Comment: The meaning and uses of models

Abstract

In a paper with the above title in this Journal (Suppes, 1960), Suppes raises certain objections to Mach's definition of mass in Newtonian mechanics. He argues (p. 296) that 'it may be proved by Padoa's principle that a proper definition of mass is not possible,' and that 'if the number of interacting bodies is greater than seven a knowledge of the mutually induced accelerations of the particles is not sufficient for unique deter mination of the ratios of the masses of the particles.' Both claims appear incorrect, unless they are substantially weakened or modified. First, as I have shown elsewhere (Simon, 1947), Mach's proposal for defining masses in terms of the mutually induced accelera tions of bodies is readily developed into a rigorous axiomatization for Newtonian particle mechanics. The definition of mass does not meet the test of Padoa's principle, but the conclusion to be drawn from this fact is not that the definition is faulty but that Padoa's principle imposes too severe a test on it. The definition of mass does meet a criterion of 'defin ability almost everywhere,' which I have proposed (Simon, 1959, pp. 449 450) as a more satisfactory notion of definability than Tarski's for axiomatizing empirical theories. Second, contrary to Suppes' assertion, the mass ratios of a system of particles can be calculated from the mutually induced accelerations even if the number of particles is greater than seven (Simon, 1947, p. 898; 1954, p. 341). The indeterminacy shown by Pendse only extends to the component accelerations and not to the mass ratios (Simon, 1947, pp. 901-2). The indeterminacy of the component accelerations sub sequently disappears when we postulate a specific law of gravitational attraction - e.g., the inverse square law. For these reasons, I cannot concur in Suppes' assessment of 'the fun damental weakness of Mach's proposal.' As developed and formalized in Simon (1947), Mach's proposal appears quite sound.