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, θ, 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, ∪ x, 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 θ into the local maps, is described. The lack of experimental evidence for θ so far shows that the coupling constant of θ to matter fields must be very small compared to the fine structure constant. Also the gradient of θ 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 θ or its gradient in regions of space and/or time that are far away from our local region.