## 1. Eliminative vs. Non-Eliminative Structuralism


### 1.1 Beginnings of the Structuralism Debate in the 1960s



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



According to Benacerraf, such a set-theoretic foundationalist position
misrepresents the structuralist character of arithmetic in particular
and of mathematics more generally. To begin with, instead of working
with the finite von Neumann ordinals, we can equally well work with
the finite Zermelo ordinals (starting again with \(\emptyset\) for 0
but using the alternative successor function \(f: x \rightarrow
\{x\})\); and there are infinitely many other choices that are
equivalent. Similarly, instead of working with a set-theoretic
construction of the real numbers in terms of Dedekind cuts, we can
work with a construction based on equivalence classes of Cauchy
sequences on the rational numbers as suggested by Cantor and
others. This basic observation is hard to deny, and even set-theorist
foundationalists can agree with it (more on that below). But
Benacerraf draws some further, more controversial conclusions from
this basic observation. 



Benacerraf argues, in particular, that the natural numbers should not
be identified with any set-theoretic objects; in fact, they should not
be taken to be objects at all. Instead, numbers should be treated as
"positions in structures", e.g., in "the natural
number structure", "the real number structure",
etc. All that matters about such positions are their structural
properties, i.e., those "stem[ming] from the relations they bear
to one another in virtue of being arranged in a progression"
(1965: 70), as opposed to further set-theoretic properties of the von
Neumann ordinals, Dedekind cuts, etc. What we study and try to
characterize in modern mathematics, along such lines, are the
corresponding "abstract structures". It is in this sense
that Benacerraf suggests a structuralist position concerning
mathematics. The details of the position are left open and somewhat
vague, however, including how we should think about Benacerraf's
abstract structures in the end, except that they are not to be
identified with set-theoretic relational systems (consisting of sets
as domains, with set-theoretic relations and functions defined on
them). 



A second article from the 1960s that was influential in the rise of
structuralism is Hilary Putnam's "Mathematics without
Foundations" (1967). Like in Benacerraf's case, for Putnam
the foil was a set-theoretic foundationalist position. This position
is sometimes, although not always, understood in a realist sense
(e.g., by Godel), i.e., as the description of an independent
realm of abstract objects, namely the universe of sets characterized
by the Zermelo-Fraenkel axioms. In opposition to that position, Putnam
suggests a form of "if-then-ism" (which can be traced back
to Russell). That alternative can again be illustrated in terms of
the natural numbers. How should an arithmetic theorem, say
"\(2+3=5\)", 
be understood now? It should be analyzed as having this form: 



> 
> 
> 
> For all relational systems *M*, if *M* is a model of the
> Dedekind-Peano axioms [the basic axioms for arithmetic], then \[2\_M
> +3\_M =5\_M\]
> 
>  
> 
> 
> 



(where \(2\_M\), \(3\_M\), and \(5\_M\) are what "play the
roles" of 2, 3, and 5 in the model *M*). Similarly for the
real numbers (see Reck & Price 2000 for details).



Instead of talking about if-then-ism in this connection, one can
describe Putnam's position also as a kind of "universalist
structuralism" (see again Reck & Price 2000), since it
involves universal quantification over relevant systems and since our
two structuralist slogans above seem satisfied. Often an objection to
this position is the "non-vacuity problem". It is based on
the observation that if-then statements of the given form are
vacuously true if there is nothing that satisfies the antecedent,
e.g., if there is no model of the Dedekind-Peano axioms. (As this
would also be the case for, say, "\(2+3=6\)", the overall
result is clearly not desirable.) In response, one can invoke
axiomatic set theory as providing the needed models. But from
Putnam's point of view this has two drawbacks: First, it relies
on a realist and foundationalist view about set theory, it would seem,
thus undermining the thrust of if-then-ism. Second and more basically,
it forces us to treat set theory differently from other mathematical
theories, on pain of circularity. As a way out of these dilemmas,
Putnam suggests to employ modal logic. However, the details are again
left open and largely unexplored, especially for the case of set
theory.


### 1.2 Consolidation and Further Discussions in the 1980s



One way to understand Benacerraf's discussion in his 1965
article is that he proposes to treat the natural number structure as a
new kind of abstract entity, different from set-theoretic objects and
systems of objects. All depends then on what exactly that amounts to,
including whether one should treat such structures as objects
themselves, thus reifying them in some substantive and still to be
worked out way, or not. Benacerraf himself was reluctant to do the
latter, in line with his overall hesitancy to talk about mathematical
objects. 



One subsequent writer who picked up on Benacerraf's ideas in the
early 1980s and tried to spell them out further, while still
refraining from treating structures as full-fledged objects, is
Michael Resnik (cf. Resnik 1981, 1982, 1988, and, most systematically,
1997). For him too, modern mathematics involves a "structuralist
perspective". This includes a kind of pattern recognition; and
one of Resnik's main goals is to spell out the corresponding
epistemology further. Along Benacerraf's lines, mathematical
objects are viewed as "positions" in corresponding
patterns; and this is meant to allow us to take mathematical
statements "at face value", in the sense of seeing
'0', '1', '2', etc. as singular
terms referring to such positions. At the same time, doing so is not
supposed to require reifying the underlying structures, which would
mean specifying precise criteria of identity for them, something
Resnik avoids intentionally. (He presents himself as a Quinean on this
point, in the sense of adopting Quine's slogan: "No entity
without identity!")



Stewart Shapiro is a second philosopher of mathematics who, in the
early 1980s, attempts to build on Benacerraf's paper (see
Shapiro 1983, 1989, and most systematically, Shapiro 1997). By
focusing more on metaphysical questions and leaving behind hesitations
about structures as objects, Shapiro's goal is to defend a more
thoroughly realist version of mathematical structuralism, thus
rejecting nominalist and constructivist views (more on that
below). Such realism includes the semantic aspect mentioned above
(taking mathematical statements at face value); but Shapiro also wants
to clarify the talk about "positions in structures"
further. He distinguishes two perspectives on them. According to the
first, the positions at issue are treated as "offices",
i.e., as slots that can be filled or occupied by various objects
(e.g., the position "0" in the natural number structure is
occupied by \(\emptyset\) in the series of finite von Neumann
ordinals). According to the second perspective, the positions are
treated as "objects" themselves; and so are the abstract
structures.



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



While Resnik's and Shapiro's structuralist positions are
sometimes identified, this is somewhat misleading given the
differences already mentioned. Nevertheless, there is significant
overlap. Both recognize mathematical structures as certain patterns
with positions in them etc. (whether these patterns are then treated
as full-fledged objects, in a realist sense, or not). And for both
Resnik and Shapiro, the notion of isomorphism is crucial (or some
related, more general notion of equivalence; see Resnik 1997 and
Shapiro 1997). That is to say, for both a relevant structure/pattern
can be instantiated by any of a relevant class of isomorphic
relational systems. This corresponds to the fact that the axiomatic
systems at issue, for the natural numbers, the real numbers, and
similar cases, are all categorical (or quasi-categorical in the case
of set theory). Not every mathematical axiom system has that feature,
of course, e.g., the axiom systems for group or ring theory allow for
non-isomorphic instances or models. For Resnik as well as Shapiro,
such "algebraic" theories are to be treated in a
different, more derivative way; their structuralist point of view is
meant to apply primarily to "non-algebraic" theories,
paradigmatically arithmetic.



There is another structuralist position from the 1980s that is quite
different and explicitly not realist, namely that introduced by
Geoffrey Hellman (cf. Hellman 1989, 1996, and later articles). While
for Resnik and Shapiro the inspiration was Benacerraf's 1965
article, the starting point for Hellman is Putnam's 1967
article. In fact, Hellman's "modal structuralism" is
meant to be a systematic development of Putnam's modalized
if-then-ism. The modal aspect is now spelled out in detail and with
real ingenuity, including for the case of set theory (building on work
by Zermelo etc.). For Hellman, a sentence such as
"\(2+3
= 5\)" is analyzed as follows: 



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



To avoid the non-vacuity problem, he adds the following
assumption:



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


(We will come back to its justification below.)



As Hellman makes clear, his goal is to develop a form of
"structuralism without structures" (Hellman 1996), since
the existence of abstract structures, postulated by Resnik and
Shapiro, is replaced by the modal aspects of his position (and
corresponding assumptions about necessity and possibility). In fact,
Hellman's position is meant to be a form of nominalism, i.e., to
eliminate the appeal to any kind of abstract entity (not only abstract
structures but also sets etc.). Yet it is also not meant to rely
on *possibilia*, i.e., possible objects existing in some
shadowy sense. This leads Hellman to a specific understanding of the
modalities at issue. They are meant to be basic, i.e., the relevant
possibilities and necessities are not reducible to anything
further. On the other hand, they are specified precisely in terms of
laws of modal logic (those of the system S5).


### 1.3 Towards a First Taxonomy of Structuralist Positions



From the late 1980s on, Shapiro's and Hellman's positions
have often been treated as the two main structuralist options. (This
is still reflected in Hellman & Shapiro 2019.) As they are quite
different, this already indicates that it is wrong to see
"structuralism in the philosophy of mathematics" as a
unique position or a unified perspective, even though certain general
slogans are shared. Beyond that, other versions of structuralism
started to play a role in the late 1980s and early 1990s (including
forms of "set-theoretic structuralism" that go back
further in time than the 1960s, as we will see). Consequently, the
discussions about structuralism have become both richer and more
complex.



To clarify this situation, Charles Parsons suggested a first taxonomy,
or at least a distinction between two main kinds of positions (Parsons
1990). Namely, there are "eliminative" forms of
structuralism, as illustrated paradigmatically by Hellman; and there
are "non-eliminative" forms of structuralism, like
Shapiro's. (The elimination at issue concerns the postulation,
or avoidance thereof, of structures as abstract objects.) Or put in
Hellman's slightly later terminology, there is
"structuralism without structures", on the one hand, and
"structuralism with structures", on the other
hand. Besides Shapiro and Resnik (with the qualifications above),
another proponent of the non-eliminative form is Parsons himself (see
Parsons 1990, 2004, and, most systematically, 2008); and another
proponent of the eliminative form of structuralism is Charles Chihara
(cf. Chihara 2004). 



Nevertheless, it seems to have been tempting, and has remained fairly
widespread in the literature, to identify non-eliminative
structuralism with Shapiro's position, i.e., with his realist
and *ante rem* version. In fact, critics sometimes dismiss
"philosophical structuralism" generally as a misguided
form of metaphysics, thereby identifying such structuralism with
Shapiro's realist form. (This seems especially tempting for
mathematicians and for philosophers rooted deeply in mathematical
practice; cf. Awodey 1996 and Carter 2008 for related discussions.) A
further consideration of Parsons' form of non-eliminative
structuralism can help to show that this is too quick and inadequate
in the end. 



Unlike Shapiro, Parsons does not offer a novel, philosophically
motivated structure theory as the backing for his position. For him,
we should remain closer to mathematical practice (as it developed in
the late nineteenth and the twentieth centuries). In fact,
structuralism should be seen as growing out of that practice, rather
than being imposed on it from outside. For Parsons this means, among
others, taking abstract structures to be introduced directly by
categorical axiom systems, a practice he spells out further in a
"meta-linguistic" way inspired by Quine (see Parsons
2008). It also means that we should refrain from
"cross-structural identifications" (as they can be found
in Shapiro's early works), e.g., from identifying the natural
number 1 and the real number 1. Such putative identities should be
left indeterminate, as is done in mathematical practice.



As should be evident, Parsons' structuralist position, like
Resnik's, is less realist than Shapiro's. Moreover, he is
explicit that adopting a "structuralist view of mathematical
objects" should be seen as separable from, and orthogonal to,
the realism/nominalism dichotomy. Consequently, for Parsons one can be
a non-eliminative structuralist without being a realist in any strong
sense; his own position is a case in point. Yet Parsons'
version of structuralism is still meant to allow taking mathematical
statements at face value (as characterized above), so that it remains
realist in this minimal semantic sense.


## 2. Later Developments and a Broader Taxonomy


### 2.1 Metaphysical and Epistemological Challenges



So far we have traced the development of structuralism in the
philosophy of mathematics from Benacerraf and Putnam, in the 1960s, to
Resnik, Shapiro, Hellman, Chihara, and Parsons, in the
1980s-90s. During the last 20 years, a number of further
philosophers have started to address this topic. We now turn to the
corresponding discussions, starting with certain epistemological and
metaphysical challenges to structuralism. Some of these concern
non-eliminative structuralism alone, especially Shapiro's
position. (This again reflects the prominence of Shapiro's
position, to the point that it is often identified, misleadingly, with
"philosophical structuralism".) Others are broader,
including comparing the basic commitments of various forms of
structuralism. In what follows, we will not try to be comprehensive,
but provide a number of illustrative examples.



Non-eliminative structuralism, in Shapiro's and other versions,
involves the thesis that all that matters about mathematical objects
are their structural properties (as opposed to further intrinsic
properties). In fact, such properties are taken to determine the
objects' identities. But then, objects that are
indistinguishable in this respect--"structural
indiscernibles"--should be identified, it seems. As several
critics pointed out around the year 2000, this leads to the
"identity problem" for structuralism (cf. Keranen
2001, earlier Burgess 1999). It arises prominently for systems or
structures that are non-rigid, i.e., allow for non-trivial
automorphisms. In such cases, there are supposedly distinct objects
that are indistinguishable in the relevant sense. A widely known case
is the system of complex numbers (with the conjugate numbers *i*
and [?]*i*); but geometry and graph theory, among others,
provide further examples. The simplest example is probably an
unlabeled 2-element graph with no edges, whose two vertices are
structurally indiscernible. 



How can such cases be handled from a structuralist point of view? Is
structuralism (of this kind) simply incoherent, as some of its critics
charge? Or is it at least not applicable to non-rigid cases, which
would limit its reach significantly? Several responses to the identity
problem have been offered in the literature. One suggestion is to
"rigidify" such structures by enlarging the vocabulary
that is used, e.g., by adding the constant symbol
'*i*' for the complex numbers (either into the
original language or into the language for the "setting"
used in the background, cf. Halimi 2019). But this still seems
problematic in the case of many indiscernibles, perhaps even
uncountably many. Another suggestion is to treat identity as a
primitive notion, as is arguably part of mathematical practice. But in
that case too a number of questions remain (see Ladyman 2005, Button
2006, Leitgeb & Ladyman 2008, Shapiro 2008, Ketland 2011, and
Menzel 2018, among others).



A second, more basic problem for structuralism starts again with the
idea that all that matters about mathematical objects are their
structural properties. With respect to non-eliminative structuralism,
this is sometimes strengthened to the thesis that the positions in
mathematical structures, as well as the abstract structures
themselves, "only have structural properties". But if not
formulated more carefully, this leads to counterexamples (cf. Reck
2003). For instance, does the natural number structure not have the
property that it is a favorite example in many debates about
structuralism? And isn't it a property of the number 9 to have
long been taken to be the number of planets in the Solar System? Both
seem clearly non-structural properties. Once again, structuralism
appears incoherent or, at the very least, in need of further
clarification.



A natural response to this challenge is to refine the original
structuralist thesis, e.g., by saying that abstract structures only
have structural properties "essentially", that only such
properties are "constitutive" for them, or something
similar, while admitting that they have other properties too (see Reck
2003, Schiemer & Wigglesworth forthcoming). This leads to
questions about how exactly to make this distinction. But even if it
is taken to constitute a satisfactory response, another problem
remains: How can we distinguish "structural" from
"non-structural" properties in the first place? Several
answers to that question have been proposed in turn, e.g., that
structural properties are those definable in a certain way, or that
they are those preserved under relevant morphisms. Yet no consensus
on that issue has been reached either (see Korbmacher & Schiemer
2018, also for further references).



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



A fourth basic challenge for structuralism, mostly again in its
non-eliminative form, is how we can have "access" to
structures seen as abstract objects (cf. Hale 1996, among others). To
some degree, this revives an older, more general debate about such
objects. An initial response, by Resnik, Shapiro, and Parsons (all
following Quine here), is to talk about the "positing" of
structures, which promises to undercut the access problem. But under
what conditions is such positing legitimate (given the threat of
paradox familiar from naive set theory)? A plausible answer
points to the "coherence" of the relevant theories (which,
after Godel's incompleteness results, is seen as taking the
place of provable consistency). Yet what exactly does such coherence
amount to? An interesting outcome of the debate about this issue is
that Shapiro and Hellman, coming from very different directions, have
arrived at views that are very close to each other (see Hellman
2005). Thus, in terms of some basic commitments--ultimate
conditions for existence in Shapiro's case and for possibility
in Hellman's case--their approaches converge in surprising
ways. (This may be seen as providing support for both sides, but also
as undermining the realism/nominalism dichotomy.)



There are more challenges to structuralism in mathematics that one can
find in the literature of the last 20 years. While usually connected
to the problems just surveyed, sometimes these challenges go
further. For example, additional questions about the semantics
involved in structuralism have been raised, typically again for the
non-eliminative variant (cf. Button & Walsh 2016, Assadian 2018,
etc.). Instead of summarizing these challenges too, we hope our survey
so far provides enough illustration of the kinds of debates that have
occurred in the recent literature already.


### 2.2 Several Additional Variants of Structuralism



As mentioned earlier, in many discussions of structuralism, from the
1980s to the early 2000s and beyond, a few positions have taken center
stage: Shapiro's, Hellman's, sometimes Parsons', and
occasionally Resnik's. But other forms of structuralism have
been around for decades too. These deserve, and are starting to get,
more attention as well. Without any claim to comprehensiveness, we
want to mention several noteworthy examples. A major one, whose
origins can be traced back to before the 1960s, is
"set-theoretic structuralism" (cf. Reck & Price 2000,
also Reck & Schiemer forthcoming). To introduce it, let us
reconsider the example central to Benacerraf's 1965 paper: the
natural numbers. 



As Benacerraf argued, there is something wrong with identifying
"the natural numbers" with a particular set-theoretic
system; or at least, it seems wrong to do so in any absolute sense.
Benacerraf's conclusion was that numbers are not sets, nor
objects of any kind, but positions in a structure. Now, one can agree
with almost everything Benacerraf says and still want to identify
"the natural numbers" with some set-theoretic system in a
less absolute sense. In doing so, one can admit that any other model
of the Dedekind-Peano axioms "would do as well", i.e.,
could have been chosen instead (except perhaps for pragmatic reasons,
e.g., the ability to generalize to the transfinite). This means that
what we identify as the natural numbers depends on an initial,
provisional, and somewhat arbitrary choice. For most mathematical
purposes, such a pragmatic identification is sufficient; in fact, this
is exactly what one does in standard axiomatic set theory. The
resulting position deserves again to be counted as a form of
structuralism, as its defenders insist. What makes it structuralist is
the "indifference to identify" the natural numbers in any
more absolute sense (cf. Burgess 2015). 



The core of set-theoretic structuralism, as just described, is to
choose one of several isomorphic systems as the pragmatic referent for
"the natural numbers" (similarly for the real numbers
etc.). In a sense, our talk of "the natural numbers", then
also of "the number 0", "the number 1", etc.,
is relative to this initial choice. This is seen as unproblematic
because, no matter how we choose, we will get the same arithmetic
theorems (because of the categoricity of the axiom system, which
implies its semantic completeness). As background, we can again employ
Zermelo-Fraenkel set theory. But we can also broaden the approach
slightly by allowing for "atoms" or
"urelements", i.e., objects that are not sets. Thus, we
can include Julius Caesar or some beer mug in our domain, with the
consequence that either of them can "be" the number 2,
say, in the sense of taking up the "2-position" in the
model for arithmetic we choose to work with. Because of this feature,
we will use "relativist structuralism" for this approach
(cf. Reck & Price 2000). It also seems fair to say that this
position, particularly in its set-theoretic version, is accepted by
many mathematicians, explicitly or implicitly. In fact, it may be the
most widely favored form of structuralism.



In set-theoretic structuralism, and in relativist structuralism more
broadly, the only mathematical objects at play are those that
axiomatic set theory, possibly with urelements, allows us to
introduce. We do not need to postulate abstract structures in
addition. For that reason, the position is another form of eliminative
structuralism (while it is not fully eliminative insofar as it
countenances sets). In fact, set-theoretic relational systems (the
set-theoretic models of theories) are themselves taken to be the
relevant structures here. (It is exactly such relational systems that
are called "structures" in many mathematics textbooks.)
However, there is another option in connection with the
latter. Namely, we can also identify the structure of the natural
numbers, say, with the (higher-order) concept defined by the
Dedekind-Peano axioms; similarly for other (categorical) axiom
systems. This leads to yet another eliminative form of structuralism:
"concept structuralism" (cf. Isaacson 2010, Feferman 2014,
also Ketland 2015, Other Internet Resources).



According to concept structuralism, what matters in modern axiomatic
mathematics aren't really objects, especially not problematic
abstract objects. Rather, crucial are mathematical concepts, e.g.,
the concept "natural number system" (or "model of
the Dedekind-Peano axioms", "progression");
similarly for the concept "complete ordered field"
etc. (In Ketland 2015, Other Internet Resources, such concepts are
explicated further in terms of intensional propositional functions,
while Isaacson and Feferman leave their nature more open.) More
precisely, what matters in the end is what follows from those
concepts, in the sense of what can be derived from the corresponding
axioms. It is true, as concept structuralists might admit, that often
the way in which we reason in mathematics involves talking about
objects falling under the relevant concepts. But as they would add,
such talk can be explained away in the end (e.g., by taking up a
formalist position). In this or similar forms, concept structuralism
seems again a fairly widespread view among mathematicians and
logicians, even though it has not been very prominent in the
structuralism debates until recently.



For the sake of a more comprehensive survey, we want to go even
further. As the next step, we will introduce two forms of
structuralism that are closely related to relativist structuralism and
concept structuralism but not identical with either. (Both are forms
of "abstractionist structuralism", as we will see.) Let
us start again with a higher-order concept defined by an axiom system,
such as "natural number system", together with the
set-theoretic systems falling under it. However, instead of
identifying the corresponding structure either with that concept or
with some pragmatically chosen system falling under it, we focus on
the whole equivalence class determined by the concept.



At this point, we can go down one of two "abstractionist"
paths. First, we can simply identify the relevant structure with that
equivalence class (with the "concept in extension", as it
is sometimes put). Thus, corresponding to the concept "natural
number system" and set-theoretic systems falling under it, like
the finite von Neumann ordinals, there is the entire equivalence class
of corresponding models as a third entity. (Our primary focus is again
on categorical axiom systems, but the approach can be generalized.) It
is this class that is now called "the natural number
structure". It is not a set, to be sure, but a proper class; but
it can still be studied logico-mathematically. Looked at from a
"relativist" perspective, the new approach can also be
described as follows: Its core is to go from a particular, arbitrarily
chosen system falling under a higher-order concept to the related
equivalence class, i.e., the class of all systems isomorphic to it (in
the categorical case). And we can think of this move as involving a
kind of "abstraction", specifically in the sense of the
"principle of abstraction" in Russell (1903; also adopted
by Rudolf Carnap and others). The result is a first form of
"abstractionist structuralism".



There is a second path we can take, resulting in a second form of
abstractionist structuralism. It too has played a role in the recent
structuralism debates; but, like the first, it can also be traced back
further in time. Let us start again from a relevant higher-order
concept, or from the axiom system that defines it, together with an
arbitrarily chosen relational system falling under it (a set-theoretic
system, say, possibly with urelements). The new suggestion is to
proceed as follows: We "abstract away from the particular nature
of its elements" so as to arrive at a novel, distinguished
relational system that deserves to be called "the natural
numbers" (cf. Dedekind 1888, especially as interpreted in Reck
2003). The intention is that the objects introduced by such
abstraction only have structural properties, or better, only have
these essentially. Also, together these objects form a system
isomorphic to the one from which we started (unlike the equivalence
class we just considered). Finally, it is the latter we now consider
the relevant abstract structure.




In several respects, this second abstractionist alternative is close
to Shapiro's
*ante rem* structuralism (who occasionally appeals to
"abstraction" himself, 
e.g., in Shapiro 1997); it is also close to Parsons' 
form of non-eliminative structuralism. Yet it neither involves a separate 
structure theory a 
la Shapiro nor an appeal to Parsons'
meta-linguistic procedure. Instead, abstract structures are introduced 
"by abstraction" from
more concrete systems, e.g., from set-theoretic relational systems.
The relevant abstraction can be explicated further in
terms of an "abstraction 
operator" and a corresponding
"abstraction principle". 
Another comparison that suggests itself then is with the use of 
abstraction principles in contemporary neo-logicism. In fact, that 
connection has already been explored in Linnebo & 
Pettigrew (2014) and Reck (2018a). What we arrive at along such lines is an
abstractionist form of non-eliminative structuralism. In contrast, the first 
alternative mentioned above is an abstractionist form of eliminative
structuralism. (Given their historical roots, these positions
might be labeled "Russellian abstractionist 
structuralism"
and "Dedekindian abstractionist 
structuralism",
respectively; cf. Reck 2018a.)



The list of additional variants of structuralism does not end here
either. Let us mention five further examples very briefly, without
going into any details (and there may undoubtedly be more). First,
Uri Nodelman and Edward Zalta have introduced a form of
non-eliminative structuralism, parallel to Shapiro's, which uses
an "object theory" inspired by Meinong to account for the
abstract structures (Nodelman & Zalta 2014). Parallel to that, one
can use other fundamental theories for the introduction of abstract
structures, leading to further forms of non-eliminative
structuralism. As a second example, Hannes Leitgeb has described how
to adapt graph theory for this purpose (Leitgeb forthcoming). Third,
Leon Horsten has built on Kit Fine's theory of "arbitrary
objects" to construct a parallel form of "generic
structuralism" (Horsten forthcoming). Fourth, above we already
mentioned Charles Chihara's "eliminative" form of
structuralism, which is not identical with Hellman's (see
Chihara 2004). And fifth, there is a whole family of
"categorical forms of structuralism", based on a variety
of axiom systems from category theory, that has become prominent
recently. 



We will postpone our discussion of categorical structuralism to
section 3,
both because it is important mathematically, thus deserving treatment
in a separate section, and because it is harder to compare to the other
forms of structuralism. Before that, we want to offer a richer, more 
comprehensive taxonomy for structuralist positions.
This taxonomy will be wide enough to encompass all the positions
mentioned so far. But it will also point beyond them, starting with
the introduction of a basic dichotomy between
"metaphysical" and 
"methodological" 
structuralism.


### 2.3 A Broader Taxonomy of Structuralist Positions



All the variants of structuralism described up to this point are forms
of "philosophical structuralism", or more precisely, of
"metaphysical structuralism". What that means is that
these positions are intended to provide answers to questions about
what mathematical structures are (even when this involves an
eliminative stance), including views about their existence,
abstractness, identity, dependence, etc. Now, one can distinguish
this whole variety of philosophical positions from what is sometimes
called "mathematical structuralism", or again more
precisely, "methodological structuralism" (cf. Reck &
Price 2000, earlier Awodey 1996). In fact, we suggest that
recognizing this basic dichotomy--"metaphysical versus
methodological"--to is crucial with respect to both
systematic and historical discussions of structuralism. (Among
others, it allows us to connect more naturally with categorical
structuralism; see, e.g., Corry 2004 and Marquis 2009.) 



As its name suggests, methodological structuralism concerns the
methodology of mathematics, thus mathematical practice. Or as one
might also put it, it concerns a certain "style" of doing
mathematics. That style consists in studying whole systems or
structures of objects in terms of their global, relational, or
structural properties, while neglecting the intrinsic nature of the
objects involved. This can be done in two main ways, often intertwined
in practice: by proceeding axiomatically, i.e., by deriving theorems
from basic axioms for the systems at issue; and by considering
morphisms between them (homomorphisms, isomorphisms, etc.), together
with invariants under those morphisms. As such an approach typically
involves infinite sets, non-decidable properties, and classical logic,
it tends to be opposed to more "computational" and
"constructivist" ways of doing mathematics (cf. Reck &
Schiemer forthcoming for extensive historical background).



Such a structuralist methodology, or corresponding forms of
methodological structuralism, tends to be tied to a general assumption
about the subject matter of mathematics, namely: mathematics is the
study structures. But accepting that assumption does not, in itself,
involve any further views about the nature of those structures, at
least not in any detailed and philosophically loaded way. In contrast,
all the forms of metaphysical structuralism considered earlier are
meant to provide such views. This is exactly how they go beyond
methodological structuralism (often by building on it).




With respect to such metaphysical positions, Parsons'
"eliminative versus non-eliminative" distinction remains
helpful (although a more positive label for the latter kind of views,
along the lines of "structuralism with structures", might
be better). However, the differences among metaphysical forms of
structuralisms do not stop there. In fact, only working with
Parsons' distinction obscures some important differences. As
already mentioned, Shapiro's *ante rem* structuralism is
far from the only version of non-eliminative structuralism; nor is
Hellman's the only form of eliminative structuralism. We now
want to suggest some more fine-grained distinctions so as to introduce
more order and clarity into the debates.



Let us first look again at non-eliminative versions of
structuralism. Some positions falling under that label introduce
abstract structures by means of a basic theory. This includes
Shapiro's structure theory, but also, e.g., Nodelman &
Zalta's object theory and Leitgeb's adaptation of graph
theory. All of them are forms of *ante rem* structuralism, but
significantly different ones. In addition, there are forms of
non-eliminative structuralism that are based on abstraction principles
instead, which we called "abstractionist forms of
structuralism" above. We distinguished two variants of those,
the Russellian and the Dedekindian variant, which involve different
kinds of structures as the outcome of the abstraction process. (If we
reconstruct such abstraction as mathematical operations or functions,
their arguments are the same but their values are different.) And as
this shows, there are abstractionist and non-abstractionist versions
of non-eliminative structuralism.



If we reflect on these alternatives more, the role of a further basic
dichotomy becomes apparent (within the category of non-eliminative
structuralism): that between
*ante rem* positions and either *in
re* or *post rem* positions.
Shapiro's position is explicitly a form of 
*ante rem* structuralism. In contrast, Russellian
abstractionist structuralism can be seen as a form of
*post rem* structuralism, because the equivalence
classes used as the relevant structures are "built
out of" their elements as classes, to which they 
are thus *posterior*. In Dedekindian abstractionist
structuralism, there is a form of posteriority
involved as well. Here too, we start with more concrete relational
systems, usually systems of sets or urelements, and we introduce
abstract structures on that basis. But the
prior versus posterior relation is different now. (It is not
based on the element-class relation, but on a more basic
argument-function-value relation.) We also end up with abstract
structures that are neither standard sets nor classes.



On the side of eliminative structuralism, further sub-divisions should
be made as well. Again, there are fully eliminative positions, which
avoid commitment to any kind of abstract objects. Hellman's
modal structuralism is designed to be of this sort. But there are also
semi-eliminative positions, which avoid commitment to abstract
structures in addition to more usual, relatively concrete mathematical
objects, which are accepted. Set-theoretic structuralism is a good
illustration; universalist structuralism is another, at least when
backed by set theory. Should relativist structuralism in general, and
set-theoretic structuralism in particular, be seen as cases of
*in re* structuralism, in the sense that
abstract structure exist "in" 
their more concrete instantiations? Perhaps; but this thesis does
not seem to be forced on us (cf. Leitgeb forthcoming for more on this
issue). Note also that we would end up with a form of
non-eliminative structuralism then instead. The same
question arises for *in re* forms of structuralism 
more generally; the details will matter. (For more on such 
approaches--sometimes labeled 
"Aristotelian" as opposed to 
"Platonist"--see Pettigrew 2008 and
Franklin 2014.) 



Yet another version of eliminative structuralism that has started to
attract more attention in the literature is concept structuralism. In
cases where this position is meant to do without any appeal to
abstract objects (e.g., by basing itself on formalism), it amounts to
a fully eliminative view. Various questions remain, however, about the
appeal to concepts in it, i.e., about their existence, nature, and
identity (cf. Parsons 2018). Depending on the answers, a strict
nominalist might still find this position unacceptable, since concepts
may be seen as another problematic kind of abstract entities. If a
concept structuralist allows for abstract objects, such as sets, to
play a secondary role, this becomes a semi-eliminative
position. Structures conceived of as abstract objects are still
eliminated (by reconceiving them as concepts), but relational systems
remain. Alternatively, we might be dealing with a particular form of
methodological structuralism here, where additional metaphysical
questions are brushed aside.


## 3. Category-Theoretic Structuralism


### 3.1 Category Theory as the Study of Mathematical Structures



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



Building on these developments, the philosophical discussion of
categorical structuralism was initiated by Awodey, Landry, Marquis,
and McLarty in the 1990s. To understand their contributions better,
it helps to go back to our distinction between
"metaphysical" and "methodological
structuralism", which is drawn explicitly in Awodey (1996). Or
rather, Awodey distinguishes between the use of category theory as a
framework for "mathematical" and for "philosophical
structuralism". He describes mathematical structuralism as a
general way of "pursuing a structural approach to the
subject", i.e., a particular style of practicing mathematics
that employs structural concepts and methods. He then argues that
category theory provides the best way of capturing structural
mathematics in this sense. However, he also presents it as a framework
for philosophical structuralism, i.e., "an approach to the
ontology and epistemology of mathematics". Let us first
consider the former argument. (We will return to the latter in
 section 3.3).



Category theory, understood as a branch of pure mathematics, has
frequently been described as a "general theory of mathematical
structures", e.g., by Mac Lane (1986, 1996). But what exactly is
meant by "structure" here? At least two relevant notions
are mentioned in the literature. First, a structure can be understood
in the set- and model-theoretic sense, i.e., as a tuple consisting of
a domain and an ordered sequence of relations, functions, and
distinguished elements used for the interpretation of a formal
language. (This is the notion of "relational system" we
appealed to above more informally.) Such structures are often called
"Bourbaki structures" in this context. Their properties
are usually defined axiomatically, e.g., by the group axioms or the
Dedekind-Peano axioms for arithmetic.



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



Why might category theory be considered a more adequate framework for
a mathematical structuralism than other disciplines, in particular
traditional (Cantorian, Zermelo-Fraenkel) set theory? It is helpful to
consult Awodey (1996) on this issue. According to him, the Bourbaki
notion of structure is a direct result of the modern axiomatic
tradition of Dedekind, Hilbert, and the Bourbaki group. This tradition
eventually led to a structuralist perspective on mathematics. Yet set
theory is not an ideal framework for capturing a structuralist
understanding of mathematical objects. To begin with, set theory is
closely tied to a model-theoretic conception of mathematical theories,
including the view that such theories study their models only
"up to isomorphism". But central to a structuralist point
of view is the principle (more on which below) to "identify
isomorphic objects"; and this principle is well motivated from a
category-theoretic viewpoint, but less so if mathematical objects are
represented set-theoretically. 



A second advantage of category theory over set theory, also mentioned
in Awodey (1996), is that the categorical notion of structure is
"syntax invariant". That is to say, unlike in standard
model theory the categorical specification of objects in terms of
their mapping properties is independent of the choice of a particular
signature used for their description (a choice of basic relations,
functions, and distinguished elements). Third and most importantly,
characteristic for category theory is its focus on morphisms and
transformations between mathematical objects that preserve (some of)
their internal structure. It is the emphasis on structure-preserving
mappings in the specification of objects that is often viewed as a
central feature of the structuralist turn in modern mathematics. As
such, it was present in various parts of nineteenth- and early
twentieth-century mathematics, including Galois theory, Klein's
Erlangen program, Dedekind's foundational writings, as well as
in work on abstract algebra by the Noether school (cf. again Reck
& Schiemer forthcoming).



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


### 3.2 Categorical Foundations and Debates About Them



As has been argued repeatedly in the literature along the lines just
summarized, category theory provides a more natural framework than
traditional set theory for mathematical or methodological
structuralism in mathematics. But what about its prospects as a form
of philosophical structuralism, i.e., an alternative to the theories
of Resnik, Shapiro, Hellman, etc.? We already mentioned that Awodey
(1996) also presents it as such; but this has led to ongoing
controversies. Hellmann (2003) contains a first critical discussion
of philosophical claims such as Awodey's. That is to say,
Hellman's article raises several objections against the view
that category theory provides an adequate framework for a
structuralist account of mathematics in the philosophical sense. As we
will see, these objections are closely related to the status of
category theory as a foundational discipline.



Recently there has been much debate on which criteria a theory has to
meet in order to serve as a proper "foundation" for
mathematics. According to a helpful proposal in Tsementzis (2017), a
foundational system has to consist of three items, namely: 


1. a formal language;
2. an axiomatic theory expressed in that language; and
3. a rich universe of objects, described by the theory, in which all
mathematical structures can be located, represented, or encoded.



Zermelo-Fraenkel set theory clearly represents a foundational system
in this sense. The axioms of ZFC are usually formulated in a formal
first-order language; and they describe a comprehensive universe, the
cumulative hierarchy of sets, in which mathematical objects such as
number systems, groups, rings, topological spaces, etc. can be
represented.



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



This puts us in a position to return to Hellman's 2003
challenge. The question of whether category theory can be used to
formulate a version of philosophical structuralism is, in his view,
directly related to the presumed autonomy of these new approaches from
traditional set theory. Building on Feferman (1977), he formulates two
general objections. Following Linnebo & Pettigrew (2011), we can
call the first the "logical dependence" objection. Its
core is the argument that category theory, general topos theory,
etc. are not autonomous of set theory in the end. The reason is that
the axiomatic specification of categories and topoi presupposes the
primitive concepts of operation, collection, and function, and the
latter need to be defined in a set theory such as ZFC. Categorical
foundations are therefore dependent on non-structural set theory.



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



In contrast, category theory represents a branch of abstract algebra,
as its origin reveals. Thus it is, by its very nature, non-assertoric
in character; it lacks existence axioms conceived as truths about an
intended universe. For example, the Eilenberg-Mac Lane axioms of
category theory are not "basic truths simpliciter", but
"schematic" or "structural" in character.
They function as implicit definitions of algebraic structures, similar
to the way in which the axioms of group theory or ring theory are
"defining conditions on types of structures". This point
is related to another argument against the autonomy of category theory
that Hellmann calls the "problem of the 'home
address': where do categories come from and where do they
live?" (2003: 136). Given the "algebraic-structuralist
perspective" underlying category theory and general topos
theory, its axioms make no assertions that particular categories or
topoi actually exist. Classical set theories, such as ZFC with its
strong existence axioms, have to step in again in order to secure the
existence of such objects.



Hellman's and Feferman's arguments against the
foundational character of category theory have been examined from
various angles in the subsequent literature. One can distinguish
between two main types of responses, namely:


1. by proponents of "categorical 
foundations", who aim
to defend the autonomous character of category theory relative to
classical set theory; and
2. by "non-foundationalists", 
who call into question that category theory should be viewed as a foundational 
discipline.



A series of articles by McLarty represent the first line of response
well (e.g., McLarty 2004, 2011, 2012). Roughly speaking, his reply to
Hellman is the following: Whereas category theory and general topos
theory did originate as algebraic theories and are, thus, not feasible
as foundational systems, certain theories of particular categories and
toposes have been introduced as alternative
foundations. McLarty's central examples are Lawvere's
axiomatizations of the category of categories and his
"Elementary Theory of the Category of Sets" (ETCS).



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



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



In fact, according to Awodey the central motivation for a
category-theoretic approach to mathematics is to sidestep foundational
issues concerning the nature of mathematical objects or the study of a
single comprehensive universe in which all structures can be
represented. While topos theory, say, might well serve as a structural
foundation for mathematics, for Awodey such a foundational approach
runs against the structuralist perspective embodied in
category-theory. In his own words, "the idea of 'doing
mathematics categorically' involves a different point of view
from the customary foundational one" (Awodey 2004: 55). In light
of such fundamental, ongoing debates about categorical foundations for
mathematics, what are the implications for categorical structuralism
in the philosophical sense, along Awodey's lines and more
generally? This is what we turn to next.


### 3.3 Distinctive Features of Categorical Structuralism



Beyond the issue of methodological structuralism, the literature on
categorical structuralism during the last 20 years centers around two
questions that have already been mentioned. First, in what sense does
category theory provide a framework for philosophical structuralism?
Second, why is it better suited for that task than other frameworks,
such as set theory, Shapiro's structure theory, and
Hellman's modal logic, as has been claimed? In recent work on
these topics, one can find three related philosophical assumptions
that characterize categorical structuralism and distinguish it from
the versions of structuralism surveyed earlier. We will treat each of
them in turn, starting again with Awodey's writings.



A first characteristic assumption is that all mathematical theorems
are schematic statements that have a conditional form. This point is
explicit in Awodey (2004). We already saw that according to Awodey a
category-theoretic approach is non-foundational in character. This
includes that mathematical axioms and theorems, as expressed in
category theory, should be understood as schematic statements. They do
not express truths about the specific nature of mathematical objects,
but rather, about their respective properties and relations. In
addition, mathematical theorems are, at least in principle, all of
hypothetical form. They can be reconstructed as if-then
statements. Note that this view of the logical form of mathematical
theorems is *prima facie* similar to the if-then-ism one can
find in Putnam's works, and earlier in Russell's, as
mentioned in section 1.
 



However, there is also an important difference between standard
if-then-ism and a category-theoretic approach in terms of the
ontological commitments involved, as Awodey points out. According to
standard if-then-ism, any mathematical statement can be translated
into a universally quantified conditional statement, where the
quantifiers are effectively meta-theoretic in nature, ranging over all
set-theoretic systems of the right type. As such, the approach
presupposes a rich ontology of sets in which such systems can be
constructed. In contrast, along category-theoretic lines mathematical
theorems do not involve such ontological commitments. There is no
implicit generalization over the Bourbaki structures of a theory,
e.g., over all groups, rings, or number systems. Rather, a
mathematical theorem is "a *schematic* statement about
a *structure* [...] which can have
various *instances*" (Awodey 2004: 57). These instances
remain undetermined on purpose, unless a further specification of them
is needed for the proof of the theorem in question.



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



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



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



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



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



As McLarty and others have argued, this observation generalizes from
numbers to all other mathematical objects studied in category theory.
The claim is that any property of the objects in a given category that
is expressible in the language of category theory is structural in the
sense of being isomorphism invariant. The main consequence for a
structuralist conception of mathematical objects is summarized by
Awodey as follows: 



> 
> 
> Since all categorical properties are thus structural, the only
> properties which a given object in a given category may have, qua
> object in that category, are structural ones. (Awodey 1996: 214) 
> 
> 
> 



This would seem to illustrate a main advantage of categorical forms of
structuralism over set-theoretic and similar ones.



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



The technical results just mentioned are clearly relevant for our
discussion, but we cannot explore them further in this general survey.
(Cf. Awodey 2014 for more on the relation between structuralism and
the Univalent Foundations project.) We also have to leave a further,
more definite evaluation of the philosophical claims and arguments
concerning categorical structuralism to another occasion, interesting
and important as that debate is. Instead, we now conclude with some
more general remarks about structuralism, even beyond the philosophy
of mathematics.


## 4. Conclusion


### 4.1 The Varieties of Mathematical Structuralism



There were two main goals for this entry. The first was to introduce
readers to the general discussion of structuralism in contemporary
philosophy of mathematics. The second was to provide a richer, more
inclusive taxonomy for that discussion, including making evident that
a much larger variety of structuralist positions has played a role in
it than is usually acknowledged. To be sure, a certain amount of
variety has been recognized before, as reflected in the distinction
between eliminative and non-eliminative positions, with
Shapiro's *ante rem* structuralism and Hellman's
modal structuralism as paradigms; and categorical structuralism has
been acknowledged as a third main alternative. But the range of
positions at play is much greater than that, especially in the recent
literature. "Mathematical structuralism" is not the name
of a single position, but of a multifaceted family of them.



There are two main reasons for a more inclusive taxonomy. The first
is that, while numerous versions of structuralism for mathematics have
been proposed, several of them have not received much attention yet,
partly because their relationships to Shapiro's,
Hellman's, and category-theoretic positions have remained
unclear. The taxonomy provides some corresponding bridges. The second
reason is that several important variants played a role already before
the 1960s, i.e., before the publications by Benacerraf and Putnam that
are often seen as the starting point for debates about structuralism
in philosophy. This concerns both methodological and metaphysical
structuralism, a basic distinction we highlighted; but we only hinted
at the relevant "pre-history" of our topic in the present
survey (We refer the reader to Reck & Schiemer forthcoming for
more on it.)


### 4.2 Structuralism Beyond Mathematics



Another relevant aspect that is not explored in the present entry,
although it should be mentioned at least briefly before closing,
consists of debates about "structuralism" outside the
philosophy of mathematics. There are two main areas in which one can
find such debates (although they have reverberated further). The first
is the philosophy of physics, where structuralist positions have
played a significant role as well (including one called
"structural realism", which comes in both an
"ontic" and an "epistemic" form). The second
consists of several parts of the humanities and social sciences,
primarily linguistics and anthropology, but also psychology,
sociology, etc. There too, "structuralism" (and
"post-structuralism") has been a major topic, indeed for a
relatively long period. In both cases, there are ties to mathematical
structuralism, although sometimes these ties are only loose.



At a basic level, the debate about structuralism in the philosophy of
physics concerns how to think about the "objects" of
modern physics, given the revolutionary changes quantum mechanics and
relativity theory have brought with them. More particularly, it
concerns the suggestion to refer to "structures" in this
connection (cf. French 2014, Ladyman 2007 [2019], etc.), in a way that
is closely related to the philosophy of mathematics. At another level,
this debate concerns how to conceive of what, if anything, remains
constant through theory changes in the ontology of physics
(cf. Worrall 1989, and see the
 entry on structuralism in physics
in this encyclopedia). Third, there are debates about representation 
in science that are related to structuralism (cf. van Frassen 2008). While 
there are several points of contact one could pursue further (e.g., concerning
"structural indiscernibles" 
and versions of "set-theoretic versus 
category-theoretic structuralism" in the context 
of physics), we refrain from doing so here.



The structuralism introduced into linguistics by Ferdinand de Saussure
and Roman Jakobson, then adopted by other thinkers in the humanities
and social sciences--most prominently by Claude Levi-Strauss in
anthropology and by Jean Piaget in psychology (cf. Levi-Strauss 1958
[1963], Piaget 1968 [1970], also Caws 1988)--does have ties to
mathematical structuralism as well. However, they are looser than in
the case of the philosophy of physics. In addition, the kind of
psychological determinism often associated, or even identified, with
structuralism in the humanities and social sciences (and criticized in
post-structuralism) has no analogue on the side of mathematics (or
physics). Still, these connections are not totally uninteresting
(starting with contacts between Levi-Strauss and mathematicians in the
Bourbaki group, cf. Dosse 1991-92 [1997]). Hence they might be
worth exploring further too, in this case not so much to derive
benefits for the philosophy of mathematics, but to understand the
history of structuralism in human thought more fully.