The no information at a distance principle and local mathematics: some effects on physics and geometry

Local mathematics assumes the existence of number structures of different types, vector spaces, etc. localized at each space time point. Relations between number structures at different locations are based on two aspects: distinction between two so far conflated concepts, number and number value and the "No information at a distance" principle. This principle forbids the choice of the value of a number at one location to determine the value of the same number at another location. Value changing connections, related to a real valued field, $g,$ move numbers between structures at different locations. The effect of the $g$ field, or its exponential equivalent, $g(y)=e^{\alpha(y)},$ on numbers extends to other mathematical structures, vector spaces, etc. The presence of $\alpha$ affects theoretical descriptions of quantities in physics and geometry. Two examples are described, the effect on the Dirac Lagrangian in gauge theory, and the effect on path lengths and distances in geometry. The gradient field of $\alpha$, $\vec{A},$ appears in the Lagrangian as a spin $0$, real scalar field that couples to the fermion field. Any value for the mass of $\vec{A}$ is possible. The lack of direct experimental evidence for the presence of the $g$ or $\alpha$ field means that the field must be essentially constant within a local region of the cosmological universe. Outside the local region there are no restrictions on the field. Possible physical candidates, (inflaton, dark matter, dark energy) for $\alpha$ are noted.