Grassmann’s Dialectics and Category Theory

Abstract

In several key connections in his foundations of geometrical algebra, Grassmann makes significant use of the dialectical philosophy of 150 years ago. Now, after fifty years of development of category theory as a means for making explicit some nontrivial general arguments in geometry, it is possible to recover some of Grassmann’s insights and to express these in ways comprehensible to the modern geometer. For example, the category A of affine-linear spaces and maps (a monument to Grassmann) has a canonical adjoint functor to the category of (anti)commutative graded algebras, which as in Grassmann’s detailed description yields a sixteen-dimensional algebra when applied to a three-dimensional affine space (unlike the eight-dimensional exterior algebra of a three-dimensional vector space). The natural algebraic structure of these algebras includes a boundary operator ∂ which satisfies the (signed) Leibniz rule; for example, if A, B are points of the affine space then the product AB is the axial vector from A to B which the boundary degrades to the corresponding translation vector: ∂(AB) = B−A (since ∂A = ∂B = 1 for points). Grassmann philosophically motivated a notion of a “simple law of change,” but his editors in the 1890’s found this notion incoherent and decided he must have meant mere translations.