Representing with imaginary models: Formats matter

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Abstract

Models such as the simple pendulum, isolated populations, and perfectly rational agents, play a central role in theorising. It is now widely acknowledged that a study of scientific representation should focus on the role of such imaginary entities in scientists’ reasoning. However, the question is most of the time cast as follows: How can fictional or abstract entities represent the phenomena? In this paper, I show that this question is not well posed. First, I clarify the notion of representation, and I emphasise the importance of what I call the “format” of a representation for the inferences agents can draw from it. Then, I show that the very same model can be presented under different formats, which do not enable scientists to perform the same inferences. Assuming that the main function of a representation is to allow one to draw predictions and explanations of the phenomena by reasoning with it, I conclude that imaginary models in abstracto are not used as representations: scientists always reason with formatted representations. Therefore, the problem of scientific representation does not lie in the relationship of imaginary entities with real systems. One should rather focus on the variety of the formats that are used in scientific practice.

Introduction

Models such as, for instance, the simple pendulum or a block sliding on a frictionless plan in mechanics, perfectly rational agents in economics, and isolated populations in genetics, obviously play a central role in theorising. Scientists who inquire into the consequences of some hypothesis (or of a set of hypotheses—namely a theory), as well as students who try to grasp the meaning and to deepen their understanding of a theory, often reason with such imaginary entities.

Indeed, as many philosophers of science (Cartwright, 1983, Frigg, 2002, Frigg, 2006, Frigg, 2010, Godfrey-Smith, 2006) have underlined, scientific hypotheses do not apply, strictly speaking, to any real system in the physical world, but rather to ‘idealised versions’ (Frigg, 2010) of the phenomena. No real system strictly obeys the laws of physics (let alone the laws of economics or biology); using these laws to explain the behaviour of real systems implies various approximations, abstractions, and idealisations1, which result in representing real systems as idealised systems (e.g. representing such particular grandfather clock as a frictionless pendulum, or such particular population as an isolated population).

Such considerations have led these philosophers2 to argue that, contrary to what the so-called ‘semantic view’ of theories3 suggests, the problem of scientific representation does not lie in the relationship between abstract mathematical structures and the real-world phenomena. Instead, according to these critics, theoretical hypotheses become genuine representations through the mediation of models such as isolated populations and frictionless pendulums, which provide them with empirical content4.

How do such imaginary—or, as Godfrey-Smith, 2006, Frigg, 2010 would say, fictional—entities provide content to abstract mathematical structures? According to Frigg (2010), they do so in virtue of their concreteness. Indeed, as he and Godfrey-Smith (2006) show, these models are not reducible to the mathematical structures they might instantiate: they ‘do not exist spatio-temporally but are nevertheless not purely mathematical or structural in that they would be physical things if they were real’ (Frigg, 2010, Section 2). In other words, the mathematical structures underdetermine them, since they are, as suggested by Godfrey-Smith (2006, pp. 734–735), ‘imagined concrete things’, ‘things that are imaginary or hypothetical, but which would be concrete if they were real’5. Therefore, a study of scientific theorising should pay a particular attention, according to most philosophers I have cited, to the construction, investigation, and manipulation of this kind of entities, which I propose to call I-models (for ‘imaginary models’). Such activities involve scientists’ creative and imaginary skills, which are most often neglected by abstract approaches such as the semantic view.

In this paper, I endorse such a view on theorising, and I aim at shedding some light on the cognitive processes of representing by means of I-models. However, I shall argue against one of the main assumption of some of the critics of the semantic view, as it is expressed in Frigg’s (2010) proposal. Frigg distinguishes between two sets of questions concerning I-models (which he calls ‘model systems’), namely ontological questions about what kind of entities I-models are, and semantic questions about what kind of relationships I-models have with the phenomena they stand for. Frigg addresses the first problem as a preliminary to solve the second one: according to him, one should first elucidate the nature of these imaginary entities before asking how such entities might represent portions of the empirical world. In doing so, he assumes that the problem of scientific representation lies in the relationship between I-models and the real systems under study. Indeed, by distinguishing between the ontological and the semantic questions, he assumes that there exists something as imaginary entities, which stand in a representational relationship with the physical world.

This is what I shall reject. I will show that analyses such as Frigg’s, despite his criticising the abstractness of the semantic view, arise also from too abstract a conception of representation. Indeed, in investigating the nature of I-models as a preliminary to the examination of their relationships with the real world phenomena, he misses a crucial aspect of the use of I-models, namely the concrete inferences agents perform when reasoning with I-models. In other words, I shall argue that, if one takes seriously the idea that a study of theorising has to concentrate on the concrete practices of scientists, rather than on theories conceived as abstract mathematical structures, then one has to focus on the cognitive interactions between agents and the representational devices they reason with and manipulate. Rather than focusing on the relationship between I-models and the real-world phenomena, I shall therefore examine the relationship between agents and I-models. My central question will be: how do scientists use models to draw inferences and gain knowledge about the systems these models stand for? It will appear that the very form in which the model is displayed is crucial to the agents’ inferential processes, and that I-models in abstracto are not, by themselves, representations. As a result, the semantic question concerning I-models (How do they relate to the phenomena?), insofar as it is raised after having posed the ontological question (What are they?) will prove meaningless.

First, I propose a clarification of the very notion of representation as I understand it (2.1), and I emphasise the importance of what I call the ‘format’ of a representation (2.2) for the inferences agents can draw from it. Then, I turn to the core question and I analyse the various representational relationships that are in play in the use of I-models (3.1 Characterisations of the I-model, 3.2 Representations of the target system). I finally conclude that the problem of scientific representation does neither lie in the nature of I-models nor in their relationship to the real systems, and that the study of the representational power of models should instead focus on the variety of the formats that are used in scientific practice.

Section snippets

Representation, inferences, and formats

In this section, I shall present a conception of representation which is not to be restricted to the scientific domain, though focusing only on the knowledge-seeking (or epistemic) aspects of representing. I will introduce the notion of format, which is intended to help us capture some important phenomena at play in the epistemic use of representations. That is a necessary preliminary to a study of the use of (scientific) I-models in representing physical systems, in which I will engage in

I-models and their formats

Let me now turn to my main question: how do we use I-models to represent real world systems? When it comes to predicting and explaining the behaviour of some physical system, one appeals most of the time to an I-model, claiming that it ‘applies’, with various approximations and idealisations, to the real world phenomena: for instance, the motion of the bob of a grandfather clock can be represented by means of the simple pendulum. As mentioned in the introduction, some philosophers, such as

Conclusion

The analysis of the representational function of models was initially motivated by the project of shedding light on scientists’ ability to draw inferences in order to give predictions and explanations. I have shown that their inferential processes are not the same according to what particular format is used to express the model one is using. As soon as one adopts a pragmatic view on theorising and one is not satisfied with abstract accounts of theories and representation, one also has to

Acknowledgements

I would like to thank Anouk Barberousse, Julien Boyer, Roman Frigg, Jean Gayon, Axel Gelfert, Paul Humphreys, Philippe Huneman, Cyrille Imbert, Tarja Knuuttila, and Philippe Lusson for comments on earlier drafts and/or for helpful discussions.

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