Approximative Reduction by Completion of Empirical Uniformities

Abstract

In physics almost all important examples of theory reduction are of an approximative nature, for instance the reductions of nonrelativistic to relativistic theories of spacetime and dynamics, or of classical to corresponding quantum theories. To formalize the idea of theory approximation one needs a mathematical structure with a certain type of convergence — the Cauchy convergence, which is defined on a uniform structure (comp. [9], [10], [11]). This convergence type reflects the general fact that the approximated structures or models of the reduced theory are not models of the reducing theory, i.e. the approximation process takes you out of the approximating theory, as it were. For instance in the wellknown reduction example of Kepler’s theory of planetary motion to Newton’s theory of gravitation, Keplerian orbits are not solutions of Newton’s differential equations.