Non-commutative Calculus and Discrete Physics

This paper is an expanded version of [28] and [30] where there is presented an introduction to a point of view for discrete foundations of physics. In taking a discrete stance, we find that the initial expression of physical observation naturally occurs in a context of non-commutative algebra. In this way a formalism similar to quantum mechanics occurs first, but not necessarily with the usual interpretations. By following this line we show how the outlines of the well-known forms of physical theory arise first in non-commutative form. The exact relation of commutative and non-commutative theories raises a host of problems.