  In gauge theories, separate vector spaces, Vx, are assigned to each space
time point x with unitary operators as maps between basis vectors in
neighboring Vx. Here gauge theories are extended by replacing the single
underlying set of complex scalars, C, with separate sets, Cx, at each x, and
including scaling between the Cx. In gauge theory Lagrangians, number scaling
shows as a scalar boson field, B, with small coupling to matter fields. Freedom
of number scaling is extended to a model with separate number structures
assigned to each point x. Separate collections, Ux, of all mathematical systems
based on numbers, are assigned to each x. Mathematics available to an observer,
Ox, at x is that in Ux. The B field induces scaling between structures in the
different Ux. Effects of B scaling on some aspects of physics and geometry are
described. The lack of experimentally observed scaling means that B(z) is
essentially constant for all points, z, in a region, Z, that can be occupied by
us as observers. This restriction does not apply outside Z. The effects of B
scaling on line elements, curve lengths, and distances between points, are
examined. Oz's description, using Uz in Z, of elements at points, x, outside Z,
includes scaling from x to z. Integrals over curves include scaling factors
inside the integrals. Two examples are discussed. One shows that B(t) can be
such that mathematical, physical, and geometric quantities approach zero as the
time t approaches zero. This mimics the big bang in that distances approach
zero. Examples of black and white scaling holes are described in which B(x) is
plus or minus infinity at a point x.
