On the incompleteness of classical mechanics

Classical mechanics is often considered to be a quintessential example of a deterministic theory. I present a simple proof, using a construction mathematically analogous to that of the Pasadena game (Nover and Hájek, 2004), to show that classical mechanics is incomplete: there are uncountably many arrangements of objects in an infinite Newtonian space such that, although the system’s initial condition is fully known, it is impossible to calculate the system’s future trajectory because the total force exerted upon some objects is mathematically undefined. It is then shown how variations of this discrete system can be obtained which increasingly approximate a uniform mass distribution, similar to that underlying a related result, due to von Seeliger (1895). It is then argued that this incompleteness result, as well as that presented by the Pasadena game, has no real philosophical significance as it is a mathematical pseudoproblem shared by all models which attempt to aggregate infinitely many numerical values of a certain kind.