Formal Models and Achinstein's “Analogies”

Abstract

This article is a brief critique of the attempt by Peter Achinstein to examine the inadequacies of formal models in science by means of his appeal to analogies ([1], [2]). Achinstein's analysis is chiefly directed against what may be called for convenience the model, because it has been most notably developed by R. Braithwaite, E. Nagel, and P. Suppes ([3], [5], [7]). The B-N-S model may be summarized as follows: if two deductive systems are interpretations of the same calculus, and if in one interpretation the initial formulas of the calculus (containing theoretical terms) are epistemologically prior to the derived formulas not containing theoretical terms, whereas in the second system the derived formulas are epistemologically prior, then these philosophers would say that the former system is related to the latter as a model is to a theory. In other words, a theory and its models are formally isomorphic, but we come to know most of the propositions of the theory starting from the lowest-level ones and working up to those of greater generality, and those of the model starting from the highest-level statements and working down to increasing specificity. For example, the calculus used to develop geometric optics may be interpreted upwards from empirical generalizations in terms of light rays, reflection, etc., to yield a physical theory, or downwards in terms of the lines and planes of Euclidean geometry to yield a geometric model isomorphic with that theory. The advantage of finding a B-N-S model for a theory is that the model interpretation is usually more straightforward and familiar than the theoretical one, and consequently serves as a stimulus to thought, with the special virtue that any discoveries made using the model will have the same logical structure as their brother derivations in the theory. Hence we have the semi-positivistic claim that formal isomorphism between two sets of statements is a good description of and an even better reconstruction of the relation of models to theories in science, because such formal modelling fulfills these two important functions: