  The work is based on two premises: local availability of mathematics to an
observer at any space time location, and the observation that number systems,
as structures satisfying axioms for the number type being considered, can be
scaled by arbitrary, positive real numbers. Local availability leads to the
assignment of mathematical universes, $V_{x},$ to each point, $x,$ of space
time. $V_{x}$ contains all the mathematics that an observer, $O_{x},$ at $x,$
can know. Each $V_{x}$ contains many types of mathematical systems. These
include the different types of numbers (natural numbers, integers, rationals,
and real and complex numbers), Hilbert spaces, algebras, and many other types
of systems. Space time dependent scaling of number systems is used to define
representations, in $V_{x}$, of real and complex number systems in $V_{y}$. The
representations are scaled by a factor $r_{y,x}$ relative to the systems in
$V_{x}.$ For $y$ a neighbor point of $x,$ $r_{y,x}$ is the exponential of the
scalar product of a gauge field, $\vec{A}(x),$ and the vector from $x$ to $y.$
For $y$ distant from $x,$ $r_{y,x}$ is a path integral from $x$ to $y.$ Some
consequences of the two premises will be examined. Number scaling has no effect
on general comparisons of numbers obtained as computations or as experimental
outputs. The effect is limited to mathematical expressions that include space
or space time integrals or derivatives. The effect of $\vec{A}$ on wave packets
and canonical momenta in quantum theory, and some properties of $\vec{A}$ in
gauge theories, is described.
