Structural Realism with a Foundation in Reality: Position, Conceptual Groundwork, and Historical Verification

This paper proposes and defends a form of structural realism with a materialist character: mathematical structures are neither abstract entities in a Platonic sense nor pure symbolic conventions. Rather, they are formal systems constructed by human subjects through abstractive relationships and mathematical language, a process of construction constrained by the recalcitrance of objective reality. The objectivity of mathematics is rooted in practical constraints, the constraints of logic as the minimum operational condition of a mathematical system, and the structural consistency formed through the evolution of mathematical language. By introducing a tripartite classification of "real mathematical structures," "virtual mathematical structures," and "unconfirmed mathematical structures," supplemented by the distinctions of "invalid paradoxes" and "boundary-incomplete structures," this paper systematically clarifies the legitimacy of actual infinity, the pluralistic truth of non-Euclidean geometries, and the cognitive function of set-theoretic paradoxes. On this basis, the paper offers a unified theoretical treatment of the status of non-standard models, the classification of large cardinal axioms, and the hierarchical structure of logic. Three historical cases—complex numbers, non-Euclidean geometry, and set-theoretic paradoxes—further verify an elevation mechanism from "intra-systemic truth" to "truth of real consistency," demonstrating that mathematical truth gradually converges through practical feedback and logical constraints.