Grassmann's Dialectics and Category Theory

Grassmann in his philosophical introduction describes the two-fold division of formal sciences, that is, the science of thinking, into dialectics and mathematics.  He briefly describes dialectics as seeking the unity in all things, and he describes mathematics as the art and practice of taking each thought in its particularity and pursuing it to the end.  There is a need for an instrument which will guide students to follow in a unified way both of these activities, passing from the general to the particular and from the particular to the general.

I believe that the theory of mathematical categories (which was made explicit 50 years ago by Eilenberg and Mac Lane), can serve as such an instrument.  It was introduced and designed in response to a very particular question involving passage to the limit in calculating cohomology of certain portions of spheres, but this particular calculation necessitated an explicit recognition of the manner in which these spaces were related to all other spaces and, in particular how their motion might induce other motions.  In other words, category theory was introduced (and still serves) as "a universal geometrical calculus."