Representation

Along with a theory of models comes a theory of representation. This is a time-honored problem in the philosophy of science which includes, under the same conceptual umbrella, the vast diversity of representational accounts for scientific models. A shortlist includes representational isomorphism, partial isomorphism, structural representation, homomorphism, similarity, and the denotation, demon-stration and interpretation (DDI) account of representation as elaborated by R. I.

G. Hughes.²² The general aim of any theory of representation is to theorize the re-lationship between a model and its target system, including the extent to which the model reliably represents the target system. This last issue is of certain importance for this work and I will say something at the end of this section.

To illuminate the general aims of a theory of representation, let me briefly an-alyze structural representation as elaborated by Chris Swoyer. To his mind, “a structural representation depends on the existence of a common structure between a representation and that which it represents, and it is important because it allows us to reason directly about the representation in order to draw conclusions about the phenomenon that it depicts” (Swoyer, 1991, 449). Following this definition, the Lokta-Volterra predator-prey model, for instance, is a structure that represents an-other structure (i.e., a dynamic system in evolutionary population biology). The representational relation can be either complete, that is, when both structures share the same features; or only partial, when they only share some features.

Let me present a concrete example. The dynamics of a biological system with two members, predator-prey, evolve according to the following pair of equations:

dx

dt = x(↵ y) dy

dt = y( x)

wherexis the number of prey andyis the number of predators. Thus understood, this model represents a biological, social, and economical system, among others. At its core lies the idea of a model and a target system sharing the same structure and having the right representational relation. As for the notion of structure, one could conceive it as =< U, O, R > where U is a non-empty set of individuals called the domain or universe of the structure (i.e., x = the prey, and y = the predator), a set of operations O on U (e.g., growth rate over time as described by

the equations ^dx_dt and ^dy_dt), and a non-empty set of relationsR onU (e.g., ↵, , and as the parameters describing the interaction of the two species).²³ Since a model is understood as a state space, then the Lotka-Volterra model can independently vary along thex axis (i.e., the prey population), they axis (i.e., the predator population), and time t.²⁴ As for the representational relation, it depends partly on the theory that the philosopher is committed to, and partly on the ontological constraints of the model and what it can offer. In this example, Swoyer is committed to some version of isomorphism²⁵ since he interprets models as structures.

Undoubtedly, each theory of representation conceptualizes the representational relation differently. Nevertheless, the general motivation is the same for all of them.

That is, to justify the use of a model as a surrogate or proxy for understanding something about the target system. At first sight, this idea is appealing because it facilitates the use of a model as if it were the target system. But questions emerge from such considerations. Of particular interest is the fact that the representational relation carries with it semantic and epistemic issues. For instance, scientific models are typically idealizations and abstractions of the target system. Idealizations, for instance, consist of deliberate distortions of properties, relations, entities, etc., of the target system with the purpose of making it more tractable. Abstraction, on the other hand, consists in ‘stripping away’ certain properties of the target system that are believed to be irrelevant for studying it. Both, idealization as ‘distortion’ and abstraction as ‘simplification’ must be taken as epistemic virtues, for they facilitate the use of a model as a proxy of the target system. Unfortunately,idealization and abstraction are elusive terms and it is not always clear how far one can abstract or idealize before completely misrepresenting the target system.²⁶ The question is, therefore, how good or reliable is the model with respect to the target system?

Theories of representation, as stated before, are a time-honored philosophical problem. So are theories of scientific models. To simplify matters, and because neither theories of representation nor of models are at the core of this work, I will follow Hans Reichenbach’s maxim: “discrepancies between an idealization and a real system constitute serious objections insofar as they defeat the purpose for which the idealization is used” (Reichenbach, 1938, 6).

With these considerations firmly in mind, we can now draw a general interpreta-tion of model as a proxy for the target system: to make use of a model as a proxy for the target system means to assume that the target system behaves in the way speci-fied by the model. Let me illustrate this point with the example of Lokta-Volterra’s predator-prey model. This model is used as a proxy for an empirical biological sys-tem in order to understand (and manipulate, measure, etc.) something about the

assumed predator-prey interaction. In a nutshell, it is used as if the target system behaves in the way that is specified by the model.

Thus understood, the predator-prey model could fail to accurately represent some aspects of the biological system in many ways, for instance, the number of abstract terms could make it too unrealistic leading to a misleading representation of the empirical biological system. Making use of a model as a proxy for the target system does not entail that it is true, adequate, nor that it isomorphically corre-sponds to the target system. For this reason, the researcher must be aware of the representational limits inherited with the use of any scientific model. It is also true, however, that a history of prior success of using the same model (e.g., the history of success and reliability of the Lokta-Volterra model for biological system) compen-sates (and sometimes even justifies) discrepancies between the model and the target system. In other words, Reichenbach’s maxim. Of course, one chief aim for any theory of representation is to reduce as much as possible the ‘conceptual distance’

between the model and the target system.

The fact that this study is not intended to fully address these issues allows me to condense the results here obtained into a working definition. I will refer to what has been said so far about the representation of a model as thegoodness of representation of such a model. I define it in the following way:

Working definition: the goodness of representation of a model describes the degree of accuracy to which a model represents its intended target system. The higher the goodness of representation, the more accurate the model represents the intended target system. The more accurate the model represents, the more reliable it becomes as a proxy for that target system. The more reliable the model, the better the results it renders about the target system.²⁷

Thus understood, goodness of representation is an umbrella term that covers the entire variety of theories of representation in the philosophical literature without committing to any. The goodness of representation of a model, then, is a measure of the discrepancy in the representational relation between the model and the target system. One can easily state that the goodness of representation of a Newtonian model is higher than a Ptolemaic model of the same target system, that is, the planetary movement. In other words, the goodness of representation measures the accuracy of the variables, relations, properties, etc., used for a model with respect to the target system. The more accurate, the higher the goodness of representation, and therefore the results are more accurate than otherwise.

The working definition aims at capturing the core of every theory of representa-tion, that is, that a model can be more (or less) reliable as a proxy depending on

the degree of accuracy with which it represents the target system. Admittedly, this working definition is a bit clumsy and does not answer many of the central philo-sophical questions about representation. However, it is useful insofar as it provides the necessary epistemic guarantees that the scientific model implemented as a com-puter simulation, and therefore the comcom-puter simulation itself, represents a target system. It is not within my interests to deepen the discussion of the relation be-tween scientific models and the world, but rather take it as relatively unproblematic and focus on the implementation of a scientific model into the computer simulation.

Allow me now to move to a brief discussion on sub-classes of scientific models that are of interest for this study.