Models: An Overview

It is customary to refer to the model as the ‘source’ and the thing being modelled as the ‘target’.

The target is the phenomenon or phenomena that the scientist is trying to make sense of.¹ This terminology should be treated minimally at this stage; it implies no commitments to what types of things the source and target can be: they may be concrete or abstract, physical or mathematical, real or imaginary (Suárez 2015: 41). Models come in different types. In Models and Metaphors, Max Black (1962) distinguished between four overarching types of models found in science: scale models, analogue models, mathematical models, and theoretical model.²

Scale models enlarge or shrink certain features of their target system, such as a globe as a model of Earth (Bailer-Jones 2009: 3). Often, scale models are used to simulate the behaviour of their target systems, without seeking to explain why it behaves as it does. The San Francisco Bay Delta Model would be an example (Weisberg 2013). Analogue models function in a similar way, but involve a change of medium. The Phillips machine, for example, represents the workings of the macro-economy by the ebb and flow of coloured water in a hydraulic system (Morgan and Boumanns 2004). In turn, mathematical models make use of mathematical language and formalism to represent their target system, for example, the classical simple harmonic oscillator model of the mass-spring system. Finally, theoretical models frequently employ abstract concepts, idealizations, and theoretical principles. Familiar examples here include the billiard ball model of gases, Bohr’s model of the atom, the liquid-drop model of the nucleus, or the Lotka-Volterra model of predator-prey interaction. The use

1 Following Bogen and Woodward (1988), ‘phenomena’ refer to stable regularities in the world, as distinct from individual instances of observational or experimental data. For example, lead melting at 327.5 ˚C (phenomenon) as distinct from a series of temperature readings as a piece of lead is heated up (data).

2 This list is by no means exhaustive. I only offer it here as a useful entry point to the variety of types of models in scientific practice. See Frigg (2012) for a more expansive discussion.

of ‘theoretical’ to describe these models is intended in a reasonably broad sense and includes any case in which scientists deliberately simplify or idealise a system or phenomenon to explain or predict an aspect of its behaviour (Toon 2012: 9).

In recent years, philosophical interest in models has been driven by close attention to how models are used in scientific practice.³ An influential idea in this context is that models are partially autonomous mediators between theory and reality (Morgan and Morrison 1999). On this view, models are to be regarded neither as a merely auxiliary intermediate step in applying scientific theories, nor as constructed purely from data. It is argued that this is because it is argued that there is no algorithm for constructing adequate models from theoretical principles alone. Thus, more recent work typically rejects what Cartwright (1999) labels the ‘vending machine’ view of theories and models: the view that models are neatly contained within a theory and all that is required is the right kind of input in order to derive a particular model.

Instead, it is now widely acknowledged that scientific models are often constructed from a heterogeneous mixture of elements, such as simplifications, metaphors, policy views, empirical facts, as well as theoretical ideas. As Marcel Boumans colourfully put it, model building can be ‘like baking a cake without a recipe’ (Boumans 1999: 67). This is not to suggest that theories do not guide model construction; merely that that the construction of models, especially new models, is more like a trial-and-error process of working out how these various elements fit together.

Michael Weisberg (2007) and Peter Godfrey-Smith (2006) have proposed that model-based reasoning in science can be distinguished from other theoretical activities through its indirect nature of representation. Weisberg suggests that modelling proceeds in three stages.

First, a model is constructed, then, second, the modeller refines, analyses and articulates its properties and dynamics. It is not until the third stage that the relationship between the model and its target is assessed, ‘if such assessment is necessary’ (Weisberg 2007: 209). By calling modelling ‘indirect’, Weisberg and Godfrey-Smith draw attention to the fact that models are not primarily constructed to represent their target systems as faithfully as possible. Rather than studying real-world phenomena directly, modellers construct and study idealized surrogate

3 Typically, this move is framed as a reaction against both ‘traditional’ syntactic and semantic accounts of theories, from which earlier accounts of models were derived. On the syntactic version, a theory is a set of claims and models of the theory, such that all the claims of the theory are satisfied and no additional assumptions are imported except those which are legitimately grounded in a description of the phenomena to be represented. Semantic accounts held that theories are sets of models (e.g., Van Fraassen 1980; Giere 1988). The syntactic view has been widely discredited, but the semantic view is still defended (e.g., French and Ladyman 1999; da Costa and French 2000).

systems to which only a few select properties are attributed.⁴ The important insight of the notion of indirect representation is ‘to redirect the focus from models to the activity of modelling’ (Knuuttila and Loettgers 2017: 1010).

As an example, the Lotka-Volterra model of predator-prey interaction was constructed to study the dynamics between two idealized populations of organisms. Unlike real populations of organisms, the kind of properties stipulated to be possessed by these model populations were very few, such as intrinsic exponential growth rate for the prey in the absence of predators and a constant death rate for the predators (Weisberg 2007: 210). Even so, the model was used to infer empirically verifiable conclusions concerning the dynamics of fish populations in the Adriatic after the First World War. Often, the motivation for constructing a model is to reduce the amount of complexity that must be accounted for. With an idealized model system to hand, scientists can focus on the effects of a few choice properties or mechanisms. By studying how variations in the assumptions out of which the model is built lead to different or similar outcomes, scientists can infer conclusions about its real-world counterpart (the ‘target system’).⁵ Thus, modelling is a form of surrogative reasoning, which

‘designate[s] those cases in which someone uses one object, the vehicle of representation, to learn about some other object, the target of representation’ (Contessa 2007: 51).

It has been argued that the understanding of models as surrogate systems is analogous to the epistemic dynamics of models and experiments (Mäki 2005; Morgan 2003). According to Mäki, modelling and experimentation are both attempts at isolating the casually relevant factors. Whereas in the case of experimentation, such isolation is achieved causally through experimental controls, in the case of models, such isolation is achieved by making more or less unrealistic assumptions (Kuorikoski and Lehtinen 2009: 121). The unrealistic nature of such assumptions derives from the use of idealizations and abstractions in the construction of models. As several philosophers of science have argued (e.g., Cartwright 1983), scientific hypotheses typically do not apply, strictly speaking, to any real system in the physical world, but rather to idealized versions of the phenomena. As Roman Frigg puts it:

When studying the orbit of a planet we take both the planet and the sun to be spinning perfect spheres with homogenous mass distributions gravitationally interacting with each other but nothing else in the universe […] [W]hen studying the exchange of goods in an economy we

4 The notion of models as ‘surrogate systems’ has been discussed in Swoyer (1991) and Mäki (2005).

5 Weisberg contrasts this with ‘abstract direct representation’ which refers to the analysis of real-world phenomena (abstracted and represented in certain ways) without any mediating model. For example, Weisberg argues that Mendeleev’s ordering of chemical elements is a case of abstract direct representation, since Mendeleev worked directly with representations of real-world phenomena, the chemical elements, rather than through the construction and analysis of an idealized model of those elements (Weisberg 2007: 216).

consider a situation in which there are only two goods, two perfectly rational agents, no restrictions on available information, no transaction costs, no money, and dealings are done in no time. (Frigg 2010: 251-252)

Idealizations are typically introduced into models as conscious misrepresentations of some aspects of a target system in order to render other factors of interest more salient.⁶ The billiard ball model of gases, for instance, treats the molecules of a gas as composed of dimensionless, spherical molecules that are not acted upon by gravity and do not act upon each other in collisions. Of course, no such gas exists, but treating gases in this way allows scientists to selectively focus on specific aspects of a phenomenon, while ignoring others, and to make such aspects computationally tractable. Other examples include frictionless planes, Newtonian point masses, and, in the case of the Hodgkin-Huxley model of action potential, the representation of neurons as electrical circuits.

Both idealization and abstraction are an essential part of building a model. This is because the variables exerting some influence on a phenomenon of interest are often too numerous to take into account. Idealizations and abstractions allow the system to become more analytically tractable and computable. For instance, no real system strictly obeys the laws of physics, so using these laws to explain the behaviour of real systems requires the use of various approximations, abstractions and idealizations. By idealising selected parts of a phenomenon, scientists can draw attention to and home in on those aspects they take to be relevant to the problem at hand, while ignoring other features. In doing so, scientists make choices about which idealising assumptions should be involved in the model. The nature of these purposes and epistemic goals are dictates how the model is used and which aspects of the target system are to be focused on.

Hence, as partially autonomous surrogate systems, models provide scientists with a range of tools and can be used for a variety of different purposes. As well as enabling scientists to infer conclusions about their real-world counterparts, models are often employed for heuristic or predictive purposes. In these ways, models are used for determining which aspects of a target phenomenon to focus on, which concepts can be brought to bear on those phenomena, and what the implications of modelling the target in a specific way would be. Because of this, recent philosophical interest in modelling practice is often less concerned with what models are and more concerned with what models are used for, i.e. their various functions and purposes in scientific research (e.g., Keller 2000).

6 Two types of idealization are distinguished in the literature on scientific models, namely Aristotelian and Galilean idealizations. Aristotelian idealization refers to the process of stripping away properties of the target that are taken to be irrelevant. This kind of idealization is often called

‘abstraction’. Galilean idealization refers to the conscious and deliberate misrepresentation of a target system in order to make it more tractable for study.