Abstract

In this dissertation I address the general question, "What is the role of mathematical models in the way in which scientific theories represent the world?" The specific questions I address are these: What is the relation between the laws of a theory and its models? Do scientific theories give us coherent accounts of physical possibility? Must a theory represent a phenomenon truthfully in order to explain it? ;In the first chapter I distinguish and examine some central uses to which the term "model" has been put in philosophical discussions of scientific theories. According to the current "semantic view," theories are to be understood as consisting of families of models. I argue that standard presentations of the semantic view are guilty of confounding two distinct notions of model: that of a structure which represents the world and that of a structure which satisfies a set of sentences. ;In the second chapter I examine the relation between the general principles or "laws" of a theory and the mathematical models that scientists use to represent individual phenomena within the theory's purview. According to an intuitively plausible view, individual models satisfy the laws of the theory and can, at least in principle, be integrated into coherent 'global' representations of the phenomena in the domain of the theory. Scientific theories, according to this view, delineate coherent physical possibilities. Against this, I show in a detailed case study of classical electrodynamics that there are no possible worlds of which all the basic equations of the theory are simultaneously true under their standard interpretation. Instead of presenting us with a conceptually coherent picture of a range of possibilities, classical electrodynamics allows for the construction of a variety of models which locally represent different phenomena, but which cannot be integrated into a single unified and coherent account. ;In the third and fourth chapters, I take up the question of whether a theory must represent the phenomena within its purview truthfully in order to explain them. Classical electrodynamics could not possibly be a true theory, yet, perhaps surprisingly, physicists take the theory to provide adequate explanations. After surveying many of the major current accounts of explanation, most of which hold that truth is a necessary condition of explanatory adequacy, I propose an account of scientific explanation that does two things: it tries to do justice to the intuition embodied in these accounts that there is a close connection between explanation and truth, but at the same time it allows good scientific explanations to be at least partially untrue. To explain a phenomenon, I argue, is to exhibit it as part of a systematic pattern of alternative possibilities. I elaborate this idea and show how it can be made precise by proposing a technical framework which appeals to Tarski's notion of a relational system. I argue that we use three criteria to evaluate putative explanations: degrees of truth, richness of explanatory pattern, and simplicity; and I claim that something like the weighted average of the three criteria determines the goodness of an explanation. I show how my proposal can account for the structure of paradigmatic instances of scientific explanations, and I defend the proposal from challenges posed by causal theories of explanation