  The origin of this paper starts with the observation by Yang Mills that what
state represents a proton in isospin space at one location does not determine
what state represents a proton in isospin space at another location. This is
accounted for by the presence of a unitary gauge transformation operator,
$U(y,x)$, between vector spaces at different locations. This operator defines
the notion of same states for vector spaces at different locations. If $\psi$
is a state in a vector space at $x$ then $U(y,x)\psi$ is the same state in the
vector space at $y$. Vector spaces include scalar fields in their axiomatic
description. These appear as norms, closure under vector scalar multiplication,
etc. This leads to a conflict: local vector spaces and global scalar fields.
Here this conflict is removed by replacing global scalar fields with local
scalar fields. These are represented by $\bar{S}_{x}$ where $x$ is any location
in Euclidean space or space time. Here $S$ represents the different type of
numbers, (natural, integers, rational, real, and complex). The association of
scalar fields with vector spaces and the Yang Mills observation raises the
question, What corresponds to the Yang Mills observation for numbers? The
answer is that two different concepts, number and number meaning or value, are
conflated in the usual use of mathematics. These two concepts are distinct.
