A Geometric–Logical Reconsideration of Mathematical Foundations: Implications for Logic, Science, and Artificial Intelligence

This paper develops a structural analysis of the foundations of mathematics centered on a distinction between representation and approximation. It introduces a criterion of structural adequacy according to which a representation must preserve the defining structural feature of the domain it represents. Applying this criterion, the paper argues that standard set-theoretic and limit-based frameworks treat infinity as a completed magnitude and continuity as constructed from discrete elements, and therefore provide approximations rather than representations of unboundedness and continuous structure. Infinity is interpreted as unboundedness rather than as a quantitative extension of the finite, and continuity is treated as primitive, though dependent upon unboundedness. On this basis, discrete constructions are understood as operational tools rather than as foundational representations. The analysis further develops a distinction between logical classification and mathematical construction, and examines its implications for scientific reasoning. It argues that mathematics functions as a conditional instrument within science, while logic provides the basis for classification and inference. This analysis suggests that foundational indeterminacies arise not only from incomplete axioms, but from limitations inherent in discrete representational frameworks, and that continuous geometric approaches provide a more adequate basis for representing, rather than merely approximating, both finite and unbounded domains.