## 1. Problems Concerning Scientific Representation



In most general terms, any representation that is the product of a
scientific endeavour is a scientific representation. These
representations are a heterogeneous group comprising anything from
thermometer readings and flow charts to verbal descriptions,
photographs, X-ray pictures, digital imagery, equations, models, and
theories. How do these representations work?



The first thing that strikes the novice in the debate about scientific
representation is that there seems to be little agreement about what
the problem is. Different authors frame the problem of scientific
representation in different ways, and eventually they examine
different issues. So a discussion of scientific representation has to
begin with a clarification of the problem itself. Reviewing the
literature on the subject leads us to the conclusion that there is no
such thing as *the* problem of scientific
representation--in fact, there are at least five different
problems concerning scientific representation (Frigg and Nguyen 2020:
ch. 1). In this section we formulate these problems and articulate
five conditions of adequacy that every account of scientific
representation has to satisfy.



The first problem is: what turns something into a scientific
representation of something else? It has become customary to phrase
this problem in terms of necessary and sufficient conditions and ask:
what fills the blank in "\(S\) is a scientific representation of
\(T\) iff \_\_\_", where "\(S\)" stands for the object
doing the representing and "\(T\)" for "target
system", the part or aspect of the world the representation is
 about?[1]
 Let us call this the *Scientific Representation Problem* (or
*SR-Problem* for short).



A number of contributors to the debate have emphasised that scientific
representation is an intentional concept, depending on factors such as
a user's intentions, purposes and objectives, contextual
standards of accuracy, intended audiences, and community practices
(see, for instance, Boesch 2017; Giere 2010; Maki 2011;
Suarez 2004; and van Fraassen 2008). We will discuss these in
detail below. At this point it needs to be emphasised that framing the
problem in terms of a biconditional does not preclude such factors to
be part of the analysis. The blank can be filled with a \((n+2)\)-ary
relation \(C(S, T, x\_1, \ldots, x\_n)\) (\(C\) for
"constitutes"), where \(n \ge 0\) is a natural number and
the \(x\_i\) are factors such as intentions and purposes.



A first important condition of adequacy on any reply to this problem
is that scientific representations allow us to form hypotheses about
their target systems. An X-ray picture provides information about the
bones of the patient, and models allow investigators to discover
features of the things models stands for. Every acceptable theory of
scientific representation has to account for how reasoning conducted
on representations can yield claims about their target systems. Swoyer
(1991: 449) refers to this kind of representation-based thinking as
"surrogative reasoning" and so we call this the
*Surrogative Reasoning
 Condition*.[2]
 This condition distinguishes models from lexicographical and
indexical representations, which do not allow for surrogative
reasoning.



Unfortunately this condition does not constrain answers sufficiently
because any account of representation that fills the blank in a way
that satisfies the surrogative reasoning condition will almost
invariably also cover other kinds of representations. Speed camera
photographs give the police information about drivers breaking the
law, a cardboard model of the palace instructs us about its layout and
proportions, and a weather map shows you where to expect rain. These
representations are therefore are likely to fall under an account of
representation that explains surrogative reasoning. Hence,
representations other than scientific representations also allow for
surrogative reasoning, which raises the question: how do scientific
representations differ from other kinds of representations that allow
for surrogative reasoning? Callender and Cohen (2006: 68-69)
point out that this is a version Popper's demarcation problem,
now phrased in terms of representation, and so we refer to it as the
*Representational Demarcation Problem*.



Callender and Cohen voice scepticism about there being a solution to
this problem and suggest that the distinction between scientific and
non-scientific representations is circumstantial (2006: 83):
scientific representations are representations that are used or
developed by someone who is a scientist. Other authors do not
explicitly discuss the representational demarcation problem, but
stances similar to Callender and Cohen's are implicit in any
approach that analyses scientific representation alongside other kinds
of representation. Elgin (2010), French (2003), Frigg (2006), Hughes
(1997), Suarez (2004), and van Fraassen (2008), for instance,
all draw parallels between scientific and pictorial representation,
which would make little sense if pictorial and scientific
representation were categorically different.



Those who reject the notion that there is an essential difference
between scientific and non-scientific representation can follow a
suggestion of Contessa's (2007) and broaden the scope of the
investigation. Rather than analysing scientific representation, they
can analyse the broader category of *epistemic representation*.
This category comprises scientific representations, but it also
includes other representations that allow for surrogative reasoning.
The task then becomes to fill the blank in "\(S\) is an
epistemic representation of \(T\) iff \_\_\_". We call this the
*Epistemic Representation Problem* (*ER-Problem*, for
short), and the biconditional the *ER-Scheme*. So one can say
that the ER-Problem is to fill the blank in the ER-Scheme.



Not all representations are of the same kind, not even if we restrict
our attention to scientific representations (assuming they are found
to be relevantly different to non-scientific epistemic
representations). An X-ray photograph represents an ankle joint in a
different way than a biomechanical model, a mercury thermometer
represent the temperature of gas in a different way than statistical
mechanics does, and chemical theory represents a C60 fullerene in
different way that an electron-microscope image of the molecule. Even
when restricting attention to the same kind of representation, there
are important differences: Weizsacker's liquid drop model,
for instance, represents the nucleus of an atom in a manner that seems
to be different from the one of the shell model, and an electric
circuit model represents the brain function in a different way than a
neural network model. In brief, there seem to be different
representational styles. This raises the question: what styles are
there and how can they be
 characterised?[3]
 We call this the *Problem of
 Style*.[4]
 There is no expectation that a *complete* list of styles be
provided in response. Indeed, it is unlikely that such a list can ever
be drawn up, and new styles will be invented as science progresses.
For this reason a response to the problem of style will always be
open-ended, providing a taxonomy of what is currently available while
leaving room for later additions.



Some representations are accurate; others aren't. The quantum
mechanical model is an accurate representation of the atom (at least
by our current lights) but the Thomson model isn't. On what
grounds do we make such judgments? Morrison (2008: 70) reminds us that
it is a task for theory of representation to identify what constitutes
an accurate representation. We call this the problem of *Standards
of Accuracy*. Providing such standards is one of the issues an
account of representation has to address, which, however, is not to
say that accurate representation is the sole epistemic virtue of
scientific models. As Parker (2020) points out, there are numerous of
considerations to take into account when evaluating a model's
adequacy for its purpose, see Downes (2021: ch. 5) for further
discussion.



This problem goes hand in hand with a condition of adequacy: the
*Possibility of Misrepresentation*. Asking what makes a
representation an accurate representation *ipso facto*
presupposes that inaccurate representations are representations too.
And this is how it should be. If \(S\) does not accurately represent
\(T\), then it is a misrepresentation but not a non-representation. It
is therefore a general constraint on a theory of scientific
representation that it has to make misrepresentation
 possible.[5]



A related condition concerns models that misrepresent in the sense
that they lack target systems. Models of the ether, phlogiston,
four-sex populations, and so on, are all deemed scientific models, but
ether, phlogiston, and four-sex populations don't exist. Such
models lack (actual) target systems, and one hopes that an account of
scientific representation would allow us to understand how these
models work. This need not imply the claim that they are
representations in the same sense as models with actual targets, and,
as we discuss below, there are accounts that deny targetless models
the status of being representations.



A further condition of adequacy for an account of scientific
representation is that it must account for the directionality of
representation. As Goodman points out (1976: 5), representations are
about their targets, but (at least in general) targets are not about
their representations: a photograph represents the cracks in the wing
of aeroplane, but the wing does not represent the photograph. So there
is an essential directionality to representations, and an account of
scientific, or epistemic, representation has to identify the root of
this directionality. We call this the *Requirement of
Directionality*.



Some representations, in particular models and theories, are
mathematized and their mathematical aspects are crucial to their
cognitive and representational function. This forces us to reconsider
a time-honoured philosophical puzzle: the applicability of mathematics
in the empirical sciences. The problem can be traced back at least to
Plato's *Timaeus*, but its modern expression is due to
Wigner who challenged us to find an explanation for the enormous
usefulness of mathematics in the sciences (1960: 2). The question how
a mathematized model represents its target implies the question how
mathematics applies to a physical system (see Pincock 2012 for an
explicit discussion of the relationship between scientific
representation and the applicability of mathematics). For this reason,
our fifth and final condition of adequacy is that an account of
representation has to explain how mathematics is applied to the
physical world. We call this the *Applicability of Mathematics
Condition*.



In answering the above questions one invariably runs up against a
further problem, the *Problem of Ontology*: what kinds of
objects are representations? If representations are material objects
the answer is straightforward: photographic plates, pieces of paper
covered with ink, elliptical blocks of wood immersed in water, and so
on. But not all representations are like this. As Hacking (1983: 216)
puts it, some representations one holds in one's head rather
than one's hands. The Newtonian model of the solar system, the
Lotka-Volterra model of predator-prey interaction and the general
theory of relativity are not things you can put on your laboratory
table and look at. The problem of ontology is to come clear on our
commitments and provide a list with things that we recognise--or
don't recognise--as entities performing a representational
function and give an account of what they are in case these entities
raise questions (what exactly do we mean by something that one holds
in one's head rather than one's hands?). Contessa (2010),
Frigg (2010a,b), Godfrey-Smith (2006), Levy (2015), Thomson-Jones
(2010), Weisberg (2013), among others, have drawn attention to this
problem in different ways.



In sum, a theory of scientific representation has to respond to the
following issues:


1. Address the *Representational Demarcation Problem* (the
question how scientific representations differ from other kinds of
representations).
2. Those who demarcate scientific from non-scientific representations
have to provide an answer to the *Scientific Representation
Problem* (fill the blank in "\(S\) is a scientific
representation of \(T\) iff \_\_\_"). Those who reject the
representational demarcation problem can address the *Epistemic
Representation Problem* (fill the blank in ER-Scheme: "\(S\)
is an epistemic representation of \(T\) iff \_\_\_").
3. Respond to the *Problem of Style* (what styles are there
and how can they be characterised?).
4. Formulate *Standards of Accuracy* (how do we identify what
constitutes an accurate representation?).
5. Address the *Problem of Ontology* (what are the kind of
objects that serve as representations?).



Any satisfactory answer to these five issues will have to meet the
following five conditions of adequacy:


1. *Surrogative Reasoning* (scientific representations allow
us to generate hypotheses about their target systems).
2. *Possibility of Misrepresentation* (if \(S\) does not
accurately represent \(T\), then it is a misrepresentation but not a
non-representation).
3. *Targetless Models* (what are we to make of scientific
representations that lack targets?).
4. *Requirement of Directionality* (scientific representations
are about their targets, but targets are not about their
representations).
5. *Applicability of Mathematics* (how does the mathematical
apparatus used in some scientific representations latch onto the
physical world).



Listing the problems in this way is not to say that these are separate
and unrelated issues. This division is analytical, not factual. It
serves to structure the discussion and to assess proposals; it does
not imply that an answer to one of these questions can be dissociated
from what stance we take on the other issues.



Any attempt to tackle these questions faces an immediate
methodological problem. As per the problem of style, there are
different kinds of representations: scientific models, theories,
measurement outcomes, images, graphs, diagrams, and linguistic
assertions are all scientific representations, and even within these
groups there can be considerable variation. But every analysis has to
start somewhere, and so the problem is where. One might adopt a
universalist position, holding that the diversity of styles dissolves
under analysis and at bottom all instances of scientific/epistemic
representation function in the same way and are covered by the same
overarching account. For such a universalist the problem loses its
teeth because any starting point will lead to the same result. Those
of particularist bent deny that there is such a theory. They will
first divide the scientific/epistemic representations into relevant
subclasses and then analyse each subclass separately.



Different authors assume different stances in this debate, and we will
discuss their positions below. However, there are few, if any,
thoroughgoing universalists and so a review like the current one has
to discuss different cases. Unfortunately space constraints prevent us
from examining all the different varieties of scientific/epistemic
representation, and a selection has to be made. This invariably leads
to the neglect of some kinds of representations, and the best we can
do about this is to be explicit about our choices. We resolve to
concentrate on scientific models, and therefore replace our variable
\(S\) for the object doing the representing with the variable \(M\)
for model. This is in line both with the more recent literature on
scientific representation, which is predominantly concerned with
scientific models, and with the prime importance that current
philosophy of science attaches to models (see the SEP entry on
 models in science
 for a
 survey).[6]



It is, however, worth briefly mentioning some of the omissions that
this brings with it. Various types of images have their place in
science, and so do graphs, diagrams, and drawings. Perini (2010) and
Elkins (1999) provide discussions of visual representation in science.
Measurements also supply representations of processes in nature,
sometimes together with the subsequent condensation of measurement
results in the form of charts, curves, tables and the like (see the
SEP entry on
 measurement in science).
 Furthermore, theories represent their subject matter. At this point
the vexing problem of the nature of theories rears again (see the SEP
entry on
 the structure of scientific theories
 and also Portides (2017) for an extensive discussion). Proponents of
the semantic view of theories construe theories as families of models,
and so for them the question of how theories represent coincides with
the question of how models represent. By contrast, those who regard
theories as linguistic entities see theoretical representation as a
special kind of linguistic representation and focus on the analysis of
scientific languages, in particular the semantics of so-called
theoretical terms (see the SEP entry on
 theoretical terms in science).



Before delving into the discussion a common misconception needs to be
dispelled. The misconception is that a representation is a mirror
image, a copy, or an imitation of the thing it represents. On this
view representation is *ipso facto* realistic representation.
This is a mistake. Representations can be realistic, but they need
not. And representations certainly need not be copies of the real
thing, an observation exploited by Lewis Carroll and Jorge Luis Borges
in their satires, *Sylvie and Bruno* and *On Exactitude in
Science* respectively, about cartographers who produce maps as
large as the country itself only to see them abandoned (for a
discussion see Boesch 2021). Throughout this review we encounter
positions that make room for non-realistic representation and hence
testify to the fact that representation is a much broader notion than
 mirroring.[7]


## 2. General Griceanism and Stipulative Fiat



Callender and Cohen (2006) give a radical answer to the demarcation
problem: there is no difference between scientific representations and
other kinds of representations, not even between scientific and
artistic representation. Underlying this claim is a position they call
"General Griceanism" (GG). The core of GG is the reductive
claim that all representations owe their status as representations to
a privileged core of fundamental representations. GG then comes with a
practical prescription about how to proceed with the analysis of a
representation:



> 
> 
> the General Gricean view consists of two stages. First, it explains
> the representational powers of derivative representations in terms of
> those of fundamental representations; second, it offers some other
> story to explain representation for the fundamental bearers of
> content. (2006: 73)
> 
> 
> 



Of these stages only the second requires serious philosophical work,
and this work is done in the philosophy of mind because the
fundamental form of representation is mental representation.



Scientific representation is a derivative kind of representation
(2006: 71, 75) and hence falls under the first stage of the above
recipe. It is reduced to mental representation by an act of
stipulation. In Callender and Cohen's own example, the salt
shaker on the dinner table can represent Madagascar as long as someone
stipulates that the former represents the latter, since



> 
> 
> the representational powers of mental states are so wide-ranging that
> they can bring about other representational relations between
> arbitrary relata by dint of mere stipulation. (2006: 73-74)
> 
> 
> 



So explaining any form of representation other than mental
representation is a triviality--all it takes is an act of
"stipulative fiat" (2006: 75). This supplies an answer to
the ER-problem:



*Stipulative Fiat*: A scientific model \(M\) represents a
target system \(T\) iff a model user stipulates that \(M\) represents
\(T\).



The first problem facing *Stipulative Fiat* is whether or not
stipulation, or the bare intentions of language users, suffice to
establish representational relationships. In the philosophy of
language this gets called the "Humpty Dumpty" problem. It
concerns whether or not Lewis Carroll's Humpty Dumpty could use
the word "glory" to mean "a nice knockdown
argument" (Donnellan 1968; MacKay 1968). (We ignore the
difference between meaning and denotation here). In that context it
doesn't seem like he can, and analogous questions can be posed
in the context of scientific representation: can a scientist make any
model represent any target simply by stipulating that it does?



Even if stipulation were sufficient to establish some sort of
representational relationship, *Stipulative Fiat* fails to meet
the Surrogative Reasoning Condition: assuming a salt shaker represents
Madagascar in virtue of someone's stipulation that this is so,
this tells us nothing about how the salt shaker could be used to learn
about Madagascar in the way that scientific models are used to learn
about their targets (Liu 2015: 46-47, for a related objections
see Boesch 2017: 974-978, Bueno and French 2011: 871-873,
Gelfert 2016: 33, Ruyant 2021: 535). And appealing to additional facts
about the salt shaker (the salt shaker being to the right of the
pepper mill might allow us to infer that Madagascar is to the east of
Mozambique) in order to answer this objection goes beyond
*Stipulative Fiat*. Callender and Cohen do admit some
representations are more useful than others, but claim that



> 
> 
> the questions about the utility of these representational vehicles are
> questions about the pragmatics of things that are representational
> vehicles, not questions about their representational status *per
> se*. (2006: 75)
> 
> 
> 



But even if the Surrogative Reasoning Condition is relegated to the
realm of "pragmatics" it seems reasonable to ask for an
account of how it is met.



An important thing to note is that even if *Stipulative Fiat*
is untenable, we needn't give up on GG. GG only requires that
there be *some* explanation of how derivative representations
relate to fundamental representations; it does not require that this
explanation be of a particular kind, much less that it consist in
nothing but an act of stipulation (Toon 2010: 77-78). Ruyant
(2021) has recently proposed what he calls "true
Griceanism", which differs from the original account in that it
involves a non-trivial reduction of scientific representation to more
fundamental representations (complex sequences mental states and
purpose-directed behaviour) in a way that is also sensitive to the
communal aspects of practices in which the models are embedded. This
is a step into the direction indicated by Toon. More generally, as
Callender and Cohen note, all that it requires is that there is a
privileged class of representations and that other types of
representations owe their representational capacities to their
relationship with the primitive ones. So philosophers need an account
of how members of this privileged class of representations represent,
and how derivative representations, which includes scientific models,
relate to this class. When stated like this, many recent contributors
to the debate on scientific representation can be seen as falling
under the umbrella of GG. Indeed, As we will see below, many of the
more developed versions of the accounts of scientific representation
discussed throughout this entry invoke the intentions of model users,
albeit in a more complex manner than *Stipulative Fiat*.


## 3. The Similarity Conception



Similarity and representation initially appear to be two closely
related concepts, and invoking the former to ground the latter has a
philosophical lineage stretching back at least as far as Plato's
*The
 Republic*.[8]
 In its most basic guise the similarity conception of scientific
representation asserts that scientific models represent their targets
in virtue of being similar to them. This conception has universal
aspirations in that it is taken to account for representation across a
broad range of different domains. Paintings, statues, and drawings are
said to represent by being similar to their subjects, and Giere
proclaimed that it covers scientific models alongside "words,
equations, diagrams, graphs, photographs, and, increasingly,
computer-generated images" (2004: 243). So the similarity view
repudiates the demarcation problem and submits that the same
mechanism, namely similarity, underpins different kinds of
representation in a broad variety of contexts.



The view offers an elegant account of surrogative reasoning.
Similarities between model and target can be exploited to carry over
insights gained in the model to the target. If the similarity between
\(M\) and \(T\) is based on shared properties, then a property found
in \(M\) would also have to be present in \(T\); and if the similarity
holds between properties themselves, then \(T\) would have to
instantiate properties similar to \(M\).



However, appeal to similarity in the context of representation leaves
open whether similarity is offered as an answer to the ER-Problem or
the Problem of Style, or whether it is meant to set Standards of
Accuracy. Proponents of the similarity conception typically have
offered little guidance on this issue. So we examine each option in
turn and ask whether similarity offers a viable answer. We then turn
to the question of how the similarity view deals with the Problem of
Ontology.


### 3.1 Similarity and ER-Problem



Understood as response to the ER-Problem, the simplest similarity view
is the following:



*Similarity 1*: A scientific model \(M\) represents a target
\(T\) iff \(M\) and \(T\) are similar.



A well-known objection to this account is that similarity has the
wrong logical properties. Goodman (1976: 4-5) points out that
similarity is symmetric and reflexive yet representation isn't.
If object \(A\) is similar to object \(B\), then \(B\) is similar to
\(A\). But if \(A\) represents \(B\), then \(B\) need not (and in fact
in most cases does not) represent \(A\). Everything is similar to
itself, but most things do not represent themselves. So this account
does not meet our fourth condition of adequacy for an account of
scientific representation insofar as it does not provide a direction
to representation.



There are accounts of similarity under which similarity is not a
symmetric relation (see Tversky 1977; Weisberg 2012, 2013: ch. 8; and
Poznic 2016: sec. 4.2). This raises the question of how to analyse
similarity. We turn to this issue in the next subsection. However,
even if we concede that similarity need not always be symmetrical,
this does not solve Goodman's problem with reflexivity; nor does
it, as we will see, solve other problems of the similarity
account.



The most significant problem facing *Similarity 1* is that
without constraints on what counts as similar, any two things can be
considered similar (Aronson et al. 1995: 21; Goodman 1972:
443-444). This has the unfortunate consequence that anything
represents anything else. A natural response to this difficulty is to
delineate a set of relevant respects and degrees to which \(M\) and
\(T\) have to be similar. This idea can be moulded into the following
definition:



*Similarity 2*: A scientific model \(M\) represents a target
\(T\) iff \(M\) and \(T\) are similar in relevant respects and to the
relevant degrees.



On this definition one is free to choose one's respects and
degrees so that unwanted similarities drop out of the picture. While
this solves the last problem, it leaves the problem of logical
properties untouched: similarity in relevant respects and to the
relevant degrees is reflexive (and symmetrical, depending on
one's notion of similarity). Moreover, *Similarity 2*
faces three further problems.



Firstly, similarity, even restricted to relevant similarities, is too
inclusive a concept to account for representation. In many cases
neither one of a pair of similar objects represents the other. This
point has been brought home in Putnam's now-classical thought
experiment due to Putnam (1981: 1-3). An ant is crawling on a
patch of sand and leaves a trace that happens to resemble Winston
Churchill. Has the ant produced a picture, a representation, of
Churchill? Putnam's answer is that it didn't because the
ant has never seen Churchill, had no intention to produce an image of
him, was not causally connected to Churchill, and so on. Although
someone else might see the trace as a depiction of Churchill, the
trace itself does not represent Churchill. The fact that the trace is
similar to Churchill does not suffice to establish that the trace
represents him. And what is true of the trace and Churchill is true of
every other pair of similar items: even relevant similarity on its own
does not establish representation.



Secondly, as noted in
 Section 1,
 allowing for the possibility of misrepresentation is a key desiderata
required of any account of scientific representation. In the context
of a similarity conception it would seem that a misrepresentation is
one that portrays its target as having properties that are not similar
in the relevant respects and to the relevant degree to the true
properties of the target. But then, on *Similarity 2*, \(M\) is
not a representation at all. The account thus has difficulty
distinguishing between misrepresentation and non-representation
(Suarez 2003: 233-235).



Thirdly, there may simply be nothing to be similar to because some
representations are not about an actual object. Some paintings
represent elves or dragons, and some models represent phlogiston or
the ether. None of these exist. This is a problem for the similarity
view because models without targets cannot represent what they seem to
represent because in order for two things to be similar to each other
both have to exist. If there is no ether, then an ether model cannot
be similar to the ether.



At least some of these problems can be resolved by taking the very act
of *asserting* a specific similarity between a model and a
target as constitutive of the scientific representation. Giere (1988:
81) suggests that models come equipped with what he calls
"theoretical hypotheses", statements asserting that model
and target are similar in relevant respects and to certain degrees. He
emphasises that "scientists are intentional agents with goals
and purposes" (2004: 743) and proposes to build this insight
explicitly into an account of representation. This involves adopting
an agent-based notion of representation that focuses on "the
activity of representing" (2004). Analysing representation in
these terms amounts to analysing schemes like



> 
> 
> Agents (1) intend; (2) to use model, \(M\); (3) to represent a part of
> the world \(W\); (4) for purposes, \(P\). So agents specify which
> similarities are intended and for what purpose. (2010: 274)
> 
> 
> 



(see also Maki 2009, 2011; although see Rusanen and Lappi 2012:
317 for arguments to the contrary). This leads to the following
definition:



*Similarity 3*: A scientific model \(M\) represents a target
system \(T\) iff there is an agent \(A\) who uses \(M\) to represent a
target system \(T\) by proposing a theoretical hypothesis \(H\)
specifying a similarity (in certain respects and to certain degrees)
between \(M\) and \(T\) for purpose \(P\).



This version of the similarity view avoids problems with
misrepresentation because \(H\) being a hypothesis, there is no
expectation that the assertions made in \(H\) are true. If they are,
then the representation is accurate (or the representation is accurate
to the extent that they hold). If they do not, then the representation
is a misrepresentation. It also resolves the issue with directionality
because \(H\) can be understood as introducing an asymmetry that is
not present in the similarity relation. However, it fails to resolve
the problem with representation without a target. If there is no
ether, no hypotheses can be asserted about it, at least in any
straightforward way.



*Similarity 3*, by invoking an active role for the purposes and
actions of scientists in constituting scientific representation, marks
a significant change in emphasis for similarity-based accounts.
Suarez (2003: 226-227), drawing on van Fraassen (2002)
and Putnam (2002), defines "naturalistic" accounts of
representation as ones where



> 
> 
> whether or not representation obtains depends on facts about the world
> and does not in any way answer to the personal purposes, views or
> interests of enquirers.
> 
> 
> 



By building the purposes of model users directly into an answer to the
ER-problem, *Similarity 3* is explicitly not a naturalistic
account (in contrast to *Similarity 1*). The shift to users
performing representational actions invites the question of it means
for a scientist to perform such an action. Boesch (2019) offers an
answer which draws on Anscombe's (2000) account of intentional
action: something is a "scientific, representational
action" (Boesch 2019: 312) when a description of a
scientist's interaction with a model stands as an earlier
description towards the final description which is some scientific aim
such as explanation, predication, or theorizing.



Even though *Similarity 3* resolves a number of issues that
beset simpler versions, it does not seem to be a successful
similarity-based solution to the ER-Problem. A closer look at
*Similarity 3* reveals that the role of similarity has shifted.
As far as offering a solution to the ER-Problem is concerned, all the
heavy lifting in *Similarity 3* is done by the appeal to agents
and their intentions. Giere implicitly concedes this when he observes
that similarity is "the most important way, but probably not the
only way" for models to function representationally (2004: 747).
But if similarity is not the only way in which a model can be used as
a representation, then similarity has become otiose in a reply to the
ER-problem. In fact, being similar in the relevant respects to the
relevant degree now plays the role either of a representational style
or of a normative criterion for accurate representation, rather than
constituting representation *per se*. We assess in the next
section whether similarity offers cogent replies to the issues of
style and accuracy, and we raise a further problem for any account of
scientific representation that relies on the idea that models,
specifically non-concrete models, are similar to their targets.


### 3.2 Accuracy, Style, and Ontology



The fact that relevant properties can be delineated in different ways
could potentially provide an answer to the Problem of Style. If \(M\)
representing \(T\) involves the claim that \(M\) and \(T\) are similar
in a certain respect, the respect chosen specifies the style of the
representation; and if \(M\) and \(T\) are in fact similar in that
respect (and to the specified degree), then \(M\) accurately
represents \(T\) within that style. For example, if \(M\) and \(T\)
are proposed to be similar with respect to their causal structure,
then we might have a style of causal modelling; if \(M\) and \(T\) are
proposed to be similar with respect to structural properties, then we
might have a style of structural modelling; and so on and so
forth.



A first step in the direction of such an understanding of styles is
the explicit analysis of the notion of similarity. The standard way of
cashing out what it means for an object to be similar to another
object is to require that they co-instantiate properties. In fact,
this is the idea that Quine (1969: 117-118) and Goodman (1972:
443) had in mind in their influential critiques of similarity. The two
most prominent formal frameworks that develop this idea are the
geometric and contrast accounts (see Decock and Douven 2011 for a
discussion).



The geometric account, associated with Shepard (1980), assigns objects
a place in a multidimensional space based on values assigned to their
properties. This space is then equipped with a metric and the degree
of (dis)similarity between two objects is the distance between the
points representing the two objects in that space. This account is
based on the strong assumptions that values can be assigned to all
features relevant to similarity judgments, which is deemed unrealistic
(and to the best of our knowledge no one has developed such an account
in the context of scientific representation).



This problem is supposed to be overcome in Tversky's contrast
account (1977). This account defines a gradated notion of similarity
based on a weighted comparison of properties. Weisberg has recently
introduced this account into the philosophy of science where it serves
as the starting point for his *weighted feature matching account of
model world-relations* (for details see Weisberg 2012, 2013: ch.
8). Although the account has some advantages, questions remain whether
it can capture the distinction between what Niiniluoto (1988:
272-274) calls "likeness" and "partial
identity". Two objects are alike to the extent that they
co-instantiate similar properties (for example, a red phone box and a
red London bus might be alike with respect to their colour, despite
not instantiating the exact same shade of red). Two objects are
partially identical to the extent that they co-instantiate identical
properties. As Parker (2015: 273) notes, contrast based accounts of
similarity like Weisberg's have difficulties capturing the
former, and this is often pertinent in the context of scientific
representation where models and their targets need not co-instantiate
the exact same property. Concerns of this sort have led Khosrowi
(2020) to suggest that the notion of sharing features should be
analysed in a pluralist manner. Sharing a feature sometimes means
sharing the exact same feature; sometimes it means to share features
which are sufficiently quantitatively close to one another; and
sometimes it means having features which are themselves
"sufficiently similar".



A further question that remains for someone who uses the notion of
similarity to answer to the Problem of Style and provide standards of
accuracy in the manner under consideration here is whether it truly
captures all of scientific practice. Similarity theorists are
committed to the claim that whenever a scientific model represents its
target system, this is established in virtue of a model user
specifying a relevant similarity, and if the similarity holds, then
the representational relationship is accurate. These universal
aspirations require that the notion of similarity invoked capture the
relationship that holds between diverse entities such as a basin-model
of the San Francisco bay area, a tube map and an underground train
system, and the Lotka-Volterra equations of predator-pray interaction.
Whether all of these relationships can be captured in terms of
similarity remains an open question. In addition, this view is
commited to the idea that idealised aspects of scientific models,
understood as dissimilarities between models and their targets, are
misrepresenations and as such the view has difficulty capturing the
positive epistemic role that such aspects can play (Nguyen 2020). 



Another problem facing similarity based approaches concerns their
treatment of the ontology of models. If models are supposed to be
similar to their targets in the ways specified by theoretical
hypotheses, then they must be the *kind* of things that can be
so similar. For material models like the San Francisco Bay model
(Weisberg 2013), ball and stick models of molecules (Toon 2011), the
Phillips-Newlyn machine (Morgan and Boumans 2004), or model organisms
(Ankeny and Leonelli 2021) this seems straightforward because they are
of the same ontological kind as their respective targets. But many
interesting scientific models are not like this: they are what Hacking
aptly describes as "something you hold in your head rather than
your hands" (1983: 216). Following Thomson-Jones (2012) we call
such models *non-concrete models*. The question then is how
such models can be similar to their targets. At the very least these
models are "abstract" in the sense that they have no
spatiotemporal location. But if so, then it remains unclear how they
can instantiate the sorts of properties specified by theoretical
hypotheses, since these properties are typically *physical*,
and presumably being located in space and time is a necessary
condition on instantiating such properties. For further discussion of
this objection, and proposed solutions, see Teller (2001: 399),
Thomson-Jones (2010; 2020), and Giere (2009), and Thomasson
(2020).


## 4. The Structuralist Conception



The structuralist conception of model-representation originated in the
so-called semantic view of theories that came to prominence in the
second half of the 20th century (see the SEP entry on the
 structure of scientific theories
 for further details). The semantic view was originally proposed as an
account of theory structure rather than scientific representation. The
driving idea behind the position is that scientific theories are best
thought of as collections of models. This invites the questions: what
are these models, and how do they represent their target systems? Most
defenders of the semantic view of theories (with the notable exception
of Giere, whose views on scientific representation were discussed in
the previous section) take models to be structures, which represent
their target systems in virtue of there being some kind of
*morphism* (isomorphism, partial isomorphism, homomorphism,
...) between the two.



This conception has two *prima facie* advantages. The first
advantage is that it offers a straightforward answer to the ER-Problem
(or SR-problem if the focus is on scientific representation), and one
that accounts for surrogative reasoning: the mappings between the
model and the target allow scientists to convert truths found in the
model into claims about the target system. The second advantage
concerns the applicability of mathematics. There is a time-honoured
position in the philosophy of mathematics which sees mathematics as
the study of structures; see, for instance Resnik (1997) and Shapiro
(2000). It is a natural move for the scientific structuralist to adopt
this point of view, which then provides a neat explanation of how
mathematics is used in scientific modelling.


### 4.1 Structures and the Problem of Ontology



Almost anything from a concert hall to a kinship system can be
referred to as a "structure". So the first task for a
structuralist account of representation is to articulate what notion
of structure it employs. A number of different notions of structure
have been discussed in the literature (for a review see Thomson-Jones
2011), but by far the most common is the notion of structure one finds
in set theory and mathematical logic. A structure \(\mathcal{S}\) in
that sense (sometimes "mathematical structure" or
"set-theoretic structure") is a composite entity
consisting of the following: a non-empty set \(U\) of objects called
the domain (or universe) of the structure and an indexed set \(R\) of
relations on \(U\) (supporters of the partial structures approach,
e.g., Da Costa and French (2003) and Bueno, French, and Ladyman
(2002), use partial \(n\)-place relations, for which it may be
undefined whether or not some \(n\)-tuples are in their extension).
This definition of structure is widely used in mathematics and logic.
We note, however, that in mathematical logic structures also contain a
language and an interpretation function, interpreting symbols of the
language in terms of \(U\) (see for instance Machover 1996 and Hodges
1997), which is absent from structures in the current context. It is
convenient to write these as \(\mathcal{S}= \langle U, R \rangle\),
where "\(\langle \, , \rangle\)" denotes an ordered
tuple.



It is important to be clear on what we mean by "object"
and "relation" in this context. As regards objects, all
that matters from a structuralist point of view is that there are so
and so many of them. Whether the objects are desks or planets is
irrelevant. All we need are dummies or placeholders whose only
property is "objecthood". Similarly, when defining
relations one disregards completely what the relation is "in
itself". Whether we talk about "being the mother of"
or "standing to the left of" is of no concern in the
context of a structure; all that matters is between which objects it
holds. For this reason, a relation is specified purely extensionally:
as a class of ordered \(n\)-tuples. The relation literally is nothing
over and above this class. So a structure consists of dummy-objects
between which purely extensionally defined relations hold.



The first basic posit of the structuralist theory of representation is
that models are structures in this sense (the second is that models
represent their targets by being suitably morphic to them; we discuss
morphisms in the next subsection). Suppes has articulated this stance
clearly when he declared that "the meaning of the concept of
model is the same in mathematics and the empirical sciences"
(1960 [1969]: 12), and many have followed suit. So we are presented
with a clear answer to the Problem of Ontology: models are structures.
The remaining issue is what structures themselves are. Are they
Platonic entities, equivalence classes, modal constructs, or yet
something else? In the context of a discussion of scientific
representation one can push these questions off to the philosophy of
mathematics (see the SEP entries on the
 philosophy of mathematics,
 nominalism in the philosophy of mathematics, and
 Platonism in the philosophy of mathematics
 for further details).


### 4.2 Structuralism and the ER-Problem



The most basic structuralist conception of scientific representation
asserts that scientific models, understood as structures, represent
their target systems in virtue of being isomorphic to them. An
isomorphism between two structures \(\mathcal{S}\) and
\(\mathcal{S}'\) is a bijective function from \(U\) to \(U'\) that
preserves the relations on \(U\) (and inversely, the relations on
\(U'\)). An isomorphism associates each object in \(U\) with an object
in \(U'\) and pairs up each relation in \(R\) with a relation in
\(R'\) so that a relation holds between certain objects in \(U\) iff
the corresponding relation holds between the objects in \(U'\) that
are associated with
 them.[9]
 Assume now that the target system \(T\) exhibits the structure
\(\mathcal{S}\_T\) and the model is the structure \(\mathcal{S}\_M\).
Then the model represents the target iff it is isomorphic to the
target:



*Structuralism 1*: A scientific model \(M\) represents its
target \(T\) iff \(\mathcal{S}\_T\) is isomorphic to
\(\mathcal{S}\_M\).



It bears noting that few adherents of the structuralist account of
scientific representation, most closely associated with the semantic
view of theories, explicitly defend this position (although see Ubbink
1960: 302). Representation was not the focus of attention in the
semantic view, and the attribution of (something like)
*Structuralism 1* to its supporters is an extrapolation.
Representation became a much-debated topic in the first decade of the
21st century, and many proponents of the semantic view then
either moved away from *Structuralism 1*, or pointed out that
they never held such a view. We turn to more advanced positions
shortly, but to understand what motivates such positions it is helpful
to understand why *Structuralism 1* fails.



The first and most obvious problem is the same as with the similarity
view: isomorphism is symmetrical and reflexive (and transitive) but
representation isn't. This problem could be addressed by
replacing isomorphism with an alternative mapping. Bartels (2006),
Lloyd (1984), and Mundy (1986) suggest homomorphism; van Fraassen
(1980, 1997, 2008) and Redhead (2001) isomorphic embeddings; advocates
of the partial structures approach prefer partial isomophisms (Bueno
1997; Bueno and French 2011; Da Costa and French 1990, 2003; French
2003, 2014; French and Ladyman 1999); and Swoyer (1991) introduces
what he calls \(\Delta/\Psi\) morphisms. We refer to these
collectively as "morphisms". Pero and Suarez (2016)
provide a comparative discussion of different morphisms.



These suggestions solve some, but not all problems. While many of
these mappings are not symmetrical, they are all still reflexive. But
even if these formal issues could be resolved in one way or another, a
view based on structural mappings would still face other serious
problems. For ease of presentation we discuss these problems in the
context of the isomorphism view; *mutatis mutandis* other
formal mappings suffer from the same difficulties. Like similarity,
isomorphism is too inclusive: not all things that are isomorphic
represent each other. In the case of similarity this case was brought
home by Putnam's thought experiment with the ant crawling on the
beach; in the case of isomorphism a look at the history of science
will do the job. Many mathematical structures were discovered and
discussed long before they were used in science. Non-Euclidean
geometries were studied by mathematicians long before Einstein used
them in the context of spacetime theories, and Hilbert spaces were
studied by mathematicians prior to their use in quantum theory. If
representation was nothing over and above isomorphism, then we would
have to conclude that Riemann discovered general relativity or that
that Hilbert invented quantum mechanics. This does not seem correct,
so it doesn't seem like isomorphism on its own establishes
scientific representation (Frigg 2002: 10).



Isomorphism is more restrictive than similarity: not everything is
isomorphic to everything else. But isomorphism is still too abundant
to correctly identify what a model represents. The root of the
difficulties is that the same structures can be instantiated in
different kinds of target systems. Certain geometrical structures are
instantiated by many different systems; just think about how many
spherical things we find in the world. The \(1/r^2\) law of Newtonian
gravity is also the "mathematical skeleton" of
Coulomb's law of electrostatic attraction and the weakening of
sound or light as a function of the distance to the source. The
mathematical structure of the pendulum is also the structure of an
electric circuit with a condenser and a solenoid (Kroes 1989). The
same structure can be exhibited by more than one kind of target
system, and so isomorphism by itself is too weak to identify a
model's target.



As we have seen in the last section, a misrepresentation is one that
portrays its target as having features it doesn't have. In the
case of a structural account of representation, this means that the
model portrays the target as having structural properties that it
doesn't have. However, isomorphism demands identity of
structure: the structural properties of the model and the target must
correspond to one another exactly. So a misrepresentation won't
be isomorphic to the target. By the lights of *Structuralism 1*
it therefore is not a representation at all. Like simple similarity
accounts, *Structuralism 1* conflates misrepresentation with
non-representation (Suarez 2003: 234-235). Partial
structures can avoid a mismatch due to a target relation being omitted
in the model and hence go some way to shoring up the structuralist
account (Bueno and French 2011: 888). It remains unclear, however, how
they account for distortive representations (Pincock 2005).



Finally, like similarity accounts, *Structuralism 1* has a
problem with non-existent targets because no model can be isomorphic
to something that doesn't exist. If there is no ether, a model
can't be isomorphic to it. Hence models without target cannot
represent what they seem to represent.



Most of these problems can be resolved by making moves similar to the
ones that lead to
 *Similarity 3*:
 introduce agents and hypothetical reasoning into the account of
representation. Going through the motions one finds:



*Structuralism 2*: A scientific model \(M\) represents a target
system \(T\) iff there is an agent \(A\) who uses \(M\) to represent a
target system \(T\) by proposing a theoretical hypothesis \(H\)
specifying an isomorphism between \(\mathcal{S}\_M\) and
\(\mathcal{S}\_T\).



This is in line with van Fraassen's views on representation. He
offers the following as the "Hauptstatz" of a theory of
representation: "*There is no representation except in the
sense that some things are used, made, or taken, to represent things
as thus and so*" (2008: 23, original emphasis). Likewise,
Bueno submits that "representation is an *intentional*
act relating two objects" (2010: 94-95, original
emphasis), and Bueno and French point out that using one thing to
represent another thing is not only a function of (partial)
isomorphism but also depends on "pragmatic" factors
"having to do with the use to which we put the relevant
models" (2011: 885).



As in the shift from
 *Similarity 2*
 to
 *Similarity 3*,
 this seems like a successful move, with many (although not all) of
the aforementioned concerns being met. But, again, the role of
isomorphism has shifted. The crucial ingredient is the agent's
intention and isomorphism has in fact become either a representational
style or normative criterion for accurate representation. Let us now
assess how well isomorphism fares as a response to these problems, and
the others outlined above.


### 4.3 Demarcation, Accuracy, Style, and Target-end Structure



Structuralism's stand on the Demarcation Problem is by and large
an open question. Unlike similarity, which has been widely discussed
across different domains, structural mappings are tied closely to the
formal framework of set theory, and have been discussed only sparingly
outside the context of the mathematized sciences. An exception is
French (2003), who discusses isomorphism accounts in the context of
pictorial representation. He considers in detail Budd's (1993)
account of pictorial representation and points out that it is based on
the notion of a structural isomorphism between the structure of the
surface of the painting and the structure of the relevant visual
field. Therefore representation is the perceived isomorphism of
structure (French 2003: 1475-1476) (this point is reaffirmed by
Bueno and French (2011: 864-865); see Downes (2009:
423-425) and Isaac (2019) for critical discussions).



The Problem of Style is to identify representational styles and
characterise them. A proposed structural mapping between the model and
the target offers an obvious response to this challenge: one can
represent a system by coming up with a model that is proposed to be
appropriately morphic to it. This delivers the isomorphism-style, the
homomorphism-style, the partial-isomorphism style and so on. We can
call these "morphism-styles" when referring to them in
general. Each of these styles also offers a clear-cut condition of
accuracy: the representation is accurate if the hypothesised morphism
holds; it is inaccurate if it doesn't.



This is neat answer. The question is what status it has
*vis-a-vis* the Problem of Style. Are morphism-styles
merely a subgroup of styles or are they privileged? The former is
uncontentious. However, the emphasis many structuralists place on
structure preserving mappings suggests that they do not regard
morphisms as merely one way among others to represent something. What
they seem to have in mind is the stronger claim that a representation
*must* be of that sort, or that morphism-styles are the only
acceptable styles.



This claim seems to conflict with scientific practice in at least two
respects. Firstly, many representations are inaccurate (and known to
be) in some way. Some models distort, deform and twist properties of
the target in ways that seem to undercut isomorphism, or indeed any of
the proposed structure preserving mappings. Some models in statistical
mechanics have an infinite number of particles and the Newtonian model
of the solar system represents the sun as a perfect sphere where in
reality it is fiery ball with no well-defined surface at all. It is at
best unclear how isomorphism, partial or otherwise, or homomorphism
can account for these kinds of idealisations. So it seems that styles
of representation other than structure preserving mappings have to be
recognised.



Secondly, the structuralist view is a rational reconstruction of
scientific modelling, and as such it has some distance from the actual
practice. Some philosophers have worried that this distance is too
large and that the view is too far removed from the actual practice of
science to be able to capture what matters to the practice of
modelling (this is the thrust of many contributions to Morgan and
Morrison 1999; see also Cartwright 1999). Although some models used by
scientists may be best thought of as set theoretic structures, there
are many where this seems to contradict how scientists actually talk
about, and reason with, their models. Obvious examples include
physical models like the San Francisco Bay model (Weisberg 2013), but
also systems such as the idealized pendulum or imaginary populations
of interbreeding animals. Such models have the strange property of
being *concrete-if-real* and scientists talk about them as if
they were real systems, despite the fact that they are obviously not
(Godfrey-Smith 2006). Thomson-Jones (2010) dubs this "face value
practice", and there is a question whether structuralism can
account for that practice.



There remains a final problem to be addressed in the context of
structural accounts of scientific representation. Target systems are
physical objects: atoms, planets, populations of rabbits, economic
agents, etc. Isomorphism is a relation that holds between two
structures and claiming that a set theoretic structure is isomorphic
to a piece of the physical world is *prima facie* a category
mistake. By definition, a morphism can only hold between two
structures. If we are to make sense of the claim that the model is
isomorphic to its target we have to assume that the target somehow
exhibits a certain structure \(\mathcal{S}\_T\). But what does it mean
for a target system--a part of the physical world--to
possess a structure, and where in the target system is the structure
located?



There are two prominent suggestions in the literature. The first,
originally suggested by Suppes (1962 [1969]), is that data models are
the target-end structures represented by models. This approach faces a
question whether we should be satisfied with an account of scientific
representation that precludes phenomena being represented (see Bogen
and Woodward (1988) for a discussion of the distinction between data
and phenomena, and Brading and Landry (2006) for a discussion of the
distinction in the context of scientific representation). Van Fraassen
(2008) has addressed this problem and argues for a pragmatic
resolution: in the context of use, there is no pragmatic difference
between representing phenomena and data extracted from it (see Nguyen
2016 for a critical discussion). The alternative approach locates the
target-end structure in the target system itself. One version of this
approach sees structures as being *instantiated* in target
systems. This view seems to be implicit in many versions of the
semantic view, and it is explicitly held by authors arguing for a
structuralist answer to the problem of the applicability of
mathematics (Resnik 1997; Shapiro 1997). This approach faces
underdetermination issues in that the same target can instantiate
different structures. The issue can be seen as arising due to there
being alternative descriptions of the system (Frigg 2006) or because a
version of "Newman's Objection" also bites in the
current context (Newman 1928; see Ainsworth 2009 and Ketland 2004 for
further discussion). A more radical version simply *identifies*
targets with structures (Tegmark 2008). This approach is highly
revisionary in particular when considering target systems like
populations of breeding rabbits or economies. So the question remains
for any structuralist account of scientific representation: where are
the required target-end structures to be found?


## 5. The Inferential Conception



The core idea of the inferential conception is to analyse scientific
representation in terms of the inferential function of scientific
models. In the previous accounts discussed, a model's
inferential capacity dropped out of whatever it was that was supposed
to answer the ER-problem (or SR-problem): proposed morphisms or
similarity relations between models and their targets for example. The
accounts discussed in this section reverse this order and explain
scientific representation directly in terms of surrogative
reasoning.


### 5.1 The DDI Account



According to Hughes' *Denotation, Demonstration, and
Interpretation* (DDI) account of scientific representation (1997,
2010: ch. 5), models *denote* their targets; are such that
model users can perform *demonstrations* on them; and
*interpret* the results of such demonstrations in terms of the
target. The last step is necessary because demonstrations establish
results about the model itself, and in interpreting these results the
model user draws inferences about the target from the model (1997:
333). Unfortunately Hughes has little to say about what it means to
interpret a result of a demonstration on a model in terms of its
target system, and so one has to retreat to an intuitive (and
unanalysed) notion of drawing inferences about the target based on the
 model.[10]



Hughes is explicit that he is not attempting to answer the ER-problem,
and that he does not offer denotation, demonstration, and
interpretation as individually necessary and jointly sufficient
conditions for scientific representation. He prefers the more



> 
> 
> modest suggestion that, if we examine a theoretical model with these
> three activities in mind, we shall achieve some insight into the kind
> of representation that it provides. (1997: 339)
> 
> 
> 



This is unsatisfactory because it ultimately remains unclear what
allows scientists to use a model to draw inferences about the target,
and it raises the question of what would have to be added to the DDI
conditions to turn them into a full-fledged response to the
ER-problem. If, alternatively, the conditions were taken to be
necessary and sufficient, then the account would require further
elaboration on what establishes the conditions.


### 5.2 Deflationary Inferentialism



Suarez argues that we should adopt a "deflationary or
minimalist attitude and strategy" (2004: 770) when addressing
the problem of scientific representation. Two different notions of
deflationism are in operation in his account. The first is to abandon
the aim of seeking necessary and sufficient conditions; necessary
conditions will be good enough (2004: 771). The second notion is that
we should seek "no deeper features to representation other than
its surface features" (2004:771) or "platitudes"
(Suarez and Sole 2006: 40), and that we should deny that
an analysis of a concept "is the kind of analysis that will shed
explanatory light on our use of the concept" (Suarez
2015: 39). Suarez intends his account of scientific
representation to be deflationary in both senses, and dubs it
"inferentialism". Letting \(A\) stand for the model and
\(B\) for the target, he offers the following analysis:



*Inferentialism*: "\(A\) represents \(B\) only if (i) the
representational force of \(A\) points towards \(B\), and (ii) \(A\)
allows competent and informed agents to draw specific inferences
regarding \(B\)" (2004: 773).



The first condition addresses the Requirement of Directionality and
ensures that \(A\) and \(B\) indeed enter into a representational
relationship. On might worry that explaining representation in terms
of representational force sheds little light on the matter as long as
no analysis of representational force is offered. But Suarez
resists attempts to explicate representational force in terms of a
stronger relation, like denotation or reference, on grounds that this
would violate deflationism (2015: 41). The second condition is in fact
just the Surrogative Reasoning Condition, now taken as a necessary
condition on scientific representation. Contessa (2007: 61) points out
that it remains mysterious how these inferences are generated. An
appeal to further analysis can, again, be blocked by appeal to
deflationism because any attempt to explicate how inferences are drawn
would go beyond "surface features". So the tenability of
*Inferentialism* in effect depends on the tenability of
deflationism about scientific representation. Suarez (2015)
defends deflationism by drawing analogies with three different
deflationary theories about truth, Ramsey's
"redundancy" theory, Wright's "abstract
minimalism" and Horwich's "use theory" (for
more information of these theories see the SEP entry on
 the deflationary theory of truth).
 An alternative defence builds on Brandom's inferentialism in
the philosophy of language (1994, 2000), a line of argument that is
developed by de Donato Rodriguez and Zamora Bonilla (2009) and
Kuorikoski and Lehtinen (2009).



*Inferentialism* provides a neat explanation of the possibility
of misrepresentation because the inferences drawn about a target need
not be true (Suarez 2004: 776). In as far as one accepts
representational force as a cogent concept, targetless models are
dealt with successfully because representational force (unlike
denotation) does not require the existence of a target (2004: 772).
Inferentialism repudiates the Representational Demarcation Problem and
aims to offer an account of representation that also works in other
domains such as painting (2004: 777). The account is ontologically
non-committal because anything that has an internal structure that
allows an agent to draw inferences can be a representation. Relatedly,
since the account is supposed to apply to a wide variety of entities
including equations and mathematical structures, the account implies
that mathematics is successfully applied in the sciences, but in
keeping with the spirit of deflationism no explanation is offered
about how this is possible. The account does not directly address the
Problem of Style.


### 5.3 Inflating Inferentialism: Interpretation



In response to the difficulties with *Inferentialism* Contessa
submits that "it is not clear why we should adopt a deflationary
attitude *from the start*" (2007: 50) and provides a
"interpretational account" of scientific representation
that is inspired by Suarez's account, but without being
deflationary. Contessa introduces the notion of an
*interpretation* of a model, in terms of its target system, as
a necessary and sufficient condition on epistemic representation (see
also Ducheyne 2012 for a related account):



*Interpretation*: "A [model \(M\)] is an epistemic
representation of a certain target [\(T\)] (for a certain user) if and
only if the user adopts an interpretation of the [\(M\)] in terms of
[\(T\)]." (Contessa 2007: 57; see also Contessa 2011:
126-127)



The leading idea of an interpretation is that the model user first
identifies sets of relevant objects in the model and the target, and
then pins down sets of properties and relations these objects
instantiate both in the model and the target. The user then (a) takes
\(M\) to denote \(T\); (b) takes every identified object in the model
to denote exactly one object in the target (and every relevant object
in the target has to be so denoted); (c) takes every property and
relation in the model to denote a property or relation of the same
type in the target (and, again, and every property and relation in the
target has to be so denoted). A formal rendering of these conditions
is what Contessa calls an "analytic interpretation" (see
his 2007: 57-62 for details; he also includes an additional
condition pertaining to functions in the model and target, which we
suppress for brevity).



*Interpretation* offers a neat answer to the ER-problem. The
account also explains the directionality of representation:
interpreting a model in terms of a target does not entail interpreting
a target in terms of a model. However, it has been noted that
*Interpretation* has difficulty accounting for the possibility
of misrepresentation, since it seems to require that the relevant
objects, properties, and relations actually exist in the target (Shech
2015), although this objection turns on a very strict reading of
Contessa's account. This problem is solved in
Diez's (2020) "Ensemble-Plus-Standing-For"
account of representation, which is based on conditions that rule out
a mismatch concerning the number of objects in the collection.
Contessa does not comment on the applicability of mathematics but
since his account shares with the structuralist account an emphasis on
relations and one-to-one model-target correspondence, Contessa can
appeal to the same account of the applicability of mathematics as the
structuralist. Like Suarez, Contessa takes his account to be
universal and apply to non-scientific representations such as
portraits and maps. But it remains unclear how *Interpretation*
addresses the Problem of Style. As we have seen earlier, in particular
visual representations fall into different categories and there is a
question about how these can be classified within the interpretational
framework. With respect to the Question of Ontology,
*Interpretation* itself places few constraints on what
scientific models are. All it requires is that they consist of
objects, properties, relations, and functions (but see Contessa (2010)
for further discussion of what he takes models to be, ontologically
speaking).


## 6. The Fiction View of Models



A recent family of approaches analyses models by drawing an analogy
between models and literary fiction. This analogy can be used in two
ways, yielding two different version of the fiction view. The first is
primarily motivated by ontological considerations rather than the
question of scientific representation *per se*. Scientific
discourse is rife with passages that appear to be descriptions of
systems in a particular discipline, and the pages of textbooks and
journals are filled with discussions of the properties and the
behaviour of those systems. In mechanics, for instance, the dynamical
properties of a system consisting of three spinning spheres with
homogenous mass distributions are the focus of attention; in biology
infinite populations are investigated; and in economics perfectly
rational agents with access to perfect information exchange goods.
Their surface structure notwithstanding, no one would mistake
descriptions of such systems as descriptions of an *actual*
system: we know very well that there are no such systems.



Thomson-Jones (2010: 284) refers to such a description as a
"description of a missing system". These descriptions are
embedded in what he calls the "face value practice" (2010:
285) the practice of talking and thinking about these systems as if
they were real. The face-value practice raises a number of questions.
What account should be given of these descriptions and what sort of
objects, if any, do they describe? Are we putting forward
truth-evaluable claims when putting forward descriptions of missing
systems?



The fiction view of models provides an answer: models are akin to
places and characters in literary fiction and claims about them are
true or false in the same way in which claims about these places and
characters are true or false. Such a position has been recently
defended explicitly by some authors (Frigg 2010a,b; Frigg and Nguyen
2021; Godfrey-Smith 2006; Salis 2021), but not without opposition
(Giere 2009; Magnani 2012). It does bear noting that the analogy has
been around for a while (Cartwright 1983; McCloskey 1990; Vaihinger
1911 [1924]). This leaves the thorny issue of how to analyse fictional
places and characters. Here philosophers of science can draw on
discussions from aesthetics to fill in the details about these
questions (Friend 2007 and Salis 2013 provide useful reviews).



The second version of the fiction view explicitly focuses on
representation. Most theories of representation we have encountered so
far posit that there are model systems and construe scientific
representation as a relation between two entities, the model system
and the target system. Toon calls this the *indirect* view of
representation (2012: 43). Indeed, Weisberg (2007) views this
indirectness as the defining feature of modelling (see also Knuuttila
and Loettgers 2017). This view contrasts with what Toon (2012: 43) and
Levy (2015: 790) call a *direct* view of representation. This
view does not recognise model systems instead aims to explain
representation as a form of direct description. On this view, models
provide an "imaginative description of real things" (Levy
2012: 741) such as actual pendula, and there is no such thing as a
model system of which the pendulum description is literally true (Toon
2012: 43-44).



Both Toon (2012) and Levy (2015) articulate the direct view by drawing
on Walton's (1990) theory of make-believe. At the heart of this
theory is the notion of a game of make-believe (see the SEP entry on
 imagination
 for further discussion). We play such a game if, for instance, when
walking through a forest we imagine that stumps are bears and if we
spot a stump we imagine that we spot a bear. In Walton's
terminology the stumps are *props*, and the rule that we
imagine a bear when we see a stump is a *principle of
generation*. Together a prop and principle of generation prescribe
what is to be imagined. Walton considers a vast variety of different
props, including statues and works of literary fiction. Toon focuses
on the particular kind of game in which we are prescribed to imagine
something of a real world object. A statue showing Napoleon on
horseback (Toon 2012: 37) is a prop mandating us to imagine, for
instance, that Napoleon has a certain physiognomy and certain facial
expressions. When reading *The War of the Worlds* (2012: 39) we
are prescribed to imagine that the dome of St Paul's Cathedral
has been attacked by aliens and now has a gaping hole on its western
side.



The crucial move is to say that models are props in games of make
believe. Specifically, material models are like the statue of Napoleon
and theoretical models are like the text of *The War of the
Worlds*: both prescribe, in their own way, to imagine something
about a real object. A ball-and-stick model of a methane molecule
prescribes us to imagine particular things about methane, and a model
description describing a point mass bob bouncing on a perfectly
elastic spring represents the real ball and spring system by
prescribing imaginings about the real system. This provides the
following answer to the ER-problem (Toon 2012: 62):



*Direct Representation*: \(M\) is a scientific representation
of \(T\) iff \(M\) functions as prop in game of make-believe which
prescribes imaginings about \(T\).



This account solves some of the problems posed in
 Section 1:
 *Direct Representation* is asymmetrical, makes room for
misrepresentation, and, given its roots in aesthetics, it renounces
the Demarcation Problem. The view absolves the Problem of Ontology
since models are either physical objects or descriptions, neither or
which are problematic in this context. Toon remains silent on both the
Problem of Style, and the applicability of mathematics.



Important questions remain. According to *Direct
Representation* models prescribe us to imagine certain things
about their target system. The account remains silent, however, on the
relationship between what a model prescribes us to imagine and what a
model user should actually infer about the target system, and so it
offers no answer to the ER-problem. Levy (2015) identifies this as a
gap in Toon's account and proposes to fill it by invoking
Yablo's (2014) notion of "partial truth", the idea
being that a model user should take the imagined propositions to be
partially true of their target systems. However, as Levy admits, there
are other sorts of cases that don't fit the mould, most notably
distortive idealisations. These require a different treatment and
it's an open question what this treatment would be.



A further worry is how *Direct Representation* deals with
targetless models. If there is no target system, then what does the
model prescribe imaginings about? Toon is well aware of such models
and suggests the following solution: if a model has no target it
prescribes imaginings about a fictional character (2012: 76). This
solution, however, comes with ontological costs, and one of the
declared aims of the direct view was to avoid such costs by removing
model systems from the picture. Levy (2015) aims to salvage
ontological parsimony and proposes a radical move: there are no
targetless models. If a (purported) model has no target then it is not
a model. There remains a question, however, how this view can be
squared with scientific practice where targetless models are not only
common but also clearly acknowledged as such.


## 7. Representation-As



In Goodman's (1976) account of aesthetic representation the idea
is that a work of art does not just denote its subject, but moreover
it represents it as being thus or so (see the SEP entry on
 Goodman's aesthetics
 for further discussion). Elgin (2010) further developed this account
and, crucially, suggested that it also applies to scientific
representations. We first discuss Goodman and Elgin's notion of
representation-as and then consider a recent extension of their
framework.


### 7.1 From Art to Science



Many instances of epistemic representation are instances of what
Goodman and Elgin call "representation-as". Caricatures
are paradigmatic examples: Churchill is represented as a bulldog and
Thatcher is represented as a boxer. But the notion is more general:
Holbein's *Portrait of Henry VIII* represents Henry as
imposing and powerful and Stoddart's statue of David Hume
represents him as thoughtful and wise. Using these representations we
can learn about their targets, for example we can learn about a
politician's or philosopher's personality. The leading
idea of the views discussed in this section is that scientific
representation works in much the same way. A model of the solar system
represents it as consisting of perfect spheres; the logistic model of
growth represents the population as reproducing at fixed intervals of
time; and so on. In each instance, models can be used to attempt to
learn about their targets by determining what the former represent the
latter as being.



The locution of representation-as functions in the following way: an
object \(X\) (e.g., a picture, statue, or model) represents a subject
\(Y\) (e.g., a person or target system) as being thus or so \((Z)\).
The question then is what establishes this sort of representational
relationship. The answer requires introducing some of the concepts
Goodman and Elgin use to develop their account of
representation-as.



Goodman and Elgin draw a distinction between something being a
representation of a \(Z\), and something being a \(Z\)-representation
(Elgin 2010: 1-2; Goodman 1976: 21-26). A painting of a
unicorn is a unicorn-representation because it shows a unicorn, but it
is not a representation of a unicorn because there are no unicorns.
Being a \(Z\)-representation is a one-place predicate that categorises
representations according to their subject matter. Being a
representation *of* something is established by denotation; it
is a binary relation that holds between a symbol and the object which
it denotes. The two can, but need not, coincide. Some
dog-representations are representations of dogs, but not all are
(e.g., a caricature of Churchill), and not all representations of dogs
are dog-representations (e.g., a lightening bolt may represent the
fastest greyhound at the races).



The next notion is *exemplification*: an object \(X\)
exemplifies a property \(P\) iff \(X\) instantiates \(P\) and thereby
refers back to \(P\) (Goodman 1976: 53). In the current context
properties are to be understood in the widest possible sense. An item
can exemplify one-place properties, multi-place properties (i.e.,
relations), higher order properties, structural properties, etc.
Paradigmatic examples of this are samples. A chip of paint on a
manufacturer's sample card instantiates a certain colour and at
the same time refers to that colour (Elgin 1983: 71). Notice that
instantiation is necessary but insufficient for exemplification: the
sample card does not exemplify being rectangular for example. When a
object exemplifies a property it provides us with epistemic access to
that property.



Representation-as is then established by combining these notions
together: a \(Z\)-representation exemplifies properties associated
with
 \(Z\)s,[11]
 and if the \(Z\)-representation additionally denotes \(Y\), then
these properties can be imputed onto \(Y\) (cf. Elgin 2010: 10). This
provides the following account of epistemic representation:



*Representation-As*: \(X\) is an epistemic representation of
\(Y\) iff (i) \(X\) denotes \(Y\), (ii) \(X\) is a
\(Z\)-representation exemplifying properties \(P\_1, \ldots , P\_n\),
and (iii) \(X\) imputes \(P\_1, \ldots , P\_n\), or related properties,
onto \(Y\).



Applying this in the scientific context, i.e., by letting \(X\) range
over models and \(Y\) over target systems, we arrive at an answer to
the ER-problem. *Representation-As* also answers the other
problems introduced in
 Section 1:
 it repudiates the demarcation problem and it explains the
directionality of representation. It accounts for surrogative
reasoning in terms of the properties imputed to the target. If \(Y\)
possesses the imputed properties then the representation is accurate,
but since the target doesn't necessarily need to instantiate
them it allows for the possibility of misrepresentation. Different
styles can be accounted for by categorising representations in terms
of different \(Z\)s, or in terms of the properties they exemplify.
However, at least as stated, the account remains silent on the problem
of ontology and the applicability of mathematics. We discuss below how
to account for targetless models.


### 7.2 The DEKI Account



*Representation-As*
 raises a number of questions when applied in the scientific context.
The first concerns the notion of a \(Z\)-representation. While it has
intuitive appeal in the case of pictures, it is less clear how it
works in the context of science. Phillips and Newlyn constructed an
elaborate system of pipes and tanks, now know as the Phillips-Newlyn
machine, to model an economy (see Morgan and Boumans 2004 and Barr
2000 for useful discussions). So the machine is an
economy-representation. But what turns a system of pipes and tanks
into an economy-representation?



Frigg and Nguyen (2016: 227-8) argue that in order to turn an
object \(X\) into a scientific model, it must be interpreted in the
appropriate way (note that they do not use
"interpretation" in the way that Contessa uses it, as
discussed above): properties that \(X\) has, qua object, are paired up
with relevant \(Z\) properties and for quantitative properties, i.e.,
properties such as mass or flow, which take numerical values, there
needs to a further association between the values of the \(X\)
property and the values of the \(Z\) property to which it is mapped.
In the case of the Phillips-Newlyn machine, for example, hydraulic
properties of the machine are associated with economic properties and
a rule is given specifying that a litre of water corresponds to
certain amount of a model-economy's currency (Frigg and Nguyen
2018).



The \(X\) and \(Z\) properties that are so associated need not exhaust
the properties that \(X\) instantiates, nor do all possible \(Z\)
properties need be associated with an \(X\) property. A scientific
model can then be defined as a \(Z\)-representation, i.e., an object
under an interpretation. This notion of a model explicitly does not
presuppose a target system and hence makes room for targetless
models. Models can then be seen as instantiating
\(Z\)-properties *under the relevant interpretation*, which
explains how the model can exemplify such properties.



The next issue is that exemplified properties are rarely exactly
imputed onto target systems. According to
 *Representation-As*
 the imputed properties are either the ones exemplified by the
\(Z\)-representation, "or related ones". Frigg and Nguyen
(2016: 228), building on Frigg (2010a: 125-135), prefer to be
explicit about the relationship between the exemplified properties and
the ones to be imputed onto the target. They do this by introducing a
"key", which explicitly associates the exemplified
properties with properties to be imputed onto the target. For example,
in the case of a London Tube map, the key associates particular
colours with particular tube lines, and in the case of idealisations
the key associates de-idealised properties with model-properties.



Gathering the various pieces together leads to the following account
of representation (Frigg and Nguyen 2016: 229):




DEKI: Let \(M = \langle X, I \rangle\) be a model, where \(X\) is an
object and \(I\) an interpretation. Let \(T\) be the target system.
\(M\) represents \(T\) as \(Z\) iff all of the following conditions
are satisfied:


1. \(M\) denotes \(T\).
2. \(M\) exemplifies \(Z\)-properties \(P\_1, \ldots , P\_n\).
3. \(M\) comes with key \(K\) associating the set \(\{P\_1, \ldots ,
P\_n\}\) with a (possibly identical) set of properties \(\{Q\_1, \ldots
, Q\_m\}\).
4. \(M\) imputes at least one of the properties \(Q\_1, \ldots , Q\_m\)
to \(T\).



\(M\) is a scientific representation of \(T\) iff \(M\) represents
\(T\) as \(Z\) as defined in (i)-(iv).




The moniker "DEKI" highlights the account's key
features: denotation, exemplification, keying-up and imputation.
*DEKI* answers the problems from
 Section 1
 in much the same way as
 *Representation-As*
 did. However, it adds to the latter in at least three ways. Firstly,
the conditions given make it clear what makes scientific models
\(Z\)-representations in the first place: interpretations. Secondly,
it makes explicit that the properties exemplified by the model need
not be imputed exactly onto the target, and highlights the need to
investigate keys specifying the relationship between properties in
models and the properties that models actually impute onto their
targets. Finally, it makes explicit how to account for targetless
models. A scientific model that fails to denote a target system can
nevertheless be a \(Z\)-representation. A model of a bridge that is
never built is still a bridge-representation, which exemplifies
properties related to bridges (stability and so on), despite the fact
that it is not a representation of anything. However, as in the case
of
 *Representation-As*
 questions remain with respect to the problem of ontology and the
applicability of mathematics.