In contrast, our first version ofabstractionist structuralism can be seen as a form ofpostrem structuralism, because the equivalence classes used as thestructures in it are “built out of” their elements towhich they are thusposterior. In our second version ofabstractionist structuralism, there is a form of posteriority involvedas well. Here too, we start with a more concrete relational system,usually a system of sets perhaps containing urelements, and weintroduce an abstract structure on that basis. Thepriorversusposterior relation in that case is not based on theelement-class relation, but on a more fundamentalargument-function-value relation. And we end up with abstractstructures, as opposed to classes, as the result of the abstractionprocess.

On the side of eliminative structuralism, further distinctions can beintroduced as well. In particular, there arefullyeliminative positions, which avoid commitment to any kind ofabstract objects. Hellman’s modal structuralism is of this sort.But there are alsosemi-eliminative positions, which avoidcommitment tosui generis abstract structures, but acceptother, more concrete mathematical objects. Set-theoretic structuralismis a good illustration; universalist structuralism is another, atleast when backed up by set theory. Beyond that, should relativiststructuralism, and set-theoretic structuralism more particularly, beseen as a case ofin re structuralism, in the sense that hereabstract structures exist “in” their more concreteinstantiations? Perhaps; but this thesis does not seem forced on us(see Leitgeb 2020a, 2020b for more). The same question arises forin re forms of structuralism more generally; the details willmatter. (For more on the latter positions, sometimes labeled“Aristotelian” as opposed to “Platonist”, seePettigrew 2008 and Franklin 2014.)

Yet another version of eliminative structuralism is conceptstructuralism. If it is meant to work without any appeal to abstractobjects (e.g., by basing itself on formalism), it amounts to afully eliminative position. Questions remain, however, aboutthe appeal to concepts, including their existence, nature, andidentity (see Parsons 2018). Depending on the answers, a strictnominalist may find this position unacceptable, namely if the conceptsare seen as another problematic kind of abstract entities. On theother hand, if a concept structuralist allows for sets to play asecondary role, it becomes asemi-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 structuralismmerely as a form of methodological structuralism, in which caseadditional metaphysical questions are brushed aside as irrelevant.Here again, the details matter. But let us now turn to methodologicalstructuralism in itself.

3. Epistemological and Methodological Aspects

3.1 Patterns, Positings, Constructions, Abstractions

While discussions of structuralism in the philosophy of mathematicsoften focus on metaphysical questions, occasionally these are tied toepistemological questions. In the English-speakingliterature, an early illustration is provided by MichaelResnik’s works. As we already noted, his approach is distinctivein two ways: by emphasizing the role of structures, but not fullyobjectifying them; and by addressing, more than most, questions abouthow we canknow anything about such structures (Resnik 1982,1997). The main phenomenon Resnik points to in connection with thelatter is our ability torecognize patterns. Two simpleexamples would be intuitive patterns in geometry (involving points,lines, etc.) and numerical patterns in arithmetic (concerningnumerical sequences represented by means of numerals etc.). The basicsuggestion for the latter is, then, that knowledge about the naturalnumber structure is grounded in recognizing the beginnings of suchpatterns, usually involving finitely many numbers, and thengeneralizing to the infinite. If one considers education inmathematics, there is clearly something to this suggestion. Yetquestions about the process of generalization remain.

A second suggestion for how to account for our knowledge of structuralfacts in mathematics, including our knowledge of structures asabstract objects, has come up already as well. Namely, Resnik,Shapiro, and Parsons argue that modern mathematics involves thepositing of various structures based on corresponding axiomsystems (see section 2.1). Along such lines no access to some“Platonic heaven“, as rejected by nominalists, is required(and more positively, one can combine such positing with patternrecognition). In this case, what remains are questions about when suchpositing islegitimate. Other variants of metaphysicalstructuralism, including eliminative ones, lead to related issues.Thus in Hellman’s nominalist structuralism any claim aboutknowledge of abstract objects is avoided, while mathematical knowledgeis tied to establishing certain if-then claims by mathematical proof,thereby assuring theirnecessity. Yet an additional kind ofmodal knowledge is required as well, involving mathematicalpossibility. And as noted earlier, Hellman points to thecoherence of basic axiom systems here, parallel toShapiro’s justification for assuming the existence of abstractstructures. Here one can ask: what kind of coherence exactly; and howcan it be established?

A third approach, in our two versions of abstractionist structuralismand in related positions, is to begin with set theory. A basic kind ofmathematical knowledge consists, then, in learning how to performset-theoreticconstructions. Here too, questions remain aboutthe consistency, or perhaps again the coherence, of the set-theoreticframework, in relation to which such constructions are legitimate. Butan additional appeal toabstraction, which can take severaldifferent forms, is important. Some kind ofknowledge byabstraction is thus involved as well, including questions abouthow it is to be justified. This is not unrelated to patternrecognition, postulation, and their legitimacy; but here the detailsneed to be spelled out in terms of abstraction principles, in one wayor another. Dedekind’s version of structuralism, in particular,stays close to the methodology of modern mathematics in thisconnection, which provides the basis for more advanced kinds ofknowledge and understanding as well; and this is the topic to which weturn next.

3.2 Levels of Methodological Structuralism

With the appeal to set-theoretic constructions, related forms ofabstraction, etc., we have arrived at methodological questions aboutmathematics. Broadly speaking again, astructuralistmethodology consists in studying whole systems of objects interms of their global, relational, or structural properties, insteadof the intrinsic nature of the objects involved. Such a methodologyhas roots in traditional mathematics, which can be traced all the wayback to ancient times (see Reck 2020). But it was further developedand became more prevalent in the second half of the nineteenth and theearly twentieth centuries, especially as far asinfinitestructures are involved. Arguably the first systematic employment anddiscussion of this methodology can be found in the works of Dedekind(see Reck 2003, 2009, Ferreiros and Reck 2020), although numerousother mathematicians contributed to it as well (see Reck and Schiemer2020a).

This description of a structuralist methodology is still very general,of course. In the recent literature one can find the beginnings of amore differentiated treatment of it. In this context, we candistinguish four levels of methodological structuralism. (Probablymore can be added in the end.) The most basic level consists ofstudying an area of mathematics by firstcharacterizing themain structure at issue (a task that relies on pattern recognitionetc.), and based on that,provingtheorems about it.This is typically doneaxiomatically, especially in the senseof 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 theDedekind-Peano axioms and proved theorems about it (see Reck 2003).Structuralism at this first level is the culmination of doingmathematics moreconceptually than before (thus leading toconcept structuralism, among others), as opposed to focusing oncomputational methods (see Stein 1988, Ferreiros and Reck 2020).

A second level of proceeding structurally in mathematics, one thatbuilds on the first, involves considering not just one structure butthe relationships between several of them. This can be done by eitherinvoking abstract structures or working with more concreteinstantiations of such structures, as in standard model theory. Whatbecomes crucial at this level is the consideration of variousstructure-preservingmappings between suchstructures, especially homomorphisms and isomorphisms. Using Dedekindas an early source again, one example is his theorem that all modelsof the Dedekind-Peano axioms (all “simple infinities”) areisomorphic (the basis of his abstractionist structuralism). Perhaps amore striking example is Dedekind’s proof that the naturalnumbers, as characterized by him, can be embedded homomorphically intoany recursively constructed system (thus anticipating category theory,see McLarty 1993). Later this approach was pushed further and mademore abstract in works by Hilbert, Noether, Tarski, etc. (see Corry2004).

We reach a third level of structuralist mathematics when wedon’t just consider one or a few closely related kinds ofstructure, but attempt to characterize all the main kinds ofstructures in mathematics, including their interrelations. Paradigmexamples of such an approach can be found in the works of the Bourbakigroup, including their distinction between three kinds ofmotherstructure: algebraic, topological, and order-theoretic (seeHeinzmann and Petitot 2020). In terms of historical shifts, Bourbakiused set theory as the relevant framework initially (Bourbaki 1939),but it was later replaced by category theory (including the notion offunctor, universal mapping properties, etc.). In fact, category theorycan be seen as a systematic, even more abstract development of exactlythis kind of methodological structuralism. Moreover, it is noexaggeration to say that such structuralism, either in a set-theoreticor a category-theoretic form, has dominated large parts of mathematicssince the mid-twentieth century (see Corry 2004, especially foralgebra).

A fourth level of methodological structuralism, or perhaps anespecially interesting procedure on the second level instead (since itis more concrete again), has not found much attention in thephilosophical literature yet, while some mathematicians have startedto highlight it. Here mathematicians investigate how some mathematicaltheorem, proven in terms of structuralist background assumptions inone part of mathematics, can be transferred to other parts in fruitfulways. The terminology typically used for successful cases is that ofstructure theorems. (A relatively recent illustration isSzemerédi’s Theorem; see Tao 2006.) In this context thestructuralist methodology is not used in a way that is as global andabstract as on level three. Instead, the focus is on particulartheorems, while general assumptions remain in the background. Thisstrategy is again very significant in mathematical practice and, assuch, deserves more philosophical attention (see Ryan 2023, moregenerally also Carter 2024, for some forays into this area).

3.3 Structuralist Understanding in Mathematics

When philosophers such as Resnik, Shapiro, and Parsons appeal topattern recognition and the postulation of structures, their goal isusually to address very basic questions aboutknowledge inmathematics, questions that are highlighted in traditional philosophy.This still applies to the investigation of basic kinds ofpossibility in mathematics, e.g., involving the notion ofcoherence in Hellman and Shapiro. Usually the motivation for suchappeals consists, at least in part, in a desire to respond toskeptical or nominalistic challenges to the very possibility ofmathematical knowledge. Hence we are dealing again with philosophicalstructuralism, although its epistemological rather than itsmetaphysical dimension; and working mathematicians are seldominterested in such work, since it doesn’t affect their practicedirectly.

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

The step-by-step crystallization of such structuralist understanding,via the novel methodologies that made it possible, played a crucialrole in the emergence of modern mathematics in the late nineteenth andearly twentieth centuries. Arguably, it is also what drove the rise ofmetaphysical structuralism, as closely tied to methodologicalstructuralism, as the case of Dedekind illustrates (see again Reck andSchiemer 2020a). But even for current philosophers of mathematics whoare skeptical about “structuralism“, perhaps because oftheir (mistaken) identification of it with a purely metaphysicalposition, this kind of understanding is an aspect they should notignore. Moreover, a charitable way to interpret recent philosophicaldiscussions of structuralism, at least in some of its variants(perhaps most clearly Parsons’s, see section 1.3), is that itwas motivated by the realization that methodological structuralism hadbecome quite prevalent in mathematics by the middle of the 20thcentury, thereby raising new philosophical questions. In any case, ourdiscussion of structuralist understanding has led to epistemologicalissues that are relevant for both philosophers and mathematicians.

Another advantage of including an explicit treatment of structuralistunderstanding in this survey (more than in its previous edition) isthat this allows us to incorporatecategory-theoreticstructuralism better. As already noted, category theory grew outof reflections on the methodology introduced by Dedekind, Klein,Hilbert, Poincaré, Noether, Bourbaki, etc. As such, it isprimarily a form of methodological structuralism, one meant to enhancestructuralist understanding further. But it has also been proposed asa newfoundation for mathematics, thus leading back tometaphysical 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 toformulate a theory of mathematical structuralism based on categorytheory, thus a theory, or theories, ofcategory-theoreticstructuralism, or in slightly different terminology, ofcategorical structuralism. We are now in a better position toconsider these proposals, although we will still proceed indirectly,starting with more background. Category theory was first introduced asa branch of abstract algebra, in Samuel Eilenberg and Saunders MacLane’s famous article, “General Theory of NaturalEquivalences” (1945). It subsequently developed into anautonomous mathematical discipline, in work by Mac Lane, AlexanderGrothendieck, Daniel Kan, William Lawvere, and many others (see Corry2004, Krömer 2007), with important, wide-ranging applications inalgebraic topology and homological algebra, more recently also inlogic and computer science (see Landry & Marquis 2005, as well astheentry on category theory in this encyclopedia).

Building on these developments, the philosophical discussion ofcategorical structuralism was initiated by Steve Awodey, ElaineLandry, Jean-Pierre Marquis, and Colin McLarty in the 1990s. Tounderstand their contributions, our distinction betweenmethodological andmetaphysicalstructuralism, which is drawn explicitly by Awodey (1996),will again be helpful. Or rather, Awodey distinguishes between the useof category theory as a framework for “mathematicalstructuralism” and as a framework for “philosophicalstructuralism”. He describes mathematical structuralism as ageneral way of “pursuing a structural approach to thesubject“, i.e., a style of practicing mathematics that employsstructural concepts and methods; and he argues that category theoryprovides the best way of capturing structuralist mathematics in thissense. However, he also presents it as a framework for philosophicalstructuralism, i.e., “an approach to the ontology andepistemology of mathematics”. Let us first consider the formersuggestion and argument.

Category theory, understood as a branch of mathematics, had beendescribed as a “general theory of mathematical structures”already earlier, e.g., by Mac Lane (1986, 1996). But what exactly ismeant by “structure” here? At least two relevant notionsare mentioned in the literature concerning categorial structuralism.First, a structure can be understood in theset- andmodel-theoretic sense, which we can specify further as a tupleconsisting of a domain and an ordered sequence of relations,functions, and distinguished elements used for the interpretation of aformal language. (This is the notion of “set-theoreticrelational system” appealed to in earlier sections, especiallyin 2.2.) Such structures are also called “Bourbakistructures” in this context, because of Bourbaki’s earlyreliance on set theory as the framework (see Corry 2004). Theirproperties are usually defined axiomatically, e.g., by theDedekind-Peano axioms for arithmetic or by the group axioms.

Second and alternatively, there is acategorical notion ofstructure based on the primitive concept of morphism betweenmathematical objects as used in category theory. Typically, a categoryconsists of two types of entities, namely objects and morphismsbetween them, in the sense of mappings represented by arrows thatpreserve some of the internal structural composition of the objects atissue. An axiom system that defines the general concept of a category,along such lines, was first introduced by Eilenberg and Mac Lane intheir 1945 article. It describes a suitable composition operation onarrows, its associativity, as well as the existence of an identitymorphism for each object (see Awodey 2010 for a textbookpresentation).

Now, why might category theory be considered a more adequate frameworkfor a mathematical structuralism than other disciplines, in particulartraditional set theory? According to Awodey, the “Bourbakinotion of structure” is a direct result of the modern axiomatictradition, from Dedekind and Hilbert to the Bourbaki group. Thistradition did introduce a structuralist perspective on mathematics.Yet set theory is not an ideal framework for capturing a structuralistunderstanding of mathematical objects, as Awodey goes on to argue. Tobegin with, it is closely tied to a model-theoretic conception ofmathematical theories, including the view that such theories studytheir models only “up to isomorphism”. But according toAwodey, central to a structuralist point of view is the principle(more on which below) ofidentifying isomorphic objects; andthis principle is well motivated from a category-theoretic viewpoint,but less so if mathematical objects are constructedset-theoretically.

A second advantage of category theory over set theory, also mentionedby Awodey, is that the categorical notion of structure issyntaxinvariant. This means that, unlike in standard model theory, thecategorical specification of objects in terms of their mappingproperties is independent of the choice of a particular signature usedfor their description (a choice of basic relations, functions, anddistinguished elements).

Third and most importantly, characteristic of category theory is itsfocus on mappings between mathematical objects that preserve (some of)their internal structure. Besides the use of set theory, it is theemphasis onstructure-preserving mappings that is oftenviewed as a central feature of methodological structuralism, as we sawabove already (see also again Reck and Schiemer 2020a).

Category theory was first worked out, against the background of thesedevelopments, as a unifying framework for the study of the relationsbetween different mathematical structures (see Landry and Marquis2005, Marquis 2009). Several types of mappings were introduced forthat task. One type involves themorphisms between objects ofthe same category, e.g., group homomorphisms in the category ofgroups, or linear maps in the category of vector spaces. Anotherimportant type of mappings isfunctors between differentcategories. (Roughly, a functor between two categories is a mapping ofobjects to objects and arrows to arrows that preserves the categoricalproperties in question.) It is such functors that are the crucialtools in category theory for comparing the objects of differentmathematical categories, and thus for relating structures of differentkinds (Awodey 1996). As such, they are central in categoricalstructuralism. In particular, they allow for structuralistunderstanding of an advanced kind (as highlighted in section 3.3).

4.2 Categorical Foundations and Debates About Them

As has been argued repeatedly along such lines, category theoryprovides a more natural framework than traditional set theory formathematical or, as we prefer to call it,methodologicalstructuralism in mathematics. But what about its prospects as aform ofphilosophical ormetaphysical structuralism,i.e., an alternative to the recent theories of Resnik, Shapiro, andHellman, or those of Nodelman and Zalta, Horsten, Leitgeb, etc.? Wealready indicated that Awodey also presents it as such (Awodey 1996);but this has led to ongoing controversies. A well-known article byHellmann (2003) contains a first critical discussion of philosophicalclaims such as Awodey’s. More specifically, Hellman’spiece raises several objections against the view that category theoryprovides an adequate framework for a structuralist account ofmathematics in the philosophical sense, let alone a better suchframework. As we will see, these objections are closely related to thestatus of category theory as afoundational discipline.

In the last few years there has been much debate about which criteriaa theory has to meet in order to serve as a proper“foundation” for mathematics. According to a helpfulproposal by Tsementzis, a foundational system consists of three items:(i) a formal language; (ii) an axiomatic theory expressed in thatlanguage; and (iii) a rich universe of objects, described by thetheory, in which all mathematical structures can be located,represented, or encoded (Tsementzis 2017). Zermelo-Fraenkel set theoryclearly represents a foundational system in this sense. The axioms ofZFC are usually formulated in a formal first-order language; and theydescribe 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 itsestablishment as a fruitful methodological framework, severalaxiomatizations of specific categories have been proposed asalternative foundations for mathematics. This includes the axiomsystems describing the category of sets and functions, on the onehand, and the category of categories, on the other hand (see Lawvere1964, 1966). Both were explicitly introduced as foundational systems,and thus as alternatives to Zermelo-Fraenkel set theory. Morerecently, elementary topos theory has been developed as a form ofcategorical set theory, which can serve as a foundationalsystem in the above sense as well (see Landry and Marquis 2005,Marquis 2013).

Returning to Hellman’s 2003 challenge, the question of whethercategory theory is usable to defend a version of philosophicalstructuralism can then be directly related to the presumedautonomy 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 thelogical dependence objection (Linnebo & Pettigrew 2011).Its core is the argument that category theory, general topos theory,etc. arenot autonomous of set theory in the end. The reasongiven is that the axiomatic specification of categories and topoipresupposes the concepts of operation, collection, and function, andthe latter need to be defined in a set theory such as ZFC. Categoricalfoundations are therefore dependent on non-structural set theory.

The second argument against the autonomy of categorical foundations isthemismatch objection. It concerns the general status ofcategory theory or topos theory; and it is based on the distinctionbetween two ways of understanding mathematical axioms: as“structural”, “algebraic”,“schematic”, or “Hilbertian”, on the one hand,and as “assertoric” or “Fregean”, on the otherhand. As Hellman also argues, foundational systems need to beassertoric in character, in the sense that their axiomsdescribe a comprehensive universe of objects used for the codificationof other mathematical structures. Zermelo-Fraenkel set theory is anassertoric, “contentual” theory in that sense. Its axioms(e.g., the power set axiom or the axiom of choice) make generalexistence 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 axiomsconceived as truths about an intended universe. For example, theEilenberg-Mac Lane axioms of category theory are not “basictruths simpliciter”, but “schematic” or“structural”. They function as implicit definitions ofalgebraic structures, similar to the way in which the axioms of grouptheory or ring theory are “defining conditions on types ofstructures”. This point is related to another argument againstthe autonomy of category theory, theproblem of the homeaddress. As Hellman asks, “where do categoriescome from and where do they live?” Given the“algebraic-structuralist perspective” underlying categorytheory and general topos theory, its axioms make no assertions thatparticular categories or topoi actually exist. Classical set theories,such as ZFC with its strong existence axioms, have to step in again inorder to secure the existence of such objects.

Hellman’s and Feferman’s arguments against thefoundational character of category theory have been examined fromvarious angles in the subsequent literature. One can distinguishbetween two main types of responses, namely: (i) by proponents ofcategorical foundations, who defend the autonomous characterof category theory relative to classical set theory; and (ii) bynon-foundationalists, who call into question whether categorytheory should be viewed as a foundational discipline. A series ofarticles by McLarty represent the first line of response well (McLarty2004, 2011, 2012). Roughly speaking, his reply to Hellman is asfollows: whereas category theory and general topos theory didoriginate as algebraic theories and are, as such, not feasible asfoundational systems, certain theories of particular categories andtoposes have been introduced as alternative foundations.McLarty’s central examples are Lawvere’s axiomatizationsof the category of categories and his “Elementary Theory of theCategory of Sets” (ETCS).

According to McLarty, these theories should be understood asassertoric in Hellman’s sense. That is to say, their axioms arenot merely implicit definitions, but general existential claims aboutcategories, sets, and functions. ETCS, for instance, presents afunction-based set theory, where sets and mappings between them form atopos. In contrast to ZFC, with its primitive membership relation, inETCS a set is not specified via its internal composition, but in termsof its mapping properties with respect to other sets, formulatedindependently of ZFC. McLarty’s response to the objectionsmentioned above is, then, that categorical set theories such as ETCSdo provide a foundation for mathematics, one that is logicallyautonomous from traditional set theories. Moreover, given thatmathematical structures can only be encoded in ETCS as objects up toisomorphism, such categorical set theories provide a more adequatefoundation for modern structural mathematics than ZFC. (We will returnto this point below.)

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

In fact, according to Awodey the central motivation for acategory-theoretic approach to mathematics is to sidestep foundationalissues concerning the nature of mathematical objects or the study of asingle comprehensive universe in which all structures can berepresented. While topos theory, say, might well serve as a structuralfoundation for mathematics, for Awodey such a foundational approachruns against the structuralist perspective embodied in categorytheory. In his own words, “the idea of ‘doing mathematicscategorically’ involves a different point of view from thecustomary foundational one” (Awodey 2004: 55). In light of suchfundamental, ongoing debates about categorical foundations formathematics, what are the implications for categorical structuralismin the metaphysical sense, along Awodey’s lines and moregenerally? Or does it just amount to a form of methodologicalstructuralism, after all? We will conclude our discussion ofcategory-theoretic structuralism by addressing this question.

4.3 Philosophical Features of Category-Theoretic Structuralism

Beyond issues involving methodological structuralism (includingbroadening and strengthening structuralist understanding inmathematics), the literature on category-theoretic structuralism inthe early 21st century centers around two questions that have alreadybeen mentioned. First, in what sense does category theory provide aframework for philosophical or metaphysical structuralism? Second, whyis it supposed to be better suited for that task than otherframeworks, such as set theory, Shapiro’s structure theory,Hellman’s modal structuralism, etc.? In recent work on thesetopics, one can find three related philosophical assumptions thatcharacterize categorical structuralism and distinguish it further fromthe versions of structuralism surveyed earlier. We will treat each ofthem in turn, starting again with Awodey’s writings.

A first characteristic assumption is that all mathematical theoremsareschematic statements that have aconditionalform. (This point is explicit in Awodey 2004.) We already sawthat according to Awodey a category-theoretic approach isnon-foundational in character. This includes that mathematical axiomsand theorems, as expressed in category theory, should be understood asschematic statements. As such, they do not express truths about thespecific nature of mathematical objects. In addition, mathematicaltheorems are, at least in principle, all of hypothetical form, i.e.,they can be reconstructed as if-then statements. Note that this viewof the logical form of mathematical theorems isprima faciesimilar to the if-then-ism one can find in Putnam’s andHellman’s works, building on Russell’s earlier work.

However, according to Awodey there is an important difference betweenstandard if-then-ism and a category-theoretic approach in terms of theontological commitments involved. According to standardif-then-ism, any mathematical statement can be translated into auniversally quantified conditional statement, where the quantifiersare meta-theoretic in nature, ranging over all set-theoretic systemsof the right type. (This is essentially what we called“universalist structuralism” above.) As such, the approachpresupposes a rich ontology of sets. In contrast, alongcategory-theoretic lines mathematical theorems do not involve suchontological commitments. There is no implicit generalization over theset-theoretic structures of a theory. Rather, a mathematical theoremis “aschematic statement about astructure[…] which can have variousinstances” (Awodey2004: 57). These instances remain undetermined on purpose, unless afurther specification of them is needed for the proof of a theorem inquestion.

A second distinctive feature of categorical structuralism, both forAwodey and others, concerns a certaintop-down conception ofmathematical objects characteristic for category theory. According tostandard set theory, mathematical objects are constructed from thebottom up, in successive steps starting from some groundlevel (the empty set or also a domain of urelements). Every object isthus determined, as a set, in terms of its members. In contrast,mathematical objects in category theory are characterized in atop-down fashion, starting with the Eilenberg-Mac Lane axioms andusing the notion of morphism. Hence, the objects in a given category,such as that of rings or topological spaces, are not consideredindependently of the relevant morphisms. They are fully determined bytheir mapping properties, as expressible in the language of categorytheory. Nothing further is assumed about their inner constitution. Inparticular, questions about their set-theoretic nature are consideredredundant (see Landry and Marquis 2005.)

Third, arguably the most important feature of categoricalstructuralism is that it verifies a version of thestructuralistthesis (hinted at earlier). Recall here Benacerraf’sargument, in his 1965 paper, that numbers should not be identifiedwith particular sets, but rather, with positions in an abstractstructure. Benacerraf also emphasizes that only certain properties arerelevant in arithmetic. For him, these are the number-theoreticproperties, such as “being prime” or “beingeven”, that can be defined in terms of the primitive relationsand functions of the theory at issue. The general structuralist thesisholds, then, that all (relevant) properties of the objects treated bya mathematical theory should bestructural in a specifiedsense. (The question what “structural” means also came upearlier.)

Categorial structuralists typically argue that category theorypresents the most adequate framework for a structuralist conception ofmathematical objects, given that all the properties expressible in itslanguage turn out to be structural (see McLarty 1993, Awodey 2004,Marquis 2013). This is so, presumably, because the category-theoreticstudy of mathematical objects, such as rings or topological spaces,allows us to express just the right kind ofstructuralinformation, namely information about the structural propertiesof these objects. In this context, structural properties are usuallycharacterized in terms of the notion ofisomorphisminvariance. (Given a categoryC, a morphism \(f: A\rightarrow B\) presents an isomorphism between objectsA andB 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 propertyP of an objectA in categoryC is thenstructural if it remains invariant under the isomorphisms inC, that is, if \(P(A) \leftrightarrow P(f(A))\), for allisomorphismsf; see Awodey 1996.)

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

To elaborate this point further, number systems of basic arithmeticcan be characterized in such categorical frameworks as “naturalnumber objects”, as was first shown by Lawvere (see McLarty1993). In contrast to their ZFC-based representation, such objects arenot just isomorphic but share exactly the same properties, namelythose expressible in the language of the category of sets. In otherwords, any two natural number objects are “provablyindiscernible” (McLarty 1993). Moreover, all of these propertiesare structural in the above sense. As a consequence,Benacerraf’s dilemma of isomorphic numbers systems withdifferent set-theoretic properties does not arise in the context ofcategorical set theory. The conclusion is that numbers can, after all,be identified with sets, but with structural ones as defined inETCS.

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

A rejoinder should be added, however. McLarty’s andAwodey’s claim that all mathematical properties expressible incategorical set theory are isomorphism-invariant has been contested.In fact, it has become clear that neither ZFC nor ETCS provide fullystructuralist foundations for mathematics, since their respectivelanguages do not, after all, exclusively allow for the formulation ofinvariant properties. The reason is, very briefly, that they stillinvolve a global, untyped identity relation or something analoguous(see Tsementzis 2017 for details). Then again, both MichaelMakkai’s FOLDS system (Makkai 1998) and the UnivalentFoundations program developed in Homotopy Type Theory (UnivalentFoundations Program 2013) seem to meet this condition. Here we havereached another ongoing debate about mathematical structuralism in theliterature, one that partly depends on technical results.

Debates like the one just mentioned are clearly relevant for us, butwe cannot explore them further here. (Besides Tsementzis 2017, seeAwodey 2014 for more on the relation between structuralism andUnivalent Foundations.) Some additional questions can be raised aswell. For example, does Awodey’s anti-foundationalism reducecategory theory to an advanced form of methodological structuralism, aposition meant to have no philosophical implications; or is hisposition better understood as a deflationary form of metaphysicalstructuralism, so a version of philosophical structuralism after all?As another example, in which ways exactly does category theory involvenew forms of structuralist understanding, beyond what was availablebefore (e.g., with the notion of functor and related techniques)?However, we have to reserve answering such questions again for otheroccasions. Instead, we will conclude this survey with a few generalremarks about structuralism, in and beyond the philosophy ofmathematics.

5. Conclusion

5.1 Varieties of Mathematical Structuralism

In retrospect, we pursued two main goals in this survey. The first wasto provide an introduction to the discussions of structuralism incontemporary philosophy of mathematics, usually traced back to the1960s. The second was to broaden and deepen those discussions, bymaking clear that a much larger variety of structuralist positions hasplayed a role than is often acknowledged. To be sure, some amount ofvariety was recognized before, as reflected in Parsons’sdistinction between eliminative and non-eliminative positions, withShapiro’sante rem structuralism and Hellman’smodal structuralism as paradigms. In addition, category-theoreticforms of structuralism have been acknowledged as a third mainalternative, although their relationship to the others has neededclarification. Yet the range of positions at play encompasses muchmore than those, including older variants, like set-theoreticstructuralism, and several more recent ones. Overall,“mathematical structuralism” is not the name of a singleposition, but of a multifaceted family of them.

Three additional, more specific goals influenced the content andoverall shape of this survey. First, while numerous variants ofstructuralism for mathematics have been proposed, a number of of themhave not received adequate attention yet, including clarification oftheir relationships to Shapiro’s, Hellman’s,set-theoretic, and category-theoretic positions. Besides includingthose variants explicitly, our discussion is intended to build bridgesand to provide conceptual tools for that purpose. Second, severalimportant forms of structuralism played a role already before the1960s, i.e., before the publications by Benacerraf and Putnam withwhich debates about structuralism in the philosophy of mathematics areoften assumed to have started. We wanted to make that evident as well(and point the reader again to Reck & Schiemer 2020a for more).Third, we have repeatedly highlighted the distinction betweenmetaphysical and methodological structuralism; this has also led tonew questions about structuralist understanding, which deserve moreattention in themselves.

5.2 Structuralism Beyond Mathematics

If one widens one’s perspective even further, it becomes clearthat another theme that is generally relevant for our discussion hasnot been covered in this survey and should be mentioned at leastbriefly 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 ofwhich has had broader reverberations). The first is the philosophy ofphysics, where structuralist positions have played asignificant role as well (including a position called“structural realism”, which comes in both an“ontic” and an “epistemic” form). The secondincludes various parts of thehumanities and social sciences,primarily linguistics and anthropology, but also psychology,sociology, etc. There too, “structuralism” (and“post-structuralism”) has been a major topic, indeedalready for a relatively long time. In both cases there are systematicand historical ties to mathematical structuralism, although sometimesthose ties are only loose.

At a basic level, the debate about structuralism in the philosophy ofphysics concerns how to think about the “objects” ofmodern physics, given the revolutionary changes quantum mechanics andrelativity theory have brought about. More specifically, it concernsthe move to appeal to “structures” in this connection, ina way that is closely related to the philosophy of mathematics (seeFrench 2014, Ladyman 2007 [2019], etc.). Second and in addition, thisdebate concerns the question how to characterize what, if anything,remains constant throughtheory changes in the ontology ofphysics (see Worrall 1989, see also theentry on structuralism in physics in this encyclopedia). Third, there are debates aboutscientificrepresentation that are related tostructuralism (see van Fraassen 2008). While one could pursue severalresulting points of contact further (e.g., concerning“structural indiscernibles”, or versions of set-theoreticvs. category-theoretic structuralism in the context of physics), werefrain from doing so here.

The structuralism introduced into linguistics by Ferdinand de Saussure(Saussure 1916 [1959]) and Roman Jakobson (in writings from the 1920son, see Holenstein 1976), then adopted by various thinkers in thehumanities and social sciences—most prominently by ClaudeLévi-Strauss in anthropology and by Jean Piaget in psychology(Lévi-Strauss 1958 [1963], Piaget 1968 [1970], see also Caws1988)—does have ties to mathematical structuralism as well. Butthey are looser than in the case of physics and there are moredifferences. In particular, the kind of psychological determinismoften associated, or even identified, with structuralism in thehumanities and social sciences (and criticized in post-structuralism)has no analogue in mathematics (or physics). Still, some strikinghistorical ties exist, e.g., personal contacts betweenLévi-Strauss and the Bourbaki group (see Dosse 1991–92[1997]). And those ties might be worth exploring further too, at leastif one wants to understand the role of structuralist ideas in humanthought more fully.

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Acknowledgments

The authors wish to thank Steve Awodey, Francesca Biagioli, NorbertGratzl, Henning Heller, Johannes Korbmacher, Hans-Christoph Kotzsch,Hannes Leitgeb, Øystein Linnebo, Jean-Pierre Marquis, MichaelPrice, Patrick Ryan, Andrea Sereni, Matthew Warren, John Wigglesworth,and Edward Zalta for helpful comments and discussions. We would alsolike to thank two anonymous reviewers for constructive suggestions,Dilek Kadıoğlu for noting some infelicitous typographicalerrors, as well as Peter Momtchiloff for his help in improving variousformulations and adding references. Finally, we are grateful to theeditors forThe Stanford Encyclopedia of Philosophy for theirpatience and for their general support.

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