2022/05/27 by Nadir Maaroufi, Maaroufi, Nadir, El Hassan Zerouali +1 · 1 citation
Mathematics · Medicine · Physics and Astronomy · #Biofield Effects and Biophysics #Complex dimension #Computer science #Dimension (graph theory) #Dimension function #Dimension theory (algebra) #Effective dimension #Einstein #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Geometry #Hausdorff dimension #History and Philosophy of Physics (physics.hist-ph) #Inductive dimension #Mathematical analysis #Mathematics #Minkowski–Bouligand dimension #Physics #Point (geometry) #Pure mathematics #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Space Science and Extraterrestrial Life #Theoretical physics #math.FA #physics.hist-ph #quant-ph
paper · pdf · doi:10.48550/arxiv.2206.06271
published in arXiv (Cornell University) (Cornell University) · 72 pages, 4 figures
arxiv created 2022/05/27 · openalex publication_date 2022/05/27 · arxiv updated 2022/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.