  A stronger foundation for earlier work on the effects of number scaling, and
local mathematics is described. Emphasis is placed on the effects of scaling on
coordinate systems. Effects of scaling are represented by a scalar field,
$\theta,$ that appears in gauge theories as a spin zero boson. Gauge theory
considerations led to the concept of local mathematics, as expressed through
the use of universes, as collections of local mathematical systems at each
point, x, of a space time manifold, M. Both local and global coordinate charts
are described. These map M into either local or global coordinate systems
within a universe or between universes, respectively. The lifting of global
expressions of nonlocal physical quantities, expressed by space and or time
integrals or derivatives on M, to integrals or derivatives on coordinate
systems, is described. The assumption of local mathematics and universes makes
integrals and derivatives, on M or on global charts, meaningless. They acquire
meaning only when mapped into a local universe. The effect of scaling, by
including the effect of $\theta$ into the local maps, is described. The lack of
experimental evidence for $\theta$ so far shows that the coupling constant of
$\theta$ to matter fields must be very small compared to the fine structure
constant. Also the gradient of $\theta$ must be very small in the local region
of cosmological space and time occupied by us as observers. So far, there are
no known restrictions on $\theta$ or its gradient in regions of space and/or
time that are far away from our local region.
