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Effects of local availability of mathematics (LAM) and space time dependentnumber scaling on physics and, especially, geometry are described. LAM assumesseparate mathematical systems as structures at each space time point. Extensionof gauge theories to include freedom of choice of scaling for numberstructures, and other structures based on numbers, results in a space timedependent scaling factor based on a scalar boson field. Scaling has no effecton comparison of experimental results with one another or with theorycomputations. With LAM all theory expressions are elements of mathematics atsome reference point. Changing the reference point introduces (external)scaling. Theory expressions with integrals or derivatives over space or timeinclude scaling factors (internal scaling) that cannot be removed by referencepoint change. Line elements and path lengths, as integrals over space and/ortime, show the effect of scaling on geometry. In one example, the scalingfactor 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. Otherexamples have spherically symmetric scaling factors about some point, $x.$ Inone type, a black scaling hole, the scaling factor goes to infinity as thedistance, $d$, between any point $y$ and $x$ goes to 0. For scaling whiteholes, 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 thanthe unscaled distance from $z$ to $x.$

Effects of gauge theory based number scaling on geometry