Mathematics -- an imagined tool for rational cognition

TL;DR

The paper argues that mathematical models are junctions of axiom systems and partial internal interpretations of language, not external realities; mathematical objects are imagined constructs whose truths are specifications of conceptions rather than facts about the world. It analyzes natural numbers, real numbers, Euclidean geometry, group theory, and set theory to illustrate how axioms and internal interpretation generate useful tools for rational cognition, while addressing the applicability and ontological status of mathematics. The approach connects language, intuition, and practice, and situates mathematics within a broader philosophy that emphasizes internal constructs, partial interpretations, and the practical success of mathematical reasoning in science. Ultimately, mathematics is presented as an imagined, highly effective tool for understanding and navigating reality, whose value arises from its structured language and its context‑dependent applicability.

Abstract

Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are imagined objects, some of which, at least approximately, exist in our internal world of activities or we can realize or represent them there; (ii) mathematical truths are not truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it.