Towards a Coherent Theory of Physics and Mathematics: The Theory-Experiment Connection

The problem of how mathematics and physics are related at a foundational level is of much interest. One approach is to work towards a coherent theory of physics and mathematics together. Here steps are taken in this direction by first examining the theory experiment connection. The role of an implied theory hierarchy and use of computers in comparing theory and experiment is described. The main idea of the paper is to tighten the theory experiment connection by bringing physical theories, as mathematical structures over C, the complex numbers, closer to what is actually done in experimental measurements and computations. The method replaces C by C_{n} which is the set of pairs, R_{n},I_{n}, of n figure rational numbers in some basis. The properties of these numbers are based on the type of numbers that represent measurement outcomes for continuous variables. A model of space and time based on R_{n} is discussed. The model is scale invariant with regions of constant step size interrupted by exponential jumps. A method of taking the limit n to infinity to obtain locally flat continuum based space and time is outlined. Possibly the most interesting result is that R_{n} based space is invariant under scale transformations which correspond to expansion and contraction of space relative to a flat background. Also the location of the origin, which is a space and time singularity, does not change under these transformations. Some properties of quantum mechanics based on C_{n} and on R_{n} space are briefly investigated.