Authors: Adriaan van der Walt
The Leibnizean cosmology, where both space and discrete objects (including particles) are assumed to be formed from continuous entities called infinitesimals, replaces the Euclidean Cosmology, where continuous space is assumed to be formed from discrete entities called points. Infinitesimals as well as infinitesimal numbers are defined in this monograph and an Arithmetic for non-standard analysis is developed by generalising the concept of real number to a wider class of numbers, called here the Cauchy numbers, and by rearranging Number Theory somewhat. This is motivated in part one by showing that Cantor’s famous diagonal proof, which is an algebraic formulation of Euclidean Cosmology, rests on the fallacy that an infinite decimal fraction can be identified by specifying its finite digits. The argument of this part includes a proof that the set of equivalence classes of Cauchy sequences is countable. In part two the Leibnizean model for the infinite divisibility of space is developed by introducing the concepts of infinitesimal and infinitesimal number from the context of Calculus. An Arithmetic for Cauchy numbers is developed, including an interpretation of L’Hospital’s rules and a description of the Real Continuum. The concept of cascades of infinitesimals as directed sets is introduced and the Fundamental Theorem of the Calculus is studied as a Net defined on such a cascade. In part three the Leibnizean Cosmology is argued to be in line with the ideas of Parmenides and that space, in this cosmology, is ‘fuzzy’ - thus clarifying the paradox of the arrow. It is also pointed out that, with particles suitably defined as infinitesimals, Parmenides’ ideas are vindicated because everything is one, and motion is only an illusion because nothing comes into being where it did not exist before. Furthermore, some intractable problems of Physics, like the particle/wave duality and action at a distance, are pointed out to be direct consequences of the Euclidean Cosmology and that they all but disappear in the Leibnizean Cosmology. In part four it is pointed out that when the role of points as building blocks of space is discarded, points can be used comfortably in the Leibnizean model as indicators of locations in space. Thus Mathematics can once more become a canonical model, but without the parts that depend on the Euclidean properties of points; e.g. open and closed sets on the real line.
Comments: 33 Pages. This monograph is of a fundamental nature and its ideas should be accessible to any reader with a post graduate background in Mathematics and/or Physics (hence no references are given).
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