  Local availability of mathematics and number scaling provide an approach to a
coherent theory of physics and mathematics. Local availability of mathematics
assigns separate mathematical universes, U_{x}, to each space time point, x.
The mathematics available to an observer, O_{x}, at x is contained in U_{x}.
Number scaling is based on extending the choice freedom of vector space bases
in gauge theories to choice freedom of underlying number systems. Scaling
arises in the description, in U_{x}, of mathematical systems in U_{y}. If a_{y}
or \psi_{y} is a number or a quantum state in U_{y}, then the corresponding
number or state in U_{x} is r_{y,x}a_{x} or r_{y,x}\psi_{x}. Here a_{x} and
\psi_{x} are the same number and state in U_{x} as a_{y} and \psi_{y} are in
U_{y}. If y=x+\hat{\mu}dx is a neighbor point of x, then the scaling factor is
r_{y,x}=\exp(\vec{A}(x)\cdot\hat{\mu}dx) where \vec{A} is a vector field,
assumed here to be the gradient of a scalar field. The effects of scaling and
local availability of mathematics on quantum theory show that scaling has two
components, external and internal. External scaling is shown above for a_{y}
and \psi_{y}. Internal scaling occurs in expressions with integrals or
derivatives over space or space time. An example is the replacement of the
position expectation value, \int\psi^{*}(y)y\psi(y)dy, by
\int_{x}r_{y,x}\psi^{*}_{x}(y_{x})y_{x}\psi_{x}(y_{x})dy_{x}. This is an
integral in U_{x}. The good agreement between quantum theory and experiment
shows that scaling is negligible in a space region, L, in which experiments and
calculations can be done, and results compared. L includes the solar system,
but the speed of light limits the size of L to a few light years. Outside of
$L$, at cosmological distances, the limits on scaling are not present.
