The prospective question:
I quote from Petsche's Hermann Graßmann: Biography (emphasis mine):
The mathematical part of the book begins with the conception of the “General Theory of Forms”. Starting with a perspective on mathematics as a theory of forms, Graßmann analyses in the most abstract way possible the general structures of concrete “conjunctions of forms”. Here, he places special emphasis on “elementary conjunctions”, demanding they have module properties, that is, associativity, commutativity and an inverse and neutral element. The so-defined conjunction of the first order, or “formal addition”, is then followed up by an investigation of a conjunction of the second order (“formal multiplication”), for which he only requires distributivity with respect to formal addition. Graßmann directly posits the validity of the module properties for formal addition and distributivity for formal multiplication as the principles for constructing these conjunctions: “This generally is the way”, he wrote, “that initially, that is when no species of conjunction is yet given, such a conjunction of next higher order is defined.”
Since Graßmann does not require the forms generated by conjunctions of the second order to be embedded in the fundamental domain, he can use this form of conjunction for the formal generation of new mathematical objects in the further course of the text.
After Graßmann has laid down the foundation for all mathematical disciplines by presenting these uniquely generalized group-theoretical and structural abstractions he starts with the actual presentation of his new mathematical discipline.
What is the modern terminology for Grassmann's "General Theory of Forms"? How successful was this theory? What research work has been done in order to continue this line of thinking? Which resources could I acquaint myself with in order to answer these questions?
Is this an appropriate question for MO? I would really like someone who knows their stuff to answer it for me. I attempted to put a 200 rep bounty for it on M.SE, but people are really hung up on relating this to geometric algebra immediately, when it seems clear to me that the author is attempting to communicate that Grassmann attempted to explore something more general, before moving onto geometric algebra in particular. Am I misunderstanding the passage?