Mathematical Necessity and the Necessity Spectrum: From Euclidean Invariance to the Absolute Necessity Limit

Abstract Mathematical propositions are generally understood to possess a form of necessity distinct from empirical facts. Yet while the sum of the interior angles of a triangle is 180 degrees in the Euclidean plane, the sum of the interior angles of a triangle formed by great-circle arcs on a sphere may exceed 180 degrees. This Difference does not mean that mathematical necessity has simply been invalidated. Rather, it reveals that each proposition holds within a Structure constituted by different axioms, curvatures, Relations, and operational conditions. This study examines mathematical structuralism, the philosophical implications of non-Euclidean geometry, modal structuralism, theories of necessity based on invariance, and recent constraint-structural approaches. Existing discussions have explained how mathematical truths hold within particular Structures or possible Structures. They nevertheless differ over whether necessity should be treated as a principle prior to Structure, as a modal Relation holding within Structure, or as something identical to structural constraint itself. GGM does not establish mathematical necessity as the cause of Structure Formation or as an ontological foundation. Structural Preservation concerns the degree to which an already formed Structure maintains its Relations persistently and stably through changes and conditions, and the level at which those Relations may be recovered after damage or loss. This state of preservation becomes the object of Structural Evaluation. Necessity is neither a generative principle of Structure nor a criterion that precedes evaluation. It is a subsequent evaluative result attributed to a Structure that has already been formed and preserved. GGM accordingly presents the following order: Structure, Structural Preservation, Structural Evaluation, Necessity Spectrum, and Absolute Necessity Limit. This study analyzes the Difference between the sums of the interior angles of triangles in Euclidean and spherical geometry according to this order. The Euclidean angle sum of 180 degrees is not refuted by spherical geometry. It is attributed to a formal Relation preserved within Euclidean Structure. The sum of the interior angles of a spherical triangle likewise possesses its own formal necessity under the conditions of positive curvature and great-circle arcs. Mathematical necessity can therefore be distinguished according to the scope of the Structure within which it holds and is preserved, without being reduced to arbitrary relativism. Necessity Spectrum is the evaluative domain in which the results of Structural Evaluation appear as a continuum between Contingency and Necessity. Absolute Necessity Limit is the ideal and regulative boundary established as evaluation within Necessity Spectrum advances in the direction of Necessity. Purpose and Relation to GGM Volume I This paper is designed both as an independent study of mathematical necessity and as an accessible entry point to GGM Volume I: Generative General Methodology. Through a comparison of Euclidean and spherical geometry, it presents the sequence Structure → Structural Preservation → Structural Evaluation → Necessity Spectrum → Absolute Necessity Limit in a concrete problem context. The paper should therefore be read neither as a substitute for the original work nor as a complete presentation of the GGM system, but as a problem-centered route into its conceptual architecture. Readers seeking the official definitions, the derivational structure of the concepts, and their broader systematic relations should consult GGM Volume I. This record contains the English and Korean editions of the same preprint.