Dimensional Scaling and Geometric Epistemology: A Unified Mathematical Framework Bridging Algebraic-Geometric Duality and Symbolic Etymology

Contemporary mathematics education systematically bifurcates algebra and geometry, reducingspatial intuition into the mechanical manipulation of formal symbols. This paper proposes aunified epistemological framework grounded in 'Dimensional Scaling'—the perspective that allcore mathematical operations, from elementary arithmetic to multi-variable calculus, representcontinuous expansions of a one-dimensional baseline into higher-dimensional spaces and dynamicstate-sweeps. By integrating René Descartes' analytic duality with an etymological reconstructionof notation (e.g., lim, ∫, d/dx), we demonstrate that mathematical anxiety and 'symbol overload'stem not from cognitive limitations, but from a pedagogical detachment of symbols from their intuitive spatial origins. Furthermore, this paper critiques the industrial reliance on binary, single-answer evaluation, asserting that genuine mathematical literacy requires assessing the structural integrity and spatial logic of a learner's reasoning. Ultimately, we call for a paradigm shift inteacher expertise: transitioning from procedural computational technicians to philosophicalarchitects capable of presenting mathematics as a living, multi-dimensional language fordeciphering reality.