Mathematics -- an imagined tool for rational cognition
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.
Cite
@article{arxiv.2306.03909,
title = {Mathematics -- an imagined tool for rational cognition},
author = {Boris Čulina},
journal= {arXiv preprint arXiv:2306.03909},
year = {2025}
}