Methodology of Numerical Computations with Infinities and Infinitesimals
arXiv:1203.3165
Abstract
A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The developed approach has a pronounced applied character and is based on the principle `The part is less than the whole' introduced by Ancient Greeks. This principle is used with respect to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The point of view on infinities and infinitesimals (and in general, on Mathematics) presented in this paper uses strongly physical ideas emphasizing interrelations holding between a mathematical object under the observation and tools used for this observation. It is shown how a new numeral system allowing one to express different infinite and infinitesimal quantities in a unique framework can be used for theoretical and computational purposes. Numerous examples dealing with infinite sets, divergent series, limits, and probability theory are given.
This paper presents Honorable Lagrange Lecture delivered in Turin on 16 April 2010. It consists of 20 pages, 1 figure