Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

408 alasdair urquhartfrom the observational terms. A description and extended critical discussionof this picture, known as the ‘Received View on Theories’ can be foundin Suppe (1977). Although only small parts of classical physics were in factgiven an axiomatic form by the philosophers of science of the day, there wasconsiderable optimism that larger and larger parts of science, or at least themore formal parts, could be formulated as sets of logical postulates and rules.This approach to the philosophy of science, one that is associated with thenames of neo-positivist philosophers such as Carnap, Reichenbach, Hempel,Feigl, and others, is now almost totally out of style. There are good reasons forthis. The actual formulation of non-trivial scientific theories as logical edificesis a difficult task, demanding very good knowledge of both the scientificliterature and the mathematical concepts needed to make the methods of agiven area precise, and it is not surprising that philosophers and logicians neveraccomplished more than a very small fraction of what they had set out to do.In any case, the relationship between theory and applications is surelymore complicated than the schematic ideas propounded in the philosophicalliterature of the immediate post-war period. The ‘theory and interpretation’view is that the application of a theory can be reduced to the ‘theory andmodel’ paradigm beloved of model theorists. So, for example, we can formulatethe abstract theory of groups as a first-order theory with a single operationrepresenting group composition. An applied version of this theory would begiven by an interpretation in terms of a particular group, say, for example, thegroup of rigid motions in Euclidean 3-space.This view is attractive and elegant. However, it is hard to square with theactual practice of scientists. The world is so complex that physicists who areattempting to provide mathematical models of physical reality do not in generalbegin by a direct attempt to formulate theories in which the primitive termscan be given an immediate empirical interpretation. Rather, they very oftenconstruct idealized mathematical models in which the behaviour of certainvariables bears at least a qualitative resemblance to the real world—if theresemblance can be quantified, so much the better.¹As an example of such a model, let us look at a well-known modelof ferromagnetism, the so-called Curie–Weiss model, described in mostelementary textbooks of statistical physics (Thompson, 1972, pp. 95–105),(Yeomans, 1992, pp. 50–54). The empirical phenomenon to be explained isthe spontaneous magnetization of ferromagnetic materials. If we cool a sampleof such a material, while subjecting it to a magnetic field, then there is a sharply¹ Recent work on the role of models and idealization can be found in the two collections Morganand Morrison (1999) and Jones and Cartwright (2005).