Universal Geometric Algebra

Alfred North Whitehead promoted the idea of UNIVERSAL ALGEBRA in his monumental treatise of 1897 [1]. He proposed two candidates for this lofty title, the algebra of Boole and Grassmann’s Algebra of Extension. Boolean algebra has since secured universality status in Set Theory and Symbolic Logic, although only the former is universally known and used by mathematicians. However, Whitehead’s work on Grassmann Algebra, which ironically is much the larger portion of his treatise, has been almost totally ignored. Grassmann algebra has been making a comeback among mathematicians in recent decades, primarily in the guise of differential forms. But this has involved only the half of Grassmann’s algebra generated by his progressive product, while his regressive product remains unappreciated along with Whitehead’s elaborate treatment of it. Of course, the regressive product was designed to perform an important mathematical function which must be handled by other means in modern mathematics. As Gian-Carlo Rota and coworkers ([2],[3]) have vigorously argued, the result has been a step backward in conceptual clarity and computational efficiency. I agree that the regressive product should be revived, but I wish to go further. I claim that Grassmann and Whitehead were just one step away from a mathematical system that truly deserves to be regarded as a UNIVERSAL GEOMETRIC ALGEBRA. That system is known in mathematics as Clifford algebra. However, the true universality of Clifford algebra has remained unrecognized, even by mathematicans specializing in the subject. The main reason, I suppose, is that the demonstration of universality requires a wholescale reorganization and redesign of mathematics, integrating into a single mathematical system such superficially disparate systems as quaternion calculus, differential forms