Local mathematics and scaling field: effects on local physics and on cosmology

Abstract: 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.