Structuralism in philosophy of mathematics is the view that mathematical statements rather than describing objects, describe only structures on objects, and that mathematical entities as such are or positions in a structure, rather than objects/entities in their own right. This view has been around in some form for most of the history of contemporary philosophy of mathematics, and has been defended by figures such as Dedekind, Hilbert, Poincaré, Quine, Benacerraf, and others (Colyvan 2012, 40).
However, following (Benacerraf 1965), interest in structuralism was renewed due to its potential as a solution to Benacerraf’s underdeterminacy problem. This problem, simply put, consists in the fact that in the standard foundation of mathematics, the Zermelo/Fraenkel axioms (ZFC), all mathematical objects are reducible to sets, and yet there are often multiple equally valid choices for which sets a collection of objects should be reducible to. Famously, Benacerraf used the example of the Zermelo and Von Neumann constructions for the natural numbers. Both systems identify the natural number $0$ with the empty set, but the constructions differ in their choice of successor function. In the Zermelo construction $\textbf{succ}(x)=\{ x \}$ and in the Von Neumann construction $\textbf{succ}(x)=x\bigcup\{ x \}$. This yields the interesting consequence that although these constructions are elementarily equivariant in the language of Peano Arithmetic - both systems prove exactly the same theorems - one system, the Von Neumann construction, would have it that $1\in 42$, the other not. This underdeterminacy problem presents a substantial barrier for any prospect of an ontological reduction of numbers into sets, and has largely caused the view that numbers are sets to fall into disrepute (Colyvan 2012).
Structuralism promises an exceedingly elegant and straightforward solution to the problem. Instead of identifying numbers with any particular object - abstract or not - they should instead be thought of as referring to positions in the structure of the natural numbers, which in turn is instantiated in both the Zermelo and Von Neumann constructions. Being the number $2$ is to stand in a certain relation to a surrounding structure, and both $\{ \{ \emptyset \} \}$ and $\{ \emptyset,\{ \emptyset \} \}$ stand as such to their respective constructions. Not only does this avoid underdeterminacy in the choice of which structure should be the one, but it seems at the face of it that only some sort of structuralism is able to avoid having to justify one construction or another as the canonically correct choice, which constitutes an inherent advantage over other accounts of the ontology of mathematics. Furthermore, structuralism is also uniquely harmonious with mathematical practice. As noted by (Colyvan 2012)

; "Structuralism is able to explain why mathematicians are typically only interested in describing the objects they study up to isomorphism - for that is all there is to describe."
Structuralism is, of course, not without issues of its own, but exactly what these are can be hard to pin down given the sheer variety of different versions of structuralism that have been proposed. Broadly speaking, most fall into one of two camps; realist or ante rem structuralism a la (Shapiro 1997) or eliminative/in re structuralism a la Hellmann (G. Hellman 1989) (Reck and Schiemer 2025). Both positions of course have to deal with the usual problems that realist (respectively anti-realist) philosophies of mathematics have to contend with, e.g. the dilemma posed in (Benacerraf 1973). And yet, in both cases the arguably bigger issue lies in spelling out what exactly counts as a structure and when two structures should be considered the same. It should therefore come as no surprise that alternative foundations for mathematics such as category theory and homotopy type theory, that are inherently structuralist in nature have recently been of great interest in the contemporary discourse on the philosophy of mathematics.

The Discussion So Far

The question was discussed in (Geoffrey Hellman 2003). Just one year later an answer was given in (Awodey 2004) - that answer being: ¹. The reason for the terseness of this reply being that Hellman’s doubts stem largely from concerns about whether category theory makes for an autonomous foundation for mathematics. A question that -(Awodey 2004) points out - is separate from the question of whether category theory can give a precise notion of the term on top of which a formal structuralist system can be built. This, (Awodey 2004) concludes, should be relatively obvious, and indeed a structuralist need not argue for an outright replacement of the ZFC with a structuralist alternative in order for their argument to succeed.
However, a successful argument for the replacement of ZFC with one of the proposed structuralist alternatives would still be a strong point in structuralism’s favor, and as such I claim the questions raised by Hellman are still relevant for structuralists to consider. The issue of whether category theory specifically makes for such an autonomous foundation has been discussed at length ever since the publication of Lawvere’s Elementary Theory of the Category of Sets (ETCS) (Lawvere 1964) and has more recently been given a comprehensive treatment by (Linnebo and Pettigrew 2011). They argue that a (worthwhile) alternative foundation to set theory must be logically, conceptually, and justificatorily autonomous from ZFC. They conclude that the ETCS - as well as a few other proposed categorical foundations - indeed enjoy both logical and conceptual autonomy, but that their justificatory autonomy hinges on the extent to which the practice of working mathematicians may be appealed to as justification for its adoption.
More recently (Tsementzis 2017) has made an intriguing case that categorical foundations such as the ETCS are not actually structuralist in a strict sense and has argued convincingly that the Univalent Foundations (UF) based on homotopy type theory provide a better alternative foundation that is truly structuralist. He subsequently outlines a formal method for translating standard results from mathematics into the structural language by means of UF. This method does not allow us to translate all the mathematics, as indeed UF is a constructive system unlike ZFC, but it does translate quite a lot, and according to (Tsementzis 2017) it is open ended enough that it can always be expanded in the future. Although there will inevitably be some compared to ZFC, this is an intended consequence of using a constructive system, and many mathematicians will probably consider this a point in UF’s favor.
In the essay this article is based on, I try to evaluate where the ETCS and UF respectively stand in light of the ongoing discussion, but due to this format I will not be able to do that adequately here. Instead, I will spend the remainder of the article making it a bit more explicit what it means for a mathematical framework to be structuralist.

What Does it Mean to be Structuralist?

As noted already, there is a panoply of different structuralist positions that all conceptualize structures slightly differently, making it impossible to consider them all here. However, both (Shapiro 1997) and (Tsementzis 2017) have adopted a view inspired by the Quine slogan: according to which the relevant issue is not what structures are, but when structures should be considered identical. And in this regard, we are in the fortunate position that mathematicians already have a rigorous notion of structural identity: isomorphism.

Building on this idea, (Tsementzis 2017) argues that whether or not a system is structuralist is not dependent on what our ontology of structures are, but whether or not our formal system considers isomorphic objects to be identical (SFOM). By this there can be little doubt that UF is a structuralist system. Indeed, this is baked essentially into the core of the program in the form of the axiom of univalence. This axiom states;

Or in words; for any two types $A,B$ the type of ‘proofs of identity between $A$ and $B$’ is isomorphic to the type of ‘proofs that $A$ and $B$ are isomorphic’, or as (Tsementzis 2017) succinctly puts it: . In UF, equality and identity are simply equivalent. Tsementzis further notes that this notion of identity is so strongly structuralist that at first it appears ungrammatical. Indeed, how can what appears to be not a structure, but a statement - that $A=B$ - be isomorphic to anything? The answer being simply that, in UF, $(A=_\mathcal{U}B)$ just is a structure. The statement is simply the same as . In UF, structure is quite literally all there is to describe.
The Elementary Theory of the Category of Sets of (Lawvere 1964) does not contain anything quite like the axiom of univalence, but it is still often considered a structuralist foundation. Briefly put, the ETCS is a recreation of standard set theory within a category theoretical framework. As such, sets are not determined by their members in the ETCS, but rather characterized in terms of morphisms and universal properties. Starting with the latter, a universal property is something that specifies an object/construction up to isomorphism. For instance; the object 1 is identified not as some particular set, but rather defined in terms of the properties shared only by sets with exactly 1 element. Such sets are called terminal objects and are defined thus

::: definition Definition 1 (from (Leinster 2012)). A set T is called terminal if for every set $X$, there is a unique function $X\longrightarrow T$. :::

Here, it should be quickly apparent that, though there are many sets that obey the definition of terminal objects, all of these will be mutually isomorphic to each other. The significance of this is that the object 1 is precisely the category of things isomorphic to 1 (or $\{ \emptyset \}$ or $\{ \circ \}$ and so on). Further, the notions of sum and product are similarly specified in category theory by means of universal properties.
\

:::: minipage ::: definition Definition 2 (Categorical product). *Let $X,Y$ be sets. A product of $X$ and $Y$ is a set $X\times Y$ together with a pair of projection functions $X\longleftarrow_{p_1}X\times Y \longrightarrow_{p_2} Y$ such that for any set and pair of functions $X\longleftarrow_{f_1} I \longrightarrow_{f_{2}} Y$ the following diagram commutes
* ::: ::::

:::: minipage ::: definition Definition 3 (Categorical sum (or co-product)). *Let $X,Y$ be sets. A sum of $X$ and $Y$ is a set $X\coprod Y$ together with a pair of injections $X\longleftarrow_{in_1} X\times Y \longrightarrow_{in_2} Y$ such that for any set and pair of functions $X\longrightarrow_{f_1} Q \longleftarrow_{f_2} Y$ the following diagram commutes
* ::: ::::

It is worth mentioning that the cartesian product and disjoint union operations are categorical products and sums, respectively. Once again, notice that this specifies the respective notions up to isomorphism, so for instance the set 2 is specified up to isomorphism as $\textbf{1}\coprod \textbf{1}$, which means that the number 2 is similarly the class of objects isomorphic to $2$ (or $\{ \emptyset,\{ \emptyset \} \}$)², and subsequently all the natural numbers may be similarly defined. This certainly seems like a thoroughly structuralist system, so are we safe to conclude that the ETCS is a structuralist foundation for mathematics?

As alluded to earlier, the main source of doubt on this matter again comes from (Tsementzis 2017), who argues that not only does the ETCS not in fact satisfy the SFOM, it in principle cannot. This is for the simple reason that the ETCS is, by design, a reconstruction of standard ZFC within a category theoretic framework, which means that violations of SFOM in ZFC carry over into the ETCS. For an easy example of such a violation, he gives the example that while $\mathbb{Z}$ and $2\mathbb{Z}$ are isomorphic, we of course have it that $\textbf{ZFC}\models 1\in \mathbb{Z}$ but $\textbf{ZFC}\not\models 1\in 2\mathbb{Z}$. Therefore, set theory as a foundation is indeed not structuralist in the sense of (Tsementzis 2017) because isomorphic structures can and do validate different formulations in the language of set theory, and as such, structural identity is different from actual identity. But these examples carry over almost verbatim to the ETCS.
Does this mean that category theory is not a structuralist foundation for mathematics? Not in the sense used by Tsementzis, but it should be noted that there are valid reasons to think that his sense is a bit too strict. First, the ETCS is at least structuralist, in the sense that the objects of the theory - the objects that everything else is defined in terms of - are indeed objects as they are specified only up to isomorphism. Arguably, being structuralist in this sense is sufficient for a philosophical case for structuralism.

Secondly; it is not entirely obvious to me that (Tsementzis 2017) is right to claim that a structuralist system must always consider all isomorphic objects as elementarily equivalent. Tsementzis is of course right to point out that isomorphism sometimes results in a loss of information in ETCS, and that this does seem unexpected in a structuralist system, but I believe that it is worth pointing out that these cases are restricted to situations where a reduct of the language has taken place. What I mean by this is that we know from model theory that if two $\mathcal{L}$-structures $\mathcal{M},\mathcal{N}$ are isomorphic, then they are elementarily equivalent, i.e. for any $\mathcal{L}$-sentence $\varphi$ we have $\mathcal{M}\models \varphi$ if and only if $\mathcal{N}\models \varphi$. In other words, when the two structures in the same language are isomorphic, they preserve information as we would expect them to, with the caveat that it is only the information expressible in $\mathcal{L}$. In other words, the SFOM is recoverable for ETCS if we add this single stipulation. I believe this is a rather small concession.
In summary; we have two promising candidates for a structuralist foundation for mathematics. One constructivist, the other of the same proof theoretic strength as ZFC. One structuralist in a quite strong sense, where isomorphic objects are literally indistinguishable, the other in a slightly more fine grained sense that allows for some language specific discrepancies, but still specifies its primitive objects only up to isomorphism. Both of which would lend themselves to quite a strong case for mathematical structuralism.

Bibliography

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———. 1973. “Mathematical Truth.” Journal of Philosophy 70 (19): 661–79. https://doi.org/10.2307/2025075 .

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