Orthogonality as a Structural Selection in the Cartesian Representation of Space

This article examines the structural role of orthogonality in the Cartesian representation of Euclidean space. In classical analytic geometry, coordinate systems are typically constructed using mutually perpendicular axes, a convention that has become so widespread that it is often perceived as an intrinsic property of geometric space itself. The present work proposes a reinterpretation of this orthogonality as a privileged structural selection within the Euclidean representational regime rather than as an ontological necessity of space. By distinguishing between linear independence (an algebraic property) and orthogonality (a metric relation defined through an inner product), the paper shows that directional independence does not inherently depend on the right angle. Orthogonality emerges only after the introduction of a metric structure. The article also examines the historical development of coordinate systems and vector analysis, discussing the roles of Descartes, Lamé, Hamilton, and Gibbs in shaping the mathematical framework in which orthogonal coordinate systems became standard. Particular attention is given to the metric dependence of orthogonality and to the structural simplifications produced by orthogonal bases. Within this perspective, Cartesian orthogonality can be interpreted as the result of a representational angular collapse, whereby a continuum of possible directional relations is represented through a specific angular configuration that simplifies both the metric structure and algebraic computations. This work forms part of a broader research program investigating structural restrictions in classical mathematical representations, including previous analyses of the real line and quasi-numerical structures. The goal of this program is to clarify how certain canonical mathematical frameworks arise as stabilized representational regimes within more general conceptual possibilities. This work forms part of the research program Model of General Quasi-Coherence (MGQC).