2.1. Representation

Scientific models represent their target systems. This much is uncontroversial. How they do so is not. Two popular views are that scientific models are, in some sense, intended copies of their target systems. For the likes of Giere (Reference Giere2004) and Weisberg (Reference Weisberg2013), model users intend that their models are similar (in the relevant respects, to the appropriate degree) to their targets. If they are so similar, then the model is an accurate representation of its targets, at least with respect to the intended features and to the appropriate degree. For the likes of van Fraassen (Reference van Fraassen2008) along with, perhaps,Footnote ¹ French and his collaborators (e.g., Da Costa and French Reference Da Costa and French2003; French Reference French2003, Reference French2014; Bueno and French Reference Bueno and French2011), models are mathematical structures, and model users intend them to be appropriately morphic to their target systems or data gathered from them. If such a morphism exists, then the models are accurate representations of their targets, at least with respect to the structural features of the target covered by the morphism in question.

Both approaches face serious challenges (Suárez Reference Suárez2003; Frigg Reference Frigg2006). And accounts of scientific representation that diverge from the copy-making accounts have emerged as attractive alternatives (Hughes Reference Hughes1997; Suárez Reference Suárez2004; Contessa Reference Contessa2007; Frigg Reference Frigg, Frigg and Hunter2010). According to Hughes’s (Reference Hughes1997, S328) “DDI” account, models denote their targets and allow model users to perform demonstrations on models, whose results are then interpreted in terms of their target systems. Suárez (Reference Suárez2004, 773) submits that models represent their targets only if their representational force “points” toward their targets and they allow competent model users to draw inferences about their targets. Contessa (Reference Contessa2007, 57–58) argues that models denote their targets and that parts of models are interpreted as (purported) parts of their targets. Frigg (Reference Frigg, Frigg and Hunter2010, 126–32) agrees about denotation and adds that models come with keys that allow users to “translate” model facts into target claims.

Although there are important differences between these ‘interpretational’ accounts, for my current purposes what is important is that they all agree that the primary purpose of scientific models is to license inferences about their target systems, and the way they do this is a function on model facts (Hughes’s demonstration, the premises of Suárez’s inferences, the model part of Contessa’s interpretation, and the arguments of Frigg’s keys) combined with intentional acts of model users interpreting these facts in terms of their target systems (Hughes’s and Contessa’s interpretations, Suárez’s inferential schema used, and Frigg’s keys). I do not argue for them here; I want to explore their insights for questions that invoke the representational capacities of scientific models.

2.2. Equivalence

When are two scientific theories, T ₁ and T ₂, equivalent? This, obviously, depends on what scientific theories are. If they are sets of formulas, formalized in an appropriate language, then they are logically equivalent if and only if (iff) their semantic closures are identical. But this requires that they are formulated in the same signature and thus might be too strict a criterion. Alternatively, they are definitionally equivalent iff for each term in the signature of T ₁, a sentence can be constructed defining it in terms of the signature of T ₂, and mutatis mutandis for each term in the signature of T ₂, and the result of adding the respective definitions to each theory provides two expanded theories that are logically equivalent (Glymour Reference Glymour1970; see also Barrett and Halvorson Reference Barrett and Halvorson2016).

Alternatively, if two theories are construed as collections (in a sense to be specified below) of models, then an alternative approach is required. Halvorson (Reference Halvorson2012) provides three options available to someone who wants to define theoretical equivalence in this context. Two theories T ₁ and T ₂, construed as sets of models, are theoretically equivalent iff (a) T ₁ and T ₂ have the same cardinality, (b) there is a pointwise isomorphism between T ₁ and T ₂ (i.e., a bijection f such that for each model $M\in T_1$, $f\left(M\right)=M^{\prime }$, where $M^{\prime }\in T_2$, and M and M′ are isomorphic), or (c) $T_1=T_2$. Only b is remotely plausible, but it requires specifying the sense in which M and M′ can be isomorphic to each other (see Halvorson Reference Halvorson2012; Glymour Reference Glymour2013; Lutz Reference Lutz2017, for useful discussions about this and whether it can be done without reference to a linguistic signature).

Halvorson (Reference Halvorson2012, Reference Halvorson and Humphreys2016) has also recently suggested an exciting third sense in which theories can be seen as theoretically equivalent.Footnote ² This can be applied in cases in which theories are construed linguistically or as collections of models. The idea in both cases is to take theories to be categories consisting of objects (formulas, models) and arrows (deductive relationships, homomorphisms, or elementary embeddings). Two theories T ₁ and T ₂ thus construed are categorically equivalent iff there is a functor F from T ₁ to T ₂ such that F is full, faithful, and essentially surjective.

Although there are clear differences between these approaches, what is important to note is that all of them focus on the formal features of the theories (construed as sets of models or formulas or categories) in question.Footnote ³

4.1. Weatherall and the Hole Argument

The hole argument runs as follows. Take a Lorentzian manifold, a model from general relativity. This is a structure of the form $〈ℳ,g_{\mathrm{ab}}〉$, where $ℳ$ is a four-dimensional differentiable manifold and g_ab a Lorentz-signature metric tensor. We suppose that $ℳ$ has a Cauchy surface S and a time orientation. Take an open region U to the time-like future of S, and let $\psi :ℳ\to ℳ$ be a diffeomorphism that is the identity for $ℳ$\U but is not the identity for U. Using the push forward of ψ to define a new metric tensor, ψ_* g_ab, we have two isometric Lorentzian manifolds $〈ℳ,g_{\mathrm{ab}}〉$ and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$. We can now ask about what these structures represent. The problem arises if we take the following conditions to hold: $〈ℳ,g_{\mathrm{ab}}〉$ and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$ both represent a space-time ST, but $〈ℳ,g_{\mathrm{ab}}〉$ represents ST accurately, and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$ does not. This substantivalist position introduces indeterminism into general relativity because we can suppose that both models are accurate representations of the past (i.e., the area of space-time represented by the “past” of S, in the sense determined by the time orientation), and yet they represent the future in incompatible ways (they cannot both be accurate representations thereof). Someone who denies the substantivalist position takes it that if two isometric Lorentzian manifolds represent the same space-time, then they do so equally accurately.

In an attempt to diffuse this argument, Weatherall (Reference Weatherallforthcoming) argues that as long as we are careful in how we understand how mathematical structures like Lorentzian manifolds represent, we can resist the idea of two Lorentzian manifolds such that one accurately represents ST and the other does not. To do so he invokes the premise that “interpretations of our physical theories should be guided by the formalism of those theories … [and] … insofar as they are so guided, we need to be sure that we are using the formalism correctly, consistently, and according to our best understanding of the mathematics” (2). He further claims that “if a particular mathematical model may be used to represent a given physical situation, then any isomorphic model may be used to represent that situation equally well” (4).Footnote ⁶ Applying this observation to the Lorentzian manifolds defined above, we arrive at the following conclusion: “once one asserts that spacetime is represent[ed] by a Lorentzian manifold, one is committed to taking isometric spacetimes to have the capacity to represent the same physical situations, since isometry is the standard of isomorphism given in the mathematical theory of Lorentzian manifolds” (15).

One way of interpreting this claim in the terminology of this article is that if two isometric Lorentzian manifolds represent the same physical situations, then one accurately represents that situation iff the other does. Under this reading, the argument runs as follows:

1. 1. If two models M and M′ represent the same target system(s) T, then if M and M′ are equivalent in the sense specified by the mathematical theory of M and M′, then M and M′ are equally accurate representations of T (premise)

2. 2. $〈ℳ,g_{\mathrm{ab}}〉$ and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$ are equivalent (isometric) in the sense specified by the mathematical theory of Lorentzian manifolds (premise)

3. 3. $〈ℳ,g_{\mathrm{ab}}〉$ and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$ both represent ST (suppose)

4. 4. $〈ℳ,g_{\mathrm{ab}}〉$ and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$ are equally accurate representations of ST (from 1, 2, and 3)

Premises 2 and 3 are uncontroversial. Premise 2 is a mathematical fact, and the idea that both $〈ℳ,g_{\mathrm{ab}}〉$ and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$ represent ST seems benign.Footnote ⁷

We might, however, put pressure on premise 1. To do so it is worth unpacking what it means for two models to be equally accurate representations of T. According to the interpretational accounts of scientific representation, this amounts to them licensing the same conclusions about T, which depends on the interpretational schema used to generate claims about T from facts about M and M′. In this framework the assumption underpinning premise 1 is that since $〈ℳ,g_{\mathrm{ab}}〉$ and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$ are isometric they should be interpreted in the same way. That is, given that they are isometric, we should restrict the domain of the interpretation schemas to the facts that they share and map these facts to the same claims about ST.

I think this claim is true, but only in contexts in which all we care about are the isometric invariant facts about ST. But as Roberts (Reference Roberts2014) notes, if this argument is supposed to block the hole argument, then it must be the case that the disagreement between the substantivalist and her interlocutor cannot be formulated: “In particular [this] prohibits isometric Lorentzian manifolds from [accurately] representing distinct physical situations for any purpose whatsoever, not even to express manifold substantivalism for the purposes of reductio. It is this presumption, that isometric Lorentzian manifolds represent the same situation equally well ‘for all purposes,’ which is supposed to prohibit the (reductio) hypothesis that two isometric spacetimes can ever [accurately] represent different things” (6).

We are now in a position to deny premise 1. The claim is that (supposing that M and M′ represent the same target system(s) T) if M and M′ are equivalent in the sense specified by the mathematical theory of M and M′, then M and M′ are equally accurate representations of T in all contexts and for all purposes. But this requires that since $〈ℳ,g_{\mathrm{ab}}〉$ and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$ are isometric, any way of interpreting them must deliver the same claims about ST. And this need not be true. Even the very same structure can be interpreted in one way in one context and another way in another.

A familiar example illustrates this. Suppose that you are in London attempting to navigate using the tube map. Someone familiar with the tube map will adopt an interpretational schema toward the map that includes the map’s topological facts in its domain but no metric facts (of the map). Someone unfamiliar with the tube map may adopt an interpretational schema that includes these metric facts as well. Under the former, the map is an accurate representation of London with respect to its topological properties and remains silent with respect to its metric ones. Under the latter, the map is an accurate representation of London with respect to its topological properties and inaccurate with respect to its metric ones. There is a clear sense in which the very same object—the tube map—can be associated with different interpretational schema, such that under one it is an inaccurate representation (with respect to metric facts about London) and under another it is not (in virtue of remaining silent about them). Requiring that equivalent, let alone identical, mathematical structures be associated with the same interpretational schema is too demanding, and thus premise 1 is false.

Weatherall (Reference Weatherallforthcoming, sec. 5) seems sensitive to this objection. He states that in order to makes sense of the claim that isometric Lorentzian manifolds differ with respect to their representational accuracy the substantivalist “needs to stipulate what the additional structure might be and why we should think it matters. And it is difficult to see how this could be done in a mathematically natural or philosophically satisfying way [footnote omitted]” (16–17). I take it that ‘additional structure’ here refers to facts about $〈ℳ,g_{\mathrm{ab}}〉$ and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$ that are not preserved by the isometry, facts that would have to appear in the domain of the substantivalist’s interpretation schemas associated with them (such facts do exist, $〈ℳ,g_{\mathrm{ab}}〉$ and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$ are not the same mathematical object by construction). But under this reading, what Weatherall’s argument establishes is as follows. Although in the context of doing relativistic physics $〈ℳ,g_{\mathrm{ab}}〉$ and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$ will be interpreted in the same way, in the context of the hole argument, the substantivalist needs to specify the facts in which $〈ℳ,g_{\mathrm{ab}}〉$ and $〈ℳ,{\psi }_{*}{g}_{\mathrm{ab}}〉$ come apart and how, via the claims they are associated with by their respective interpretational schemas, they allow for one to represent ST accurately while the other does not. When phrased like this, Weatherall’s argument is recast as challenge for the substantivalist, rather than ‘blocking’ the hole argument per se.Footnote ⁸

4.2. Formalism and the Pragmatics of Representation

I hope to have shown that thinking about interpretational schemas associated with models, rather than focusing solely on their formal properties, will allow us to properly appreciate arguments that turn on how they represent. I now briefly discuss how the notion of theoretical equivalence I introduced in section 3 relates to those outlined in section 2.2.

My point in this subsection concerns what ‘models’ should be taken to be in the context of thinking about theoretical equivalence in terms of the formal properties of models associated with theories. Recall the claim that an object (mathematical structure) is a scientific model only if it is used as such, and this requires offering an interpretational schema from model facts to target claims. In order to generate such a schema, model users have to access facts about their models. In the case of physical models, like the Phillips-Newlyn machine (Morgan Reference Morgan2012, chap. 5), this is relatively straightforward. In the case of mathematical models, less so. Facts about mathematical structures have to be determined in some way; they are not self-evident. One (but maybe not the only) way to think about how this is done is in terms of the deductive closure of the axioms that are used to pick them out. In the first-order case, since the logic is sound and complete, two sets of axioms $A$ and $B$ have the same deductive closure iff they are satisfied by the same class of structures $S$. This allows us to identify the ‘theory’ with $S$ itself and ignore the axioms used to pick them out as alternative ‘intrinsic characterizations’ of the same theory (French and Saatsi Reference French and Saatsi2006).

But this approach breaks down once we move to higher-order theories. Since (e.g.) second-order logic is not complete, there exist alternative axiomatizations of the same class of structures that have different deductive closure. Corcoran (Reference Corcoran1980) provides the following example of two alternative categorical (in the model-theoretic sense) axiomatizations of $〈\mathbb{N},s〉$.

Let $ℒ$ be a first-order language with identity including a constant ‘0’ and a monadic function ‘s’. Augment $ℒ$ with a single monadic second-order variable P. Then let $T=\left\{\left(P0\wedge \forall x\left(\mathrm{Px}\to \mathrm{Psx}\right)\right)\to \forall \mathrm{yPy},\forall x\forall y\left(sx=sy\to x=y\right),\forall x\left(\mathrm{sx}\ne 0\right)\right\}$. The set of sentences $T$ is categorically satisfied by $〈\mathbb{N},s〉$, where the interpretation function takes ‘0’ to 0 and ‘s’ to the successor function.Footnote ⁹ It is a categorical characterization of the natural numbers. Let $T^{\prime }=\{(P0\wedge \forall x(\mathrm{Px}\to \mathrm{Psx}\left)\right)\to \forall yPy,0=0,\dots ,s^{n}0=s^{n}0,0\ne s0,\dots ,s^{n}0\ne s^{m}0\}$ for all n, and $m\in \mathbb{N}$ such that $n\ne m$. Corcoran (Reference Corcoran1980) shows that $T$ is also categorically satisfied by $〈\mathbb{N},s〉$ (with the same interpretation function as before).

Although $T$ is incomplete (it does not prove all sentences satisfied by $〈\mathbb{N},s〉$), we can still use it to derive all sorts of interesting sentences that are satisfied by that structure. The set of sentences $T^{\prime }$, however, cannot even prove basic arithmetical truths. For example, $T^{\prime }⊬\forall x\left(\mathrm{sx}\ne 0\right)$ since there is no finite derivation from the axioms in $T^{\prime }$ to this sentence.Footnote ¹⁰ But this means that these alternative axiomatizations provide access to different facts about the same mathematical structure. And since the inferences that $〈\mathbb{N},s〉$ licenses about a target system depend on the facts about $〈\mathbb{N},s〉$ to which we have access, this means that under different axiomatizations $〈\mathbb{N},s〉$ could license different inferences about its target system(s). Which means that $〈\mathbb{N},s〉$ under one axiomatization is not, in general, equivalent with respect to its representational content to $〈\mathbb{N},s〉$ under another. As Corcoran puts it: “[there is a] vast difference between characterizing an interpretation [structure] and axiomatising its set of truths” (Reference Corcoran1980, 204).

As a corollary, a theory consisting of the singleton $\left\{〈\mathbb{N},s〉\right\}$ should not be identified with the structure (or set containing the structure) itself, at least from a representational perspective, since under different axiomatizations the structure licenses different inferences.Footnote ¹¹

This generates a problem for the adherent of the semantic view of theories, and I take it to be a serious problem rather than “tilting at windmills” (van Fraassen Reference van Fraassen2014, 279). Suppose one identified a theory as a class of models. Of course, to present such a theory, one will have to use language. And, if you like, models can be associated with an interpretation function for the language used to present them. So isomorphisms between models in different theories can be defined straightforwardly in terms of the relations and functions that interpret the same syntactic symbols. This avoids the issues raised by Halvorson (Reference Halvorson2012). The question, however, is what attitude we should adopt to the formulas that are used to present these models.

Glymour writes: “to be at all plausible, the semantic view must distinguish between the content of a theory (its class of models) and the means of characterizing that class—the theory expressed in some language” (Reference Glymour2013, 288). But then the content of a theory is identified with the class of models, and the theory, as expressed in a language used to present this class, is accorded the role of presenting theoretical content, rather than being included in it. I think that the above example shows that even this position is untenable in the context in which higher-order logics are used. If theoretical content somehow depends on representational capacity, and representational capacity somehow depends on the interpretational schema, and this somehow depends on model facts we can access, then it remains that we cannot guarantee that the interpretational schema associated with $〈\mathbb{N},s〉$ under one axiomatization is also applicable to $〈\mathbb{N},s〉$ under another axiomatization. Sometimes we can present a class of models without being able to access all of the truths satisfied by those models. And these truths correspond to potential elements in the domain of interpretational schema. So it is not only that language must be used to present a class of models, but the language used to present that very class of models can constrain the interpretational schema available and therefore the representational content of the theory. Presentation matters, and it should be accorded a proper role even in a formal account of theoretical equivalence.