Observer's mathematics - mathematics of relativity

Abstract

Observer dependent ascending chain of embedded sets of decimal fractions and their Cartesian products is considered. For every set, arithmetic operations are defined (these operations locally coincide with standard operations), which transform every set into a local ring. The basic problems of Algebra, Geometry, Topology, Logic, and Functional Analysis are considered for this chain. Definition of Dimension of these sets is introduced. In particular, the dimension of each of these sets is greater than or equal to seven. Euclidean, Lobachevsky, and Riemannian Geometries become particular cases of the developed Geometry, although many others are possible. For example, we proved that two lines in a plane may intersect in 0 (without being parallel in the usual sense), 1, 2, 10, or even 100 points. Two of the classical Geometries depend on a particular neighborhood of a given line. For example, Euclidean Geometry works in sufficiently small neighborhood of the given line, but when we enlarge the neighborhood, Lobachevsky Geometry takes over. Developed Topology gives birth to Time, and Time becomes a function of Space. Also, the Axiom of Choice becomes invalid in the new model of Mathematics. The application of the new model to Einstein's special physical theory of relativity is considered. The existence of Time and Space quantums is proved. We also construct a new system of coordinate transformations that substitute Lorenz transformations. We also consider the application of the new model to data-mining. © 2006 Elsevier Inc. All rights reserved.