IJPAM: Volume 76, No. 2 (2012)
Department of Physics
ST. Xavier's College
30, Mother Teresa Sarani, Kolkata, 700016, INDIA
Abstract. In this article we will review a few basic concepts of geometry and coordinatization. We will first give a precise definition of geometric continuity based on the concept of homogeneity and extension. A few logical arguments lead us to propose that a line-element can not be a collection of points. Line-elements, area-elements and volume-elements and their higher dimensional generalizations are as fundamental as points. This also lead us to a corresponding definition of dimension of a geometric continuum different from the conventional definition. We will generalize the well-known fact that rational numbers forms a dense set to real numbers. That is for real numbers only the concept of "two numbers separated by an interval" or "two numbers are same" have meaning but "two numbers adjacent" is not defined. We will illustrate the relationship of the above two aspects in the context of coordinatization. We will also state a corresponding version of the completeness of the real numbers. We will discuss in brief the the relationships of these aspects and the three spatial dimensions with the the kinematical and dynamical properties of the elementary particles and Electromagnetic fields. We will conclude this article with a few consequences of these aspects for differential geometry, classical statistical mechanics and set theory.
Received: November 12, 2011
AMS Subject Classification: 03A05
Key Words and Phrases: geometric continuum, coordinatization, dimension, phase-space, sets
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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395