Mathematical Models in Philosophy of Science

Abstract

The concept of mathematical models occupies a central position in the science of today. It has replaced the received view of theories, according to which a scientific theory is a deductively organized (axiomatic) body of empirical statements, whose theoretical terms are partially interpreted via a set of correspondence rules. Many deficiencies in this conception were also a contributing factor in the development of the ‘Weltanschauungen’ analyses of Kuhn, Lakatos, and others, and led to a denigration of formalist methodology in general. Around the beginning of the 1970s, several mathematical definitions of the notion of scientific theory have been proposed. The reason for increased interest in the subject was the failure of the earlier developed syntactic approach. The main objective here is to outline the current understanding of three prominent approaches to the notion of scientific theory, data and their relationships to empirical phenomena. These are the set-theoretical predicate approach, state space view, and the structuralist program. The essence of the set-theoretical predicate approach is captured by the idea that to axiomatize a scientific theory is the same thing as to define a set-theoretical predicate, that is, to specify a collection of mathematical models in which the axioms are true. Formally, set-theoretical predicates are very similar to the classical Bourbaki notion of species of structures. In general terms, the state space view is a special case of the set-theoretical approach, focusing on theories of time-dependent systems. Accordingly, instead of formulating a theory in terms of traditional differential or difference equations of motion, one proceeds geometrically by specifying a topological state space, together with a dynamical structure. Under this program, many traditional notions of science receive a novel geometric meaning. The classic example is the interpretation of laws of motion as phase portraits in the underlying state space. The state space apparatus can be used to reason about the behavior of many kinds of systems, including the familiar dissipative, conservative, nonlinear, and chaotic systems. The structuralist program provides a full-fledged methodology aimed at reconstructing a large variety of actual scientific theories and their applications, all in terms of set-theoretic structures. From the structuralist perspective a scientific theory is specified in terms of possible, actual and partial models, and their constraints. This program captures a large number of aspects of how theoretical concepts and experimental methods are deployed in science. Simple and familiar examples are used to illustrate the use of these modeling techniques in empirical science.