Functional Conceptual Substratum as a New Cognitive Mechanism for Mathematical Creation

TL;DR

This paper introduces the concept of functional conceptual substratum as a new cognitive mechanism in mathematical creation, formalizes it, and explores its implications for proofs, definitions, and inference rules.

Contribution

It provides the first formalization of the functional conceptual substratum and links it to classic notions, offering new inference rules for mathematical definitions.

Findings

Formalization of functional conceptual substratum

Relation to primitive positive definability and Diophantiveness

Soundness and completeness of new inference rules

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.