  Effects of local availability of mathematics (LAM) and space time dependent
number scaling on physics and, especially, geometry are described. LAM assumes
separate mathematical systems as structures at each space time point. Extension
of gauge theories to include freedom of choice of scaling for number
structures, and other structures based on numbers, results in a space time
dependent scaling factor based on a scalar boson field. Scaling has no effect
on comparison of experimental results with one another or with theory
computations. With LAM all theory expressions are elements of mathematics at
some reference point. Changing the reference point introduces (external)
scaling. Theory expressions with integrals or derivatives over space or time
include scaling factors (internal scaling) that cannot be removed by reference
point change. Line elements and path lengths, as integrals over space and/or
time, show the effect of scaling on geometry. In one example, the scaling
factor goes to 0 as the time goes to 0, the big bang time. All path lengths,
and values of physical quantities, are crushed to 0 as $t$ goes to 0. Other
examples have spherically symmetric scaling factors about some point, $x.$ In
one type, a black scaling hole, the scaling factor goes to infinity as the
distance, $d$, between any point $y$ and $x$ goes to 0. For scaling white
holes, the scaling factor goes to 0 as $d$ goes to 0. For black scaling holes,
path lengths from a reference point, $z$, to $y$ become infinite as $y$
approaches $x.$ For white holes, path lengths approach a value much less than
the unscaled distance from $z$ to $x.$
