Scientific representation and theoretical equivalence

James Nguyen∗†

May 8, 2017

Abstract

In this paper I connect two debates in the philosophy of science; the questions
of scientific representation and both model and theoretical equivalence. I argue
that by paying attention to how a model is used to draw inferences about its target
system, we can define a notion of theoretical equivalence that turns on whether their
models licence the same claims about the same target systems. I briefly consider
the implications of this for two questions that have recently been discussed in the
context of the formal philosophy of science.

1 Introduction

Here is one question: in virtue of what do models represent their target systems? Or
more specifically, in virtue of what do models represent their target systems in a way
that allows us to reason about target systems via reasoning about their models? Here is
another: under what conditions are two scientific theories (whatever they are) equivalent?
Despite the heated debates surrounding these questions, investigations into how they relate
– e.g. how insights from one can be utilised in the context of the other – are rare, if not
non-existent. This is surprising. If scientific theories are, in some sense, composed of
models that represent the world, then investigating how models from two distinct theories
represent would seem to have implications on whether or not the theories are equivalent.
I investigate these implications here.

∗jnguyen4@nd.edu, History and Philosophy of Science, University of Notre Dame and Centre for Phi-
losophy of Natural and Social Science, London School of Economics and Political Science.
†I have benefited from numerous fruitful discussions with Roman Frigg and Bryan Roberts on the

material contained in this paper. I am also grateful to Marton Gömöri, Ashton Green, Xavi Lanao, Pablo
Ruiz de Olano, and Jim Weatherall for very useful comments on previous drafts. Thanks also to audiences
at the University of Notre Dame, and the 2016 Philosophy of Science Association meeting for constructive
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In Section 2 I provide some context by briefly outlining each of the questions men-
tioned above. In Section 3 I argue that what I take to be the most promising accounts
of scientific representation provide a natural sense in which two models have the same
‘representational content’, and that such content is a function on facts about models com-
bined with interpretational schema from these facts to claims to be imputed onto their
target systems. I generalise this to the representational content of theories. In Section
4 I discuss the impact of introducing these, model-fact to target-claim, schema into an
account of model and theoretical equivalence. I point out two implications: first how we
should understand Weatherall’s (forthcoming) recent attempt to diffuse the Hole Argu-
ment with reference to the representational capacities of Lorentzian manifolds; and second
how recent attempts to provide formal accounts of theoretical equivalence should be un-
derstood (Barrett and Halvorson, 2016; Halvorson, 2012, 2016; Weatherall, 2016a,b). The
overall purpose of this paper is to encourage a dialogue between philosophers working on
formal aspects of scientific theories, and philosophers working on more general questions
concerning scientific representation.

2 Representation and Equivalence

2.1 Representation

Scientific models represent their target systems. This much is uncontroversial. How they
do so isn’t. Two popular views are that scientific models are, in some sense, intended copies
of their target systems. For the likes of Giere (2004) and Weisberg (2013), 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 (2008) along, perhaps1, with French and his collaborators
(e.g. French, 2003, 2014; Da Costa and French, 2003; Bueno and French, 2011), 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, 2003; Frigg, 2006). And accounts of
scientific representation that diverge from the copy making accounts have emerged as at-
tractive alternatives (Hughes, 1997; Suárez, 2004; Contessa, 2007; Frigg, 2010). According
to Hughes’ (1997, S328) ‘DDI’ account: models denote their targets; allow model users to
perform demonstrations on models; whose results are then interpreted in terms of their

1I say ‘perhaps’ because they express reservations regarding whether user intentions should be built
directly into an account of scientific representation (French, 2003; Bueno and French, 2011).

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target systems. Suárez (2004, 773) submits that models represent their targets only if
their representational force ‘points’ towards their targets and they allow competent model
users to draw inferences about their targets. Contessa (2007, 57-58) argues that models
denote their targets; and that parts of models are interpreted as (purported) parts of their
targets. Frigg (2010, 126-132) 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 licence inferences about their target systems, and the way they
do this is a function on model-facts (Hughes’ 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’ and Contessa’s interpretations, Suárez’s inferential schema used,
and Frigg’s keys). I don’t 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, T1 and T2, equivalent? This, obviously, depends on what
scientific theories are. If they are sets of formulae, formalised in an appropriate language,
then they are logically equivalent 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 definitionaly equivalent iff for each term in the signature of T1,
a sentence can be constructed defining it in terms of the signature of T2, and mutatis
mutandis for each term in the signature of T2, and the result of adding the respective
definitions to each theory provides two expanded theories that are logically equivalent
(Glymour 1970; see also Barrett and Halvorson 2016).

Alternatively, if two theories are construed as collections (in a sense to be specified
below) of models, then an alternative approach is required. Halvorson (2012) provides
three options available to someone who wants to define theoretical equivalence in this
context. Two theories T1 and T2, construed as sets of models, are theoretically equivalent
iff a) T1 and T2 have the same cardinality; b) there is a pointwise isomorphism between T1
and T2 (i.e. a bijection f such that for each model M ∈ T1, f(M) = M ′ where M ′ ∈ T2,
and M and M’ are isomorphic); or c) T1 = T2. Only b) is remotely plausible, but it
requires specifying the sense in which M and M’ can be isomorphic to one another (see
Halvorson, 2012; Glymour, 2013; Lutz, 2015, for useful discussions about this, and whether
it can be done without reference to a linguistic signature).

Halvorson (2012, 2016) has also recently suggested an exciting third sense in which
theories can be seen as theoretically equivalent.2 This can be applied in cases where

2This idea has been applied to specific instances of theoretical equivalence by Rosenstock et al. (2015)

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theories are construed linguistically or as collections of models. The idea in both cases
is to take theories to be categories consisting of objects (formulae, models), and arrows
(deductive relationships, homomorphisms or elementary embeddings). Two theories T1
and T2 thus construed are categorically equivalent iff there is a functor F from T1 to T2
such that F is full, faithful, and essentially surjective.

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

3 Representational Equivalence

The accounts of how models represent discussed in subsection 2.1 can be used to provide
an account of equivalence that diverges (at least prima facie) from the accounts outlined
in subsection 2.2. However, this divergence should not be taken to indicate an essential
tension between those accounts and the one outlined below (which I take to supplement
rather than supplant the formal investigation). My aim in this paper is more modest: I
only hope to show the benefit of keeping interpretive questions in mind when thinking
about theoretical equivalence.

We begin by considering how two models can be equivalent with respect to their
representational content. I suggest that this has two individually necessary and jointly
sufficient conditions. Firstly: M and M’ have the same target system(s). Although
this seems obvious, even this condition may be missed if one focuses solely on the formal
properties of the models in question. As van Fraassen (2014, 278) points out, we shouldn’t
rule out cases where two models satisfy the same equation, but one of them is targeted at
gas diffusion and the other at temperature distribution through time. Or we can consider
models that satisfy an inverse square law; but one targets gravitational interaction between
two massive bodies, and the other electrostatic interaction between two charged points.
Whether or not two models have the same target systems already depends on factors
beyond the formal properties of the models in question.

In addition to having the same target system(s) two models need to ‘tell us the same
things’ about those systems in order to be equivalent. In the accounts of representation
discussed previously, this gets cashed out as M and M’ licencing the same conclusions
about those systems, where the set of conclusions licenced by a model M depends on: 1)
facts about M ; and 2) the interpretational schema used by the model user (or community

and Weatherall (2016a,b).
3Rosenstock et al. (2015) and Weatherall (2016a,b) do refer to ‘empirical equivalence’ in addition

to the formal relations between theories construed as categories, but their emphasis is still on the latter.
Moreover the sort of equivalence I am considering here goes beyond equivalence of empirical content (Sklar
(1982) provides a useful discussion of the relationship between empirical and theoretical equivalence). I
am grateful to Jim Weatherall for drawing my attention to this.

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thereof). An interpretational schema can be thought of as a function that takes model-
facts to target-claims.4 This function does not have to include every fact about M in
its domain, nor does it have to be a (proposed) morphism or specified similarity relation
between M and its target system(s).

A crucial implication of this is that in order to define, and use, an interpretational
schema between model-facts and claims to be imputed onto a target system, we need to
be able to access facts about the model. It’s no good to have a schema that says ‘if the
model is such and such, then it represents its target as such and such’ without being able
to determine the truth of the antecedent. Since if a ‘model’ is not used to infer things
about its target, then it is not really a model. This is the lesson we learn from both
the ‘intended copy making’ and the interpretational accounts of scientific representation
discussed above. As I discuss below, this has important implications for what adherents
to the semantic view of theories should take models ‘to be’.

So in order to be equivalent with respect to their representational content, two models,
via their interpretational schema, have to licence the same conclusions about the same
target systems:

Model equivalence: two models M and M’ are equivalent with respect to their repre-
sentational content iff (i) M and M’ have the same targets; and (ii) they licence the
same claim about those targets via their interpretational schema.

Assuming that, in some sense, theories should be thought of as collections (possibly cate-
gories) of models, this can be generalised to account for theoretical equivalence well:

Theoretical equivalence: Two theories T and T ′ are representationally equivalent iff,
for each claim about a target licensed by a model of T there is a model of T ′ that
licences the same conclusion about the same target under the same circumstances,
and (ii) mutatis mutandis for every claim about a target licensed by a model of T ′.5

This seems so easy one might object that these definitions cannot be informative. Of
course whether two theories or models are equivalent depends on whether they ‘tell us’ the
same things about the same target systems; working out when this is the case is precisely
the point in investigating the notion of theoretical equivalence.

This objection misses the point. Firstly, the condition that two models have the same
target system(s) requires a detailed investigation into what establishes the representational

4I use ‘model-fact’ liberally. These can include things like ‘object a is in the extension of the relation
R’, ‘R1 and R2 are the only relations defined on the model’, or ‘there is a morphism between the model
and some other model in the theory considered as a category’.

5This way of approaching theoretical equivalence has much in common with Coffey (2014). The current
discussion builds on his by emphasising how recent discussions of scientific representation lead to a similar
conclusion.

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relationship (e.g. denotation or ‘representational force’) between models and their targets.
But this does not depend on the formal properties of models alone, and as a fact of the
matter, some models target some systems, and others do not. Although the question of
whether two models have the same targets seems like an obvious one, it places a substantial
condition on the notions of equivalence defined above.

Secondly, whether two models licence the same claims about a given target system
depends, at least on the accounts outlined above, on the interpretational schema associated
with the models. The notion of an interpretational schema is defined in the abstract: a
function from model-facts to target-claims. Understanding how a specific model, or theory,
works will require a detailed investigation into the relevant model-facts, and the details of
the schema itself (how does it associate model-facts with target-claims?).

The point is that if we are interested in notions of equivalence that are stated at
such a level of generality as to apply to diverse scientific fields then our conditions on
equivalence best be stated at the appropriate level of generality. I think the conditions
above are, and moreover, are substantial enough that the question of whether they are met
in any particular case is non-vacuous and requires detailed investigation into the scientific
practice associated with the case.

The resulting accounts of equivalence go beyond formal investigations. They provide
a standard of equivalence that is sensitive to the pragmatic features of using models to
represent the world. I think this is how it should be. To co-opt van Fraassen’s (2014)
example, an account of model equivalence that focused purely on the formal aspects of
models would be analogous to an account of aesthetic equivalence that focused purely on
the material constitution of works of art. Of course, in some sense, what makes the Mona
Lisa the Mona Lisa is determined by facts about how oil paint is distributed across a
specific block of wood. But this does not suffice to establish the representational content
of the picture. Moreover, a copy of the picture, an object with the same oil-on-wood facts,
would only be equivalent (in the sense of representing the same subject in the same way)
to the Mona Lisa if the same interpretational schema was used to interpret it. If, for
example, the copy was demonstrated in the context of a forgery class, then it could be
interpreted in a very different manner delivering very different facts about the world.

Having said that, I do not take my proposed accounts of equivalence to be in competi-
tion with the formal accounts of theoretical equivalence. In one sense they might be taken
to address different questions: one concerning the technicalities of formal equivalence, the
other an interpretive question concerning representational equivalence. But I take it that
the former question isn’t motivated by technical considerations alone. The underlying
idea seems to be that once we have established an account of formal equivalence, then
this can be put to work to help answer the interpretive question. This idea can be seen
in the likes of North (2009) and Curiel (2014) who disagree about whether Lagrangian or
Hamiltonian mechanics is most appropriate for representing classical systems. My claim
here is simply that to fully answer the interpretive question, in addition to the question of

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formal equivalence, we should keep interpretational schema, and coincidence with respect
to target systems, in mind.

4 Fruitful Dialogue

In this section I discuss how paying attention to the fact that models have to be interpreted
when thinking about whether they are equivalent will prove fruitful in contexts where their
representational capacities are important.

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,gab〉 where M is a four dimensional differ-
entiable manifold and gab 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 timelike future
of S, and let ψ : M → M, be a diffeomorphism that is the identity for M\ U, but is
not the identity for U. Using the pushforward of ψ to define a new metric tensor, ψ∗gab,
we have two isometric Lorentzian manifolds 〈M,gab〉 and 〈M,ψ ∗ gab〉. We can now ask
about what these structures represent. The problem arises if we take the following to
hold: 〈M,gab〉 and 〈M,ψ ∗ gab〉 both represent a spacetime ST, but 〈M,gab〉 represents
ST accurately, and 〈M,ψ ∗ gab〉 does not. This substantivalist position introduces inde-
terminism into general relativity because we can suppose that both models are accurate
representations of the past (i.e. the area of spacetime represented by the “past” of S, in
the sense determined by the time orientation) and yet they represent the future in incom-
patible 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 spacetime, then they do so equally accurately.

In an attempt to diffuse this argument, Weatherall (forthcoming) argues that as long
as we are careful in how we understand how mathematical structures like Lorentzian mani-
folds 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 ‘interpre-
tations 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 formal-
ism correctly, consistently, and according to our best understanding of the mathematics’
(Weatherall, forthcoming, 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’ (Weatherall, forthcoming, 4).6 Applying this

6To clarify terminology here, I interpret Weatherall as claiming that if a particular mathematical
model is used to represent a given physical situation, then any isomorphic model that also represents that

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observation to the Lorentzian manifolds defined above we arrive at the following: ‘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 situa-
tions, since isometry is the standard of isomorphism given in the mathematical theory of
Lorentzian manifolds’ (Weatherall, forthcoming, 15).

One way of interpreting this claim in the terminology of this paper 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. 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. 〈M,gab〉 and 〈M,ψ ∗ gab〉 are equivalent (isometric) in the sense specified by the
mathematical theory of Lorentzian manifolds (premise)

3. 〈M,gab〉 and 〈M,ψ ∗gab〉 both represent ST (suppose)

4. 〈M,gab〉 and 〈M,ψ ∗gab〉 are equally accurate representations of ST (from 1,2,3)

Premises 2 and 3 are uncontroversial. 2 is a mathematical fact, and the idea that both
〈M,gab〉 and 〈M,ψ ∗gab〉 represent ST seems benign.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 licencing 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 under-
pinning premise 1 is that since 〈M,gab〉 and 〈M,ψ ∗ gab〉 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 where all we care about are the isometric
invariant facts about ST. But as Roberts (2014) notes, if this argument is supposed to

physical situation may do so equally accurately. I sharply distinguish between ‘represent’ and ‘accurately
represent’, but I think this is what Weatherall has in mind.

7To avoid confusion, sometimes the the debate is stated in terms of whether ‘isometric Lorentzian
manifolds represent the same physical situations’ (cf.Earman and Norton, 1987, 520), but I take it that
we are supposed to interpret this claim as ‘if isometric Lorentzian manifolds represent the same physical
situations, then they do so equally accurately’. Once an account of scientific representation is adopted
where models represent only if model users use them to do so, then the claim that any of the infinite
structures isometric to 〈M, gab〉, including those never considered by anyone, represents ST becomes
implausible.

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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 pre-
sumption, 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 dif-
ferent things’ (Roberts, 2014, 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,gab〉 and 〈M,ψ ∗ gab〉 are isometric, any way of interpreting them must deliver the
same claims about ST. And this needn’t 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’re in London attempting to
navigate using the tube map. Someone familiar with the tube map will adopt an interpre-
tational schema towards 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 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 (forthcoming, Section 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,gab〉
and 〈M,ψ∗gab〉 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,gab〉 and 〈M,ψ∗gab〉 are not the same mathematical object by construction).

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But under this reading, what Weatherall’s argument establishes is the following. Although
in the context of doing relativistic physics, 〈M,gab〉 and 〈M,ψ ∗ gab〉 will be interpreted
in the same way, in the context of the Hole argument, the substantivalist needs to specify
the facts in which 〈M,gab〉 and 〈M,ψ ∗ gab〉 come apart, and how, via the claims they
are associated with by their respective interpretational schemas, they allow for one to
represent ST accurately whilst 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.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 subsection
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, 2012, Chapter 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 which
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 characterisations’ of the same
theory (French and Saatsi, 2006).

But this approach breaks down once we move to higher order theories. Since (e.g.)
second-order logic is not complete, there exist alternative axiomatisations of the same class
of structures that have different deductive closure. Corcoran (1980) provides the following
example of two alternative categorical (in the model-theoretic sense) axiomatisations of
〈N,s〉.

Let L be a first-order language with identity including a constant ‘0’, and a monadic
8I am grateful to Jim Weatherall for confirming that this is his current position in personal communi-

cation. One might argue that this ‘blocks’ the Hole argument as traditionally understood. I do not take
a stand on this issue.

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function ‘s’. Augment L with a single monadic second-order variable P . Then let T =
{(P0∧∀x(Px → Psx)) →∀yPy,∀x∀y(sx = sy → x = y),∀x(sx 6= 0)}. T is categorically
satisfied by 〈N,s〉 where the interpretation function9 takes ‘0’ to 0 and ‘s’ to the successor
function. It is a categorical characterisation of the natural numbers. Let T ′ = {(P0 ∧
∀x(Px → Psx)) → ∀yPy, 0 = 0, ...,sn0 = sn0, 0 6= s0, ...,sn0 6= sm0} for all n,m ∈ N
such that n 6= m. Corcoran (1980) shows that T ′ is also categorically satisfied by 〈N,s〉
(with the same interpretation function as before).

Although T is incomplete (it doesn’t prove all sentences satisfied by 〈N,s〉), we can
still use it to derive all sorts of interesting sentences that are satisfied by that structure.
T ′, on the other hand, cannot even prove basic arithmetical truths. For example, T ′ 0
∀x(sx 6= 0), since there is no finite derivation from the axioms in T ′ to this sentence.10 But
this means that these alternative axiomatisations provide access to different facts about
the same mathematical structure. And since the inferences that 〈N,s〉 licences about a
target system depends on the facts about 〈N,s〉 to which we have access, this means that
under different axiomatisations, 〈N,s〉 could licence different inferences about its target
system(s). Which means that 〈N,s〉 under one axiomatisation is not, in general, equivalent
with respect to its representational content to 〈N,s〉 under another. As Corcoran puts
it: ‘[there is a] vast difference between characterizing an interpretation [structure] and
axiomatising its set of truths’ (1980, 204).

As a corollary, a theory consisting of the singleton {〈N,s〉} should not be identified
with the structure (or set containing the structure) itself, at least from a representa-
tional perspective, since under different axiomatisations the structure licences different
inferences.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, 2014,
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 (2012). The question, however, is what attitude we should adopt to the
formulae that are used to present these models.

9Notice that an ‘interpretation function’ is a very different thing to an ‘interpretation schema’. The
former is a function from syntax to structure, the latter a function from model-facts to target-claims.

10What governs ` in second-order logic is contentious, but the result holds for any sound recursive
system of deductions in the language under consideration. All this observation relies upon is that ` does
not allow for infinite proofs. Since scientists don’t have the time for them, this seems justified to me.

11And the matter won’t be improved if we consider theories as categories of structures, since 〈N, s〉
is the same under both axiomatisations. This category is straightforwardly categorically equivalent to
itself. But, for the aforementioned reason, the inferences that this category licences about target systems
depends on the axiomatisation used to pick it out.

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Glymour writes: ‘[t]o 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’ (2013, 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 where 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
〈N,s〉 under one axiomatisation is also applicable to 〈N,s〉 under another axiomatisation.
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’s 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 should be accorded a proper role even in a formal account of
theoretical equivalence.

5 Conclusion

Mathematical structures, and models more generally, don’t represent without some sort
of intentional act of model users associating model-facts with target-claims. Once this
aspect of scientific representation is made explicit, we are able to ask how it impacts
questions that rely on how models represent. I have argued that, at least under one
way of reading it, Weatherall’s recent attempt to dissolve the Hole argument requires
too strong a restriction on how we associate interpretational schema with mathematical
structures. I have also suggested that formal approaches to theoretical equivalence could
be augmented by investigating the interpretational schema associated with mathematical
models, and moreover, that once we pay due heed to the limitations of how we present
classes of models, the linguistic presentations themselves can be included in a theory’s
‘theoretical content’.

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