Boundaries, Contingencies and Rigor

Abstract

A case study in the history of transonic aerodynamics, circa 1950, is used as a basis for reflecting on the character of the distinction between pure and applied mathematics, including the mathematics used by engineers. The case study is set against an historical background of disciplinary confrontation led by such eminent representatives of mathematics and aerodynamics as Garrett Birkhoff and Theodore von Kármán. The successful attempt to construct an adequate account of the aerodynamics of the transonic realm highlighted some sharp differences in the procedures and preferences of mathematical practitioners operating in different fields. The existence and general character of these differences is already widely acknowledged but the task of exploring them from a sociological standpoint still requires much work. The nature of the disciplinary distinctions between different areas of mathematics is examined using Barnes’ theory of (idealized) natural-and social-kind terms (so-called N-and S-predicates). Although the ultimate status of the disciplinary boundaries turns out, uncontroversially, to be ‘conventional’, the attempt to make out and exhibit the conventionality in detail proves to be a non-trivial exercise. It transpires that a thorough study of the issues turns on deep questions about the nature of mathematical rigor and a process that might be called ‘the exploitation of contingency’. These points are illustrated in detail by reference to the technical work of the area (in which one of the authors was an active participant).