Point-Dimension Theory (Part I): The Point-Extended Box Dimension

TL;DR

This paper introduces the point-dimension theory, proposing new ways to conceptualize dimension by involving a hypothetical elementary particle and entropy comparisons, extending the box dimension concept to topological vector spaces.

Contribution

It develops the novel point-extended box dimension framework, challenging traditional zero-dimensional point concept and proposing a unifying approach with potential applications in quantum mechanics and relativity.

Findings

Proposes the existence of a Dimensionad particle conferring dimension.

Extends box dimension to topological vector spaces beyond metric spaces.

Establishes a foundation for infinite-dimensional analysis.

Abstract

This article is an introductory work to a larger research project devoted to pure, applied and philosophical aspects of dimension theory. It concerns a novel approach toward an alternate dimension theory foundation: the point-dimension theory. For this purpose, historical research on this notion and related concepts, combined with critical analysis and philosophical development proved necessary. Hence, our main objective is to challenge the conventional zero dimension assigned to the point. This reconsideration allows us to propose two new ways of conceiving the notion of dimension, which are the two sides of the same coin. First as an organization; accordingly, we suggest the existence of the Dimensionad, an elementary particle conferring dimension to objects and space-time. The idea of the existence of this particle could possibly adopted as a projection to create an alternative way to unify quantum mechanics and Einstein's general relativity. Secondly, in connection with Boltzmann and Shannon entropies, dimension appears essentially as a comparison between entropies of sets. Thus, we started from the point and succeeded in constructing a point-dimension notion allowing us to extend the principle of box dimension in many directions. More precisely, we introduce the notion of point-extended box dimension in the large framework of topological vector spaces, freeing it from the notion of metric. This general setting permits us to treat the case of finite, infinite and invisible dimensions. This first part of our research project focuses essentially on general properties and is particularly oriented towards establishing a well founded framework for infinite dimension. Among others, one prospect is to test the possibility of using other types of spaces as a setting for quantum mechanics, instead of limiting it to the exclusive Hilbertian framework.