Mathematics -- an imagined tool for rational cognition

TL;DR

This paper argues that mathematical objects are imagined constructs rather than external entities, and mathematics serves as a tool for rational cognition and exploration of nature, based on analyzing various mathematical models.

Contribution

It provides a philosophical analysis showing that mathematical models are internal constructs and mathematics is primarily an imagined tool for understanding the world.

Findings

Mathematical objects are imagined, not external.

Mathematical truths are about conceptions, not external reality.

Mathematics is a tool for exploring and understanding nature.

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.