## 1. Semantics: Models and Representation



Many scientific models are representational models: they represent a
selected part or aspect of the world, which is the model's
target system. Standard examples are the billiard ball model of a gas,
the Bohr model of the atom, the Lotka-Volterra model of predator-prey
interaction, the Mundell-Fleming model of an open economy, and the
scale model of a bridge.



This raises the question what it means for a model to represent a
target system. This problem is rather involved and decomposes into
various subproblems. For an in-depth discussion of the issue of
representation, see the entry on
 scientific representation.
 At this point, rather than addressing the issue of what it means for
a model to represent, we focus on a number of different kinds of
representation that play important roles in the practice of
model-based science, namely scale models, analogical models, idealized
models, toy models, minimal models, phenomenological models,
exploratory models, and models of data. These categories are not
mutually exclusive, and a given model can fall into several categories
at once.



*Scale models*. Some models are down-sized or enlarged copies
of their target systems (Black 1962). A typical example is a small
wooden car that is put into a wind tunnel to explore the actual
car's aerodynamic properties. The intuition is that a scale
model is a naturalistic replica or a truthful mirror image of the
target; for this reason, scale models are sometimes also referred to
as "true models" (Achinstein 1968: Ch. 7). However, there
is no such thing as a perfectly faithful scale model; faithfulness is
always restricted to some respects. The wooden scale model of the car
provides a faithful portrayal of the car's shape but not of its
material. And even in the respects in which a model is a faithful
representation, the relation between model-properties and
target-properties is usually not straightforward. When engineers use,
say, a 1:100 scale model of a ship to investigate the resistance that
an actual ship experiences when moving through the water, they cannot
simply measure the resistance the model experiences and then multiply
it with the scale. In fact, the resistance faced by the model does not
translate into the resistance faced by the actual ship in a
straightforward manner (that is, one cannot simply scale the water
resistance with the scale of the model: the real ship need not have
one hundred times the water resistance of its 1:100 model). The two
quantities stand in a complicated nonlinear relation with each other,
and the exact form of that relation is often highly nontrivial and
emerges as the result of a thoroughgoing study of the situation
(Sterrett 2006, forthcoming; Pincock forthcoming).



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



Analogies can also be based on the sameness or resemblance of
relations between parts of two systems rather than on their monadic
properties. It is in this sense that the relation of a father to his
children is asserted to be analogous to the relation of the state to
its citizens. The analogies mentioned so far have been what Hesse
calls "material analogies". We obtain a more formal notion
of analogy when we abstract from the concrete features of the systems
and only focus on their formal set-up. What the analogue model then
shares with its target is not a set of features, but the same pattern
of abstract relationships (i.e., the same structure, where structure
is understood in a formal sense). This notion of analogy is closely
related to what Hesse calls "formal analogy". Two items
are related by formal analogy if they are both interpretations of the
same formal calculus. For instance, there is a formal analogy between
a swinging pendulum and an oscillating electric circuit because they
are both described by the same mathematical equation.



A further important distinction due to Hesse is the one between
positive, negative, and neutral analogies. The positive analogy
between two items consists in the properties or relations they share
(both gas molecules and billiard balls have mass); the negative
analogy consists in the properties they do not share (billiard balls
are colored, gas molecules are not); the neutral analogy comprises the
properties of which it is not known (yet) whether they belong to the
positive or the negative analogy (do billiard balls and molecules have
the same cross section in scattering processes?). Neutral analogies
play an important role in scientific research because they give rise
to questions and suggest new hypotheses. For this reason several
authors have emphasized the heuristic role that analogies play in
theory and model construction, as well as in creative thought
(Bailer-Jones and Bailer-Jones 2002; Bailer-Jones 2009: Ch. 3; Hesse
1974; Holyoak and Thagard 1995; Kroes 1989; Psillos 1995; and the
essays collected in Helman 1988). See also the entry on
 analogy and analogical reasoning.



It has also been discussed whether using analogical models can in some
cases 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 models
cannot be confirmatory in a Bayesian sense because the information
encapsulated in an analogical model is part of the relevant background
knowledge, which has the consequence that the posterior probability of
a hypothesis about a target system cannot change as a result of
observing the analogy. Analogical models can therefore only establish
the *plausibility* of a conclusion in the sense of justifying a
non-negligible prior probability assignment (Bartha 2010:
SS8.5).



More recently, these questions have been discussed in the context of
so-called analogue experiments, which promise to provide knowledge
about an experimentally inaccessible target system (e.g., a black
hole) by manipulating another system, the source system (e.g., a
Bose-Einstein condensate). Dardashti, Thebault, and Winsberg
(2017) and Dardashti, Hartmann et al. (2019) have argued that, given
certain conditions, an analogue simulation of one system by another
system can confirm claims about the target system (e.g., that black
holes emit Hawking radiation). See Crowther et al. (forthcoming) for a
critical discussion, and also the entry on
 computer simulations in science.



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



Philosophical debates over idealization have focused on two general
kinds of idealizations: so-called Aristotelian and Galilean
idealizations. Aristotelian idealization amounts to "stripping
away", in our imagination, all properties from a concrete object
that we believe are not relevant to the problem at hand. There is
disagreement on how this is done. Jones (2005) and Godfrey-Smith
(2009) offer an analysis of abstraction in terms of truth: while an
abstraction remains silent about certain features or aspects of the
system, it does not say anything false and still offers a true (albeit
restricted) description. This allows scientists to focus on a limited
set of properties in isolation. An example is a classical-mechanics
model of the planetary system, which describes the position of an
object as a function of time and disregards all other properties of
planets. Cartwright (1989: Ch. 5), Musgrave (1981), who uses the term
"negligibility assumptions", and Maki (1994), who
speaks of the "method of isolation", allow abstractions to
say something false, for instance by neglecting a causally relevant
factor.



Galilean idealizations are ones that involve deliberate distortions:
physicists build models consisting of point masses moving on
frictionless planes; economists assume that agents are omniscient;
biologists study isolated populations; and so on. Using
simplifications of this sort whenever a situation is too difficult to
tackle was characteristic of Galileo's approach to science. For
this 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 of
motion 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 controlled
idealizations, i.e., as ones that allow for de-idealization by
successive removal of the distorting assumptions (McMullin 1985;
Weisberg 2007). Thus construed, Galilean idealizations don't
cover all distortive idealizations. Batterman (2002, 2011) and Rice
(2015, 2019) discuss distortive idealizations that are ineliminable in
that they cannot be removed from the model without dismantling the
model altogether.



What does a model involving distortions tell us about reality? Laymon
(1991) formulated a theory which understands idealizations as ideal
limits: imagine a series of refinements of the actual situation which
approach the postulated limit, and then require that the closer the
properties of a system come to the ideal limit, the closer its
behavior has to come to the behavior of the system at the limit
(monotonicity). If this is the case, then scientists can study the
system at the limit and carry over conclusions from that system to
systems distant from the limit. But these conditions need not always
hold. In fact, it can happen that the limiting system does not
approach the system at the limit. If this happens, we are faced with a
singular limit (Berry 2002). In such cases the system at the limit can
exhibit behavior that is different from the behavior of systems
distant from the limit. Limits of this kind appear in a number of
contexts, most notably in the theory of phase transitions in
statistical mechanics. There is, however, no agreement over the
correct interpretation of such limits. Batterman (2002, 2011) sees
them as indicative of emergent phenomena, while Butterfield (2011a,b)
sees them as compatible with reduction (see also the entries on
 intertheory relations in physics
 and
 scientific reduction).



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



A concept that is closely related to idealization is approximation. In
a broad sense, *A* can be called an approximation of *B* if
*A* is somehow close to *B*. This, however, is too broad
because it makes room for any likeness to qualify as an approximation.
Rueger and Sharp (1998) limit approximations to *quantitative*
closeness, and Portides (2007) frames it as an essentially
mathematical concept. On that notion *A* is an approximation of
*B* iff *A* is close to *B* in a specifiable
mathematical sense, where the relevant sense of "close"
will be given by the context. An example is the approximation of one
curve with another one, which can be achieved by expanding a function
into a power series and only keeping the first two or three terms. In
different situations we approximate an equation with another one by
letting a control parameter tend towards zero (Redhead 1980). This
raises the question of how approximations are different from
idealizations, which can also involve mathematical closeness. Norton
(2012) sees the distinction between the two as referential: an
approximation is an inexact description of the target while an
idealization introduces a secondary system (real or fictitious) which
stands for the target system (while being distinct from it). If we say
that the period of the pendulum on the wall is roughly two seconds,
then this is an approximation; if we reason about the real pendulum by
assuming that the pendulum bob is a point mass and that the string is
massless (i.e., if we assume that the pendulum is a so-called ideal
pendulum), then we use an idealization. Separating idealizations and
approximations in this way does not imply that there cannot be
interesting relations between the two. For instance, an approximation
can be justified by pointing out that it is the mathematical
expression of an acceptable idealization (e.g., when we neglect a
dissipative term in an equation of motion because we make the
idealizing assumption that the system is frictionless).



*Toy models*. Toy models are extremely simplified and strongly
distorted renderings of their targets, and often only represent a
small number of causal or explanatory factors (Hartmann 1995;
Reutlinger et al. 2018; Nguyen forthcoming). Typical examples are the
Lotka-Volterra model in population ecology (Weisberg 2013) and the
Schelling model of segregation in the social sciences (Sugden
2000). Toy models usually
do not perform well in terms of prediction and empirical adequacy, and
they seem to serve other epistemic goals (more on these in
 Section 3).
 This raises the question whether they should be regarded as
representational at all (Luczak 2017).



Some toy models are characterized as "caricatures"
(Gibbard and Varian 1978; Batterman and Rice 2014). Caricature models
isolate a small number of salient characteristics of a system and
distort them into an extreme case. A classic example is
Akerlof's (1970) model of the car market ("the market for
lemons"), which explains the difference in price between new and
used cars solely in terms of asymmetric information, thereby
disregarding all other factors that may influence the prices of cars
(see also Sugden 2000).
However, it is controversial whether such highly idealized models can
still be regarded as informative representations of their target
systems. For a discussion of caricature models, in particular in
economics, see Reiss (2006).



*Minimal models*. Minimal models are closely related to toy
models in that they are also highly simplified. They are so simplified
that some argue that they are non-representational: they lack any
similarity, isomorphism, or resemblance relation to the world
(Batterman and Rice 2014). It has been argued that many economic
models are of this kind (Grune-Yanoff 2009). Minimal economic models are
also unconstrained by natural laws, and do not isolate any real
factors (*ibid*.). And yet, minimal models help us to learn
something about the world in the sense that they function as
surrogates for a real system: scientists can study the model to learn
something about the target. It is, however, controversial whether
minimal models can assist scientists in learning something about the
world if they do not represent anything (Fumagalli 2016). Minimal
models that purportedly lack any similarity or representation are also
used in different parts of physics to explain the macro-scale behavior
of various systems whose micro-scale behavior is extremely diverse
(Batterman and Rice 2014; Rice 2018, 2019; Shech 2018). Typical
examples are the features of phase transitions and the flow of fluids.
Proponents of minimal models argue that what provides an explanation
of the macro-scale behavior of a system in these cases is not a
feature that system and model have in common, but the fact that the
system and the model belong to the same *universality class* (a
class of models that exhibit the same limiting behavior even though
they show very different behavior at finite scales). It is, however,
controversial whether explanations of this kind are possible without
reference to at least some common features (Lange 2015; Reutlinger
2017).



*Phenomenological models*. Phenomenological models have been
defined in different, although related, ways. A common definition
takes them to be models that only represent observable properties of
their targets and refrain from postulating hidden mechanisms and the
like (Bokulich 2011). Another approach, due to McMullin (1968),
defines phenomenological models as models that are independent of
theories. This, however, seems to be too strong. Many phenomenological
models, while failing to be derivable from a theory, incorporate
principles and laws associated with theories. The liquid-drop model of
the atomic nucleus, for instance, portrays the nucleus as a liquid
drop and describes it as having several properties (surface tension
and charge, among others) originating in different theories
(hydrodynamics and electrodynamics, respectively). Certain aspects of
these theories--although usually not the full theories--are
then used to determine both the static and dynamical properties of the
nucleus. Finally, it is tempting to identify phenomenological models
with *models of a phenomenon*. Here, "phenomenon"
is an umbrella term covering all relatively stable and general
features of the world that are interesting from a scientific point of
view. The weakening of sound as a function of the distance to the
source, the decay of alpha particles, the chemical reactions that take
place when a piece of limestone dissolves in an acid, the growth of a
population of rabbits, and the dependence of house prices on the base
rate of the Federal Reserve are phenomena in this sense. For further
discussion, see Bailer-Jones (2009: Ch. 7), Bogen and Woodward (1988),
and the entry on
 theory and observation in science.



*Exploratory models*. Exploratory models are models which are
not proposed in the first place to learn something about a specific
target system or a particular experimentally established phenomenon.
Exploratory models function as the starting point of further
explorations in which the model is modified and refined. Gelfert
(2016) points out that exploratory models can provide
proofs-of-principle and suggest how-possibly explanations (2016: Ch. 4). As an example, Gelfert mentions early models in theoretical
ecology, such as the Lotka-Volterra model of predator-prey
interaction, which mimic the qualitative behavior of speed-up and
slow-down in population growth in an environment with limited
resources (2016: 80). Such models do not give an accurate account of
the behavior of any actual population, but they provide the starting
point 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 a
given theory.



*Models of data.* A model of data (sometimes also "data
model") is a corrected, rectified, regimented, and in many
instances idealized version of the data we gain from immediate
observation, the so-called raw data (Suppes 1962). Characteristically,
one first eliminates errors (e.g., removes points from the record that
are due to faulty observation) and then presents the data in a
"neat" way, for instance by drawing a smooth curve through
a set of points. These two steps are commonly referred to as
"data reduction" and "curve fitting". When we
investigate, for instance, the trajectory of a certain planet, we
first eliminate points that are fallacious from the observation
records and then fit a smooth curve to the remaining ones. Models of
data play a crucial role in confirming theories because it is the
*model* 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. It
requires sophisticated statistical techniques and raises serious
methodological as well as philosophical questions. How do we decide
which points on the record need to be removed? And given a clean set
of data, what curve do we fit to it? The first question has been dealt
with mainly within the context of the philosophy of experiment (see,
for instance, Galison 1997 and Staley 2004). At the heart of the
latter question lies the so-called curve-fitting problem, which is
that the data themselves dictate neither the form of the fitted curve
nor what statistical techniques scientists should use to construct a
curve. The choice and rationalization of statistical techniques is the
subject matter of the philosophy of statistics, and we refer the
reader to the entry
 Philosophy of Statistics
 and to Bandyopadhyay and Forster (2011) for a discussion of these
issues. Further discussions of models of data can be found in
Bailer-Jones (2009: Ch. 7), Brewer and Chinn (1994), Harris (2003),
Hartmann (1995), Laymon (1982), Mayo (1996, 2018), and Suppes
(2007).



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


## 2. Ontology: What Are Models?



What are models? That is, what kind of object are scientists dealing
with when they work with a model? A number of authors have voiced
skepticism that this question has a meaningful answer, because models
do not belong to a distinctive ontological category and anything can
be a model (Callender and Cohen 2006; Giere 2010; Suarez 2004;
Swoyer 1991; Teller 2001). Contessa (2010) replies that this is a
*non sequitur*. Even if, from an ontological point of view,
anything can be a model and the class of things that are referred to
as models contains a heterogeneous collection of different things, it
does not follow that it is either impossible or pointless to develop
an ontology of models. This is because even if not all models are of a
particular ontological kind, one can nevertheless ask to what
ontological kinds the things that are *de facto* used as models
belong. There may be several such kinds and each kind can be analyzed
in its own right. What sort of objects scientists use as models has
important repercussions for how models perform relevant functions such
as representation and explanation, and hence this issue cannot be
dismissed as "just sociology".



The objects that commonly serve as models indeed belong to different
ontological kinds: physical objects, fictional objects, abstract
objects, set-theoretic structures, descriptions, equations, or
combinations of some of these, are frequently referred to as models,
and some models may fall into yet other classes of things. Following
Contessa's advice, the aim then is to develop an ontology for
each of these. Those with an interest in ontology may see this as a
goal in its own right. It pays noting, however, that the question has
reverberations beyond ontology and bears on how one understands the
semantics and the epistemology of models.


### 2.1 Physical objects



Some models are physical objects. Such models are commonly referred to
as "material models". Standard examples of models of this
kind are scale models of objects like bridges and ships (see
 Section 1),
 Watson and Crick's metal model of DNA (Schaffner 1969),
Phillips and Newlyn's hydraulic model of an economy (Morgan and
Boumans 2004), the US Army Corps of Engineers' model of the San
Francisco Bay (Weisberg 2013), Kendrew's plasticine model of
myoglobin (Frigg and Nguyen 2016), and model organisms in the life
sciences (Leonelli and Ankeny 2012; Leonelli 2010; Levy and Currie
2015). All these are material objects that serve as models. Material
models do not give rise to ontological difficulties over and above the
well-known problems in connection with objects that metaphysicians
deal with, for instance concerning the nature of properties, the
identity of objects, parts and wholes, and so on.



However, many models are *not* material models. The Bohr model
of the atom, a frictionless pendulum, or an isolated population, for
instance, are in the scientist's mind rather than in the
laboratory and they do not have to be physically realized and
experimented upon to serve as models. These "non-physical"
models raise serious ontological questions, and how they are best
analyzed is debated controversially. In the remainder of this section
we review some of the suggestions that have attracted attention in the
recent literature on models.


### 2.2 Fictional objects and abstract objects



What has become known as the *fiction view of models* sees
models as akin to the imagined objects of literary fiction--that
is, as akin to fictional characters like Sherlock Holmes or fictional
places like Middle Earth (Godfrey-Smith 2007). So when Bohr introduced
his model of the atom he introduced a fictional object of the same
kind as the object Conan Doyle introduced when he invented Sherlock
Holmes. This view squares well with scientific practice, where
scientists often talk about models as if they were objects and often
take themselves to be describing *imaginary* atoms, populations,
or economies. It also squares well with philosophical views that see
the construction and manipulation of models as essential aspects of
scientific investigation (Morgan 1999), even if models are not
material objects, because these practices seem to be directed toward
*some* kind of object.



What philosophical questions does this move solve? Fictional discourse
and fictional entities face well-known philosophical questions, and
one may well argue that simply likening models to fictions amounts to
explaining *obscurum per obscurius* (for a discussion of these
questions, see the entry on
 fictional entities).
 One way to counter this objection and to motivate the fiction view of
models is to point to the view's heuristic power. In this vein
Frigg (2010b) identifies five specific issues that an ontology of
models has to address and then notes that these issues arise in very
similar ways in the discussion about fiction (the issues are the
identity conditions, property attribution, the semantics of
comparative statements, truth conditions, and the epistemology of
imagined objects). Likening models to fiction then has heuristic value
because there is a rich literature on fiction that offers a number of
solutions to these issues.



Only a small portion of the options available in the extensive
literature on fictions have actually been explored in the context of
scientific models. Contessa (2010) formulates what he calls the
"dualist account", according to which a model is an
abstract object that stands for a possible concrete object. The
Rutherford model of the atom, for instance, is an abstract object that
acts as a stand-in for one of the possible systems that contain an
electron orbiting around a nucleus in a well-defined orbit.
Barberousse and Ludwig (2009) and Frigg (2010b) take a different route
and develop an account of models as fictions based on Walton's
(1990) pretense theory of fiction. According to this view the
sentences of a passage of text introducing a model should be seen as a
prop in a game of make-believe, and the model is the product of an act
of pretense. This is an antirealist position in that it takes talk of
model "objects" to be figures of speech because ultimately
there are no model objects--models only live in scientists'
imaginations. Salis (forthcoming) reformulates this view to become
what she calls the "the new fiction view of models". The
core difference lies in the fact that what is considered as the model
are the model descriptions and their content rather than the
imaginings that they prescribe. This is a realist view of models,
because descriptions exist.



The fiction view is not without critics. Giere (2009), Magnani (2012),
Pincock (2012), Portides (2014), and Teller (2009) reject the fiction
approach and argue, in different ways, that models should not be
regarded as fictions. Weisberg (2013) argues for a middle position
which sees fictions as playing a heuristic role but denies that they
should be regarded as forming part of a scientific model. The common
core of these criticisms is that the fiction view misconstrues the
epistemic standing of models. To call something a fiction, so the
charge goes, is tantamount to saying that it is false, and it is
unjustified to call an entire model a fiction--and thereby claim
that it fails to capture how the world is--just because the model
involves certain false assumptions or fictional elements. In other
words, a representation isn't automatically counted as fiction
just because it has some inaccuracies. Proponents of the fiction view
agree with this point but deny that the notion of fiction should be
analyzed in terms of falsity. What makes a work a fiction is not its
falsity (or some ratio of false to true claims): neither is everything
that is said in a novel untrue (Tolstoy's *War and Peace*
contains many true statements about Napoleon's Franco-Russian
War), nor does every text containing false claims qualify as fiction
(false news reports are just that, they are not fictions). The
defining feature of a fiction is that readers are supposed to
*imagine* the events and characters described, not that they
are false (Frigg 2010a; Salis forthcoming).



Giere (1988) advocated the view that "non-physical" models
are abstract entities. However, there is little agreement on the
nature of abstract objects, and Hale (1988: 86-87) lists no less
than twelve different possible characterizations (for a review of the
available options, see the entry on
 abstract 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 abstract artifacts. This view agrees with the pretense
theory that the content of text that introduces a fictional character
or a model should be understood as occurring in pretense, but at the
same time insists that in producing such descriptions authors create
abstract cultural artifacts that then exist independently of either
the author or the readers. Artifactualism agrees with Platonism that
abstract objects exist, but insists, *contra* Platonism, that
abstract objects are brought into existence through a creative act and
are not eternal. This allows the artifactualist to preserve the
advantages of pretense theory while at the same time holding the
realist view that fictional characters and models actually exist.


### 2.3 Set-theoretic structures



An influential point of view takes models to be set-theoretic
structures. This position can be traced back to Suppes (1960) and is
now, with slight variants, held by most proponents of the so-called
semantic view of theories (for a discussion of this view, see the
entry on
 the structure of scientific theories).
 There are differences between the versions of the semantic view, but
with the exception of Giere (1988) all versions agree that models are
structures of one sort or another (Da Costa and French 2000).



This view of models has been criticized on various grounds. One
pervasive criticism is that many types of models that play an
important role in science are not structures and cannot be
accommodated within the structuralist view of models, which can
neither account for how these models are constructed nor for how they
work in the context of investigation (Cartwright 1999; Downes 1992;
Morrison 1999). Examples for such models are interpretative models and
mediating models, discussed later in
 Section 4.2.
 Another charge held against the set-theoretic approach is that
set-theoretic structures by themselves cannot be representational
models--at least if that requires them to share some structure
with the target--because the ascription of a structure to a
target system which forms part of the physical world relies on a
substantive (non-structural) description of the target, which goes
beyond what the structuralist approach can afford (Nguyen and Frigg
forthcoming).


### 2.4 Descriptions and equations



A time-honored position has it that a model is a stylized description
of a target system. It has been argued that this is what scientists
display in papers and textbooks when they present a model (Achinstein
1968; Black 1962). This view has not been subject to explicit
criticism. However, some of the criticisms that have been marshaled
against the so-called syntactic view of theories equally threaten a
linguistic understanding of models (for a discussion of this view, see the
entry on
 the structure of scientific theories).
 First, a standard criticism of the syntactic view is that by
associating a theory with a particular formulation, the view
misconstrues theory identity because any change in the formulation
results in a new theory (Suppe 2000). A view that associates models
with descriptions would seem to be open to the same criticism. Second,
models have different properties than descriptions: the Newtonian
model of the solar system consists of orbiting spheres, but it makes
no sense to say this about its description. Conversely, descriptions
have properties that models do not have: a description can be written
in English and consist of 517 words, but the same cannot be said of a
model. One way around these difficulties is to associate the model
with the content of a description rather than with the description
itself. For a discussion of a position on models that builds on the
content of a description, see Salis (forthcoming).



A contemporary version of descriptivism is Levy's (2012, 2015)
and Toon's (2012) so-called direct-representation view. This
view shares with the fiction view of models
 (Section 2.2)
 the reliance on Walton's pretense theory, but uses it in a
different way. The main difference is that the views discussed earlier
see modeling as introducing a vehicle of representation, the model,
that is distinct from the target, and they see the problem as
elucidating what kind of thing the model is. On the
direct-representation view there are no models distinct from the
target; there are only model-descriptions and targets, with no models
in-between them. Modeling, on this view, consists in providing an
imaginative description of real things. A model-description prescribes
imaginings about the real system; the ideal pendulum, for instance,
prescribes model-users to imagine the real spring as perfectly elastic
and the bob as a point mass. This approach avoids the above problems
because the identity conditions for models are given by the conditions
for games of make-believe (and not by the syntax of a description) and
property 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 how
models thus understood deal with idealizations. For a discussion of
these points, see Frigg and Nguyen (2016), Poznic (2016), and Salis
(forthcoming).



A closely related approach sees models as equations. This is a version
of the view that models are descriptions, because equations are
syntactic items that describe a mathematical structure. The issues
that this view faces are similar to the ones we have already
encountered: First, one can describe the same situation using
different kinds of coordinates and as a result obtain different
equations but without thereby also obtaining a different model.
Second, the model and the equation have different properties. A
pendulum contains a massless string, but the equation describing its
motion does not; and an equation may be inhomogeneous, but the system
it describes is not. It is an open question whether these issues can
be 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 in
science is that they perform a number of cognitive functions. For
example, models are vehicles for learning about the world. Significant
parts of scientific investigation are carried out on models rather
than on reality itself because by studying a model we can discover
features of, and ascertain facts about, the system the model stands
for: models allow for "surrogative reasoning" (Swoyer
1991). For instance, we study the nature of the hydrogen atom, the
dynamics of a population, or the behavior of a polymer by studying
their respective models. This cognitive function of models has been
widely acknowledged in the literature, and some even suggest that
models give rise to a new style of reasoning, "model-based
reasoning", according to which "inferences are made by
means of creating models and manipulating, adapting, and evaluating
them" (Nersessian 2010: 12; see also Magnani, Nersessian, and
Thagard 1999; Magnani and Nersessian 2002; and Magnani and Casadio
2016).


### 3.1 Learning about models



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



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



Not so with fictional and abstract models. What constraints are there
to the construction of fictional and abstract models, and how do we
manipulate them? A natural response seems to be that we do this by
performing a thought experiment. Different authors (e.g., Brown 1991;
Gendler 2000; Norton 1991; Reiss 2003; Sorensen 1992) have explored
this line of argument, but they have reached very different and often
conflicting conclusions about how thought experiments are performed
and what the status of their outcomes is (for details, see the entry
on
 thought experiments).



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



Computer simulations are also heuristically important. They can
suggest new theories, models, and hypotheses, for example, based on a
systematic exploration of a model's parameter space (Hartmann
1996). But computer simulations also bear methodological perils. For
example, they may provide misleading results because, due to the
discrete nature of the calculations carried out on a digital computer,
they only allow for the exploration of a part of the full parameter
space, and this subspace need not reflect every important feature of
the model. The severity of this problem is somewhat mitigated by the
increasing power of modern computers. But the availability of more
computational power can also have adverse effects: it may encourage
scientists to swiftly come up with increasingly complex but
conceptually premature models, involving poorly understood assumptions
or mechanisms and too many additional adjustable parameters (for a
discussion of a related problem in the social sciences, see Braun and
Saam 2015: Ch. 3). This can lead to an increase in empirical
adequacy--which may be welcome for certain forecasting
tasks--but not necessarily to a better understanding of the
underlying mechanisms. As a result, the use of computer simulations
can change the weight we assign to the various goals of science.
Finally, the availability of computer power may seduce scientists into
making calculations that do not have the degree of trustworthiness one
would expect them to have. This happens, for instance, when computers
are used to propagate probability distributions forward in time, which
can turn out to be misleading (see Frigg et al. 2014). So it is
important not to be carried away by the means that new powerful
computers offer and lose sight of the actual goals of research. For a
discussion of further issues in connection with computer simulations,
we refer the reader to the entry on
 computer 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 is
at this point that the representational function of models becomes
important again: if a model represents, then it can instruct us about
reality because (at least some of) the model's parts or aspects
have corresponding parts or aspects in the world. But if learning is
connected to representation and if there are different kinds of
representations (analogies, idealizations, etc.), then there are also
different kinds of learning. If, for instance, we have a model we take
to be a realistic depiction, the transfer of knowledge from the model
to the target is accomplished in a different manner than when we deal
with an analogue, or a model that involves idealizing assumptions. For
a discussion of the different ways in which the representational
function of models can be exploited to learn about the target, we
refer the reader to the entry
 Scientific Representation.


### 3.3 Explaining with models



Some models explain. But how can they fulfill this function given that
they typically involve idealizations? Do these models explain
*despite* or *because* *of* the idealizations
they involve? Does an explanatory use of models presuppose that they
represent, or can non-representational models also explain? And what
kind of explanation do models provide?



There is a long tradition requesting that the explanans of a
scientific explanation must be true. We find this requirement in the
deductive-nomological model (Hempel 1965) as well as in the more
recent literature. For instance, Strevens (2008: 297) claims that "no causal
account of explanation ... allows nonveridical models to
explain". For further discussions, see also Colombo et al.
(2015).



Authors working in this tradition deny that idealizations make a
positive contribution to explanation and explore how models can
explain despite being idealized. McMullin (1968, 1985) argues that a
causal explanation based on an idealized model leaves out only
features which are irrelevant for the respective explanatory task (see
also Salmon 1984 and
Piccinini and Craver 2011 for a discussion of mechanism sketches).
Friedman (1974) argues
that a more realistic (and hence less idealized) model explains better
on the unification account. The idea is that idealizations can (at
least in principle) be de-idealized (for a critical discussion of this
claim in the context of the debate about scientific explanations, see
Batterman 2002; Bokulich 2011; Morrison 2005, 2009; Jebeile and
Kennedy 2015; and Rice 2015). Strevens (2008) argues that an explanatory
causal model has to provide an accurate representation of the relevant
causal relationships or processes which the model shares with the
target system. The idealized assumptions of a model do not make a
difference for the phenomenon under consideration and are therefore
explanatorily irrelevant. In contrast, both Potochnik (2017) and Rice
(2015) argue that models that explain can directly *distort*
many difference-making causes.



According to Woodward's (2003) theory, models are tools to find
out about the causal relations that hold between certain facts or
processes, and it is these relations that do the explanatory work.
More specifically, explanations provide information about patterns of
counterfactual dependence between the explanans and the explanandum
which 



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



Accounts of causal explanation have also led to various claims about
how idealized models can provide explanations, exploring to what
extent idealization allows for the misrepresentation of irrelevant
causal 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 systems continues to play the essential role in showing how
idealized models can be explanatory.



But is it really the truth of the explanans that makes the model
explanatory? Other authors pursue a more radical line and argue that
false models explain not only *despite* their falsity, but in
fact *because* of their falsity. Cartwright (1983: 44)
maintains that "the truth doesn't explain much". In
her so-called "simulacrum account of explanation", she
suggests that we explain a phenomenon by constructing a model that
fits 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 us
with the question of what notion of explanation is at work (see also
Elgin and Sober 2002) and of what explanatory function idealizations
play 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 the
details that characterize specific systems do not matter for the
explanation. Bokulich (2008, 2009, 2011, 2012) pursues a similar line
of reasoning and sees the explanatory power of models as being closely
related to their fictional nature. Bokulich (2009) and Kennedy (2012)
present non-representational accounts of model explanation (see also
Jebeile and Kennedy 2015). Reiss (2012) and Woody (2004) provide general discussions of the relationship between
representation and explanation.


### 3.4 Understanding with models



Many authors have pointed out that understanding is one of the central
goals of science (see, for instance, de Regt 2017; Elgin 2017; Khalifa
2017; Potochnik 2017). In some cases, we want to understand a certain
phenomenon (e.g., why the sky is blue); in other cases, we want to
understand a specific scientific theory (e.g., quantum mechanics) that
accounts for a phenomenon in question. Sometimes we gain understanding
of a phenomenon by understanding the corresponding theory or model.
For instance, Maxwell's theory of electromagnetism helps us
understand why the sky is blue. It is, however, controversial whether
understanding a phenomenon *always* presupposes an
understanding 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 particular
importance here (de Regt et al. 2009; Morrison 2009; Potochnik 2017;
Rice 2016). This insight
can be traced back at least to Lord Kelvin who, in his famous 1884
*Baltimore Lectures on Molecular Dynamics and the Wave Theory of
Light*, maintained that "the test of 'Do we or do we
not understand a particular subject in physics?' is 'Can
we make a mechanical model of it?'" (Kelvin 1884 [1987: 111]; see also Bailer-Jones 2009: Ch. 2; and de Regt 2017: Ch. 6). 



But why do models play such a crucial role in the understanding of a
subject matter? Elgin (2017) argues that this is not despite, but
because, of models being literally false. She views false models as
"felicitous falsehoods" that occupy center stage in the
epistemology of science, and mentions the ideal-gas model in
statistical mechanics and the Hardy-Weinberg model in genetics as
examples for literally false models that are central to their
respective disciplines. Understanding is holistic and it concerns a
topic, a discipline, or a subject matter, rather than isolated claims
or facts. Gaining understanding of a context means to have 



> 
> 
> an epistemic commitment to a comprehensive, systematically linked body
> of information that is grounded in fact, is duly responsive to reasons
> or evidence, and enables nontrivial inference, argument, and perhaps
> action regarding the topic the information pertains to (Elgin 2017:
> 44) 
> 
> 
> 



and models can play a crucial role in the pursuit of these epistemic
commitments. For a discussion of Elgin's account of models and
understanding, see Baumberger and Brun (2017) and Frigg and Nguyen
(forthcoming).



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



> 
> 
> an individual has scientific understanding of a phenomenon just in
> case 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 of
epistemic access an individual scientist has to an explanation. For
Strevens this aspect is "grasping", while for de Regt
(2017) it is "intelligibility". It is important to note
that both Strevens and de Regt hold that such "subjective"
aspects are a worthy topic for investigations in the philosophy of
science. This contrasts with the traditional view (see, e.g., Hempel
1965) that delegates them to the realm of psychology. See Friedman
(1974), Trout (2002), and
Reutlinger et al. (2018) for further discussions of understanding.


### 3.5 Other cognitive functions



Besides the functions already mentioned, it has been emphasized
variously that models perform a number of other cognitive functions.
Knuuttila (2005, 2011) argues that the epistemic value of models is
not limited to their representational function, and develops an account
that views models as epistemic artifacts which allow us to gather
knowledge in diverse ways. Nersessian (1999, 2010) stresses the role
of analogue models in concept-formation and other cognitive processes.
Hartmann (1995) and Leplin (1980) discuss models as tools for theory
construction and emphasize their heuristic and pedagogical value.
Epstein (2008) lists a number of specific functions of models in the
social sciences. Peschard (2011) investigates the way in which models
may be used to construct other models and generate new target systems.
And Isaac (2013) discusses non-explanatory uses of models which do not
rely on their representational capacities.


## 4. Models and Theory



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


### 4.1 Models as subsidiaries to theory



To discuss the relation between models and theories in science it is
helpful to briefly recapitulate the notions of a model and of a theory
in logic. A *theory* is taken to be a (usually deductively
closed) set of sentences in a formal language. A *model* is a
structure (in the sense introduced in
 Section 2.3)
 that makes all sentences of a theory true when its symbols are
interpreted as referring to objects, relations, or functions of a
structure. The structure is a *model of* the theory in the
sense that it is correctly described by the theory (see Bell and
Machover 1977 or Hodges 1997 for details). Logical models are
sometimes also referred to as "models of theory" to
indicate that they are interpretations of an abstract formal
system.



Models in science sometimes carry over from logic the idea of being
the interpretation of an abstract calculus (Hesse 1967). This is
salient in physics, where general laws--such as Newton's
equation of motion--lie at the heart of a theory. These laws are
applied to a particular system--e.g., a pendulum--by
choosing a special force function, making assumptions about the mass
distribution of the pendulum etc. The resulting model then is an
interpretation (or realization) of the general law.



It is important to keep the notions of a logical and a
representational model separate (Thomson-Jones 2006): these are
distinct concepts. Something can be a logical model without being a
representational model, and *vice versa*. This, however, does
not mean that something cannot be a model in both senses at once. In
fact, as Hesse (1967) points out, many models in science are both
logical and representational models. Newton's model of planetary
motion is a case in point: the model, consisting of two homogeneous
perfect spheres located in otherwise empty space that attract each
other gravitationally, is simultaneously a logical model (because it
makes the axioms of Newtonian mechanics true when they are interpreted
as referring to the model) and a representational model (because it
represents the real sun and earth).



There are two main conceptions of scientific theories, the so-called
syntactic view of theories and the so-called semantic view of theories
(see the entry on
 the structure of scientific theories).
 On both conceptions models play a subsidiary role to theories, albeit
in very different ways. The syntactic view of theories (see entry
section on
 the syntactic view)
 retains the logical notions of a model and a theory. It construes a
theory as a set of sentences in an axiomatized logical system, and a
model 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 theory
of gases and reinterpret the terms of this calculus in a way that
makes them refer to billiard balls, the billiard balls are a model of
the kinetic theory of gases in the sense that all sentences of the
theory come out true. The model is meant to be something that we are
familiar with, and it serves the purpose of making an abstract formal
calculus more palpable. A given theory can have different models, and
which model we choose depends both on our aims and our background
knowledge. Proponents of the syntactic view disagree about the
importance of models. Carnap and Hempel thought that models only serve
a pedagogic or aesthetic purpose and are ultimately dispensable
because all relevant information is contained in the theory (Carnap
1938; Hempel 1965; see also Bailer-Jones 1999). Nagel (1961) and
Braithwaite (1953), on the other hand, emphasize the
*heuristic* role of models, and Schaffner (1969) submits that
theoretical terms get at least part of their meaning from models.



The semantic view of theories (see entry section on
 the semantic view)
 dispenses with sentences in an axiomatized logical system and
construes a theory as a family of models. On this view, a theory
literally is a class, cluster, or family of models--models are the
building blocks of which scientific theories are made up. Different
versions of the semantic view work with different notions of a model,
but, as noted in
 Section 2.3,
 in the semantic view models are mostly construed as set-theoretic
structures. For a discussion of the different options, we refer the
reader to the relevant entry in this encyclopedia (linked at the
beginning of this paragraph).


### 4.2 Models as independent from theories



In both the syntactic and the semantic view of theories models are
seen as subordinate to theory and as playing no role outside the
context of a theory. This vision of models has been challenged in a
number of ways, with authors pointing out that models enjoy various
degrees of freedom from theory and function autonomously in many
contexts. Independence can take many forms, and large parts of the
literature on models are concerned with investigating various forms of
independence.



*Models as completely independent of theory*. The most radical
departure from a theory-centered analysis of models is the realization
that there are models that are completely independent from any theory.
An example of such a model is the Lotka-Volterra model. The model
describes the interaction of two populations: a population of
predators and one of prey animals (Weisberg 2013). The model was
constructed using only relatively commonsensical assumptions about
predators and prey and the mathematics of differential equations.
There was no appeal to a theory of predator-prey interactions or a
theory of population growth, and the model is independent of theories
about its subject matter. If a model is constructed in a domain where
no theory is available, then the model is sometimes referred to as a
"substitute model" (Groenewold 1961), because the model
substitutes a theory.



*Models as a means to explore theory*. Models can also be used
to explore theories (Morgan and Morrison 1999). An obvious way in
which this can happen is when a model is a logical model of a theory
(see
 Section 4.1).
 A logical model is a set of objects and properties that make a formal
sentence true, and so one can see in the model how the axioms of the
theory play out in a particular setting and what kinds of behavior
they dictate. But not all models that are used to explore theories are
logical models, and models can represent features of theories in other
ways. As an example, consider chaos theory. The equations of
non-linear systems, such as those describing the three-body
problem, have solutions that are too complex to study with
paper-and-pencil methods, and even computer simulations are limited in
various ways. Abstract considerations about the qualitative behavior
of solutions show that there is a mechanism that has been dubbed
"stretching and folding" (see the entry
 Chaos).
 To obtain an idea of the complexity of the dynamics exhibiting
stretching and folding, Smale proposed to study a simple model of the
flow--now known as the "horseshoe map" (Tabor 1989)--which
provides important insights into the nature of stretching and folding.
Other examples of models of that kind are the Kac ring model that is
used to study equilibrium properties of systems in statistical
mechanics (Lavis 2008)
and Norton's dome in Newtonian mechanics (Norton 2003).



*Models as complements of theories*. A theory may be
incompletely specified in the sense that it only imposes certain
general constraints but remains silent about the details of concrete
situations, which are provided by a model (Redhead 1980). A special
case of this situation is when a qualitative theory is known and the
model introduces quantitative measures (Apostel 1961). Redhead's
example of a theory that is underdetermined in this way is axiomatic
quantum field theory, which only imposes certain general constraints
on quantum fields but does not provide an account of particular
fields. Harre (2004) notes that models can complement theories by providing
mechanisms for processes that are left unspecified in the theory even
though they are responsible for bringing about the observed
phenomena.



Theories may be too complicated to handle. In such cases a model can
complement a theory by providing a simplified version of the
theoretical scenario that allows for a solution. Quantum
chromodynamics, for instance, cannot easily be used to investigate the
physics of an atomic nucleus even though it is the relevant
fundamental theory. To get around this difficulty, physicists
construct tractable phenomenological models (such as the MIT bag
model) which effectively describe the relevant degrees of freedom of
the system under consideration (Hartmann 1999, 2001). The advantage of
these models is that they yield results where theories remain silent.
Their drawback is that it is often not clear how to understand the
relationship between the model and the theory, as the two are, strictly
speaking, contradictory.



*Models as preliminary theories*. The notion of a model as a
substitute for a theory is closely related to the notion of a
*developmental model*. This term was coined by Leplin (1980),
who pointed out how useful models were in the development of early
quantum theory, and it is now used as an umbrella notion covering
cases in which models are some sort of a preliminary exercise to
theory.



Also closely related is the notion of a *probing model* (or
"study model"). Models of this kind do not perform a
representational function and are not expected to instruct us about
anything beyond the model itself. The purpose of these models is to
test new theoretical tools that are used later on to build
representational models. In field theory, for instance, the so-called
ph4-model was studied extensively, not because it was
believed to represent anything real, but because it served several
heuristic functions: the simplicity of the ph4-model
allowed physicists to "get a feeling" for what quantum
field theories are like and to extract some general features that this
simple model shared with more complicated ones. Physicists could study
complicated techniques such as renormalization in a simple setting,
and it was possible to get acquainted with important
mechanisms--in this case symmetry-breaking--that could later
be used in different contexts (Hartmann 1995). This is true not only
for physics. As Wimsatt (1987, 2007) points out, a false model in
genetics can perform many useful functions, among them the following:
the false model can help answering questions about more realistic
models, provide an arena for answering questions about properties of
more complex models, "factor out" phenomena that would not
otherwise be seen, serve as a limiting case of a more general model
(or two false models may define the extremes of a continuum of cases
on which the real case is supposed to lie), or lead to the
identification of relevant variables and the estimation of their
values.



*Interpretative models*. Cartwright (1983, 1999) argues that
models do not only aid the application of theories that are somehow
incomplete; she claims that models are also involved *whenever*
a theory with an overarching mathematical structure is applied. The
main theories in physics--classical mechanics, electrodynamics,
quantum mechanics, and so on--fall into this category. Theories of
that kind are formulated in terms of abstract concepts that need to be
concretized for the theory to provide a description of the target
system, and concretizing the relevant concepts, idealized objects and
processes are introduced. For instance, when applying classical
mechanics, the abstract concept of force has to be replaced with a
concrete force such as gravity. To obtain tractable equations, this
procedure has to be applied to a simplified scenario, for instance
that of two perfectly spherical and homogeneous planets in otherwise
empty space, rather than to reality in its full complexity. The result
is an *interpretative model*, which grounds the application of
mathematical theories to real-world targets. Such models are
independent from theory in that the theory does not determine their
form, and yet they are necessary for the application of the theory to
a concrete problem.



*Models as mediators*. The relation between models and theories
can be complicated and disorderly. The contributors to a programmatic
collection of essays edited by Morgan and Morrison (1999) rally around
the idea that models are instruments that mediate between theories and
the world. Models are "autonomous agents" in that they are
independent from both theories and their target systems, and it is
this independence that allows them to mediate between the two.
Theories do not provide us with algorithms for the construction of a
model; they are not "vending machines" into which one can
insert a problem and a model pops out (Cartwright 1999). The
construction of a model often requires detailed knowledge about
materials, approximation schemes, and the setup, and these are not
provided by the corresponding theory. Furthermore, the inner workings
of a model are often driven by a number of different theories working
cooperatively. 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 one
theory, but the voices of different theories when put to use in chorus
with each other in one model.



In complex cases like the study of a laser system or the global
climate, models and theories can get so entangled that it becomes
unclear where a line between the two should be drawn: where does the
model end and the theory begin? This is not only a problem for
philosophical analysis; it also arises in scientific practice.
Bailer-Jones (2002) interviewed a group of physicists about their
understanding of models and their relation to theories, and reports
widely diverging views: (i) there is no substantive difference between
model and theory; (ii) models *become* theories when their
degree of confirmation increases; (iii) models contain simplifications
and omissions, while theories are accurate and complete; (iv) theories
are more general than models, and modeling is about applying general
theories to specific cases. The first suggestion seems to be too
radical to do justice to many aspects of practice, where a distinction
between models and theories is clearly made. The second view is in
line with common parlance, where the terms "model" and
"theory" are sometimes used to express someone's
attitude towards a particular hypothesis. The phrase "it's
just a model" indicates that the hypothesis at stake is asserted
only tentatively or is even known to be false, while something is
awarded the label "theory" if it has acquired some degree
of general acceptance. However, this use of "model" is
different from the uses we have seen in
 Sections 1 to 3
 and is therefore of no use if we aim to understand the relation
between scientific models and theories (and, incidentally, one can
equally dismiss speculative claims as being "just a
theory"). The third proposal is correct in associating models
with idealizations and simplifications, but it overshoots by
restricting this to models; in fact, also theories can contain
idealizations and simplifications. The fourth view seems closely
aligned with interpretative models and the idea that models are
mediators, but being more general is a gradual notion and hence does
not provide a clear-cut criterion to distinguish between theories and
models.


## 5. Models and Other Debates in the Philosophy of Science



The debate over scientific models has important repercussions for
other issues in the philosophy of science (for a historical account of
the 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 scientific knowledge. Once models
are acknowledged as occupying an important place in the edifice of
science, these issues have to be reconsidered with a focus on models.
The question is whether, and if so how, discussions of these issues
change when we shift focus from theories to models. Up to now, no
comprehensive model-based account of any of these issues has emerged,
but models have left important traces in the discussions of these
topics.


### 5.1 Models, realism, and laws of nature



As we have seen in
 Section 1,
 models typically provide a distorted representation of their targets.
If one sees science as primarily model-based, this could be taken to
suggest an antirealist interpretation of science. Realists, however,
deny that the presence of idealizations in models renders a realist
approach to science impossible and point out that a good model, while
not literally true, is usually at least approximately true, and/or
that it can be improved by de-idealization (Laymon 1985; McMullin
1985; Nowak 1979; Brzezinski and Nowak 1992).



Apart from the usual worries about the elusiveness of the notion of
approximate truth (for a discussion, see the entry on
 truthlikeness),
 antirealists have taken issue with this reply for two (related)
reasons. First, as Cartwright (1989) points out, there is no reason to
assume that one can always improve a model by adding de-idealizing
corrections. Second, it seems that de-idealization is not in
accordance with scientific practice because it is unusual that
scientists invest work in repeatedly de-idealizing an existing model.
Rather, they shift to a different modeling framework once the
adjustments to be made get too involved (Hartmann 1998). The various
models of the atomic nucleus are a case in point: once it was realized
that shell effects are important to understand various subatomic
phenomena, the (collective) liquid-drop model was put aside and the
(single-particle) shell model was developed to account for the
corresponding findings. A further difficulty with de-idealization is
that most idealizations are not "controlled". For example,
it is not clear in what way one could de-idealize the MIT bag model to
eventually arrive at quantum chromodynamics, the supposedly correct
underlying theory.



A further antirealist argument, the "incompatible-models
argument", takes as its starting point the observation that
scientists often successfully use several incompatible models of
*one and the same* target system for predictive purposes
(Morrison 2000). These models seemingly contradict each other, as they
ascribe different properties to the same target system. In nuclear
physics, for instance, the liquid-drop model explores the analogy of
the atomic nucleus with a (charged) fluid drop, while the shell model
describes nuclear properties in terms of the properties of protons and
neutrons, the constituents of an atomic nucleus. This practice appears
to cause a problem for scientific realism: Realists typically hold
that there is a close connection between the predictive success of a
theory and its being at least approximately true. But if several
models of the same system are predictively successful and if these
models are mutually inconsistent, then it is difficult to maintain
that they are all approximately true.



Realists can react to this argument in various ways. First, they can
challenge the claim that the models in question are indeed
predictively successful. If the models are not good predictors, then
the argument is blocked. Second, they can defend a version of
"perspectival realism" (Giere 2006; Massimi 2017; Rueger
2005). Proponents of this position (which is sometimes also called
"perspectivism") situate it somewhere between
"standard" scientific realism and antirealism, and where
exactly the right middle position lies is the subject matter of active
debate (Massimi 2018a,b; Saatsi 2016; Teller 2018; and the
contributions to Massimi and McCoy 2019). Third, realists can deny
that there is a problem in the first place, because scientific models,
which are always idealized and therefore strictly speaking false, are
just the wrong vehicle to make a point about realism (which should be
discussed in terms of theories).



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



One possible response to these questions is to argue that laws of
nature govern entities and processes in a *model* rather than
in the world. Fundamental laws, on this approach, do not state facts
about the *world* but hold true of entities and processes in
the model. This view has been advocated in different variants:
Cartwright (1983) argues that all laws are *ceteris paribus*
laws. Cartwright (1999) makes use of "capacities" (which
she 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 of
> regular behavior that we represent in our scientific laws. (1999: 50;
> see also the entry on
>  ceteris 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 as
open-ended statements that can be filled in various ways in the
process of building more specific scientific models. Similar positions
have also been defended by Teller (2001) and van Fraassen (1989).


### 5.2 Models and reductionism



The multiple-models problem mentioned in
 Section 5.1
 also raises the question of how different models are related.
Evidently, multiple models for the same target system do not generally
stand in a deductive relationship, as they often contradict each
other. Some (Cartwright 1999; Hacking 1983) have suggested a picture
of science according to which there are no systematic relations that
hold between different models. Some models are tied together because
they represent the same target system, but this does not imply that
they enter into any further relationships (deductive or otherwise). We
are confronted with a patchwork of models, all of which hold
*ceteris paribus* in their specific domains of
applicability.



Some argue that this picture is at least partially incorrect because
there are various interesting relations that hold between different
models or theories. These relations range from thoroughgoing reductive
relations (Scheibe 1997, 1999, 2001: esp. Chs. V.23 and V.24) and
controlled approximations over singular limit relations (Batterman
2001 [2016]) to structural relations (Gahde 1997) and rather
loose relations called "stories" (Hartmann 1999; see also
Bokulich 2003; Teller 2002; and the essays collected in Part III of
Hartmann et al. 2008). These suggestions have been made on the basis
of case studies, and it remains to be seen whether a more general
account of these relations can be given and whether a deeper
justification for them can be provided, for instance, within a
Bayesian framework (first steps towards a Bayesian understanding of
reductive relations can be found in Dizadji-Bahmani et al. 2011;
Liefke and Hartmann 2018; and Tesic 2019).



Models also figure in the debate about reduction and emergence in
physics. Here, some authors argue that the modern approach to
renormalization challenges Nagel's (1961) model of reduction or
the broader doctrine of reductions (for a critical discussion, see,
for instance, Batterman 2002, 2010, 2011; Morrison 2012; and Saatsi and Reutlinger 2018).
Dizadji-Bahmani et al. (2010) provide a defense of the Nagel-Schaffner
model of reduction, and Butterfield (2011a,b, 2014) argues that
renormalization is consistent with Nagelian reduction. Palacios (2019)
shows that phase transitions are compatible with reductionism, and
Hartmann (2001) argues that the effective-field-theories research
program is consistent with reductionism (see also Bain 2013 and
Franklin forthcoming). Rosaler (2015) argues for a "local"
form of reduction which sees the fundamental relation of reduction
holding between models, not theories, which is, however, compatible
with the Nagel-Schaffner model of reduction. See also the entries on
 intertheory relations in physics
 and
 scientific reduction.



In the social sciences, agent-based models (ABMs) are increasingly
used (Klein et al. 2018). These models show how surprisingly complex
behavioral patterns at the macro-scale can emerge from a small number
of simple behavioral rules for the individual agents and their
interactions. This raises questions similar to the questions mentioned
above about reduction and emergence in physics, but so far one only
finds scattered remarks about reduction in the literature. See
Weisberg and Muldoon (2009) and Zollman (2007) for the application of
ABMs to the epistemology and the social structure of science, and
Colyvan (2013) for a discussion of methodological questions raised by
normative models in general.