Structural Realism on Materialist Grounds: Resistance, Practical Equivalence Classes, and the Fourfold Classification of Mathematical Structures

This paper defends a structural realism on materialist grounds. Mathematical structures are neither Platonic nor conventional: they are formal systems constructed under the resistive constraints of objective reality. Their objectivity derives from practical constraints, logical constraints, and the structural consistency emerging in language evolution. The paper introduces three innovations: resistance as a pre structural constraint, practical equivalence class as the bridge from operational feedback to mathematical structure, and a fourfold classification (real/virtual/unconfirmed/invalid) that includes permanently virtual structures. Three historical cases—complex numbers, non Euclidean geometry, and set theoretic paradoxes—verify the elevation from “truth within a system” to “truth of consistency with reality.