Models are of central importance in many scientific contexts. Thecentrality of models such as inflationary models in cosmology,general-circulation models of the global climate, the double-helixmodel of DNA, evolutionary models in biology, agent-based models inthe social sciences, and general-equilibrium models of markets intheir respective domains is a case in point (theOther Internet Resources section at the end of this entry contains links to online resourcesthat discuss these models). Scientists spend significant amounts oftime building, testing, comparing, and revising models, and muchjournal space is dedicated to interpreting and discussing theimplications of models.

As a result, models have attracted philosophers’ attention andthere are now sizable bodies of literature about various aspects ofscientific modeling. A tangible result of philosophical engagementwith models is a proliferation of model types recognized in thephilosophical literature.Probing models,phenomenological models,computational models,developmental models,explanatory models,impoverished models,testing models,idealizedmodels,theoretical models,scale models,heuristic models,caricature models,exploratorymodels,didactic models,fantasy models,minimal models,toy models,imaginarymodels,mathematical models,mechanisticmodels,substitute models,iconic models,formal models,analogue models, andinstrumentalmodels are but some of the notions that are used to categorizemodels. While at first glance this abundance is overwhelming, it canbe brought under control by recognizing that these notions pertain todifferent problems that arise in connection with models. Models raisequestions in semantics (how, if at all, do models represent?),ontology (what kind of things are models?), epistemology (how do welearn and explain with models?), and, of course, in other domainswithin philosophy of science.

• 1. Semantics: Models and Representation
• 2. Ontology: What Are Models?
  • 2.1 Physical objects
  • 2.2 Fictional objects and abstract objects
  • 2.3 Set-theoretic structures
  • 2.4 Descriptions and equations
• 3. Epistemology: The Cognitive Functions of Models
  • 3.1 Learning about models
  • 3.2 Learning about target systems
  • 3.3 Explaining with models
  • 3.4 Understanding with models
  • 3.5 Other cognitive functions
• 4. Models and Theory
  • 4.1 Models as subsidiaries to theory
  • 4.2 Models as independent from theories
• 5. Models and Other Debates in the Philosophy of Science
  • 5.1 Models, realism, and laws of nature
  • 5.2 Models and reductionism
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Semantics: Models and Representation

Many scientific models are representational models: they represent aselected part or aspect of the world, which is the model’starget system. Standard examples are the billiard ball model of a gas,the Bohr model of the atom, the Lotka–Volterra model ofpredator–prey interaction, the Mundell–Fleming model of anopen economy, and the scale model of a bridge. Elliott-Graves (2020)discusses how targets are individuated.

This raises the question what it means for a model to represent atarget system. This problem is rather involved and decomposes intovarious subproblems. For detailed overviews of the issue ofrepresentation, see Nguyen and Frigg (2022) and the entry onscientific representation. At this point, rather than addressing the issue of what it means fora model to represent, we focus on a number of different kinds ofrepresentation that play important roles in the practice ofmodel-based science, namely scale models, analogical models, idealizedmodels, toy models, minimal models, phenomenological models,exploratory models, and models of data. These categories are notmutually exclusive, and a given model can fall into several categoriesat once.

Scale models. Some models are down-sized or enlarged copiesof their target systems (Black 1962). A typical example is a smallwooden car that is put into a wind tunnel to explore the actualcar’s aerodynamic properties. The intuition is that a scalemodel is a naturalistic replica or a truthful mirror image of thetarget; for this reason, scale models are sometimes also referred toas “true models” (Achinstein 1968: Ch. 7). However, thereis no such thing as a perfectly faithful scale model; faithfulness isalways restricted to some respects. The wooden scale model of the carprovides a faithful portrayal of the car’s shape but not of itsmaterial. And even in the respects in which a model is a faithfulrepresentation, the relation between model-properties andtarget-properties is usually not straightforward. When engineers use,say, a 1:100 scale model of a ship to investigate the resistance thatan actual ship experiences when moving through the water, they cannotsimply measure the resistance the model experiences and then multiplyit with the scale. In fact, the resistance faced by the model does nottranslate into the resistance faced by the actual ship in astraightforward manner (that is, one cannot simply scale the waterresistance with the scale of the model: the real ship need not haveone hundred times the water resistance of its 1:100 model). The twoquantities stand in a complicated nonlinear relation with each other,and the exact form of that relation is often highly nontrivial andemerges as the result of a thoroughgoing study of the situation(Sterrett 2006, 2020; Pincock 2022).

Analogical models. Standard examples of analogical modelsinclude the billiard ball model of a gas, the hydraulic model of aneconomic system, and the dumb hole model of a black hole. At the mostbasic level, two things are analogous if there are certain relevantsimilarities between them. In a classic text, Hesse (1963)distinguishes different types of analogies according to the kinds ofsimilarity relations into which two objects enter. A simple type ofanalogy is one that is based on shared properties. There is an analogybetween the earth and the moon based on the fact that both are large,solid, opaque, spherical bodies that receive heat and light from thesun, revolve around their axes, and gravitate towards other bodies.But sameness of properties is not a necessary condition. An analogybetween two objects can also be based on relevant similarities betweentheir properties. In this more liberal sense, we can say that there isan analogy between sound and light because echoes are similar toreflections, loudness to brightness, pitch to color, detectability bythe ear to detectability by the eye, and so on.

Analogies can also be based on the sameness or resemblance ofrelations between parts of two systems rather than on their monadicproperties. It is in this sense that the relation of a father to hischildren is asserted to be analogous to the relation of the state toits citizens. The analogies mentioned so far have been what Hessecalls “material analogies”. We obtain a more formal notionof analogy when we abstract from the concrete features of the systemsand only focus on their formal set-up. What the analogue model thenshares with its target is not a set of features, but the same patternof abstract relationships (i.e., the same structure, where structureis understood in a formal sense). This notion of analogy is closelyrelated to what Hesse calls “formal analogy”. Two itemsare related by formal analogy if they are both interpretations of thesame formal calculus. For instance, there is a formal analogy betweena swinging pendulum and an oscillating electric circuit because theyare both described by the same mathematical equation.

A further important distinction due to Hesse is the one betweenpositive, negative, and neutral analogies. The positive analogybetween two items consists in the properties or relations they share(both gas molecules and billiard balls have mass); the negativeanalogy consists in the properties they do not share (billiard ballsare colored, gas molecules are not); the neutral analogy comprises theproperties of which it is not known (yet) whether they belong to thepositive or the negative analogy (do billiard balls and molecules havethe same cross section in scattering processes?). Neutral analogiesplay an important role in scientific research because they give riseto questions and suggest new hypotheses. For this reason severalauthors have emphasized the heuristic role that analogies play intheory and model construction, as well as in creative thought(Bailer-Jones and Bailer-Jones 2002; Bailer-Jones 2009: Ch. 3; Hesse1974; Holyoak and Thagard 1995; Kroes 1989; Psillos 1995; and theessays collected in Helman 1988). See also the entry onanalogy and analogical reasoning.

It has also been discussed whether using analogical models can in somecases be confirmatory in a Bayesian sense. Hesse (1974: 208–219)argues that this is possible if the analogy is a material analogy.Bartha (2010, 2013 [2019]) disagrees and argues that analogical modelscannot be confirmatory in a Bayesian sense because the informationencapsulated in an analogical model is part of the relevant backgroundknowledge, which has the consequence that the posterior probability ofa hypothesis about a target system cannot change as a result ofobserving the analogy. Analogical models can therefore only establishtheplausibility of a conclusion in the sense of justifying anon-negligible prior probability assignment (Bartha 2010:§8.5).

More recently, these questions have been discussed in the context ofso-called analogue experiments, which promise to provide knowledgeabout an experimentally inaccessible target system (e.g., a blackhole) by manipulating another system, the source system (e.g., aBose–Einstein condensate). Dardashti, Thébault, andWinsberg (2017) and Dardashti, Hartmann et al. (2019) have arguedthat, given certain conditions, an analogue simulation of one systemby another system can confirm claims about the target system (e.g.,that black holes emit Hawking radiation). See Crowther et al. (2021)for a critical discussion, and also the entry oncomputer simulations in science.

Idealized models. Idealized models are models that involve adeliberate simplification or distortion of something complicated withthe objective of making it more tractable or understandable.Frictionless planes, point masses, completely isolated systems,omniscient and fully rational agents, and markets in perfectequilibrium are well-known examples. Idealizations are a crucial meansfor science to cope with systems that are too difficult to study intheir full complexity (Potochnik 2017).

Philosophical debates over idealization have focused on two generalkinds of idealizations: so-called Aristotelian and Galileanidealizations. Aristotelian idealization amounts to “strippingaway”, in our imagination, all properties from a concrete objectthat we believe are not relevant to the problem at hand. There isdisagreement on how this is done. Jones (2005) and Godfrey-Smith(2009) offer an analysis of abstraction in terms of truth: while anabstraction remains silent about certain features or aspects of thesystem, it does not say anything false and still offers a true (albeitrestricted) description. This allows scientists to focus on a limitedset of properties in isolation. An example is a classical-mechanicsmodel of the planetary system, which describes the position of anobject as a function of time and disregards all other properties ofplanets. Cartwright (1989: Ch. 5), Musgrave (1981), who uses the term“negligibility assumptions”, and Mäki (1994), whospeaks of the “method of isolation”, allow abstractions tosay something false, for instance by neglecting a causally relevantfactor.

Galilean idealizations are ones that involve deliberate distortions:physicists build models consisting of point masses moving onfrictionless planes; economists assume that agents are omniscient;biologists study isolated populations; and so on. Usingsimplifications of this sort whenever a situation is too difficult totackle was characteristic of Galileo’s approach to science. Forthis reason it is common to refer to ‘distortive’idealizations of this kind as “Galilean idealizations”(McMullin 1985). An example for such an idealization is a model ofmotion on an ice rink that assumes the ice to be frictionless, when,in reality, it has low but non-zero friction.

Galilean idealizations are sometimes characterized as controlledidealizations, i.e., as ones that allow for de-idealization bysuccessive removal of the distorting assumptions (McMullin 1985;Weisberg 2007). Thus construed, Galilean idealizations don’tcover all distortive idealizations. Batterman (2002, 2011) and Rice(2019, 2021) discuss distortive idealizations that are ineliminable inthat they cannot be removed from the model without dismantling themodel altogether.

What does a model involving distortions tell us about reality? Laymon(1991) formulated a theory which understands idealizations as ideallimits: imagine a series of refinements of the actual situation whichapproach the postulated limit, and then require that the closer theproperties of a system come to the ideal limit, the closer itsbehavior has to come to the behavior of the system at the limit(monotonicity). If this is the case, then scientists can study thesystem at the limit and carry over conclusions from that system tosystems distant from the limit. But these conditions need not alwayshold. In fact, it can happen that the limiting system does notapproach the system at the limit. If this happens, we are faced with asingular limit (Berry 2002). In such cases the system at the limit canexhibit behavior that is different from the behavior of systemsdistant from the limit. Limits of this kind appear in a number ofcontexts, most notably in the theory of phase transitions instatistical mechanics. There is, however, no agreement over thecorrect interpretation of such limits. Batterman (2002, 2011) seesthem as indicative of emergent phenomena, while Butterfield (2011a,b)sees them as compatible with reduction (see also the entries onintertheory relations in physics andscientific reduction).

Galilean and Aristotelian idealizations are not mutually exclusive,and many models exhibit both in that they take into account a narrowset of properties and distort them. Consider again theclassical-mechanics model of the planetary system: the model onlytakes a narrow set of properties into account and distorts them, forinstance by describing planets as ideal spheres with arotation-symmetric mass distribution.

A concept that is closely related to idealization is approximation. Ina broad sense,A can be called an approximation ofB ifA is somehow close toB. This, however, is too broadbecause it makes room for any likeness to qualify as an approximation.Rueger and Sharp (1998) limit approximations toquantitativecloseness, and Portides (2007) frames it as an essentiallymathematical concept. On that notionA is an approximation ofB iffA is close toB in a specifiablemathematical sense, where the relevant sense of “close”will be given by the context. An example is the approximation of onecurve with another one, which can be achieved by expanding a functioninto a power series and only keeping the first two or three terms. Indifferent situations we approximate an equation with another one byletting a control parameter tend towards zero (Redhead 1980). Thisraises the question of how approximations are different fromidealizations, which can also involve mathematical closeness. Norton(2012) sees the distinction between the two as referential: anapproximation is an inexact description of the target while anidealization introduces a secondary system (real or fictitious) whichstands for the target system (while being distinct from it). If we saythat the period of the pendulum on the wall is roughly two seconds,then this is an approximation; if we reason about the real pendulum byassuming that the pendulum bob is a point mass and that the string ismassless (i.e., if we assume that the pendulum is a so-called idealpendulum), then we use an idealization. Separating idealizations andapproximations in this way does not imply that there cannot beinteresting relations between the two. For instance, an approximationcan be justified by pointing out that it is the mathematicalexpression of an acceptable idealization (e.g., when we neglect adissipative term in an equation of motion because we make theidealizing assumption that the system is frictionless).

Idealized models also pose a challenge to Bayesian philosophy ofscience. How can an agent have a positive degree of belief in astrongly idealized model? This is a pressing question becauseidealized models are ubiquitous in science and (Bayesian) statisticsis widely used. For a discussion of different replies, see Sprengerand Hartmann (2019: ch. 12).

Toy models. Toy models are extremely simplified and stronglydistorted renderings of their targets, and often only represent asmall number of causal or explanatory factors (Hartmann 1995;Reutlinger et al. 2018; Nguyen 2020). Typical examples are theLotka–Volterra model in population ecology (Weisberg 2013) andthe Schelling model of segregation in the social sciences (Sugden2000). Toy models usually do not perform well in terms of predictionand empirical adequacy, and they seem to serve other epistemic goals(more on these inSection 3). This raises the question whether they should be regarded asrepresentational at all (Luczak 2017).

Some toy models are characterized as “caricatures”(Gibbard and Varian 1978; Batterman and Rice 2014). Caricature modelsisolate a small number of salient characteristics of a system anddistort them into an extreme case. A classic example isAkerlof’s (1970) model of the car market (“the market forlemons”), which explains the difference in price between new andused cars solely in terms of asymmetric information, therebydisregarding all other factors that may influence the prices of cars(see also Sugden 2000). However, it is controversial whether suchhighly idealized models can still be regarded as informativerepresentations of their target systems. For a discussion ofcaricature models, in particular in economics, see Reiss (2006).

Minimal models. Minimal models are closely related to toymodels in that they are also highly simplified. They are so simplifiedthat some argue that they are non-representational: they lack anysimilarity, isomorphism, or resemblance relation to the world(Batterman and Rice 2014). It has been argued that many economicmodels are of this kind (Grüne-Yanoff 2009). Minimal economicmodels are also unconstrained by natural laws, and do not isolate anyreal factors (ibid.). And yet, studying minimal models canhelp us to learn something about the world. It is, however,controversial whether minimal models can assist scientists in learningsomething about the world if they do not represent anything (Fumagalli2016). Minimal models that purportedly lack any similarity orrepresentation are also used in different parts of physics to explainthe macro-scale behavior of various systems whose micro-scale behavioris extremely diverse (Batterman and Rice 2014; Rice 2018, 2019; Shech2018). Typical examples are the features of phase transitions and theflow of fluids. Proponents of minimal models argue that what providesan explanation of the macro-scale behavior of a system in these casesis not a feature that system and model have in common, but the factthat the system and the model belong to the sameuniversalityclass (a class of models that exhibit the same limiting behavioreven though they show very different behavior at finite scales). Itis, however, controversial whether explanations of this kind arepossible without reference to at least some common features (Lange2015; Reutlinger 2017).

Surrogate models. Surrogate models are used in science tostudy systems that are difficult or impossible to investigatedirectly. These models act as stand-ins for their target systems,allowing researchers to draw inferences about the target based on thebehavior of the surrogate. A classic example is the use of modelorganisms in biology, such asDrosophila melanogaster (fruitflies) orMus musculus (mice), which serve as proxies forstudying biological processes in humans. The choice of a surrogatemodel involves careful consideration of its representational adequacyand of the extent to which findings can be generalized to the targetsystem (Bolker 2009; Burian 1993). Philosophers have debated theepistemological challenges of using surrogate models, particularlywhen the surrogate and target systems differ significantly, as is thecase in model organisms, which often have different underlyingmechanisms or evolutionary histories than their targets (Dietrich etal. 2020; Ankeny & Leonelli 2020). For a discussion of modelorganisms, see the entry onexperiment in biology.

Phenomenological models. Phenomenological models have beendefined in different, although related, ways. A common definitiontakes them to be models that only represent observable properties oftheir targets and refrain from postulating hidden mechanisms and thelike (Bokulich 2011). Another approach, due to McMullin (1968),defines phenomenological models as models that are independent oftheories. This, however, seems to be too strong. Many phenomenologicalmodels, while failing to be derivable from a theory, incorporateprinciples and laws associated with theories. The liquid-drop model ofthe atomic nucleus, for instance, portrays the nucleus as a liquiddrop and describes it as having several properties (surface tensionand charge, among others) originating in different theories(hydrodynamics and electrodynamics, respectively). Certain aspects ofthese theories—although usually not the full theories—arethen used to determine both the static and dynamical properties of thenucleus. Finally, it is tempting to identify phenomenological modelswithmodels of a phenomenon. Here, “phenomenon”is an umbrella term covering all relatively stable and generalfeatures of the world that are interesting from a scientific point ofview. The weakening of sound as a function of the distance to thesource, the decay of alpha particles, the chemical reactions that takeplace when a piece of limestone dissolves in an acid, the growth of apopulation of rabbits, and the dependence of house prices on the baserate of the Federal Reserve are phenomena in this sense. For furtherdiscussion, see Bailer-Jones (2009: Ch. 7), Bogen and Woodward (1988),and the entry ontheory and observation in science.

Exploratory models. Exploratory models are models which arenot proposed in the first place to learn something about a specifictarget system or a particular experimentally established phenomenon.Exploratory models function as the starting point of furtherexplorations in which the model is modified and refined. Gelfert(2016) points out that exploratory models can provideproofs-of-principle and suggest how-possibly explanations (2016: Ch.4). As an example, Gelfert mentions early models in theoreticalecology, such as the Lotka–Volterra model of predator–preyinteraction, which mimic the qualitative behavior of speed-up andslow-down in population growth in an environment with limitedresources (2016: 80). Such models do not give an accurate account ofthe behavior of any actual population, but they provide the startingpoint for the development of more realistic models. Massimi (2019)notes that exploratory models provide modal knowledge. Fisher (2006)sees these models as tools for the examination of the features of agiven theory. Massimi (2019) notes that exploratory models providemodal knowledge. Sjölin Wirling and Grüne-Yanoff (2021)provide a survey of different kinds of modal knowledge that models arereported to provide.

Models of data. A model of data (sometimes also “datamodel”) is a corrected, rectified, regimented, and in manyinstances idealized version of the data we gain from immediateobservation, the so-called raw data (Suppes 1962). Characteristically,one first eliminates errors (e.g., removes points from the record thatare due to faulty observation) and then presents the data in a“neat” way, for instance by drawing a smooth curve througha set of points. These two steps are commonly referred to as“data reduction” and “curve fitting”. When weinvestigate, for instance, the trajectory of a certain planet, wefirst eliminate points that are fallacious from the observationrecords and then fit a smooth curve to the remaining ones. Models ofdata play a crucial role in confirming theories because it is themodel of data, and not the often messy and complex raw data,that theories are tested against.

The construction of a model of data can be extremely complicated. Itrequires sophisticated statistical techniques and raises seriousmethodological as well as philosophical questions. How do we decidewhich points on the record need to be removed? And given a clean setof data, what curve do we fit to it? The first question has been dealtwith mainly within the context of the philosophy of experiment (see,for instance, Galison 1997 and Staley 2004). At the heart of thelatter question lies the so-called curve-fitting problem, which isthat the data themselves dictate neither the form of the fitted curvenor what statistical techniques scientists should use to construct acurve. The choice and rationalization of statistical techniques is thesubject matter of the philosophy of statistics, and we refer thereader to the entryPhilosophy of Statistics, as well as to Bandyopadhyay and Forster (2011) and Otsuka (2023) fora discussion of these issues. Further discussions of models of datacan be found in Bailer-Jones (2009: Ch. 7), Brewer and Chinn (1994),Harris (2003), Hartmann (1995), Laymon (1982), Mayo (1996, 2018), andSuppes (2007).

The gathering, processing, dissemination, analysis, interpretation,and storage of data raise many important questions beyond therelatively narrow issues pertaining to models of data. Leonelli (2016,2019) investigates the status of data in science, argues that datashould be defined not by their provenance but by their evidentialfunction, and studies how data travel between different contexts.

2. Ontology: What Are Models?

What are models? That is, what kind of object are scientists dealingwith when they work with a model? A number of authors have voicedskepticism that this question has a meaningful answer, because modelsdo not belong to a distinctive ontological category and anything canbe a model (Callender and Cohen 2006; Giere 2010; Suárez 2024;Swoyer 1991; Teller 2001). Contessa (2010) replies that this is anon sequitur. Even if, from an ontological point of view,anything can be a model and the class of things that are referred toas models contains a heterogeneous collection of different things, itdoes not follow that it is either impossible or pointless to developan ontology of models. This is because even if not all models are of aparticular ontological kind, one can nevertheless ask to whatontological kinds the things that arede facto used as modelsbelong. There may be several such kinds and each kind can be analyzedin its own right. What sort of objects scientists use as models hasimportant repercussions for how models perform relevant functions suchas representation and explanation, and hence this issue cannot bedismissed as “just sociology”.

The objects that commonly serve as models indeed belong to differentontological kinds: physical objects, fictional objects, abstractobjects, set-theoretic structures, descriptions, equations, orcombinations of some of these, are frequently referred to as models,and some models may fall into yet other classes of things. FollowingContessa’s advice, the aim then is to develop an ontology foreach of these. Those with an interest in ontology may see this as agoal in its own right. It pays noting, however, that the question hasreverberations beyond ontology and bears on how one understands thesemantics and the epistemology of models.

2.1 Physical objects

Some models are physical objects. Such models are commonly referred toas “material models”. Standard examples of models of thiskind are scale models of objects like bridges and ships (seeSection 1), Watson and Crick’s metal model of DNA (Schaffner 1969),Phillips and Newlyn’s hydraulic model of an economy (Morgan andBoumans 2004), the US Army Corps of Engineers’ model of the SanFrancisco Bay (Weisberg 2013), Kendrew’s plasticine model ofmyoglobin (Frigg and Nguyen 2016), and model organisms in the lifesciences (Ankeny and Leonelli 2020; Leonelli 2010; Levy and Currie2015; Sartori forthcoming). All these are material objects that serveas models. Material models do not give rise to ontologicaldifficulties over and above the well-known problems in connection withobjects that metaphysicians deal with, for instance concerning thenature of properties, the identity of objects, parts and wholes, andso on.

However, many models arenot material models. The Bohr modelof the atom, a frictionless pendulum, or an isolated population, forinstance, are in the scientist’s mind rather than in thelaboratory and they do not have to be physically realized andexperimented upon to serve as models. These “non-physical”models raise serious ontological questions, and how they are bestanalyzed is debated controversially. In the remainder of this sectionwe review some of the suggestions that have attracted attention in therecent literature on models.

2.2 Fictional objects and abstract objects

What has become known as thefiction view of models seesmodels as akin to the imagined objects of literary fiction—thatis, as akin to fictional characters like Sherlock Holmes or fictionalplaces like Middle Earth (Godfrey-Smith 2007). So when Bohr introducedhis model of the atom he introduced a fictional object of the samekind as the object Conan Doyle introduced when he invented SherlockHolmes. This view squares well with scientific practice, wherescientists often talk about models as if they were objects and oftentake themselves to be describingimaginary atoms,populations, or economies. It also squares well with philosophicalviews that see the construction and manipulation of models asessential aspects of scientific investigation (Morgan 1999), even ifmodels are not material objects, because these practices seem to bedirected towardsome kind of object.

What philosophical questions does this move solve? Fictional discourseand fictional entities face well-known philosophical questions, andone may well argue that simply likening models to fictions amounts toexplainingobscurum per obscurius (for a discussion of thesequestions, see the entry onfictional entities). One way to counter this objection and to motivate the fiction view ofmodels is to point to the view’s heuristic power. In this veinFrigg (2010b) identifies five specific issues that an ontology ofmodels has to address and then notes that these issues arise in verysimilar ways in the discussion about fiction (the issues are theidentity conditions, property attribution, the semantics ofcomparative statements, truth conditions, and the epistemology ofimagined objects). Likening models to fiction then has heuristic valuebecause there is a rich literature on fiction that offers a number ofsolutions to these issues.

Only a small portion of the options available in the extensiveliterature on fictions have actually been explored in the context ofscientific models. Contessa (2010) formulates what he calls the“dualist account”, according to which a model is anabstract object that stands for a possible concrete object. TheRutherford model of the atom, for instance, is an abstract object thatacts as a stand-in for one of the possible systems that contain anelectron orbiting around a nucleus in a well-defined orbit.Barberousse and Ludwig (2009) and Frigg (2010b) take a different routeand develop an account of models as fictions based on Walton’s(1990) pretense theory of fiction. According to this view thesentences of a passage of text introducing a model should be seen as aprop in a game of make-believe, and the model is the product of an actof pretense. This is an antirealist position in that it takes talk ofmodel “objects” to be figures of speech because ultimatelythere are no model objects—models only live in scientists’imaginations. Salis (2021) reformulates this view to become what shecalls “the new fiction view of models”. The coredifference lies in the fact that what is considered as the model arethe model descriptions and their content rather than the imaginingsthat they prescribe. This is a realist view of models, becausedescriptions exist.

The fiction view is not without critics. Giere (2009), Magnani (2012),Pincock (2012), Portides (2014), and Teller (2009) reject the fictionapproach and argue, in different ways, that models should not beregarded as fictions. Weisberg (2013) argues for a middle positionwhich sees fictions as playing a heuristic role but denies that theyshould be regarded as forming part of a scientific model. The commoncore of these criticisms is that the fiction view misconstrues theepistemic standing of models. To call something a fiction, so thecharge goes, is tantamount to saying that it is false, and it isunjustified to call an entire model a fiction—and thereby claimthat it fails to capture how the world is—just because the modelinvolves certain false assumptions or fictional elements. In otherwords, a representation isn’t automatically counted as fictionjust because it has some inaccuracies. Proponents of the fiction viewagree with this point but deny that the notion of fiction should beanalyzed in terms of falsity. What makes a work a fiction is not itsfalsity (or some ratio of false to true claims): neither is everythingthat is said in a novel untrue (Tolstoy’sWar and Peacecontains many true statements about Napoleon’s Franco-RussianWar), nor does every text containing false claims qualify as fiction(false news reports are just that, they are not fictions). Thedefining feature of a fiction is that readers are supposed toimagine the events and characters described, not that theyare false (Frigg 2010a; Salis 2021).

Giere (1988) advocated the view that “non-physical” modelsare abstract entities. However, there is little agreement on thenature of abstract objects, and Hale (1988: 86–87) lists no lessthan twelve different possible characterizations (for a review of theavailable options, see the entry onabstract objects). In recent publications, Thomasson (2020) and Thomson-Jones (2020)develop what they call an “artifactualist view” of models,which is based on Thomasson’s (1999) theory of abstractartifacts. This view agrees with the pretense theory that the contentof text that introduces a fictional character or a model should beunderstood as occurring in pretense, but at the same time insists thatin producing such descriptions authors create abstract culturalartifacts that then exist independently of either the author or thereaders. Artifactualism agrees with Platonism that abstract objectsexist, but insists,contra Platonism, that abstract objectsare brought into existence through a creative act and are not eternal.This allows the artifactualist to preserve the advantages of pretensetheory while at the same time holding the realist view that fictionalcharacters and models actually exist.

2.3 Set-theoretic structures

An influential point of view takes models to be set-theoreticstructures. This position can be traced back to Suppes (1960) and isnow, with slight variants, held by most proponents of the so-calledsemantic view of theories; for a discussion of this view, see theentry onthe structure of scientific theories and Frigg (2023). There are differences between the versions of thesemantic view, but with the exception of Giere (1988) all versionsagree that models are structures of one sort or another (Da Costa andFrench 2000).

This view of models has been criticized on various grounds. Onepervasive criticism is that many types of models that play animportant role in science are not structures and cannot beaccommodated within the structuralist view of models, which canneither account for how these models are constructed nor for how theywork in the context of investigation (Cartwright 1999; Downes 1992;Morrison 1999). Examples for such models are interpretative models andmediating models, discussed later inSection 4.2. Another charge held against the set-theoretic approach is thatset-theoretic structures by themselves cannot be representationalmodels—at least if that requires them to share some structurewith the target—because the ascription of a structure to atarget system which forms part of the physical world relies on asubstantive (non-structural) description of the target, which goesbeyond what the structuralist approach can afford (Nguyen and Frigg2021).

2.4 Descriptions and equations

A time-honored position has it that a model is a stylized descriptionof a target system. It has been argued that this is what scientistsdisplay in papers and textbooks when they present a model (Achinstein1968; Black 1962). This view has not been subject to explicitcriticism. However, some of the criticisms that have been marshaledagainst the so-called syntactic view of theories equally threaten alinguistic understanding of models (for a discussion of this view, seethe entry onthe structure of scientific theories). First, a standard criticism of the syntactic view is that byassociating a theory with a particular formulation, the viewmisconstrues theory identity because any change in the formulationresults in a new theory (Suppe 2000). A view that associates modelswith descriptions would seem to be open to the same criticism. Second,models have different properties than descriptions: the Newtonianmodel of the solar system consists of orbiting spheres, but it makesno sense to say this about its description. Conversely, descriptionshave properties that models do not have: a description can be writtenin English and consist of 517 words, but the same cannot be said of amodel. One way around these difficulties is to associate the modelwith the content of a description rather than with the descriptionitself. For a discussion of a position on models that builds on thecontent of a description, see Salis (2021).

A contemporary version of descriptivism is Levy’s (2012, 2015)and Toon’s (2012) so-called direct-representation view. Thisview shares with the fiction view of models (Section 2.2) the reliance on Walton’s pretense theory, but uses it in adifferent way. The main difference is that the views discussed earliersee modeling as introducing a vehicle of representation, the model,that is distinct from the target, and they see the problem aselucidating what kind of thing the model is. On thedirect-representation view there are no models distinct from thetarget; there are only model-descriptions and targets, with no modelsin-between them. Modeling, on this view, consists in providing animaginative description of real things. A model-description prescribesimaginings about the real system; the ideal pendulum, for instance,prescribes model-users to imagine the real spring as perfectly elasticand the bob as a point mass. This approach avoids the above problemsbecause the identity conditions for models are given by the conditionsfor games of make-believe (and not by the syntax of a description) andproperty ascriptions take place in pretense. There are, however,questions about how this account deals with models that have no target(like models of the ether or four-sex populations), and about howmodels thus understood deal with idealizations. For a discussion ofthese points, see Frigg and Nguyen (2016), Poznic (2016), and Salis(2021).

A closely related approach sees models as equations. This is a versionof the view that models are descriptions, because equations aresyntactic items that describe a mathematical structure. The issuesthat this view faces are similar to the ones we have alreadyencountered: First, one can describe the same situation usingdifferent kinds of coordinates and as a result obtain differentequations but without thereby also obtaining a different model.Second, the model and the equation have different properties. Apendulum contains a massless string, but the equation describing itsmotion does not; and an equation may be inhomogeneous, but the systemit describes is not. It is an open question whether these issues canbe avoided by appeal to a pretense account.

3. Epistemology: The Cognitive Functions of Models

One of the main reasons why models play such an important role inscience is that they perform a number of cognitive functions. Forexample, models are vehicles for learning about the world. Significantparts of scientific investigation are carried out on models ratherthan on reality itself because by studying a model we can discoverfeatures of, and ascertain facts about, the system the model standsfor: models allow for “surrogative reasoning” (Swoyer1991). For instance, we study the nature of the hydrogen atom, thedynamics of a population, or the behavior of a polymer by studyingtheir respective models. This cognitive function of models has beenwidely acknowledged in the literature, and some even suggest thatmodels give rise to a new style of reasoning, “model-basedreasoning”, according to which “inferences are made bymeans of creating models and manipulating, adapting, and evaluatingthem” (Nersessian 2010: 12; see also Magnani, Nersessian, andThagard 1999; Magnani and Nersessian 2002; and Magnani and Casadio2016).

3.1 Learning about models

Learning about a model happens in two places: in the construction ofthe model and in its manipulation (Morgan 1999). There are no fixedrules or recipes for model building and so the very activity offiguring out what fits together, and how, affords an opportunity tolearn about the model. Once the model is built, we do not learn aboutits properties by looking at it; we have to use and manipulate themodel in order to elicit its secrets.

Depending on what kind of model we are dealing with, building andmanipulating a model amount to different activities demandingdifferent methodologies. Material models seem to be straightforwardbecause they are used in common experimental contexts (e.g., we putthe model of a car in the wind tunnel and measure its air resistance).Hence, as far as learning about the model is concerned, materialmodels do not give rise to questions that go beyond questionsconcerning experimentation more generally.

Not so with fictional and abstract models. What constraints are thereto the construction of fictional and abstract models, and how do wemanipulate them? A natural response seems to be that we do this byperforming a thought experiment. Different authors (e.g., Brown 1991;Gendler 2000; El Skaf and Stuart 2024; Norton 1991; Reiss 2003;Sorensen 1992; Sartori 2023) have explored this line of argument, butthey have reached very different and often conflicting conclusionsabout how thought experiments are performed and what the status oftheir outcomes is (for details, see the entry onthought experiments).

An important class of models is computational in nature. For somemodels it is possible to derive results or solve equations of amathematical model analytically. But quite often this is not the case.It is at this point that computers have a great impact, because theyallow us to solve problems that are otherwise intractable. Hence,computational methods provide us with knowledge about (theconsequences of) a model where analytical methods remain silent. Manyparts of current research in both the natural and social sciences relyon computer simulations, which help scientists to explore theconsequences of models that cannot be investigated otherwise. Theformation and development of stars and galaxies, the dynamics ofhigh-energy heavy-ion reactions, the evolution of life, outbreaks ofwars, the progression of an economy, moral behavior, and theconsequences of decision procedures in an organization are exploredwith computer simulations, to mention only a few examples.

Computer simulations are also heuristically important. They cansuggest new theories, models, and hypotheses, for example, based on asystematic exploration of a model’s parameter space (Hartmann1996). But computer simulations also bear methodological perils. Forexample, they may provide misleading results because, due to thediscrete nature of the calculations carried out on a digital computer,they only allow for the exploration of a part of the full parameterspace, and this subspace need not reflect every important feature ofthe model. The severity of this problem is somewhat mitigated by theincreasing power of modern computers. But the availability of morecomputational power can also have adverse effects: it may encouragescientists to swiftly come up with increasingly complex butconceptually premature models, involving poorly understood assumptionsor mechanisms and too many additional adjustable parameters (for adiscussion of a related problem in the social sciences, see Braun andSaam 2015: Ch. 3). This can lead to an increase in empiricaladequacy—which may be welcome for certain forecastingtasks—but not necessarily to a better understanding of theunderlying mechanisms. As a result, the use of computer simulationscan change the weight we assign to the various goals of science.Finally, the availability of computer power may seduce scientists intomaking calculations that do not have the degree of trustworthiness onewould expect them to have. This happens, for instance, when computersare used to propagate probability distributions forward in time, whichcan turn out to be misleading (see Frigg et al. 2014). So it isimportant not to be carried away by the means that new powerfulcomputers offer and lose sight of the actual goals of research. For adiscussion of further issues in connection with computer simulations,we refer the reader to the entry oncomputer simulations in science.

3.2 Learning about target systems

Once we have knowledge about the model, this knowledge has to be“translated” into knowledge about the target system. It isat this point that the representational function of models becomesimportant again: if a model represents, then it can instruct us aboutreality because (at least some of) the model’s parts or aspectshave corresponding parts or aspects in the world. But if learning isconnected to representation and if there are different kinds ofrepresentations (analogies, idealizations, etc.), then there are alsodifferent kinds of learning. If, for instance, we have a model we taketo be a realistic depiction, the transfer of knowledge from the modelto the target is accomplished in a different manner than when we dealwith an analogue, or a model that involves idealizing assumptions. Fora discussion of the different ways in which the representationalfunction of models can be exploited to learn about the target, werefer the reader to the entryScientific Representation.

3.3 Explaining with models

Some models explain. But how can they fulfill this function given thatthey typically involve idealizations? Do these models explaindespite orbecauseof the idealizationsthey involve? Does an explanatory use of models presuppose that theyrepresent, or can non-representational models also explain? And whatkind of explanation do models provide?

There is a long tradition requesting that the explanans of ascientific explanation must be true. We find this requirement in thedeductive-nomological model (Hempel 1965) as well as in the morerecent literature. For instance, Strevens (2008: 297) claims that“no causal account of explanation … allows nonveridicalmodels to explain”. For further discussions, see also Colombo etal. (2015).

Authors working in this tradition deny that idealizations make apositive contribution to explanation and explore how models canexplain despite being idealized. McMullin (1968, 1985) argues that acausal explanation based on an idealized model leaves out onlyfeatures which are irrelevant for the respective explanatory task (seealso Salmon 1984 and Piccinini and Craver 2011 for a discussion ofmechanism sketches). Friedman (1974) argues that a more realistic (andhence less idealized) model explains better on the unificationaccount. The idea is that idealizations can (at least in principle) bede-idealized (for a critical discussion of this claim in the contextof the debate about scientific explanations, see Batterman 2002;Bokulich 2011; Morrison 2005, 2009; Jebeile and Kennedy 2015; and Rice2015). Strevens (2008) argues that an explanatory causal model has toprovide an accurate representation of the relevant causalrelationships or processes which the model shares with the targetsystem. The idealized assumptions of a model do not make a differencefor the phenomenon under consideration and are therefore explanatorilyirrelevant. In contrast, both Potochnik (2017) and Rice (2015) arguethat models that explain can directlydistort manydifference-making causes.

According to Woodward’s (2003) theory, models are tools to findout about the causal relations that hold between certain facts orprocesses, and it is these relations that do the explanatory work.More specifically, explanations provide information about patterns ofcounterfactual dependence between the explanans and the explanandumwhich

enable us to see what sort of difference it would have made for theexplanandum if the factors cited in the explanans had been differentin various possible ways. (Woodward 2003: 11)

Accounts of causal explanation have also led to various claims abouthow idealized models can provide explanations, exploring to whatextent idealization allows for the misrepresentation of irrelevantcausal factors by the explanatory model (Elgin and Sober 2002;Strevens 2004, 2008; Potochnik 2007; Weisberg 2007, 2013). However,having the causally relevant features in common with real systemscontinues to play the essential role in showing how idealized modelscan be explanatory.

But is it really the truth of the explanans that makes the modelexplanatory? Other authors pursue a more radical line and argue thatfalse models explain not onlydespite their falsity, but infactbecause of their falsity. Cartwright (1983: 44)maintains that “the truth doesn’t explain much”. Inher so-called “simulacrum account of explanation”, shesuggests that we explain a phenomenon by constructing a model thatfits the phenomenon into the basic framework of a grand theory (1983:Ch. 8). On this account, the model itself is the explanation we seek.This squares well with basic scientific intuitions, but it leaves uswith the question of what notion of explanation is at work (see alsoElgin and Sober 2002) and of what explanatory function idealizationsplay in model explanations (Rice 2018, 2019). Wimsatt (2007: Ch. 6)stresses the role of false models as means to arrive at true theories.Batterman and Rice (2014) argue that models explain because thedetails that characterize specific systems do not matter for theexplanation. Bokulich (2008, 2009, 2011, 2012) pursues a similar lineof reasoning and sees the explanatory power of models as being closelyrelated to their fictional nature. Bokulich (2009) and Kennedy (2012)present non-representational accounts of model explanation (see alsoJebeile and Kennedy 2015). Reiss (2012) and Woody (2004) providegeneral discussions of the relationship between representation andexplanation.

3.4 Understanding with models

Many authors have pointed out that understanding is one of the centralgoals of science (see, for instance, de Regt 2017; Elgin 2017; Khalifa2017; Potochnik 2017). In some cases, we want to understand a certainphenomenon (e.g., why the sky is blue); in other cases, we want tounderstand a specific scientific theory (e.g., quantum mechanics) thataccounts for a phenomenon in question. Sometimes we gain understandingof a phenomenon by understanding the corresponding theory or model.For instance, Maxwell’s theory of electromagnetism helps usunderstand why the sky is blue. It is, however, controversial whetherunderstanding a phenomenonalways presupposes anunderstanding of the corresponding theory (de Regt 2009: 26).

Although there are many different ways of gaining understanding,models and the activity of scientific modeling are of particularimportance here (de Regt et al. 2009; Morrison 2009; Potochnik 2017;Rice 2016). This insight can be traced back at least to Lord Kelvinwho, in his famous 1884Baltimore Lectures on Molecular Dynamicsand the Wave Theory of Light, maintained that “the test of‘Do we or do we not understand a particular subject inphysics?’ is ‘Can we make a mechanical model ofit?’” (Kelvin 1884 [1987: 111]; see also Bailer-Jones2009: Ch. 2; and de Regt 2017: Ch. 6).

But why do models play such a crucial role in the understanding of asubject matter? Elgin (2017) argues that this is not despite, butbecause, of models being literally false. She views false models as“felicitous falsehoods” that occupy center stage in theepistemology of science, and mentions the ideal-gas model instatistical mechanics and the Hardy–Weinberg model in geneticsas examples for literally false models that are central to theirrespective disciplines. Understanding is holistic and it concerns atopic, a discipline, or a subject matter, rather than isolated claimsor facts. Gaining understanding of a context means to have

an epistemic commitment to a comprehensive, systematically linked bodyof information that is grounded in fact, is duly responsive to reasonsor evidence, and enables nontrivial inference, argument, and perhapsaction regarding the topic the information pertains to (Elgin 2017:44)

and models can play a crucial role in the pursuit of these epistemiccommitments. For a discussion of Elgin’s account of models andunderstanding, see Baumberger and Brun (2017) and Frigg and Nguyen(2021).

Elgin (2017), Lipton (2009), and Rice (2016) all argue that models canbe used to understand independently of their ability to provide anexplanation. Other authors, among them Strevens (2008, 2013), arguethat understanding presupposes a scientific explanation and that

an individual has scientific understanding of a phenomenon just incase they grasp a correct scientific explanation of that phenomenon.(Strevens 2013: 510; see, however, Sullivan and Khalifa 2019)

On this account, understanding consists in a particular form ofepistemic access an individual scientist has to an explanation. ForStrevens this aspect is “grasping”, while for de Regt(2017) it is “intelligibility”. It is important to notethat both Strevens and de Regt hold that such “subjective”aspects are a worthy topic for investigations in the philosophy ofscience. This contrasts with the traditional view (see, e.g., Hempel1965) that delegates them to the realm of psychology. See Friedman(1974), Trout (2002), and Reutlinger et al. (2018) for furtherdiscussions of understanding.

3.5 Other cognitive functions

Besides the functions already mentioned, it has been emphasizedvariously that models perform a number of other cognitive functions.Knuuttila (2005, 2011) argues that the epistemic value of models isnot limited to their representational function, and develops anaccount that views models as epistemic artifacts which allow us togather knowledge in diverse ways. Nersessian (1999, 2010) stresses therole of analogue models in concept-formation and other cognitiveprocesses. Hartmann (1995) and Leplin (1980) discuss models as toolsfor theory construction and emphasize their heuristic and pedagogicalvalue. Epstein (2008) lists a number of specific functions of modelsin the social sciences. Peschard (2011) investigates the way in whichmodels may be used to construct other models and generate new targetsystems. And Isaac (2013) discusses non-explanatory uses of modelswhich do not rely on their representational capacities.

Trade-offs are an inherent part of scientific modeling, as models mustbalance competing desiderata such as accuracy, generality, andsimplicity. Richard Levins (1966) articulated this challenge in hisessay “The Strategy of Model Building in PopulationBiology,” arguing that no single model can simultaneouslymaximize all three desiderata. Instead, scientists must make strategicchoices about which aspects of a system to prioritize in their models.For example, a model might sacrifice some realism to achieve greatergenerality, or it might simplify complex dynamics to make the modelcomputationally tractable.

Levins’s framework has sparked extensive philosophicaldiscussion, with some arguing that trade-offs are unavoidable inmodeling (Weisberg 2006; Odenbaugh 2006), while others have proposedintegrative approaches that combine multiple models to achieve a morecomprehensive understanding of complex systems (Orzack and Sober 1993;Matthewson 2011). Yoshida (2021) has emphasized the importance ofmultiple-models juxtaposition, where different models are compared andcontrasted to gain a more nuanced understanding of the phenomena underinvestigation.

These discussions highlight the epistemic challenges of model-buildingacross the sciences. Whether in physics, biology, economics, orclimate science, trade-offs are always present, and scientists mustcarefully consider the strengths and limitations of their models. Byacknowledging these trade-offs, researchers can make more informeddecisions about how to represent and study complex systems, andphilosophers can better understand the role of models in scientificinquiry.

4. Models and Theory

An important question concerns the relation between models andtheories. There is a full spectrum of positions ranging from modelsbeing subordinate to theories to models being independent oftheories.

4.1 Models as subsidiaries to theory

To discuss the relation between models and theories in science it ishelpful to briefly recapitulate the notions of a model and of a theoryin logic. Atheory is taken to be a (usually deductivelyclosed) set of sentences in a formal language. Amodel is astructure (in the sense introduced inSection 2.3) that makes all sentences of a theory true when its symbols areinterpreted as referring to objects, relations, or functions of astructure. The structure is amodel of the theory in thesense that it is correctly described by the theory (see Bell andMachover 1977 or Hodges 1997 for details). Logical models aresometimes also referred to as “models of theory” toindicate that they are interpretations of an abstract formalsystem.

Models in science sometimes carry over from logic the idea of beingthe interpretation of an abstract calculus (Hesse 1967). This issalient in physics, where general laws—such as Newton’sequation of motion—lie at the heart of a theory. These laws areapplied to a particular system—e.g., a pendulum—bychoosing a special force function, making assumptions about the massdistribution of the pendulum etc. The resulting model then is aninterpretation (or realization) of the general law.

It is important to keep the notions of a logical and arepresentational model separate (Thomson-Jones 2006): these aredistinct concepts. Something can be a logical model without being arepresentational model, andvice versa. This, however, doesnot mean that something cannot be a model in both senses at once. Infact, as Hesse (1967) points out, many models in science are bothlogical and representational models. Newton’s model of planetarymotion is a case in point: the model, consisting of two homogeneousperfect spheres located in otherwise empty space that attract eachother gravitationally, is simultaneously a logical model (because itmakes the axioms of Newtonian mechanics true when they are interpretedas referring to the model) and a representational model (because itrepresents the real sun and earth).

There are two main conceptions of scientific theories, the so-calledsyntactic view of theories and the so-called semantic view of theories(see the entry onthe structure of scientific theories). On both conceptions models play a subsidiary role to theories, albeitin very different ways. The syntactic view of theories (see entrysection onthe syntactic view) retains the logical notions of a model and a theory. It construes atheory as a set of sentences in an axiomatized logical system, and amodel as an alternative interpretation of a certain calculus(Braithwaite 1953; Campbell 1920 [1957]; Nagel 1961; Spector 1965).If, for instance, we take the mathematics used in the kinetic theoryof gases and reinterpret the terms of this calculus in a way thatmakes them refer to billiard balls, the billiard balls are a model ofthe kinetic theory of gases in the sense that all sentences of thetheory come out true. The model is meant to be something that we arefamiliar with, and it serves the purpose of making an abstract formalcalculus more palpable. A given theory can have different models, andwhich model we choose depends both on our aims and our backgroundknowledge. Proponents of the syntactic view disagree about theimportance of models. Carnap and Hempel thought that models only servea pedagogic or aesthetic purpose and are ultimately dispensablebecause all relevant information is contained in the theory (Carnap1938; Hempel 1965; see also Bailer-Jones 1999). Nagel (1961) andBraithwaite (1953), on the other hand, emphasize theheuristic role of models, and Schaffner (1969) submits thattheoretical terms get at least part of their meaning from models.

The semantic view of theories (see entry section onthe semantic view) dispenses with sentences in an axiomatized logical system andconstrues a theory as a family of models. On this view, a theoryliterally is a class, cluster, or family of models—models arethe building blocks of which scientific theories are made up.Different versions of the semantic view work with different notions ofa model, but, as noted inSection 2.3, in the semantic view models are mostly construed as set-theoreticstructures. For a discussion of the different options, we refer thereader to the relevant entry in this encyclopedia (linked at thebeginning of this paragraph).

4.2 Models as independent from theories

In both the syntactic and the semantic view of theories models areseen as subordinate to theory and as playing no role outside thecontext of a theory. This vision of models has been challenged in anumber of ways, with authors pointing out that models enjoy variousdegrees of freedom from theory and function autonomously in manycontexts. Independence can take many forms, and large parts of theliterature on models are concerned with investigating various forms ofindependence.

Models as completely independent of theory. The most radicaldeparture from a theory-centered analysis of models is the realizationthat there are models that are completely independent from any theory.An example of such a model is the Lotka–Volterra model. Themodel describes the interaction of two populations: a population ofpredators and one of prey animals (Weisberg 2013). The model wasconstructed using only relatively commonsensical assumptions aboutpredators and prey and the mathematics of differential equations.There was no appeal to a theory of predator–prey interactions ora theory of population growth, and the model is independent oftheories about its subject matter. If a model is constructed in adomain where no theory is available, then the model is sometimesreferred to as a “substitute model” (Groenewold 1961),because the model substitutes a theory.

Models as a means to explore theory. Models can also be usedto explore theories (Morgan and Morrison 1999). An obvious way inwhich this can happen is when a model is a logical model of a theory(seeSection 4.1). A logical model is a set of objects and properties that make a formalsentence true, and so one can see in the model how the axioms of thetheory play out in a particular setting and what kinds of behaviorthey dictate. But not all models that are used to explore theories arelogical models, and models can represent features of theories in otherways. As an example, consider chaos theory. The equations ofnon-linear systems, such as those describing the three-body problem,have solutions that are too complex to study with paper-and-pencilmethods, and even computer simulations are limited in various ways.Abstract considerations about the qualitative behavior of solutionsshow that there is a mechanism that has been dubbed “stretchingand folding” (see the entryChaos). To obtain an idea of the complexity of the dynamics exhibitingstretching and folding, Smale proposed to study a simple model of theflow—now known as the “horseshoe map” (Tabor1989)—which provides important insights into the nature ofstretching and folding. Other examples of models of that kind are theKac ring model that is used to study equilibrium properties of systemsin statistical mechanics (Lavis 2008) and Norton’s dome inNewtonian mechanics (Norton 2003).

Models as complements of theories. A theory may beincompletely specified in the sense that it only imposes certaingeneral constraints but remains silent about the details of concretesituations, which are provided by a model (Redhead 1980). A specialcase of this situation is when a qualitative theory is known and themodel introduces quantitative measures (Apostel 1961). Redhead’sexample of a theory that is underdetermined in this way is axiomaticquantum field theory, which only imposes certain general constraintson quantum fields but does not provide an account of particularfields. Harré (2004) notes that models can complement theoriesby providing mechanisms for processes that are left unspecified in thetheory even though they are responsible for bringing about theobserved phenomena.

Theories may be too complicated to handle. In such cases a model cancomplement a theory by providing a simplified version of thetheoretical scenario that allows for a solution. Quantumchromodynamics, for instance, cannot easily be used to investigate thephysics of an atomic nucleus even though it is the relevantfundamental theory. To get around this difficulty, physicistsconstruct tractable phenomenological models (such as the MIT bagmodel) which effectively describe the relevant degrees of freedom ofthe system under consideration (Hartmann 1999, 2001). The advantage ofthese models is that they yield results where theories remain silent.Their drawback is that it is often not clear how to understand therelationship between the model and the theory, as the two are,strictly speaking, contradictory.

Models as preliminary theories. The notion of a model as asubstitute for a theory is closely related to the notion of adevelopmental model. This term was coined by Leplin (1980),who pointed out how useful models were in the development of earlyquantum theory, and it is now used as an umbrella notion coveringcases in which models are some sort of a preliminary exercise totheory.

Also closely related is the notion of aprobing model (or“study model”). Models of this kind do not perform arepresentational function and are not expected to instruct us aboutanything beyond the model itself. The purpose of these models is totest new theoretical tools that are used later on to buildrepresentational models. In field theory, for instance, the so-calledφ⁴-model was studied extensively, not because it wasbelieved to represent anything real, but because it served severalheuristic functions: the simplicity of the φ⁴-modelallowed physicists to “get a feeling” for what quantumfield theories are like and to extract some general features that thissimple model shared with more complicated ones. Physicists could studycomplicated techniques such as renormalization in a simple setting,and it was possible to get acquainted with importantmechanisms—in this case symmetry-breaking—that could laterbe used in different contexts (Hartmann 1995). This is true not onlyfor physics. As Wimsatt (1987, 2007) points out, a false model ingenetics can perform many useful functions, among them the following:the false model can help answering questions about more realisticmodels, provide an arena for answering questions about properties ofmore complex models, “factor out” phenomena that would nototherwise be seen, serve as a limiting case of a more general model(or two false models may define the extremes of a continuum of caseson which the real case is supposed to lie), or lead to theidentification of relevant variables and the estimation of theirvalues.

Interpretative models. Cartwright (1983, 1999) argues thatmodels do not only aid the application of theories that are somehowincomplete; she claims that models are also involvedwhenevera theory with an overarching mathematical structure is applied. Themain theories in physics—classical mechanics, electrodynamics,quantum mechanics, and so on—fall into this category. Theoriesof that kind are formulated in terms of abstract concepts that need tobe concretized for the theory to provide a description of the targetsystem, and concretizing the relevant concepts, idealized objects andprocesses are introduced. For instance, when applying classicalmechanics, the abstract concept of force has to be replaced with aconcrete force such as gravity. To obtain tractable equations, thisprocedure has to be applied to a simplified scenario, for instancethat of two perfectly spherical and homogeneous planets in otherwiseempty space, rather than to reality in its full complexity. The resultis aninterpretative model, which grounds the application ofmathematical theories to real-world targets. Such models areindependent from theory in that the theory does not determine theirform, and yet they are necessary for the application of the theory toa concrete problem.

Models as mediators. The relation between models and theoriescan be complicated and disorderly. The contributors to a programmaticcollection of essays edited by Morgan and Morrison (1999) rally aroundthe idea that models are instruments that mediate between theories andthe world. Models are “autonomous agents” in that they areindependent from both theories and their target systems, and it isthis independence that allows them to mediate between the two.Theories do not provide us with algorithms for the construction of amodel; they are not “vending machines” into which one caninsert a problem and a model pops out (Cartwright 1999). Theconstruction of a model often requires detailed knowledge aboutmaterials, approximation schemes, and the setup, and these are notprovided by the corresponding theory. Furthermore, the inner workingsof a model are often driven by a number of different theories workingcooperatively. In contemporary climate modeling, for instance,elements of different theories—among them fluid dynamics,thermodynamics, electromagnetism—are put to work cooperatively.What delivers the results is not the stringent application of onetheory, but the voices of different theories when put to use in choruswith each other in one model.

In complex cases like the study of a laser system or the globalclimate, models and theories can get so entangled that it becomesunclear where a line between the two should be drawn: where does themodel end and the theory begin? This is not only a problem forphilosophical analysis; it also arises in scientific practice.Bailer-Jones (2002) interviewed a group of physicists about theirunderstanding of models and their relation to theories, and reportswidely diverging views: (i) there is no substantive difference betweenmodel and theory; (ii) modelsbecome theories when theirdegree of confirmation increases; (iii) models contain simplificationsand omissions, while theories are accurate and complete; (iv) theoriesare more general than models, and modeling is about applying generaltheories to specific cases. The first suggestion seems to be tooradical to do justice to many aspects of practice, where a distinctionbetween models and theories is clearly made. The second view is inline with common parlance, where the terms “model” and“theory” are sometimes used to express someone’sattitude towards a particular hypothesis. The phrase “it’sjust a model” indicates that the hypothesis at stake is assertedonly tentatively or is even known to be false, while something isawarded the label “theory” if it has acquired some degreeof general acceptance. However, this use of “model” isdifferent from the uses we have seen inSections 1 to 3 and is therefore of no use if we aim to understand the relationbetween scientific models and theories (and, incidentally, one canequally dismiss speculative claims as being “just atheory”). The third proposal is correct in associating modelswith idealizations and simplifications, but it overshoots byrestricting this to models; in fact, also theories can containidealizations and simplifications. The fourth view seems closelyaligned with interpretative models and the idea that models aremediators, but being more general is a gradual notion and hence doesnot provide a clear-cut criterion to distinguish between theories andmodels.

5. Models and Other Debates in the Philosophy of Science

The debate over scientific models has important repercussions forother issues in the philosophy of science (for a historical account ofthe philosophical discussion about models, see Bailer-Jones 1999).Traditionally, the debates over, say, scientific realism,reductionism, and laws of nature were couched in terms of theories,because theories were seen as the main carriers of scientificknowledge. Once models are acknowledged as occupying an importantplace in the edifice of science, these issues have to be reconsideredwith a focus on models. The question is whether, and if so how,discussions of these issues change when we shift focus from theoriesto models. Up to now, no comprehensive model-based account of any ofthese issues has emerged, but models have left important traces in thediscussions of these topics.

5.1 Models, realism, and laws of nature

As we have seen inSection 1, models typically provide a distorted representation of their targets.If one sees science as primarily model-based, this could be taken tosuggest an antirealist interpretation of science. Realists, however,deny that the presence of idealizations in models renders a realistapproach to science impossible and point out that a good model, whilenot literally true, is usually at least approximately true, and/orthat it can be improved by de-idealization (Laymon 1985; McMullin1985; Nowak 1979; Brzezinski and Nowak 1992).

Apart from the usual worries about the elusiveness of the notion ofapproximate truth (for a discussion, see the entry ontruthlikeness), antirealists have taken issue with this reply for two (related)reasons. First, as Cartwright (1989) points out, there is no reason toassume that one can always improve a model by adding de-idealizingcorrections. Second, it seems that de-idealization is not inaccordance with scientific practice because it is unusual thatscientists invest work in repeatedly de-idealizing an existing model.Rather, they shift to a different modeling framework once theadjustments to be made get too involved (Hartmann 1998). The variousmodels of the atomic nucleus are a case in point: once it was realizedthat shell effects are important to understand various subatomicphenomena, the (collective) liquid-drop model was put aside and the(single-particle) shell model was developed to account for thecorresponding findings. A further difficulty with de-idealization isthat most idealizations are not “controlled”. For example,it is not clear in what way one could de-idealize the MIT bag model toeventually arrive at quantum chromodynamics, the supposedly correctunderlying theory.

A further antirealist argument, the “incompatible-modelsargument”, takes as its starting point the observation thatscientists often successfully use several incompatible models ofone and the same target system for predictive purposes(Morrison 2000). These models seemingly contradict each other, as theyascribe different properties to the same target system. In nuclearphysics, for instance, the liquid-drop model explores the analogy ofthe atomic nucleus with a (charged) fluid drop, while the shell modeldescribes nuclear properties in terms of the properties of protons andneutrons, the constituents of an atomic nucleus. This practice appearsto cause a problem for scientific realism: Realists typically holdthat there is a close connection between the predictive success of atheory and its being at least approximately true. But if severalmodels of the same system are predictively successful and if thesemodels are mutually inconsistent, then it is difficult to maintainthat they are all approximately true.

Realists can react to this argument in various ways. First, they canchallenge the claim that the models in question are indeedpredictively successful. If the models are not good predictors, thenthe argument is blocked. Second, they can defend a version of“perspectival realism” (Giere 2006; Massimi 2022; Rueger2005). Proponents of this position (which is sometimes also called“perspectivism”) situate it somewhere between“standard” scientific realism and antirealism, and whereexactly the right middle position lies is the subject matter of activedebate (Massimi 2018a,b; Saatsi 2016; Teller 2018; and thecontributions to Massimi and McCoy 2019). Third, realists can denythat there is a problem in the first place, because scientific models,which are always idealized and therefore strictly speaking false, arejust the wrong vehicle to make a point about realism (which should bediscussed in terms of theories).

Sober (2015) relates the debate about model selection in statistics tothe debate between realists and instrumentalists in the philosophy ofscience. While the realist sees the goal of science as finding truetheories, Sober’s instrumentalist sees the goal of science asfinding predictively accurate theories. In the end, Sober defends theslogan “instrumentalism for models, realism for fittedmodels” (2015: 144). See also the entryPhilosophy of Statistics.

A particular focal point of the realism debate are laws of nature,where the questions arise what laws are and whether they aretruthfully reflected in our scientific representations. According tothe two currently dominant accounts, the best-systems approach and thenecessitarian approach, laws of nature are understood to be universalin scope, meaning that they apply to everything that there is in theworld (for discussion of laws, see the entry onlaws of nature). This take on laws does not seem to sit well with a view that placesmodels at the center of scientific research. What role do general lawsplay in science if it is models that represent what is happening inthe world? And how are models and laws related?

One possible response to these questions is to argue that laws ofnature govern entities and processes in amodel rather thanin the world. Fundamental laws, on this approach, do not state factsabout theworld but hold true of entities and processes inthe model. This view has been advocated in different variants:Cartwright (1983) argues that all laws areceteris paribuslaws. Cartwright (1999) makes use of “capacities” (whichshe considers to be prior to laws) and introduces the notion of a“nomological machine”. This is

a fixed (enough) arrangement of components, or factors, with stable(enough) capacities that in the right sort of stable (enough)environment will, with repeated operation, give rise to the kind ofregular behavior that we represent in our scientific laws. (1999: 50;see also the entry onceteris paribus laws)

Giere (1999) argues that the laws of a theory are better thought of,not as encoding general truths about the world, but rather asopen-ended statements that can be filled in various ways in theprocess of building more specific scientific models. Similar positionshave also been defended by Teller (2001) and van Fraassen (1989).

5.2 Models and reductionism

The multiple-models problem mentioned inSection 5.1 also raises the question of how different models are related.Evidently, multiple models for the same target system do not generallystand in a deductive relationship, as they often contradict eachother. Some (Cartwright 1999; Hacking 1983) have suggested a pictureof science according to which there are no systematic relations thathold between different models. Some models are tied together becausethey represent the same target system, but this does not imply thatthey enter into any further relationships (deductive or otherwise). Weare confronted with a patchwork of models, all of which holdceteris paribus in their specific domains ofapplicability.

Some argue that this picture is at least partially incorrect becausethere are various interesting relations that hold between differentmodels or theories. These relations range from thoroughgoing reductiverelations (Scheibe 1997, 1999, 2001: esp. Chs. V.23 and V.24) andcontrolled approximations over singular limit relations (Batterman2001 [2016]) to structural relations (Gähde 1997) and ratherloose relations called “stories” (Hartmann 1999; see alsoBokulich 2003; Teller 2002; and the essays collected in Part III ofHartmann et al. 2008). These suggestions have been made on the basisof case studies, and it remains to be seen whether a more generalaccount of these relations can be given and whether a deeperjustification for them can be provided, for instance, within aBayesian framework (first steps towards a Bayesian understanding ofreductive relations can be found in Dizadji-Bahmani et al. 2011;Liefke and Hartmann 2018; and Tešić 2019).

Models also figure in the debate about reduction and emergence inphysics. Here, some authors argue that the modern approach torenormalization challenges Nagel’s (1961) model of reduction orthe broader doctrine of reductions (for a critical discussion, see,for instance, Batterman 2002, 2010, 2011; Morrison 2012; and Saatsiand Reutlinger 2018). Dizadji-Bahmani et al. (2010) provide a defenseof the Nagel–Schaffner model of reduction, and Butterfield(2011a,b, 2014) argues that renormalization is consistent withNagelian reduction. Palacios (2019) shows that phase transitions arecompatible with reductionism, and Hartmann (2001) argues that theeffective-field-theories research program is consistent withreductionism (see also Bain 2013 and Franklin 2020). Rosaler (2015)argues for a “local” form of reduction which sees thefundamental relation of reduction holding between models, nottheories, which is, however, compatible with the Nagel–Schaffnermodel of reduction. See also the entries onintertheory relations in physics andscientific reduction.

In the social sciences, agent-based models (ABMs) are increasinglyused (Klein et al. 2018). These models show how surprisingly complexbehavioral patterns at the macro-scale can emerge from a small numberof simple behavioral rules for the individual agents and theirinteractions. This raises questions similar to the questions mentionedabove about reduction and emergence in physics, but so far one onlyfinds scattered remarks about reduction in the literature. SeeWeisberg and Muldoon (2009) and Zollman (2007) for the application ofABMs to the epistemology and the social structure of science, andColyvan (2013) for a discussion of methodological questions raised bynormative models in general.

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Acknowledgments

We would like to thank Joe Dewhurst, James Nguyen, AlexanderReutlinger, Collin Rice, Dunja Šešelja, and Paul Tellerfor helpful comments on the drafts of the revised version in 2019.When writing the original version back in 2006 we benefitted fromcomments and suggestions by Nancy Cartwright, Paul Humphreys, JulianReiss, Elliott Sober, Chris Swoyer, and Paul Teller.