Orthogonalizing the Abstract: A General Theory of Cartesian Coordinates

This paper introduces a general axiomatic theory for Cartesian coordinate systems, moving beyond their traditional foundation in Euclidean geometry and inner product spaces. The central thesis is the development of a framework for establishing coordinate systems in abstract mathematical structures, such as modules over a ring or vector spaces over finite fields, where a metric or an inner product is not naturally defined. We propose the concept of an 'abstract orthogonality operator,' which is derived from the intrinsic algebraic or topological properties of the space rather than from a pre-defined metric. This approach allows for the construction of orthogonal bases in a much broader class of mathematical objects. The methodology involves defining a set of axioms for a General Cartesian System (GCS) and demonstrating how this structure can be instantiated in various domains. We show that our general theory successfully recovers the standard Cartesian and polar coordinates in Euclidean spaces as specific cases. Furthermore, we apply the framework to construct meaningful and consistent orthogonal coordinate systems in non-Euclidean manifolds and in finite-dimensional vector spaces over finite fields, domains where traditional geometric intuition fails. The results demonstrate that the principle of orthogonality is a fundamental structural property that can be decoupled from metric-based definitions, offering new analytical tools and insights into the nature of abstract spaces. This work unifies disparate notions of orthogonality under a single theoretical umbrella and provides a robust method for imposing coordinate structures on abstract sets, thereby 'orthogonalizing the abstract.'