Abstract physical structure and a local ordering function



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In this paper physical theory is built axiomatically beginning with the fundamental concepts of physical science. It is shown that values of properties of a physical system may be represented as a basic mathematical structure and that these properties are measured by ordering functions. Physical theory is built by adding (or varying) rules (constraints) to the structure of a physical system and the ordering function(s) defined on it.This is demonstrated first by logically adding rules and constraints to the basic structure (or space) of a physical system and the ordering function, thus defining the local Riemann line element ds on a smooth manifold. As a second demonstration the rules and constraints on the structure and ordering function defining ds are relaxed (varied) and thus a new more general function [Lambda][lowered L] is defined. In order to physically interpret this new function [Lambda][lowered L], a comparison is initiated between [Lambda][lowered L] and the original equation ds. For this comparison both [Lambda][lowered L] and ds are expressed as differential 1-forms. 1-Forms are appropriate equations for this comparison because, when integrated and extremized, they generate Hamiltonian formulations. Such Hamiltonian formulations, then, are the result of defining a local ordering function on the abstract structure of a physical system.