58 Программный реализм в физике и основания математики * Часть 1: Классическая наука

Abstract

Why mathematics can adequately describe the physical reality? How one can rationally explain the “unreasonable effectiveness” of mathematics in physics and other natural sciences? In the fi rst part of this work we propose an answer to this question within the context of Classical physics and mathematics. Following Hilbert we distinguish between the real and the ideal semantics of syntactic operations in mathematics and show how the excessiveness of mathematical syntax allows one to complement the real semantics with the ideal one. Then on the basis of our analysis of Kepler’s astronomy we introduce the notion of realistic physical theory and show that the “unreasonable effectiveness of mathematics” in such theories amounts to the possibility (not granted a priori but often realized in experiments) to replace a part of the standard ideal semantics of mathematical syntax with an appropriate real semantics