Scientific Representation

First published Mon Oct 10, 2016; substantive revision Thu Nov 4, 2021

Science provides us with representations of atoms, elementaryparticles, polymers, populations, pandemics, economies, rationaldecisions, aeroplanes, earthquakes, forest fires, irrigation systems,and the world’s climate. It’s through theserepresentations that we learn about the world. This entry exploresvarious different accounts of scientific representation, with aparticular focus on how scientificmodels represent theirtarget systems. As philosophers of science are increasinglyacknowledging the importance, if not the primacy, of scientific modelsas representational units of science, it’s important to stressthat how they represent plays a fundamental role in how we are toanswer other questions in the philosophy of science (for instance inthe scientific realism debate). This entry begins by disentangling“the” problem of scientific representation, and thencritically evaluates the current options available in theliterature.

1. Problems Concerning Scientific Representation

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

The first thing that strikes the novice in the debate about scientificrepresentation is that there seems to be little agreement about whatthe problem is. Different authors frame the problem of scientificrepresentation in different ways, and eventually they examinedifferent issues. So a discussion of scientific representation has tobegin with a clarification of the problem itself. Reviewing theliterature on the subject leads us to the conclusion that there is nosuch thing asthe problem of scientificrepresentation—in fact, there are at least five differentproblems concerning scientific representation (Frigg and Nguyen 2020:ch. 1). In this section we formulate these problems and articulatefive conditions of adequacy that every account of scientificrepresentation has to satisfy.

The first problem is: what turns something into a scientificrepresentation of something else? It has become customary to phrasethis 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 objectdoing the representing and “\(T\)” for “targetsystem”, the part or aspect of the world the representation is about?^[1] Let us call this theScientific Representation Problem (orSR-Problem for short).

A number of contributors to the debate have emphasised that scientificrepresentation is an intentional concept, depending on factors such asa user’s intentions, purposes and objectives, contextualstandards of accuracy, intended audiences, and community practices(see, for instance, Boesch 2017; Giere 2010; Mäki 2011;Suárez 2004; and van Fraassen 2008). We will discuss these indetail below. At this point it needs to be emphasised that framing theproblem in terms of a biconditional does not preclude such factors tobe part of the analysis. The blank can be filled with a \((n+2)\)-aryrelation \(C(S, T, x_1, \ldots, x_n)\) (\(C\) for“constitutes”), where \(n \ge 0\) is a natural number andthe \(x_i\) are factors such as intentions and purposes.

A first important condition of adequacy on any reply to this problemis that scientific representations allow us to form hypotheses abouttheir target systems. An X-ray picture provides information about thebones of the patient, and models allow investigators to discoverfeatures of the things models stands for. Every acceptable theory ofscientific representation has to account for how reasoning conductedon 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 theSurrogative Reasoning Condition.^[2] This condition distinguishes models from lexicographical andindexical representations, which do not allow for surrogativereasoning.

Unfortunately this condition does not constrain answers sufficientlybecause any account of representation that fills the blank in a waythat satisfies the surrogative reasoning condition will almostinvariably also cover other kinds of representations. Speed cameraphotographs give the police information about drivers breaking thelaw, a cardboard model of the palace instructs us about its layout andproportions, and a weather map shows you where to expect rain. Theserepresentations are therefore are likely to fall under an account ofrepresentation that explains surrogative reasoning. Hence,representations other than scientific representations also allow forsurrogative reasoning, which raises the question: how do scientificrepresentations differ from other kinds of representations that allowfor 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 theRepresentational Demarcation Problem.

Callender and Cohen voice scepticism about there being a solution tothis problem and suggest that the distinction between scientific andnon-scientific representations is circumstantial (2006: 83):scientific representations are representations that are used ordeveloped by someone who is a scientist. Other authors do notexplicitly discuss the representational demarcation problem, butstances similar to Callender and Cohen’s are implicit in anyapproach that analyses scientific representation alongside other kindsof representation. Elgin (2010), French (2003), Frigg (2006), Hughes(1997), Suárez (2004), and van Fraassen (2008), for instance,all draw parallels between scientific and pictorial representation,which would make little sense if pictorial and scientificrepresentation were categorically different.

Those who reject the notion that there is an essential differencebetween scientific and non-scientific representation can follow asuggestion of Contessa’s (2007) and broaden the scope of theinvestigation. Rather than analysing scientific representation, theycan analyse the broader category ofepistemic representation.This category comprises scientific representations, but it alsoincludes other representations that allow for surrogative reasoning.The task then becomes to fill the blank in “\(S\) is anepistemic representation of \(T\) iff ___”. We call this theEpistemic Representation Problem (ER-Problem, forshort), and the biconditional theER-Scheme. So one can saythat the ER-Problem is to fill the blank in the ER-Scheme.

Not all representations are of the same kind, not even if we restrictour attention to scientific representations (assuming they are foundto be relevantly different to non-scientific epistemicrepresentations). An X-ray photograph represents an ankle joint in adifferent way than a biomechanical model, a mercury thermometerrepresent the temperature of gas in a different way than statisticalmechanics does, and chemical theory represents a C60 fullerene indifferent way that an electron-microscope image of the molecule. Evenwhen restricting attention to the same kind of representation, thereare important differences: Weizsäcker’s liquid drop model,for instance, represents the nucleus of an atom in a manner that seemsto be different from the one of the shell model, and an electriccircuit model represents the brain function in a different way than aneural network model. In brief, there seem to be differentrepresentational styles. This raises the question: what styles arethere and how can they be characterised?^[3] We call this theProblem of Style.^[4] There is no expectation that acomplete list of styles beprovided in response. Indeed, it is unlikely that such a list can everbe drawn up, and new styles will be invented as science progresses.For this reason a response to the problem of style will always beopen-ended, providing a taxonomy of what is currently available whileleaving room for later additions.

Some representations are accurate; others aren’t. The quantummechanical model is an accurate representation of the atom (at leastby our current lights) but the Thomson model isn’t. On whatgrounds do we make such judgments? Morrison (2008: 70) reminds us thatit is a task for theory of representation to identify what constitutesan accurate representation. We call this the problem ofStandardsof Accuracy. Providing such standards is one of the issues anaccount of representation has to address, which, however, is not tosay that accurate representation is the sole epistemic virtue ofscientific models. As Parker (2020) points out, there are numerous ofconsiderations to take into account when evaluating a model’sadequacy for its purpose, see Downes (2021: ch. 5) for furtherdiscussion.

This problem goes hand in hand with a condition of adequacy: thePossibility of Misrepresentation. Asking what makes arepresentation an accurate representationipso factopresupposes 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. Itis therefore a general constraint on a theory of scientificrepresentation that it has to make misrepresentation possible.^[5]

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

A further condition of adequacy for an account of scientificrepresentation is that it must account for the directionality ofrepresentation. As Goodman points out (1976: 5), representations areabout their targets, but (at least in general) targets are not abouttheir representations: a photograph represents the cracks in the wingof aeroplane, but the wing does not represent the photograph. So thereis an essential directionality to representations, and an account ofscientific, or epistemic, representation has to identify the root ofthis directionality. We call this theRequirement ofDirectionality.

Some representations, in particular models and theories, aremathematized and their mathematical aspects are crucial to theircognitive and representational function. This forces us to reconsidera time-honoured philosophical puzzle: the applicability of mathematicsin the empirical sciences. The problem can be traced back at least toPlato’sTimaeus, but its modern expression is due toWigner who challenged us to find an explanation for the enormoususefulness of mathematics in the sciences (1960: 2). The question howa mathematized model represents its target implies the question howmathematics applies to a physical system (see Pincock 2012 for anexplicit discussion of the relationship between scientificrepresentation and the applicability of mathematics). For this reason,our fifth and final condition of adequacy is that an account ofrepresentation has to explain how mathematics is applied to thephysical world. We call this theApplicability of MathematicsCondition.

In answering the above questions one invariably runs up against afurther problem, theProblem of Ontology: what kinds ofobjects are representations? If representations are material objectsthe answer is straightforward: photographic plates, pieces of papercovered with ink, elliptical blocks of wood immersed in water, and soon. But not all representations are like this. As Hacking (1983: 216)puts it, some representations one holds in one’s head ratherthan one’s hands. The Newtonian model of the solar system, theLotka-Volterra model of predator-prey interaction and the generaltheory of relativity are not things you can put on your laboratorytable and look at. The problem of ontology is to come clear on ourcommitments and provide a list with things that we recognise—ordon’t recognise—as entities performing a representationalfunction and give an account of what they are in case these entitiesraise questions (what exactly do we mean by something that one holdsin 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 thisproblem in different ways.

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

Address theRepresentational Demarcation Problem (thequestion how scientific representations differ from other kinds ofrepresentations).
Those who demarcate scientific from non-scientific representationshave to provide an answer to theScientific RepresentationProblem (fill the blank in “\(S\) is a scientificrepresentation of \(T\) iff ___”). Those who reject therepresentational demarcation problem can address theEpistemicRepresentation Problem (fill the blank in ER-Scheme: “\(S\)is an epistemic representation of \(T\) iff ___”).
Respond to theProblem of Style (what styles are thereand how can they be characterised?).
FormulateStandards of Accuracy (how do we identify whatconstitutes an accurate representation?).
Address theProblem of Ontology (what are the kind ofobjects that serve as representations?).

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

Surrogative Reasoning (scientific representations allowus to generate hypotheses about their target systems).
Possibility of Misrepresentation (if \(S\) does notaccurately represent \(T\), then it is a misrepresentation but not anon-representation).
Targetless Models (what are we to make of scientificrepresentations that lack targets?).
Requirement of Directionality (scientific representationsare about their targets, but targets are not about theirrepresentations).
Applicability of Mathematics (how does the mathematicalapparatus used in some scientific representations latch onto thephysical world).

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

Any attempt to tackle these questions faces an immediatemethodological problem. As per the problem of style, there aredifferent kinds of representations: scientific models, theories,measurement outcomes, images, graphs, diagrams, and linguisticassertions are all scientific representations, and even within thesegroups there can be considerable variation. But every analysis has tostart somewhere, and so the problem is where. One might adopt auniversalist position, holding that the diversity of styles dissolvesunder analysis and at bottom all instances of scientific/epistemicrepresentation function in the same way and are covered by the sameoverarching account. For such a universalist the problem loses itsteeth because any starting point will lead to the same result. Thoseof particularist bent deny that there is such a theory. They willfirst divide the scientific/epistemic representations into relevantsubclasses and then analyse each subclass separately.

Different authors assume different stances in this debate, and we willdiscuss their positions below. However, there are few, if any,thoroughgoing universalists and so a review like the current one hasto discuss different cases. Unfortunately space constraints prevent usfrom examining all the different varieties of scientific/epistemicrepresentation, and a selection has to be made. This invariably leadsto the neglect of some kinds of representations, and the best we cando about this is to be explicit about our choices. We resolve toconcentrate 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 onscientific representation, which is predominantly concerned withscientific models, and with the prime importance that currentphilosophy of science attaches to models (see the SEP entry onmodels in science for a survey).^[6]

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

Before delving into the discussion a common misconception needs to bedispelled. The misconception is that a representation is a mirrorimage, a copy, or an imitation of the thing it represents. On thisview representation isipso facto realistic representation.This is a mistake. Representations can be realistic, but they neednot. And representations certainly need not be copies of the realthing, an observation exploited by Lewis Carroll and Jorge Luis Borgesin their satires,Sylvie and Bruno andOn Exactitude inScience respectively, about cartographers who produce maps aslarge as the country itself only to see them abandoned (for adiscussion see Boesch 2021). Throughout this review we encounterpositions that make room for non-realistic representation and hencetestify 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 demarcationproblem: there is no difference between scientific representations andother kinds of representations, not even between scientific andartistic representation. Underlying this claim is a position they call“General Griceanism” (GG). The core of GG is the reductiveclaim that all representations owe their status as representations toa privileged core of fundamental representations. GG then comes with apractical prescription about how to proceed with the analysis of arepresentation:

the General Gricean view consists of two stages. First, it explainsthe representational powers of derivative representations in terms ofthose of fundamental representations; second, it offers some otherstory to explain representation for the fundamental bearers ofcontent. (2006: 73)

Of these stages only the second requires serious philosophical work,and this work is done in the philosophy of mind because thefundamental 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 aboverecipe. It is reduced to mental representation by an act ofstipulation. In Callender and Cohen’s own example, the saltshaker on the dinner table can represent Madagascar as long as someonestipulates that the former represents the latter, since

the representational powers of mental states are so wide-ranging thatthey can bring about other representational relations betweenarbitrary relata by dint of mere stipulation. (2006: 73–74)

So explaining any form of representation other than mentalrepresentation is a triviality—all it takes is an act of“stipulative fiat” (2006: 75). This supplies an answer tothe ER-problem:

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

The first problem facingStipulative Fiat is whether or notstipulation, or the bare intentions of language users, suffice toestablish representational relationships. In the philosophy oflanguage this gets called the “Humpty Dumpty” problem. Itconcerns whether or not Lewis Carroll’s Humpty Dumpty could usethe word “glory” to mean “a nice knockdownargument” (Donnellan 1968; MacKay 1968). (We ignore thedifference between meaning and denotation here). In that context itdoesn’t seem like he can, and analogous questions can be posedin the context of scientific representation: can a scientist make anymodel represent any target simply by stipulating that it does?

Even if stipulation were sufficient to establish some sort ofrepresentational relationship,Stipulative Fiat fails to meetthe Surrogative Reasoning Condition: assuming a salt shaker representsMadagascar in virtue of someone’s stipulation that this is so,this tells us nothing about how the salt shaker could be used to learnabout Madagascar in the way that scientific models are used to learnabout their targets (Liu 2015: 46–47, for a related objectionssee Boesch 2017: 974–978, Bueno and French 2011: 871–873,Gelfert 2016: 33, Ruyant 2021: 535). And appealing to additional factsabout the salt shaker (the salt shaker being to the right of thepepper mill might allow us to infer that Madagascar is to the east ofMozambique) in order to answer this objection goes beyondStipulative Fiat. Callender and Cohen do admit somerepresentations are more useful than others, but claim that

the questions about the utility of these representational vehicles arequestions about the pragmatics of things that are representationalvehicles, not questions about their representational statusperse. (2006: 75)

But even if the Surrogative Reasoning Condition is relegated to therealm of “pragmatics” it seems reasonable to ask for anaccount of how it is met.

An important thing to note is that even ifStipulative Fiatis untenable, we needn’t give up on GG. GG only requires thatthere besome explanation of how derivative representationsrelate to fundamental representations; it does not require that thisexplanation be of a particular kind, much less that it consist innothing but an act of stipulation (Toon 2010: 77–78). Ruyant(2021) has recently proposed what he calls “trueGriceanism”, which differs from the original account in that itinvolves a non-trivial reduction of scientific representation to morefundamental representations (complex sequences mental states andpurpose-directed behaviour) in a way that is also sensitive to thecommunal aspects of practices in which the models are embedded. Thisis a step into the direction indicated by Toon. More generally, asCallender and Cohen note, all that it requires is that there is aprivileged class of representations and that other types ofrepresentations owe their representational capacities to theirrelationship with the primitive ones. So philosophers need an accountof 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 contributorsto the debate on scientific representation can be seen as fallingunder the umbrella of GG. Indeed, As we will see below, many of themore developed versions of the accounts of scientific representationdiscussed throughout this entry invoke the intentions of model users,albeit in a more complex manner thanStipulative Fiat.

3. The Similarity Conception

Similarity and representation initially appear to be two closelyrelated concepts, and invoking the former to ground the latter has aphilosophical lineage stretching back at least as far as Plato’sThe Republic.^[8] In its most basic guise the similarity conception of scientificrepresentation asserts that scientific models represent their targetsin virtue of being similar to them. This conception has universalaspirations in that it is taken to account for representation across abroad range of different domains. Paintings, statues, and drawings aresaid to represent by being similar to their subjects, and Giereproclaimed that it covers scientific models alongside “words,equations, diagrams, graphs, photographs, and, increasingly,computer-generated images” (2004: 243). So the similarity viewrepudiates the demarcation problem and submits that the samemechanism, namely similarity, underpins different kinds ofrepresentation 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 overinsights gained in the model to the target. If the similarity between\(M\) and \(T\) is based on shared properties, then a property foundin \(M\) would also have to be present in \(T\); and if the similarityholds between properties themselves, then \(T\) would have toinstantiate properties similar to \(M\).

However, appeal to similarity in the context of representation leavesopen whether similarity is offered as an answer to the ER-Problem orthe Problem of Style, or whether it is meant to set Standards ofAccuracy. Proponents of the similarity conception typically haveoffered little guidance on this issue. So we examine each option inturn and ask whether similarity offers a viable answer. We then turnto the question of how the similarity view deals with the Problem ofOntology.

3.1 Similarity and ER-Problem

Understood as response to the ER-Problem, the simplest similarity viewis 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 thewrong logical properties. Goodman (1976: 4–5) points out thatsimilarity 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 factin most cases does not) represent \(A\). Everything is similar toitself, but most things do not represent themselves. So this accountdoes not meet our fourth condition of adequacy for an account ofscientific representation insofar as it does not provide a directionto representation.

There are accounts of similarity under which similarity is not asymmetric relation (see Tversky 1977; Weisberg 2012, 2013: ch. 8; andPoznic 2016: sec. 4.2). This raises the question of how to analysesimilarity. 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 doesit, as we will see, solve other problems of the similarityaccount.

The most significant problem facingSimilarity 1 is thatwithout constraints on what counts as similar, any two things can beconsidered similar (Aronson et al. 1995: 21; Goodman 1972:443–444). This has the unfortunate consequence that anythingrepresents anything else. A natural response to this difficulty is todelineate a set of relevant respects and degrees to which \(M\) and\(T\) have to be similar. This idea can be moulded into the followingdefinition:

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

On this definition one is free to choose one’s respects anddegrees so that unwanted similarities drop out of the picture. Whilethis solves the last problem, it leaves the problem of logicalproperties untouched: similarity in relevant respects and to therelevant degrees is reflexive (and symmetrical, depending onone’s notion of similarity). Moreover,Similarity 2faces three further problems.

Firstly, similarity, even restricted to relevant similarities, is tooinclusive a concept to account for representation. In many casesneither one of a pair of similar objects represents the other. Thispoint has been brought home in Putnam’s now-classical thoughtexperiment due to Putnam (1981: 1–3). An ant is crawling on apatch of sand and leaves a trace that happens to resemble WinstonChurchill. Has the ant produced a picture, a representation, ofChurchill? Putnam’s answer is that it didn’t because theant has never seen Churchill, had no intention to produce an image ofhim, was not causally connected to Churchill, and so on. Althoughsomeone else might see the trace as a depiction of Churchill, thetrace itself does not represent Churchill. The fact that the trace issimilar to Churchill does not suffice to establish that the tracerepresents him. And what is true of the trace and Churchill is true ofevery other pair of similar items: even relevant similarity on its owndoes not establish representation.

Secondly, as noted inSection 1, allowing for the possibility of misrepresentation is a key desideratarequired of any account of scientific representation. In the contextof a similarity conception it would seem that a misrepresentation isone that portrays its target as having properties that are not similarin the relevant respects and to the relevant degree to the trueproperties of the target. But then, onSimilarity 2, \(M\) isnot a representation at all. The account thus has difficultydistinguishing between misrepresentation and non-representation(Suárez 2003: 233–235).

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

At least some of these problems can be resolved by taking the very actofasserting a specific similarity between a model and atarget as constitutive of the scientific representation. Giere (1988:81) suggests that models come equipped with what he calls“theoretical hypotheses”, statements asserting that modeland target are similar in relevant respects and to certain degrees. Heemphasises that “scientists are intentional agents with goalsand purposes” (2004: 743) and proposes to build this insightexplicitly into an account of representation. This involves adoptingan agent-based notion of representation that focuses on “theactivity of representing” (2004). Analysing representation inthese terms amounts to analysing schemes like

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

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

Similarity 3: A scientific model \(M\) represents a targetsystem \(T\) iff there is an agent \(A\) who uses \(M\) to represent atarget 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 withmisrepresentation because \(H\) being a hypothesis, there is noexpectation that the assertions made in \(H\) are true. If they are,then the representation is accurate (or the representation is accurateto the extent that they hold). If they do not, then the representationis a misrepresentation. It also resolves the issue with directionalitybecause \(H\) can be understood as introducing an asymmetry that isnot present in the similarity relation. However, it fails to resolvethe problem with representation without a target. If there is noether, no hypotheses can be asserted about it, at least in anystraightforward way.

Similarity 3, by invoking an active role for the purposes andactions of scientists in constituting scientific representation, marksa significant change in emphasis for similarity-based accounts.Suárez (2003: 226–227), drawing on van Fraassen (2002)and Putnam (2002), defines “naturalistic” accounts ofrepresentation as ones where

whether or not representation obtains depends on facts about the worldand does not in any way answer to the personal purposes, views orinterests of enquirers.

By building the purposes of model users directly into an answer to theER-problem,Similarity 3 is explicitly not a naturalisticaccount (in contrast toSimilarity 1). The shift to usersperforming representational actions invites the question of it meansfor a scientist to perform such an action. Boesch (2019) offers ananswer which draws on Anscombe’s (2000) account of intentionalaction: something is a “scientific, representationalaction” (Boesch 2019: 312) when a description of ascientist’s interaction with a model stands as an earlierdescription towards the final description which is some scientific aimsuch as explanation, predication, or theorizing.

Even thoughSimilarity 3 resolves a number of issues thatbeset simpler versions, it does not seem to be a successfulsimilarity-based solution to the ER-Problem. A closer look atSimilarity 3 reveals that the role of similarity has shifted.As far as offering a solution to the ER-Problem is concerned, all theheavy lifting inSimilarity 3 is done by the appeal to agentsand their intentions. Giere implicitly concedes this when he observesthat similarity is “the most important way, but probably not theonly way” for models to function representationally (2004: 747).But if similarity is not the only way in which a model can be used asa representation, then similarity has become otiose in a reply to theER-problem. In fact, being similar in the relevant respects to therelevant degree now plays the role either of a representational styleor of a normative criterion for accurate representation, rather thanconstituting representationper se. We assess in the nextsection whether similarity offers cogent replies to the issues ofstyle and accuracy, and we raise a further problem for any account ofscientific 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 wayscould potentially provide an answer to the Problem of Style. If \(M\)representing \(T\) involves the claim that \(M\) and \(T\) are similarin a certain respect, the respect chosen specifies the style of therepresentation; and if \(M\) and \(T\) are in fact similar in thatrespect (and to the specified degree), then \(M\) accuratelyrepresents \(T\) 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 \(T\) areproposed to be similar with respect to structural properties, then wemight have a style of structural modelling; and so on and soforth.

A first step in the direction of such an understanding of styles isthe explicit analysis of the notion of similarity. The standard way ofcashing out what it means for an object to be similar to anotherobject 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 twomost prominent formal frameworks that develop this idea are thegeometric and contrast accounts (see Decock and Douven 2011 for adiscussion).

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

This problem is supposed to be overcome in Tversky’s contrastaccount (1977). This account defines a gradated notion of similaritybased on a weighted comparison of properties. Weisberg has recentlyintroduced this account into the philosophy of science where it servesas the starting point for hisweighted feature matching account ofmodel world-relations (for details see Weisberg 2012, 2013: ch.8). Although the account has some advantages, questions remain whetherit can capture the distinction between what Niiniluoto (1988:272–274) calls “likeness” and “partialidentity”. Two objects are alike to the extent that theyco-instantiate similar properties (for example, a red phone box and ared London bus might be alike with respect to their colour, despitenot instantiating the exact same shade of red). Two objects arepartially identical to the extent that they co-instantiate identicalproperties. As Parker (2015: 273) notes, contrast based accounts ofsimilarity like Weisberg’s have difficulties capturing theformer, and this is often pertinent in the context of scientificrepresentation where models and their targets need not co-instantiatethe exact same property. Concerns of this sort have led Khosrowi(2020) to suggest that the notion of sharing features should beanalysed in a pluralist manner. Sharing a feature sometimes meanssharing the exact same feature; sometimes it means to share featureswhich are sufficiently quantitatively close to one another; andsometimes it means having features which are themselves“sufficiently similar”.

A further question that remains for someone who uses the notion ofsimilarity to answer to the Problem of Style and provide standards ofaccuracy in the manner under consideration here is whether it trulycaptures all of scientific practice. Similarity theorists arecommitted to the claim that whenever a scientific model represents itstarget system, this is established in virtue of a model userspecifying a relevant similarity, and if the similarity holds, thenthe representational relationship is accurate. These universalaspirations require that the notion of similarity invoked capture therelationship that holds between diverse entities such as a basin-modelof the San Francisco bay area, a tube map and an underground trainsystem, and the Lotka-Volterra equations of predator-pray interaction.Whether all of these relationships can be captured in terms ofsimilarity remains an open question. In addition, this view iscommited to the idea that idealised aspects of scientific models,understood as dissimilarities between models and their targets, aremisrepresenations and as such the view has difficulty capturing thepositive epistemic role that such aspects can play (Nguyen 2020).

Another problem facing similarity based approaches concerns theirtreatment of the ontology of models. If models are supposed to besimilar to their targets in the ways specified by theoreticalhypotheses, then they must be thekind of things that can beso similar. For material models like the San Francisco Bay model(Weisberg 2013), ball and stick models of molecules (Toon 2011), thePhillips-Newlyn machine (Morgan and Boumans 2004), or model organisms(Ankeny and Leonelli 2021) this seems straightforward because they areof the same ontological kind as their respective targets. But manyinteresting scientific models are not like this: they are what Hackingaptly describes as “something you hold in your head rather thanyour hands” (1983: 216). Following Thomson-Jones (2012) we callsuch modelsnon-concrete models. The question then is howsuch models can be similar to their targets. At the very least thesemodels are “abstract” in the sense that they have nospatiotemporal location. But if so, then it remains unclear how theycan instantiate the sorts of properties specified by theoreticalhypotheses, since these properties are typicallyphysical,and presumably being located in space and time is a necessarycondition on instantiating such properties. For further discussion ofthis 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 theso-called semantic view of theories that came to prominence in thesecond half of the 20^th century (see the SEP entry on thestructure of scientific theories for further details). The semantic view was originally proposed as anaccount of theory structure rather than scientific representation. Thedriving idea behind the position is that scientific theories are bestthought of as collections of models. This invites the questions: whatare these models, and how do they represent their target systems? Mostdefenders of the semantic view of theories (with the notable exceptionof Giere, whose views on scientific representation were discussed inthe previous section) take models to be structures, which representtheir target systems in virtue of there being some kind ofmorphism (isomorphism, partial isomorphism, homomorphism,…) between the two.

This conception has twoprima facie advantages. The firstadvantage is that it offers a straightforward answer to the ER-Problem(or SR-problem if the focus is on scientific representation), and onethat accounts for surrogative reasoning: the mappings between themodel and the target allow scientists to convert truths found in themodel into claims about the target system. The second advantageconcerns the applicability of mathematics. There is a time-honouredposition in the philosophy of mathematics which sees mathematics asthe study of structures; see, for instance Resnik (1997) and Shapiro(2000). It is a natural move for the scientific structuralist to adoptthis point of view, which then provides a neat explanation of howmathematics is used in scientific modelling.

4.1 Structures and the Problem of Ontology

Almost anything from a concert hall to a kinship system can bereferred to as a “structure”. So the first task for astructuralist account of representation is to articulate what notionof structure it employs. A number of different notions of structurehave been discussed in the literature (for a review see Thomson-Jones2011), but by far the most common is the notion of structure one findsin set theory and mathematical logic. A structure \(\mathcal{S}\) inthat sense (sometimes “mathematical structure” or“set-theoretic structure”) is a composite entityconsisting of the following: a non-empty set \(U\) of objects calledthe domain (or universe) of the structure and an indexed set \(R\) ofrelations 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 beundefined 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 alanguage and an interpretation function, interpreting symbols of thelanguage in terms of \(U\) (see for instance Machover 1996 and Hodges1997), which is absent from structures in the current context. It isconvenient to write these as \(\mathcal{S}= \langle U, R \rangle\),where “\(\langle \, , \rangle\)” denotes an orderedtuple.

It is important to be clear on what we mean by “object”and “relation” in this context. As regards objects, allthat matters from a structuralist point of view is that there are soand so many of them. Whether the objects are desks or planets isirrelevant. All we need are dummies or placeholders whose onlyproperty is “objecthood”. Similarly, when definingrelations one disregards completely what the relation is “initself”. Whether we talk about “being the mother of”or “standing to the left of” is of no concern in thecontext of a structure; all that matters is between which objects itholds. For this reason, a relation is specified purely extensionally:as a class of ordered \(n\)-tuples. The relation literally is nothingover and above this class. So a structure consists of dummy-objectsbetween which purely extensionally defined relations hold.

The first basic posit of the structuralist theory of representation isthat models are structures in this sense (the second is that modelsrepresent their targets by being suitably morphic to them; we discussmorphisms in the next subsection). Suppes has articulated this stanceclearly when he declared that “the meaning of the concept ofmodel is the same in mathematics and the empirical sciences”(1960 [1969]: 12), and many have followed suit. So we are presentedwith a clear answer to the Problem of Ontology: models are structures.The remaining issue is what structures themselves are. Are theyPlatonic entities, equivalence classes, modal constructs, or yetsomething else? In the context of a discussion of scientificrepresentation one can push these questions off to the philosophy ofmathematics (see the SEP entries on thephilosophy of mathematics,nominalism in the philosophy of mathematics, andPlatonism in the philosophy of mathematics for further details).

4.2 Structuralism and the ER-Problem

The most basic structuralist conception of scientific representationasserts that scientific models, understood as structures, representtheir target systems in virtue of being isomorphic to them. Anisomorphism between two structures \(\mathcal{S}\) and\(\mathcal{S}'\) is a bijective function from \(U\) to \(U'\) thatpreserves the relations on \(U\) (and inversely, the relations on\(U'\)). An isomorphism associates each object in \(U\) with an objectin \(U'\) and pairs up each relation in \(R\) with a relation in\(R'\) so that a relation holds between certain objects in \(U\) iffthe corresponding relation holds between the objects in \(U'\) thatare 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 thetarget:

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

It bears noting that few adherents of the structuralist account ofscientific representation, most closely associated with the semanticview of theories, explicitly defend this position (although see Ubbink1960: 302). Representation was not the focus of attention in thesemantic 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 the21^st century, and many proponents of the semantic view theneither moved away fromStructuralism 1, or pointed out thatthey never held such a view. We turn to more advanced positionsshortly, but to understand what motivates such positions it is helpfulto understand whyStructuralism 1 fails.

The first and most obvious problem is the same as with the similarityview: isomorphism is symmetrical and reflexive (and transitive) butrepresentation isn’t. This problem could be addressed byreplacing isomorphism with an alternative mapping. Bartels (2006),Lloyd (1984), and Mundy (1986) suggest homomorphism; van Fraassen(1980, 1997, 2008) and Redhead (2001) isomorphic embeddings; advocatesof the partial structures approach prefer partial isomophisms (Bueno1997; Bueno and French 2011; Da Costa and French 1990, 2003; French2003, 2014; French and Ladyman 1999); and Swoyer (1991) introduceswhat he calls \(\Delta/\Psi\) morphisms. We refer to thesecollectively as “morphisms”. Pero and Suárez (2016)provide a comparative discussion of different morphisms.

These suggestions solve some, but not all problems. While many ofthese mappings are not symmetrical, they are all still reflexive. Buteven if these formal issues could be resolved in one way or another, aview based on structural mappings would still face other seriousproblems. For ease of presentation we discuss these problems in thecontext of the isomorphism view;mutatis mutandis otherformal mappings suffer from the same difficulties. Like similarity,isomorphism is too inclusive: not all things that are isomorphicrepresent each other. In the case of similarity this case was broughthome by Putnam’s thought experiment with the ant crawling on thebeach; in the case of isomorphism a look at the history of sciencewill do the job. Many mathematical structures were discovered anddiscussed long before they were used in science. Non-Euclideangeometries were studied by mathematicians long before Einstein usedthem in the context of spacetime theories, and Hilbert spaces werestudied by mathematicians prior to their use in quantum theory. Ifrepresentation was nothing over and above isomorphism, then we wouldhave to conclude that Riemann discovered general relativity or thatthat Hilbert invented quantum mechanics. This does not seem correct,so it doesn’t seem like isomorphism on its own establishesscientific representation (Frigg 2002: 10).

Isomorphism is more restrictive than similarity: not everything isisomorphic to everything else. But isomorphism is still too abundantto correctly identify what a model represents. The root of thedifficulties is that the same structures can be instantiated indifferent kinds of target systems. Certain geometrical structures areinstantiated by many different systems; just think about how manyspherical things we find in the world. The \(1/r^2\) law of Newtoniangravity is also the “mathematical skeleton” ofCoulomb’s law of electrostatic attraction and the weakening ofsound or light as a function of the distance to the source. Themathematical structure of the pendulum is also the structure of anelectric circuit with a condenser and a solenoid (Kroes 1989). Thesame structure can be exhibited by more than one kind of targetsystem, and so isomorphism by itself is too weak to identify amodel’s target.

As we have seen in the last section, a misrepresentation is one thatportrays its target as having features it doesn’t have. In thecase of a structural account of representation, this means that themodel portrays the target as having structural properties that itdoesn’t have. However, isomorphism demands identity ofstructure: the structural properties of the model and the target mustcorrespond to one another exactly. So a misrepresentation won’tbe isomorphic to the target. By the lights ofStructuralism 1it therefore is not a representation at all. Like simple similarityaccounts,Structuralism 1 conflates misrepresentation withnon-representation (Suárez 2003: 234–235). Partialstructures can avoid a mismatch due to a target relation being omittedin the model and hence go some way to shoring up the structuralistaccount (Bueno and French 2011: 888). It remains unclear, however, howthey account for distortive representations (Pincock 2005).

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

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

Structuralism 2: A scientific model \(M\) represents a targetsystem \(T\) iff there is an agent \(A\) who uses \(M\) to represent atarget 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. Heoffers the following as the “Hauptstatz” of a theory ofrepresentation: “There is no representation except in thesense that some things are used, made, or taken, to represent thingsas thus and so” (2008: 23, original emphasis). Likewise,Bueno submits that “representation is anintentionalact relating two objects” (2010: 94–95, originalemphasis), and Bueno and French point out that using one thing torepresent 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 relevantmodels” (2011: 885).

As in the shift fromSimilarity 2 toSimilarity 3, this seems like a successful move, with many (although not all) ofthe aforementioned concerns being met. But, again, the role ofisomorphism has shifted. The crucial ingredient is the agent’sintention and isomorphism has in fact become either a representationalstyle or normative criterion for accurate representation. Let us nowassess how well isomorphism fares as a response to these problems, andthe others outlined above.

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

Structuralism’s stand on the Demarcation Problem is by and largean open question. Unlike similarity, which has been widely discussedacross different domains, structural mappings are tied closely to theformal framework of set theory, and have been discussed only sparinglyoutside the context of the mathematized sciences. An exception isFrench (2003), who discusses isomorphism accounts in the context ofpictorial representation. He considers in detail Budd’s (1993)account of pictorial representation and points out that it is based onthe notion of a structural isomorphism between the structure of thesurface of the painting and the structure of the relevant visualfield. Therefore representation is the perceived isomorphism ofstructure (French 2003: 1475–1476) (this point is reaffirmed byBueno 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 andcharacterise them. A proposed structural mapping between the model andthe target offers an obvious response to this challenge: one canrepresent a system by coming up with a model that is proposed to beappropriately morphic to it. This delivers the isomorphism-style, thehomomorphism-style, the partial-isomorphism style and so on. We cancall these “morphism-styles” when referring to them ingeneral. Each of these styles also offers a clear-cut condition ofaccuracy: the representation is accurate if the hypothesised morphismholds; it is inaccurate if it doesn’t.

This is neat answer. The question is what status it hasvis-à-vis the Problem of Style. Are morphism-stylesmerely a subgroup of styles or are they privileged? The former isuncontentious. However, the emphasis many structuralists place onstructure preserving mappings suggests that they do not regardmorphisms as merely one way among others to represent something. Whatthey seem to have in mind is the stronger claim that a representationmust be of that sort, or that morphism-styles are the onlyacceptable styles.

This claim seems to conflict with scientific practice in at least tworespects. Firstly, many representations are inaccurate (and known tobe) in some way. Some models distort, deform and twist properties ofthe target in ways that seem to undercut isomorphism, or indeed any ofthe proposed structure preserving mappings. Some models in statisticalmechanics have an infinite number of particles and the Newtonian modelof the solar system represents the sun as a perfect sphere where inreality it is fiery ball with no well-defined surface at all. It is atbest unclear how isomorphism, partial or otherwise, or homomorphismcan account for these kinds of idealisations. So it seems that stylesof representation other than structure preserving mappings have to berecognised.

Secondly, the structuralist view is a rational reconstruction ofscientific modelling, and as such it has some distance from the actualpractice. Some philosophers have worried that this distance is toolarge and that the view is too far removed from the actual practice ofscience to be able to capture what matters to the practice ofmodelling (this is the thrust of many contributions to Morgan andMorrison 1999; see also Cartwright 1999). Although some models used byscientists may be best thought of as set theoretic structures, thereare many where this seems to contradict how scientists actually talkabout, and reason with, their models. Obvious examples includephysical models like the San Francisco Bay model (Weisberg 2013), butalso systems such as the idealized pendulum or imaginary populationsof interbreeding animals. Such models have the strange property ofbeingconcrete-if-real and scientists talk about them as ifthey were real systems, despite the fact that they are obviously not(Godfrey-Smith 2006). Thomson-Jones (2010) dubs this “face valuepractice”, and there is a question whether structuralism canaccount for that practice.

There remains a final problem to be addressed in the context ofstructural accounts of scientific representation. Target systems arephysical objects: atoms, planets, populations of rabbits, economicagents, etc. Isomorphism is a relation that holds between twostructures and claiming that a set theoretic structure is isomorphicto a piece of the physical world isprima facie a categorymistake. By definition, a morphism can only hold between twostructures. If we are to make sense of the claim that the model isisomorphic to its target we have to assume that the target somehowexhibits a certain structure \(\mathcal{S}_T\). But what does it meanfor a target system—a part of the physical world—topossess a structure, and where in the target system is the structurelocated?

There are two prominent suggestions in the literature. The first,originally suggested by Suppes (1962 [1969]), is that data models arethe target-end structures represented by models. This approach faces aquestion whether we should be satisfied with an account of scientificrepresentation that precludes phenomena being represented (see Bogenand Woodward (1988) for a discussion of the distinction between dataand phenomena, and Brading and Landry (2006) for a discussion of thedistinction in the context of scientific representation). Van Fraassen(2008) has addressed this problem and argues for a pragmaticresolution: in the context of use, there is no pragmatic differencebetween representing phenomena and data extracted from it (see Nguyen2016 for a critical discussion). The alternative approach locates thetarget-end structure in the target system itself. One version of thisapproach sees structures as beinginstantiated in targetsystems. This view seems to be implicit in many versions of thesemantic view, and it is explicitly held by authors arguing for astructuralist answer to the problem of the applicability ofmathematics (Resnik 1997; Shapiro 1997). This approach facesunderdetermination issues in that the same target can instantiatedifferent structures. The issue can be seen as arising due to therebeing alternative descriptions of the system (Frigg 2006) or because aversion of “Newman’s Objection” also bites in thecurrent context (Newman 1928; see Ainsworth 2009 and Ketland 2004 forfurther discussion). A more radical version simplyidentifiestargets with structures (Tegmark 2008). This approach is highlyrevisionary in particular when considering target systems likepopulations of breeding rabbits or economies. So the question remainsfor any structuralist account of scientific representation: where arethe required target-end structures to be found?

5. The Inferential Conception

The core idea of the inferential conception is to analyse scientificrepresentation in terms of the inferential function of scientificmodels. In the previous accounts discussed, a model’sinferential capacity dropped out of whatever it was that was supposedto answer the ER-problem (or SR-problem): proposed morphisms orsimilarity relations between models and their targets for example. Theaccounts discussed in this section reverse this order and explainscientific representation directly in terms of surrogativereasoning.

5.1 The DDI Account

According to Hughes’Denotation, Demonstration, andInterpretation (DDI) account of scientific representation (1997,2010: ch. 5), modelsdenote their targets; are such thatmodel users can performdemonstrations on them; andinterpret the results of such demonstrations in terms of thetarget. The last step is necessary because demonstrations establishresults about the model itself, and in interpreting these results themodel user draws inferences about the target from the model (1997:333). Unfortunately Hughes has little to say about what it means tointerpret a result of a demonstration on a model in terms of itstarget system, and so one has to retreat to an intuitive (andunanalysed) 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, andinterpretation as individually necessary and jointly sufficientconditions for scientific representation. He prefers the more

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

This is unsatisfactory because it ultimately remains unclear whatallows 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 DDIconditions to turn them into a full-fledged response to theER-problem. If, alternatively, the conditions were taken to benecessary and sufficient, then the account would require furtherelaboration on what establishes the conditions.

5.2 Deflationary Inferentialism

Suárez argues that we should adopt a “deflationary orminimalist attitude and strategy” (2004: 770) when addressingthe problem of scientific representation. Two different notions ofdeflationism are in operation in his account. The first is to abandonthe aim of seeking necessary and sufficient conditions; necessaryconditions will be good enough (2004: 771). The second notion is thatwe should seek “no deeper features to representation other thanits surface features” (2004:771) or “platitudes”(Suárez and Solé 2006: 40), and that we should deny thatan analysis of a concept “is the kind of analysis that will shedexplanatory light on our use of the concept” (Suárez2015: 39). Suárez intends his account of scientificrepresentation 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) therepresentational force of \(A\) points towards \(B\), and (ii) \(A\)allows competent and informed agents to draw specific inferencesregarding \(B\)” (2004: 773).

The first condition addresses the Requirement of Directionality andensures that \(A\) and \(B\) indeed enter into a representationalrelationship. On might worry that explaining representation in termsof representational force sheds little light on the matter as long asno analysis of representational force is offered. But Suárezresists attempts to explicate representational force in terms of astronger relation, like denotation or reference, on grounds that thiswould violate deflationism (2015: 41). The second condition is in factjust the Surrogative Reasoning Condition, now taken as a necessarycondition on scientific representation. Contessa (2007: 61) points outthat it remains mysterious how these inferences are generated. Anappeal to further analysis can, again, be blocked by appeal todeflationism because any attempt to explicate how inferences are drawnwould go beyond “surface features”. So the tenability ofInferentialism in effect depends on the tenability ofdeflationism about scientific representation. Suárez (2015)defends deflationism by drawing analogies with three differentdeflationary theories about truth, Ramsey’s“redundancy” theory, Wright’s “abstractminimalism” and Horwich’s “use theory” (formore information of these theories see the SEP entry onthe deflationary theory of truth). An alternative defence builds on Brandom’s inferentialism inthe philosophy of language (1994, 2000), a line of argument that isdeveloped by de Donato Rodríguez and Zamora Bonilla (2009) andKuorikoski and Lehtinen (2009).

Inferentialism provides a neat explanation of the possibilityof misrepresentation because the inferences drawn about a target neednot be true (Suárez 2004: 776). In as far as one acceptsrepresentational force as a cogent concept, targetless models aredealt with successfully because representational force (unlikedenotation) does not require the existence of a target (2004: 772).Inferentialism repudiates the Representational Demarcation Problem andaims to offer an account of representation that also works in otherdomains such as painting (2004: 777). The account is ontologicallynon-committal because anything that has an internal structure thatallows an agent to draw inferences can be a representation. Relatedly,since the account is supposed to apply to a wide variety of entitiesincluding equations and mathematical structures, the account impliesthat mathematics is successfully applied in the sciences, but inkeeping with the spirit of deflationism no explanation is offeredabout how this is possible. The account does not directly address theProblem of Style.

5.3 Inflating Inferentialism: Interpretation

In response to the difficulties withInferentialism Contessasubmits that “it is not clear why we should adopt a deflationaryattitudefrom the start” (2007: 50) and provides a“interpretational account” of scientific representationthat is inspired by Suárez’s account, but without beingdeflationary. Contessa introduces the notion of aninterpretation of a model, in terms of its target system, asa necessary and sufficient condition on epistemic representation (seealso Ducheyne 2012 for a related account):

Interpretation: “A [model \(M\)] is an epistemicrepresentation of a certain target [\(T\)] (for a certain user) if andonly 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 firstidentifies sets of relevant objects in the model and the target, andthen pins down sets of properties and relations these objectsinstantiate both in the model and the target. The user then (a) takes\(M\) to denote \(T\); (b) takes every identified object in the modelto denote exactly one object in the target (and every relevant objectin the target has to be so denoted); (c) takes every property andrelation in the model to denote a property or relation of the sametype in the target (and, again, and every property and relation in thetarget has to be so denoted). A formal rendering of these conditionsis what Contessa calls an “analytic interpretation” (seehis 2007: 57–62 for details; he also includes an additionalcondition pertaining to functions in the model and target, which wesuppress for brevity).

Interpretation offers a neat answer to the ER-problem. Theaccount also explains the directionality of representation:interpreting a model in terms of a target does not entail interpretinga target in terms of a model. However, it has been noted thatInterpretation has difficulty accounting for the possibilityof misrepresentation, since it seems to require that the relevantobjects, properties, and relations actually exist in the target (Shech2015), although this objection turns on a very strict reading ofContessa’s account. This problem is solved inDíez’s (2020) “Ensemble-Plus-Standing-For”account of representation, which is based on conditions that rule outa mismatch concerning the number of objects in the collection.Contessa does not comment on the applicability of mathematics butsince his account shares with the structuralist account an emphasis onrelations and one-to-one model-target correspondence, Contessa canappeal to the same account of the applicability of mathematics as thestructuralist. Like Suárez, Contessa takes his account to beuniversal and apply to non-scientific representations such asportraits and maps. But it remains unclear howInterpretationaddresses the Problem of Style. As we have seen earlier, in particularvisual representations fall into different categories and there is aquestion about how these can be classified within the interpretationalframework. With respect to the Question of Ontology,Interpretation itself places few constraints on whatscientific models are. All it requires is that they consist ofobjects, properties, relations, and functions (but see Contessa (2010)for further discussion of what he takes models to be, ontologicallyspeaking).

6. The Fiction View of Models

A recent family of approaches analyses models by drawing an analogybetween models and literary fiction. This analogy can be used in twoways, yielding two different version of the fiction view. The first isprimarily motivated by ontological considerations rather than thequestion of scientific representationper se. Scientificdiscourse is rife with passages that appear to be descriptions ofsystems in a particular discipline, and the pages of textbooks andjournals are filled with discussions of the properties and thebehaviour of those systems. In mechanics, for instance, the dynamicalproperties of a system consisting of three spinning spheres withhomogenous mass distributions are the focus of attention; in biologyinfinite populations are investigated; and in economics perfectlyrational agents with access to perfect information exchange goods.Their surface structure notwithstanding, no one would mistakedescriptions of such systems as descriptions of anactualsystem: 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 areembedded in what he calls the “face value practice” (2010:285) the practice of talking and thinking about these systems as ifthey were real. The face-value practice raises a number of questions.What account should be given of these descriptions and what sort ofobjects, if any, do they describe? Are we putting forwardtruth-evaluable claims when putting forward descriptions of missingsystems?

The fiction view of models provides an answer: models are akin toplaces and characters in literary fiction and claims about them aretrue or false in the same way in which claims about these places andcharacters are true or false. Such a position has been recentlydefended explicitly by some authors (Frigg 2010a,b; Frigg and Nguyen2021; Godfrey-Smith 2006; Salis 2021), but not without opposition(Giere 2009; Magnani 2012). It does bear noting that the analogy hasbeen around for a while (Cartwright 1983; McCloskey 1990; Vaihinger1911 [1924]). This leaves the thorny issue of how to analyse fictionalplaces and characters. Here philosophers of science can draw ondiscussions from aesthetics to fill in the details about thesequestions (Friend 2007 and Salis 2013 provide useful reviews).

The second version of the fiction view explicitly focuses onrepresentation. Most theories of representation we have encountered sofar posit that there are model systems and construe scientificrepresentation as a relation between two entities, the model systemand the target system. Toon calls this theindirect view ofrepresentation (2012: 43). Indeed, Weisberg (2007) views thisindirectness as the defining feature of modelling (see also Knuuttilaand Loettgers 2017). This view contrasts with what Toon (2012: 43) andLevy (2015: 790) call adirect view of representation. Thisview does not recognise model systems instead aims to explainrepresentation as a form of direct description. On this view, modelsprovide an “imaginative description of real things” (Levy2012: 741) such as actual pendula, and there is no such thing as amodel system of which the pendulum description is literally true (Toon2012: 43–44).

Both Toon (2012) and Levy (2015) articulate the direct view by drawingon Walton’s (1990) theory of make-believe. At the heart of thistheory is the notion of a game of make-believe (see the SEP entry onimagination for further discussion). We play such a game if, for instance, whenwalking through a forest we imagine that stumps are bears and if wespot a stump we imagine that we spot a bear. In Walton’sterminology the stumps areprops, and the rule that weimagine a bear when we see a stump is aprinciple ofgeneration. Together a prop and principle of generation prescribewhat is to be imagined. Walton considers a vast variety of differentprops, including statues and works of literary fiction. Toon focuseson the particular kind of game in which we are prescribed to imaginesomething of a real world object. A statue showing Napoleon onhorseback (Toon 2012: 37) is a prop mandating us to imagine, forinstance, that Napoleon has a certain physiognomy and certain facialexpressions. When readingThe War of the Worlds (2012: 39) weare prescribed to imagine that the dome of St Paul’s Cathedralhas been attacked by aliens and now has a gaping hole on its westernside.

The crucial move is to say that models are props in games of makebelieve. Specifically, material models are like the statue of Napoleonand theoretical models are like the text ofThe War of theWorlds: both prescribe, in their own way, to imagine somethingabout a real object. A ball-and-stick model of a methane moleculeprescribes us to imagine particular things about methane, and a modeldescription describing a point mass bob bouncing on a perfectlyelastic spring represents the real ball and spring system byprescribing imaginings about the real system. This provides thefollowing answer to the ER-problem (Toon 2012: 62):

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

This account solves some of the problems posed inSection 1:Direct Representation is asymmetrical, makes room formisrepresentation, and, given its roots in aesthetics, it renouncesthe Demarcation Problem. The view absolves the Problem of Ontologysince models are either physical objects or descriptions, neither orwhich are problematic in this context. Toon remains silent on both theProblem of Style, and the applicability of mathematics.

Important questions remain. According toDirectRepresentation models prescribe us to imagine certain thingsabout their target system. The account remains silent, however, on therelationship between what a model prescribes us to imagine and what amodel user should actually infer about the target system, and so itoffers no answer to the ER-problem. Levy (2015) identifies this as agap in Toon’s account and proposes to fill it by invokingYablo’s (2014) notion of “partial truth”, the ideabeing that a model user should take the imagined propositions to bepartially true of their target systems. However, as Levy admits, thereare other sorts of cases that don’t fit the mould, most notablydistortive idealisations. These require a different treatment andit’s an open question what this treatment would be.

A further worry is howDirect Representation deals withtargetless models. If there is no target system, then what does themodel prescribe imaginings about? Toon is well aware of such modelsand suggests the following solution: if a model has no target itprescribes imaginings about a fictional character (2012: 76). Thissolution, however, comes with ontological costs, and one of thedeclared aims of the direct view was to avoid such costs by removingmodel systems from the picture. Levy (2015) aims to salvageontological parsimony and proposes a radical move: there are notargetless models. If a (purported) model has no target then it is nota model. There remains a question, however, how this view can besquared with scientific practice where targetless models are not onlycommon but also clearly acknowledged as such.

7. Representation-As

In Goodman’s (1976) account of aesthetic representation the ideais that a work of art does not just denote its subject, but moreoverit represents it as being thus or so (see the SEP entry onGoodman’s aesthetics for further discussion). Elgin (2010) further developed this accountand, crucially, suggested that it also applies to scientificrepresentations. We first discuss Goodman and Elgin’s notion ofrepresentation-as and then consider a recent extension of theirframework.

7.1 From Art to Science

Many instances of epistemic representation are instances of whatGoodman and Elgin call “representation-as”. Caricaturesare paradigmatic examples: Churchill is represented as a bulldog andThatcher is represented as a boxer. But the notion is more general:Holbein’sPortrait of Henry VIII represents Henry asimposing and powerful and Stoddart’s statue of David Humerepresents him as thoughtful and wise. Using these representations wecan learn about their targets, for example we can learn about apolitician’s or philosopher’s personality. The leadingidea of the views discussed in this section is that scientificrepresentation works in much the same way. A model of the solar systemrepresents it as consisting of perfect spheres; the logistic model ofgrowth represents the population as reproducing at fixed intervals oftime; and so on. In each instance, models can be used to attempt tolearn about their targets by determining what the former represent thelatter as being.

The locution of representation-as functions in the following way: anobject \(X\) (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 representationalrelationship. The answer requires introducing some of the conceptsGoodman and Elgin use to develop their account ofrepresentation-as.

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

The next notion isexemplification: an object \(X\)exemplifies a property \(P\) iff \(X\) instantiates \(P\) and therebyrefers back to \(P\) (Goodman 1976: 53). In the current contextproperties are to be understood in the widest possible sense. An itemcan 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 amanufacturer’s sample card instantiates a certain colour and atthe same time refers to that colour (Elgin 1983: 71). Notice thatinstantiation is necessary but insufficient for exemplification: thesample card does not exemplify being rectangular for example. When aobject exemplifies a property it provides us with epistemic access tothat property.

Representation-as is then established by combining these notionstogether: a \(Z\)-representation exemplifies properties associatedwith \(Z\)s,^[11] and if the \(Z\)-representation additionally denotes \(Y\), thenthese properties can be imputed onto \(Y\) (cf. Elgin 2010: 10). Thisprovides 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 \(X\) rangeover models and \(Y\) over target systems, we arrive at an answer tothe ER-problem.Representation-As also answers the otherproblems introduced inSection 1: it repudiates the demarcation problem and it explains thedirectionality of representation. It accounts for surrogativereasoning 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 instantiatethem it allows for the possibility of misrepresentation. Differentstyles can be accounted for by categorising representations in termsof different \(Z\)s, or in terms of the properties they exemplify.However, at least as stated, the account remains silent on the problemof ontology and the applicability of mathematics. We discuss below howto 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 hasintuitive appeal in the case of pictures, it is less clear how itworks in the context of science. Phillips and Newlyn constructed anelaborate system of pipes and tanks, now know as the Phillips-Newlynmachine, to model an economy (see Morgan and Boumans 2004 and Barr2000 for useful discussions). So the machine is aneconomy-representation. But what turns a system of pipes and tanksinto an economy-representation?

Frigg and Nguyen (2016: 227–8) argue that in order to turn anobject \(X\) into a scientific model, it must be interpreted in theappropriate way (note that they do not use“interpretation” in the way that Contessa uses it, asdiscussed above): properties that \(X\) has, qua object, are paired upwith relevant \(Z\) properties and for quantitative properties, i.e.,properties such as mass or flow, which take numerical values, thereneeds 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, hydraulicproperties of the machine are associated with economic properties anda rule is given specifying that a litre of water corresponds tocertain amount of a model-economy’s currency (Frigg and Nguyen2018).

The \(X\) and \(Z\) properties that are so associated need not exhaustthe properties that \(X\) instantiates, nor do all possible \(Z\)properties need be associated with an \(X\) property. A scientificmodel can then be defined as a \(Z\)-representation, i.e., an objectunder an interpretation. This notion of a model explicitly does notpresuppose a target system and hence makes room for targetlessmodels. Models can then be seen as instantiating\(Z\)-propertiesunder the relevant interpretation, whichexplains how the model can exemplify such properties.

The next issue is that exemplified properties are rarely exactlyimputed onto target systems. According toRepresentation-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 beexplicit about the relationship between the exemplified properties andthe ones to be imputed onto the target. They do this by introducing a“key”, which explicitly associates the exemplifiedproperties with properties to be imputed onto the target. For example,in the case of a London Tube map, the key associates particularcolours with particular tube lines, and in the case of idealisationsthe key associates de-idealised properties with model-properties.

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

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

\(M\) denotes \(T\).
\(M\) exemplifies \(Z\)-properties \(P_1, \ldots , P_n\).
\(M\) comes with key \(K\) associating the set \(\{P_1, \ldots ,P_n\}\) with a (possibly identical) set of properties \(\{Q_1, \ldots, Q_m\}\).
\(M\) 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 keyfeatures: denotation, exemplification, keying-up and imputation.DEKI answers the problems fromSection 1 in much the same way asRepresentation-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 neednot be imputed exactly onto the target, and highlights the need toinvestigate keys specifying the relationship between properties inmodels and the properties that models actually impute onto theirtargets. Finally, it makes explicit how to account for targetlessmodels. A scientific model that fails to denote a target system cannevertheless be a \(Z\)-representation. A model of a bridge that isnever built is still a bridge-representation, which exemplifiesproperties related to bridges (stability and so on), despite the factthat it is not a representation of anything. However, as in the caseofRepresentation-As questions remain with respect to the problem of ontology and theapplicability of mathematics.

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