Theories between theories: Asymptotic limiting intertheoretic relations

Abstract

This paper addresses a relatively common “scientific” (as opposed to philosophical) conception of intertheoretic reduction between physical theories. This is the sense of reduction in which one (typically newer and more refined) theory is said to reduce to another (typically older and “coarser”) theory in the limit as some small parameter tends to zero. Three examples of such reductions are discussed: First, the reduction of Special Relativity (SR) to Newtonian Mechanics (NM) as (v/c)2→0; second, the reduction of wave optics to geometrical optics as λ → 0; and third, the reduction of Quantum Mechanics (QM) to Classical Mechanics (CM) asħ→0. I argue for the following two claims. First, the case of SR reducing to NM is an instance of a genuine reductive relationship while the latter two cases are not. The reason for this concerns the nature of the limiting relationships between the theory pairs. In the SR/NM case, it is possible to consider SR as a regular perturbation of NM; whereas in the cases of wave and geometrical optics and QM/CM, the perturbation problem is singular. The second claim I wish to support is that as a result of the singular nature of the limits between these theory pairs, it is reasonable to maintain that third theories exist describing the asymptotic limiting domains. In the optics case, such a theory has been called “catastrophe optics”. In the QM/CM case, it is semiclassical mechanics. Aspects of both theories are discussed in some detail.