Functional Conceptual Substratum as a New Cognitive Mechanism for Mathematical Creation

Abstract

We describe a new cognitive ability, i.e., functional conceptual substratum, used implicitly in the generation of several mathematical proofs and definitions. Furthermore, we present an initial (first-order) formalization of this mechanism together with its relation to classic notions like primitive positive definability and Diophantiveness. Additionally, we analyze the semantic variability of functional conceptual substratum when small syntactic modifications are done. Finally, we describe mathematically natural inference rules for definitions inspired by functional conceptual substratum and show that they are sound and complete w.r.t. standard calculi.