I have been studying Grassman and Clifford algebras a bit, and it is fascinating to see how, for example, the rules defining the inner product operator are enough to the capture something of the "relative direction" between vectors/bivectors/etc..
I am very interested in exploring a resource that discusses why this is so, and other such examples where function seems to pop out of algebraic form ("relational" abstract algebra because it tries to understand the relationship between what function pops out of an algebraic form) -- what would be a good resource for me to check out? What is this phenomenon called by mathematicians?