  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.
