A logical study of the correspondence relation
This paper develops a detailed logical analysis of the limiting relations that frequently hold between a scientific theory and its successor. The central relation studied is termed limiting case correspondence. In order to define it precisely, the authors distinguish the standard from the nonstandard models of a theory, and they generalise the concept of standard part (of a set or function) so that in a natural way one can speak of one model being the standard approximation of another. Limiting case correspondence can then be defined as a mapping of a certain sort which takes suitable nonstandard models to their standard approximations, together with a dual syntactic translation which maps formulas in the language of the latter structures into formulas in the language of the former. The above method is illustrated with a case study. Certain core fragments of classical and special relativistic particle mechanics (respectively, CM and RM) are reconstructed and it is shown how a correspondence of the classical to the relativistic theory can be defined. In this example, suitable nonstandard models of RM in which particle velocities and accelerations are infinitesimally small compared to the speed of light are correlated with their standard approximations which are models of CM. The relevant classes of models are defined in the infinitary logic \(L_{\omega_ 1\omega}\), and the dual mapping translates \(L_{\omega_ 1\omega}\) formulas of CM into the language of RM. It is also shown how infinitesimal Lorentz transformations in RM can be correlated with Galilean transformations in CM, so that, in a precise sense, the structure of the symmetries of the two theories is preserved (this idea is reconstructed using the notions of category and functor). The authors conjecture that the method of nonstandard analysis, as employed here, should be widely applicable in the study of limiting relations between theories, and that several logical and philosophical problems associated with intertheory reduction and explanation in science can be clarified with its help.
core fragments of classical and special relativistic particle mechanics
scientific article; zbMATH DE number 3867287
scientific article