The Axiomatization of Classical Mechanics

The purpose of this note is to examine a recent axiomatization of classical particle mechanics, and its relation to an alternative axiomatization I had earlier proposed.' A comparison of the two proposals casts some interesting light on the problems of operationalism in classical celestial mechanics. 1. Comparison of the Two Axiomatizations. The basic differences between the two proposals arise from the nature of the undefined terms. Both systems take the set of particles, time, and position as primitive notions. Both systems assume that there exists a set of particles having continuous, twice-differentiable paths over some time interval. In addition, CPM takes mass and force as primitive notions, and assumes that with each particle there is associated a mass and a set of forces such that Newton's Second Law is satisfied (Axiom P6). A system with these properties is called in CPM "a system of particle mechanics." If, in addition, the set of forces in the system satisfies Newton's Third Law, the system is called in CPM "Newtonian." In NM mass is defined. If a set of numbers, (mi), can be found (its existence is not assumed), such that the system of particles preserves its total momentum and angular momentum when these numbers, (mi), are interpreted as the mi that appear in the usual definitions of momentum and angular momentum, then the mi are defined as the masses of the particles, and the motion of the system is defined as isolated. Finally, Newton's Second Law is used to define the resultant force on each particle. Hence, the "isolated motion" of NM is a "Newtonian system" of CPM and vice versa. The "forces" of CPM are the "component forces" of NM. In the latter only "balanced" forces in the sense of CPM are considered, since only isolated systems are under discussion. In NM, the component forces are defined as numbers having certain properties. In general, their existence is guaranteed, but not their uniqueness. With respect to masses, the reverse holds: given the paths of the particles and the frame of reference, the masses in NM are, in general, unique if they exist. ("In general" is meant in the sense that a matrix "in general" is not singular.) 2. Syntactical and Semantical Aspects of Axiom Systems. These brief comments may provide the reader with a guide to the similarities and differences between the two axiomatizations. But what is their significance? What distinguishes the two axiomatizations at a more fundamental level is the difference in purposeand the difference in purpose reveals itself in the selection of undefined primitives. The authors of CPM state that: "Our sole aim has been to present an old subject in a mathematically rigorous way." My aim in NM was to present an old subject in a mathematically rigorous and operationally meaningful way.