Abstract

The development of the real number system represents both a milestone and a cornerstone in the foundation of modern mathematics. We go further and suggest that the real number system should be completed to include the concept of direction. Some of this work has already been done by the invention of the complex numbers, quaternions, and vectors. What has been lacking, however, is a general geometric number system. In 1878, William Kingdom Clifford invented his "geometric algebra", based upon the earlier work of Grassmann and Hamilton. Geometric algebra is the completion of the real number system to include new anticommuting square roots of plus and minus one, each such root representing an orthogonal direction in successively higher dimensions. All of the usual rules of the real number system remain valid, except that the commutative law of multiplication is no longer universally valid. The book, "New Foundations in Mathematics: The Geometric Concept of Number" by the author, represents an attempt to show how many ideas of modern mathematics can be developed within this new framework, including modular number systems, complex and hyperbolic numbers, geometric algebra of Euclidean and pseudo-Euclidean spaces, linear and multilinear algebra, Hermitian inner product spaces, the theory of special relativity, representations of the symmetric group, calculus and differential geometry of n-dimensional surfaces, Lie groups and Lie algebras, and other topics.