The Mathematical Representation of Physical Entities

Mathematics comprises abstract operations upon abstract elements. For the application of mathematics to the sciences, we not only must know the rules for manipulating symbols, but also must define the correspondences between the mathematical abstractions and the concepts to which application is made. Certain basic postulates about physical observables yield the structure of their mathematical representation. An understanding of this structure yields an understanding of measure equations, quantity equations, measurements units, abstract units, and the mathematical nature of dimensional analysis. Wrting equations in a dimensionally homogeneous form is often convenient, but not necessary; sometimes an inhomogeneous formulation is more useful.