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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?

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    $\begingroup$ I think this would be a very interesting question. I only hope there's someone who feels he/she could do the question justice, most preferably someone who has read Grassmann's writings and has thought deeply about them. (If Bill Lawvere were reading this question, I'd love to hear his response.) $\endgroup$ – Todd Trimble Jun 6 '14 at 17:17
  • $\begingroup$ @ToddTrimble Would you recommend that I request a migrate from M.SE to MO, or would you recommend that I simply rewrite the question on MO? $\endgroup$ – user89 Jun 6 '14 at 18:53
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    $\begingroup$ I seem to recall that a question of this form on MO was closed some time ago, though I am not computer-literate enough to track it down. $\endgroup$ – Andy Putman Jun 6 '14 at 19:11
  • $\begingroup$ @AndyPutman You're right! I did try to post this here before.. How bizarre...I am not sure what to make of it. $\endgroup$ – user89 Jun 7 '14 at 4:59
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    $\begingroup$ @user89 : So you somehow forgot? $\endgroup$ – Andy Putman Jun 7 '14 at 18:33
  • $\begingroup$ @AndyPutman I did. Most of all though, I am not sure why I listened to you the last time. I should waited for some more feedback. $\endgroup$ – user89 Jun 7 '14 at 20:08
  • $\begingroup$ @AndyPutman What's interesting too is that the question didn't get closed, as you suggested. It got a single downvote, and I got worried, and thus deleted it myself, so that I could solicit feedback. $\endgroup$ – user89 Jun 7 '14 at 20:18
  • $\begingroup$ Might Category Theory be similar? $\endgroup$ – AmagicalFishy Dec 2 '15 at 10:55
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Of all that you present, the part that I think is most likely to be received well on MO is "What resources could I acquaint myself with in order to answer these questions?"

Even tbough you are looking for a modern version, I see this more as a history of mathematics question which domain has a few but not many fans here. Also, the proper answer is "citation index", as the research you hope exists will give nod to Grassmann's work.

I think it will fly best as a reference request, especially if you mention what you have tried to track down references and good reasons for lack of success. All the other questions serve as motivation: mention them, but make clear that the specific question you want MO to answer is about modern references. You should also explain why e.g. literature on Grassmann algebras won't do for an answer.

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    $\begingroup$ I don't say that you are misunderstanding the passage so much as putting too much hope in the generality. If you axiomatize exterior algebras, you will have a more general framework, which won't buy you much unless you take a model-theoretic or mathematical-logic perspective: the structures you study will still smell like exterior algebras with that axiomatization. $\endgroup$ – The Masked Avenger Jun 8 '14 at 20:11
  • $\begingroup$ Thanks a lot for this constructive answer. I will put some effort into doing some proper research on the matter, and maybe in the end, I will end up finding what you suspect. If not, I will then at least be able to post a better MO question. $\endgroup$ – user89 Jun 9 '14 at 1:11

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