Models

The word "model" is highly ambiguous, and there is no uniform
terminology used by either scientists or philosophers. Here, a model
is considered to be a representation of some object, behavior, or
system that one wants to understand. This article presents the most
common type of models found in science as well as the different
relations—traditionally called "analogies"—between models and between
a given model and its subject. Although once considered merely
heuristic devices, they are now seen as indispensable to modern
science. There are many different types of models used across the
scientific disciplines, although there is no uniform terminology to
classify them. The most familiar are physical models such as scale
replicas of bridges or airplanes. These, like all models, are used
because of their "analogies" to the subjects of the models. A scale
model airplane has a structural similarity or "material analogy" to
the full scale version. This correspondence allows engineers to infer
dynamic properties of the airplane based on wind tunnel experiments on
the replica. Physical models also include abstract representations
which often include idealizations such as frictionless planes and
point masses. Another, but completely different type of model, is
constituted by sets of equations. These mathematical models were not
always deemed legitimate models by philosophers. Model-to-subject and
model-to-model relations are described using several different types
of analogies: positive, negative, neutral, material, and formal.

Like unobservable entities, models have been the subject of debate
between scientific realists and antirealists. One's position often
depends on what one considers the truth-bearers in science to be.
Those who take fundamental laws and/or theories to be true believe
that models are true in inverse proportion to the degree of
idealization used. Highly idealized models would therefore be (in some
sense) less true. Others take models to be true only insofar as they
describe the behavior of empirically observable systems. This
empiricism leads some to believe that models built from the bottom-up
are realistic, while those derived in a top-down manner from abstract
laws are not.

Models also play a key role in the semantic view of theories. What
counts as a model on this approach, however, is more closely related
to the sense of models in mathematical logic than in science itself.

1. Models in Science

The word "model" is highly ambiguous, and there is no uniform
terminology used by either scientists or philosophers. This article
presents the most common type of models found in science as well as
the different relations—traditionally called "analogies"—between
models and between a given model and its subject. For most of the 20th
century, the use of models in science was a neglected topic in
philosophy. Far more attention was given to the nature of scientific
theories and laws. Except for a few philosophers in the 1960's, Mary
Hesse in particular, most did not think the topic was particularly
important. The philosophically interesting parts of science were
thought to lie elsewhere. As a result, few articles on models were
published in twenty-five years following Hesse's (1966). [These
include (Redhead, 1980) and (Wimsatt, 1987), and parts of (Bunge,
1973) and (Cartwright, 1983.] The situation is now quite different. As
philosophers of science have come to pay greater attention to actual
scientific practice, the use of models has become an import area of
philosophical analysis.
2. Physical Models

One familiar type of model is the physical model: a material,
pictorial, or analogical representation of (at least some part of) an
actual system. "Physical" here is not meant to convey an ontological
claim. As we shall see, some physical models are material objects;
others are not. Hesse classifies many of these as either replicas or
analogue models. Examples of the former are scale models used in wind
tunnel experiments. There is what she calls a "material analogy"
between the model and its subject, that is, a pretheoretic similarity
in how their observable properties are related. Replicas are often
used when the laws governing the subject of the model are either
unknown or too computationally complex to derive predictions. When a
material analogy is present, one assumes that a "formal analogy" also
exists between the subject and the model. In a formal analogy, the
same laws govern the relevant parts of both the subject and model.

Analogue models, in contrast, have a formal analogy with the subject
of the model but no material analogy. In other words, the same laws
govern both the subject and the model, although the two are physically
quite different. For example, ping-pong balls blowing around in a box
(like those used in some state lotteries) constitute an analogue model
for an ideal gas. Some analogue models were important before the age
of digital computers when simple electric circuits were used as
analogues of mechanical systems. Consider a mass M on a frictionless
plane that is subject to a time varying force f(t) (Figure 1). This
system can be simulated by a circuit with a capacitor C and a time
varying voltage source v(t). The voltage across C at time t
corresponds to the velocity of M.

Figure 1: Analogue Machine

Today engineers and physicists are more familiar with simplifying
models. These are constructed by abstracting away properties and
relations that exist in the subject. Here we find the usual zoo of
physical idealizations: frictionless planes, perfectly elastic bodies,
point masses, etc. Consider a textbook mass-spring system with only
one degree of freedom (that is, the spring oscillates perfectly along
one dimension) shown in Figure 2. This particular system is physically
possible, but nonactual. Real springs always wobble just a bit. If by
chance a spring did oscillate in one dimension for some time, the
event would be unlikely but would not violate any physical laws.
Frictionless planes, on the other hand, are nonphysical rather than
merely nonactual.

Figure 2: Physical Water Drop Model

Simplifying models provide a context for Hesse's other relations known
as positive, negative, and neutral analogies. Positive analogies are
the ways in which the subject and model are alike—the properties and
relations they share. Negative analogies occur when there is a
mismatch between the two. The idealizations mentioned in the previous
paragraph are negatively analogous to their real-world subjects. In a
scale-model airplane (a replica), the length of the wing relative to
the length of the tail is a positively analogous since the ratio is
the same in the subject and the model. The wood used to make the model
is negatively analogous since the real airplane would use different
materials. Neutral analogies are relations that are in fact either
positive or negative, but it is not yet known which. The number of
neutral analogies is inversely related to our knowledge of the model
and its subject. One uses a physical model with strong, positive
analogies in order to probe its neutral analogies for more
information. Ideally, all neutral analogies will be sorted into either
positive or negative. The early success of the Bohr model of the atom
showed that it had positive analogies to real hydrogen atoms. In
Hesse's terms, the neutral analogies proved to be negative when the
model was applied to atoms with more than one electron.

The use of "analogy" in this regard has declined somewhat in recent
years. "Idealization" has replaced "negative analogy" when these
simplifications are built into physical models from the start. The
degree to which a model has positive analogies is more typically
described by how "realistic" the model is. One might also use the
notion of "approximate truth"—a term long recognized as more
suggestive than precise. The rough idea is that more realistic
models—those with stronger positive analogies—contain more truth than
others. "Negative analogy" contains an ambiguity. Some are used at the
beginning of the model-building process. The modeler recognizes the
false properties for what they are and uses them for a specific
purpose—usually to simplify the mathematics. Other negative analogies,
known as "artifacts," are unintended consequences of idealizations,
data collection, research methods, and limitations of the medium used
to construct the model. Some artifacts are benign and obvious.
Consider the wooden models of molecules used in high school chemistry
classes. Three balls held together by sticks can represent a water
molecule, but the color of the balls is an artifact. (As the early
moderns were fond of pointing out, atoms are colorless.) Other
artifacts are produced by measuring devices. It is impossible, for
example, to fully shield an oscilloscope from the periodic signal
produced by its AC current source. This produces a periodic component
in the output signal not present in the source itself.

The heavy emphasis here on models in the physical sciences has more to
do with the interests of philosophers than scientific practice.
Physical models are used throughout the sciences, from immunoglobulin
models of allergic reactions to macroeconomic models of the business
cycle.
3. Mathematical Models

Philosophers have generally taken physical models as paradigm cases of
scientific models. In many branches of science, however, mathematical
models play a far more important role. There are many examples,
especially in dynamics. Equation (1) below is an ordinary differential
equation representing the motion of a frictionless pendulum. [θ is the
angle of the string from vertical, l is the length of the string, and
g is the acceleration due to gravity. The two dots in the first term
stand for the second derivative with respect to time.] Even when sets
of equations have clearly been used "to model" some behavior of a
system, philosophers were often unwilling to take these as legitimate
models. The difference is driven in part by greater familiarity with
models in mathematical logic. In the logician's realm, a model
satisfies a set of axioms; the axioms themselves are not models. To
philosophers, equations look like axioms. Referring to a set of
equations as "a model" then sounds like a category mistake.

(1)

This attitude was eroded in part by the central role mathematical
models played in the development of chaos theory. The 1980s saw a
deluge of scientific articles with equations governing nonlinear
systems as well as the state spaces that represented their evolution
over time (see section 4). Physical models, on the other hand, were
often bypassed altogether. This made it far more difficult to dismiss
"mathematical model" as a scientist's misnomer. It soon became
apparent that all of the issues regarding idealizations, confirmation,
and construction of physical models had mathematical counterparts.

Consider the physical model of the electric circuit in Figure 1. A
common idealization is to stipulate that the circuit has no
resistance. When we look to the associated differential equations—a
mathematical model—there is a corresponding simplification, in this
case the elimination of an algebraic term that represented the
resistance of the wire. Unlike this example, simplification is often
more than a mere convenience. The governing equations for many types
of phenomena are intractable as they stand. Simplifications are needed
to bridge the computational gap between the laws and phenomena they
describe. In the old (pre-1926) quantum theory, for example, it was
common to run across a Hamiltonian (an important type of function in
physics that expresses the total energy of the system) that blocked
the usual mathematical techniques—for example, separation of
variables. Instead, a perturbation parameter λ was used to convert the
problematic Hamiltonian into a power series such as in equation (2)
below. [I, θ are classical action-angle variables. See any text on
classical mechanics for more on this method.] Once in this form, one
may generate an approximate solution for to an arbitrary degree of
precision by keeping a finite number of terms and discarding the rest.
This is sometimes called a "mediating mathematical model" (Morton
1993) since it operates, in a sense, between the intractable
Hamiltonian and the phenomenon it is thought to describe.

(2)
4. State Spaces

State spaces have received scant attention in the philosophical
literature until recently. They are often used in tandem with a
mathematical model as a means for representing the possible states of
a system and its evolution. The "system" is often a physical model,
but might also be a real-world phenomenon essentially free of
idealizations. Figure 3 is the state space associate with equation
(1), the mathematical model for an ideal (frictionless) pendulum.
Since θ represents the angle of the string, a,b correspond to the two
highest points of deflection. represents velocity. [The coefficient
.] Hence c,d are the points at which the pendulum is moving the
fastest.

Figure 3: State Space for Ideal Pendulum

State spaces take a variety of forms. Quantum mechanics uses a Hilbert
space to represent the state governed by Schrödinger's equation. The
space itself might have an infinite number of dimensions with a vector
representing an individual state. The ordinary differential equations
used in dynamics require many-dimensional phase spaces. Points
represent the system states in these (usually Euclidean) spaces. As
the state evolves over time, it carves a trajectory through the space.
Every point belongs to some possible trajectory that represents the
system's actual or possible evolution. A phase space together with a
set of trajectories forms a phase portrait (Figure 4). Since the full
phase portrait cannot be captured in a diagram, only a handful of
possible trajectories are shown in textbook illustrations. If the
system allows for dissipation (for example friction), attractors can
develop in the associated phase portrait. As the name implies, an
attractor is a set of points toward which neighboring trajectories
flow, though the points themselves possess no actual attractive force.
The center of Figure 4a, known as a point attractor, might represent a
marble coming to rest at the bottom of a bowl. Simple periodic motion,
like a clock pendulum, produces limit cycles, attracting sets forming
closed curves in phase space (Figure 4b).

Figure 4: Sample Phase Portraits

Let us consider a very simple system—a leaky faucet—that illustrates
the use of each type of model mentioned. Researchers at the University
of California, Santa Cruz, believed that the time between drops does
not change randomly over time, but instead has an underlying dynamical
structure (Martien 1985). In other words, one drip interval causally
influences the next. In order to explore this hypothesis, a simplified
physical model for a drop of water was developed (the one shown above
in Figure 2). They believed that a water drop is roughly like a
one-dimensional, oscillating mass on a spring. Part of the mass
detaches when the spring extends to a critical point. The amount of
mass that detaches depends on the velocity of the block when it
reaches this point.

The mathematical model (3) for this system is relatively simple. y is
the vertical position of the drop, v is its velocity, m is its mass
prior to detachment, and Δm is the amount of mass that detaches (k, b,
and c are constants). When this model is simulated on a computer, the
resulting phase portrait is very similar to the one that was
reconstructed from the data in the lab. Although this qualitative
agreement is too weak to completely vindicate these models of the
dripping faucet, it does provide a small degree confirmation.

(3)

Going back to the physical model, there are two clear
idealizations/negative analogies. First, of course, is that water
drops are not shaped like rigid blocks. Second, the mass-spring model
only oscillates along one axis. Real liquids are not constrained in
this way. However, these idealization allow for a far simpler
mathematical model to be used than one would need for a realistic
fluid. (Without these idealizations, (3) would have to be replaced by
a difficult partial differential equation.) In addition, Peter Smith
has argued that this mathematical tractability came with a steep
price, namely, an unrecognized artifact (1998). The problem is that
the state space for this particular system contains a "strange
attractor" with a fractal structure, a geometrical structure far more
complex than the attractors in Figure 4. Smith argues that the
infinitely intricate structure of this attractor is an artifact of the
mathematics used to describe the evolution of the system. If more
realistic physical and mathematical models were used, this negative
analogy would likewise disappear.
5. Models and Realism

One of the perennial debates in the philosophy of science has to do
with realism. What aspects of science—if any—truly represent the real
world? Which devices, on the other hand, are merely heuristic?
Antirealists hold that some parts of the scientific enterprise—laws,
unobservable entities, etc.—do not correspond to anything in reality.
(Some, like van Fraassen (1980), would say that if by chance the
abstract terms used by scientists did denote something real, we have
no way of knowing it.) Scientific realists argue that the successful
use of these devices shows that they are, at least in part, truly
describing the real world. Let's now consider what role models have
played in this debate.

Whether models should be taken realistically depends on what one takes
the truth-bearers in science to be. Some hold that foundational,
scientific truths are contained either in mature theories or their
fundamental laws. If so, then idealized models are simply false. The
argument for this is straightforward (Achinstein 1965). Let's say that
theory T describes a system S in terms of properties p1, p2, and p3.
As we have seen, simplified models either modify or ignore some of the
properties found in more fundamental theories. Say that a physical
model M describes S in terms of p1 and p4. If so, then T describes S
in one way; M describes S in a logically incompatible way. The
simplifying assumptions needed to build a useful model contradict the
claims of the governing theory. Hence, if T is true, M is false.

In contrast, Nancy Cartwright has long argued that abstract laws, no
matter how "fundamental" to our understanding of nature, are not
literally true. In her earlier work (1983), she argued that it is not
models that are highly idealized, but rather the laws themselves.
Abstract laws are useful for organizing scientific knowledge, but are
not literally true when applied to concrete systems. They are "true,"
she argues, only insofar as they correctly describe simplified
physical models (or "simulacra"). Fundamental laws are
true-of-the-model, not true simpliciter. The idea is something like
being true-in-a-novel. The claim "The beast that terrorized the island
of Amity in 1975 was a squid" is false-in-the-novel Jaws. Similarly,
Newton's second law of motion plus universal gravitation are only
true-in-Newtonian-particle-models.

For most scientific realists, whether physical models are "true" or
"real" is not a simple yes-or-no question. Most would point out that
even idealizations like the frictionless plane are not simply false.
For two blocks of iron sliding past each other, neglecting friction is
a poor approximation. For skis sliding over an icy slope, it is much
better. In other words, negative analogies come in degrees. If the
idealizations are negligible, we may properly say that a physical
model is realistic.

Scientific realists have not always held similar views about
mathematical models. Textbook model building in the physical sciences
often follows a "top-down" approach: start with general laws and first
principles and then work toward the specifics of the phenomenon of
interest. Dynamics texts are filled with models that can serve as the
foundation for a more detailed mathematical treatment (for example, an
ideal damped pendulum or a point particle moving in a central field).
Philosophers have paid much less attention to models constructed from
the bottom-up, that is, models that begin with the data rather than
theory. What little attention bottom-up modeling did receive in the
older modeling literature was almost entirely negative. Conventional
wisdom seemed to be that phenomenological laws and curve-fitting
methods were devices researchers sometimes had to stoop to in order to
get a project off the ground. They were not considered models, but
rather "mathematical hypotheses designed to fit experimental data"
(Hesse 1967, 38). According to Ernan McMullin, sometimes
physicists—and other scientists presumably—simply want a function that
summarizes their observations (1967, 390-391). Curve-fitting and
phenomenological laws do just that. The question of realism is avoided
by denying the legitimacy of bottom-up mathematical models.

In her broad attack on "theory-driven" philosophy of science,
Cartwright has recently defended a nearly opposite view (1999). She
argues that top-down mathematical models are not realistic, but
bottom-up models are. Once again, this verdict follows from a more
general thesis about the truth-bearers in science. Cartwright is an
antirealist about fundamental laws and abstract theories which, she
claims, serve only to systematize scientific knowledge. Since top-down
mathematical models use these laws as first principles from which to
begin, they cannot possibly represent real systems. Bottom-up models,
on the other hand, are not derived from covering laws. They are
instead tied to experimental knowledge of particular systems. Unlike
fundamental theories and their associated top-down models, bottom-up
models are designed to represent actual objects and their behavior. It
is this grounding in empirical knowledge that allows these kinds of
mathematical models to be the primary device in science for
representing real-world systems.
6. Models and the Semantic View of Theories

This typology of models and their properties has been developed with
an eye toward scientific practice. Within the philosophy of science
itself, models have also played a central role in understanding the
nature of scientific theories. For most of the 20th century,
philosophers considered theories to be special sets of sentences.
Theories on this so-called "syntactic view" are linguistic entities.
The meaning of the theory is contained in the sentences that
constitute it, roughly the same way the meaning of this article is
contained in these sentences. The semantic view, in contrast, uses the
model-theoretic language of mathematical logic. In broad terms, a
theory just is a family of models. The theory/model distinction
collapses. Using the terminology we have already defined, a model in
this sense might be an idealized physical model, an existing system in
nature, or even a state space. The semantic content of a theory, on
this view, is found in a family of models rather than in the sentences
that describe them. If a given theory were axiomatized—a rare
occurrence—one could think of these models as those entities for which
the axioms are true. To take a toy example, say T1 is a theory whose
sole axiom is "for any two lines, at most one point lies on both."
Figure 5 is one model that constitutes T1:

Figure 5: A Model of Theory T1

A model for ideal gases would be a physical model of dilute, perfectly
elastic atoms in a closed container with an ordered set of parameters
P, V, m, M, T> that satisfies the equation . (Respectively, pressure,
volume, mass of the gases, molecular weight of the molecules, and
temperature. R is a constant). In fact two different sets of
parameters P1, V1, m1, M1, T1> and P2, V2, m1, M1, T2> constitute two
separate models in the same family.

Some advocates of the semantic view claim that the use of the term
"model" is similar in science and in logic (van Fraassen, 1980). This
similarity has been one of the motivating forces behind this
particular understanding of scientific theories. Given the
distinctions made in previous sections of this article, this
similarity seems to be questionable.

First, many things that would count as a model on the semantic view,
for example the geometric diagram in Figure 5, are not physical
models, mathematical models, or state spaces. In what sense, one
wonders, are they scientific models? Moreover, a model on the semantic
view might be an existing physical system. For example, Jupiter and
its moons would constitute another model of Newton's laws of motion
plus universal gravitation. This blurs the distinction between the
model and its subject. One may use a physical and/or mathematical
model to study celestial bodies, but such entities are not themselves
models. The scientist's use of the term is not this broad.

Second, as we have already seen, sets of equations often constitute
mathematical models. In contrast, laws and equations on the semantic
approach are said to describe and classify models, but are never
themselves taken to be models. Their relation is satisfaction, not
identity.

Some time before the semantic view became popular, Hesse issued what
still seems to be the correct verdict: "[M]ost uses of 'model' in
science do carry over from logic the idea of interpretation of a
deductive system," however, "most writers on models in the sciences
agree that there is little else in common between the scientist's and
the logician's use of the term, either in the nature of the entities
referred to or in the purpose for which they are used" (1967, 354).
7. References and Further Reading
Achinstein, P. "Theoretical Models." The British Journal for the
Philosophy of Science 16 (1965): 102-120.
Bunge, M. Method, Model and Matter. Dordrecht: Reidel, 1973.
Cartwright, N. How the Laws of Physics Lie. New York: Clarendon Press, 1983.
Cartwright, N. The Dappled World. Cambridge: Cambridge University Press, 1999.
Hesse, M. Models and Analogies in Science. Notre Dame: University of
Notre Dame Press, 1966.
Hesse, M. "Models and Analogy in Science." The Encyclopedia of
Philosophy. New York: Macmillan Publishing, 1967.
McMullin, E. "What do Physical Models Tell Us?" Logic, Methodology,
and Philosophy of Science III. Eds. B. van Rootselaar and J. F. Staal.
Amsterdam: North-Holland Publishing, 1967: 385-396.
Morrison, M. and M. Morgan, eds. Models as Mediators. Cambridge:
Cambridge University Press, 1999.
Morton, A. "Mathematical Models: Questions of Trustworthiness." The
British Journal for the Philosophy of Science 44 (1993): 659-674.
Morton, A. and M. Suàrez. "Kinds of Models." Model Validation in
Hydrological Science. Eds. P. Bates and M. Anderson. New York: John
Wiley Press, 2001.
Redhead, M. "Models in Physics." The British Journal for the
Philosophy of Science 31 (1980): 154-163.
Smith, P. Explaining Chaos. Cambridge: Cambridge University Press, 1998.
Van Fraassen, B. The Scientific Image. New York: Clarendon Press, 1980.
Wimsatt, W. "False Models as Means to Truer Theories." Neutral Models
in Biology. Eds. M. Nitecki and A. Hoffmann. New York: Oxford
University Press, 1987.