3. Discussion

I turn now to Hudetz’s claim that “it is utterly impossible that a natural number $n$ could serve the same representational purposes as a vector space of dimension $n$ ” (Hudetz Reference Hudetz2019a, 52). Let us suppose—as Hudetz suggests—that we wish to model the forces on an object by using a three-dimensional vector space $V$ , with vector summation corresponding to the superposition of forces. Can we indeed “replace” the vector space $V$ with the natural number $3$ ? Per the previous discussion, this replacement would presumably mean using $3 \times 1$ matrices to represent the various forces, with superposition of forces represented by matrix addition; does this properly “count” as using the natural number $3$ to represent the empirical situation?

Fully answering this question would mean engaging with the extensive literature on scientific representation, which is not going to be possible in a short discussion note like this.Footnote ³ To make things manageable, then, let us restrict our attention to so-called “structuralist” accounts of representation—that is, accounts that “take models to be structures, which represent their target systems in virtue of there being some kind of morphism (isomorphism, partial isomorphism, homomorphism, $\ldots$ ) between the two” (Frigg and Nguyen Reference Frigg, Nguyen and Edward2018, n.p.). The structuralist account has its critics, but it is a reasonably popular approach and certainly seems like a natural place to try to get a grip on the significance of internal structure for the purposes of representation. For the sake of simplicity (and ease of connection with the literature), I will focus on the case where the relevant morphism is taken to be an isomorphism.

Evidently, the content of the structuralist account is then going to depend on how the notions of structure and isomorphism are understood. Frigg and Nguyen (Reference Frigg, Nguyen, Magnani and Bertolotti2017) observe that these are usually explicated in a sense familiar from set theory and mathematical logic, through definitions like the following:

How the isomorphism relation is set up between the model and the target system is then a further point of controversy, with several proposed answers; perhaps the simplest suggestion is that target systems instantiate structures, by virtue of having parts that stand in appropriate relations to one another, and that the isomorphism (if it exists) holds between the model and this physically instantiated or realized structure.Footnote ⁴ In the context of our example, the “parts” of the target system are presumably the individual forces: these stand in structure-realizing relations (e.g., the ternary relation “ $x$ is the superposition of $y$ and $z$ ”) and thus form a structure to which the model is—hopefully—isomorphic.

I conjecture that this account of representation, or something like it, is at work in Hudetz’s assertion that $3$ cannot serve the same representational purposes as the three-dimensional vector space $V$ (Hudetz Reference Hudetz2019a): the problem, so the thought goes, is that $3$ is not capable of standing in the appropriate kind of isomorphism relation to the physical situation. It seems to me that there are two ways in which this assertion could be backed up. On the one hand, it might be maintained that $3$ simply is not a “structure” in the sense laid out previously (i.e., a composite entity consisting of a set of objects equipped with relations): it is an element of a certain structure or structures (e.g., the natural or real numbers) but is not itself a structure. On the other hand, one could admit that $3$ is or may be taken to be a structure but nevertheless argue that it is not a structure that is isomorphic to $V$ . For instance, perhaps we can take $3$ to have the structure of the relevant von Neumann ordinal $\{\varnothing, \{\varnothing \}, \{\varnothing, \{\varnothing \}\} \}$ or the Zermelo ordinal $\{\{\{\varnothing \}\}\}$ , but these are both finite sets, so no bijection could be set up between either of them and the infinite-cardinality structure $V$ .

Thus, in resisting Hudetz’s (Reference Hudetz2019a) assertion, we must also resist this account of representation.Footnote ⁵ But if one is attracted to category-theoretic accounts of the structure of scientific theories, then this is a very natural move: the reasoning is explicitly based on resources from set theory, after all.Footnote ⁶ From a more categorical perspective, it is just as true that the three-dimensional vector space $V$ , as an object of $\mathsf{FinVect}$ , is not a composite entity consisting of a set of objects equipped with relations. Thus, what makes the object $V$ an appropriate representation of the physical situation is not the existence of a bijection between the parts of that situation and the elements of $V$ . What does make it an appropriate representation is a somewhat tricky question; here are two possible answers.

First, the “conservative” answer. Although $V$ does not possess elements in the set-theoretic sense, it does possess elements in a generalized sense—if “elements” are understood as morphisms from some suitable “test object” (as is often done in category theory), then we can take the elements of $V$ to be the morphisms from $\mathbb{1}$ to $V$ that were employed earlier. Moreover, we have seen that the vector-space structure of $V$ is recoverable from the categorical facts about these morphisms. It is therefore coherent to require (in order that $V$ be a representation) that there be an appropriate isomorphism between the physically instantiated structure and this vector-space structure. Thus, we can borrow the structuralist’s account of what makes for representation, modulo the appeal to a more generalized understanding of what the “elements” of a structure are.

And of course, this more generalized understanding of elements opens the door to admitting the number $3$ (at least qua element of $\mathsf{Mat}$ ) as an appropriate representation of the physical situation. It admits of elements in just the same manner that $V$ does, and as we have seen earlier, those elements (namely, $3 \times 1$ matrices) are just as good at being correlated to forces for representational purposes as the elements of $V$ are. Of course, qua an object of a different category, $3$ will not carry this representational capacity. For instance, $3$ will not be suitable for representing the physical situation as an object of an $\omega$ -sequence $\mathsf{Nat}$ (which, being a partial order, we may regard as a category). But from the present perspective, this is a feature, not a bug. Regarded as an object of $\mathsf{Nat}$ , the number $3$ is also not equivalent to a three-dimensional vector space because there is no categorical equivalence between $\mathsf{Nat}$ and $\mathsf{FinVect}$ . For that matter, there is no categorical equivalence between $\mathsf{Nat}$ and $\mathsf{Mat}$ , so the number $3$ is not always equivalent to itself, nor does it always have the same representational capacities as itself—just as the sign “die” does not have the same representational capacities as a word in the English language and as a word in the German language.

This account invites the following objection: that in such a case, it is not $3$ itself that is doing the representing but rather some substructure of the category $\mathsf{Mat}$ (because the forces are being correlated to certain elements of $\mathsf{Mat}$ , i.e., the morphisms from $1$ to $3$ ).Footnote ⁷ However, this objection only works if we insist that elements must be understood in the set-theoretic, rather than the category-theoretic, fashion—and to make such an insistence is to beg the point at issue. In category-theoretic accounts of scientific theories, we should expect the representational capacities of any object to be relative to the category that it is presented as being a part of; just as in the set-theoretic account, the representational capacities of a certain set are relative to what structures and operations it has been equipped with.

Let us turn now to the second, more radical, answer to the question of what it is that could make an object of a category (e.g., $V$ or $3$ ) an appropriate representation of a certain concrete physical situation. This will be somewhat more vague and sketchy, but I find it intriguing, so I include it in case other readers of this journal find it stimulating to consider. This more radical answer gives up on the idea that the representation relation is based on any kind of correspondence between the parts, pieces, or components of the situation and the representation. Rather (in this view), the situation is appropriately represented by $V$ because it is a situation of a certain kind, and the structural features of the whole collection of situations of that kind make them apt to be represented by the category of which $V$ is an object. For instance, if we can consider all the situations featuring forces on an object as a category, then we can propose that these situations can be represented by vector spaces just in case there is an appropriate relationship between that category and the category $\mathsf{FinVect}$ . An isomorphism of categories is not a very natural candidate: it would be more natural to require that the categories be equivalent, or perhaps merely that one category be equivalent to a subcategory of the other. Evidently, if we do so, then representational appropriateness will be preserved across categorical equivalence: that is, $\mathsf{FinVect}$ will be appropriate for representation if and only if the equivalent category $\mathsf{Mat}$ is. Note that this holds independently of the considerations in §2; in other words, the reconstruction of vectors in the categories $\mathsf{FinVect}$ or $\mathsf{Mat}$ is only playing a dialectical role in the first (more conservative) answer.

This second proposal foregrounds the significance of modal structure. In effect, the idea is to consider a category of possible situations (rather than an individual situation) and hold that representation arises when that category of possibilities is appropriately similar to the category being used for representation.Footnote ⁸ We might, therefore, think of it as a special case of a more general account of how state- or possibility-spaces represent their targets. In such a space, each individual possibility is represented by something lacking internal structure: a point of a configuration space, a vector in a Hilbert space, an element of a Boolean algebra, and so forth. However, that does not make such a space representationally hopeless because the space itself can carry a rich enough structure to support our representational practice.Footnote ⁹ This structure must, however, be understood as imputing properties or relations to the individual possibilities (as wholes) rather than to whatever individuals might populate one possibility or another.Footnote ¹⁰

This then raises the question of what (if anything) might be significant about the case where that modal space is represented as a category. Here we should remember that, by virtue of the Yoneda lemma, the objects of a category are fully determined by their morphisms: as Spivak puts it, “categories can only model ‘relationally-determined’ subjects, subjects in which each object of study is ontologically determined by its relationships to the others” (2017, 393). So the modal spaces that category theory is capable of describing will be those where the relationships between possibilities encapsulate everything there is to know about those possibilities. This suggests a potentially very novel way of thinking about possibility and ontology: one where the differences between possibilities, rather than the intrinsic character of those possibilities, are primary.Footnote ¹¹ Clearly, however, working out such a view will have to be a project for another time.