Mathematics as Cognitive Formalization: An Epistemological Account of Its Scientific Effectiveness

Mathematics is discovered only in the sense that perception presents patterns to cognition; what we formalize is never reality itself, only the trace our minds can grasp. This paper argues that mathematics is a formal extension of human cognition rather than a mind-independent structure of reality. Mathematical structures emerge from evolved cognitive capacities for abstraction, categorization, recursion, and symbolic manipulation. Their effectiveness in empirical science arises from a selective alignment between these cognitive capacities and the regularities of the world, rather than from a pre-existing mathematical ontology. By addressing classical objections—including Platonism, the “unreasonable effectiveness” of mathematics, indispensability, and results beyond human intuition—this work situates mathematics as a cognitively mediated, indispensable tool for modeling and understanding the empirical world. It emphasizes that all scientific knowledge is produced through the interaction between observer and observed, constrained by the limits of cognition, and that epistemic modesty is essential for interpreting the apparent universality of mathematical structures.