This question seems really controversial for some reason, but I don't understand why (most of the time every upvote has been balanced out by some determined downvoter, now it's in the negatives).

The question has been a genuine source of confusion of mine for a really long time, and a major part of that confusion was the effort of integrating all of the sub-answers together into one consistent answer. (Basically everything is multilinear algebra, the study of tensor algebras, and in the case of objects meant to capture geometric intuition or used most frequently in mathematical physics, they are over the field of real numbers.)

What is the "taxonomy" or "hierarchy" (partial ordering) of algebraic objects used to attempt to capture geometric intuition?

I.e. sure I could have asked twenty questions and then tried to put the pieces together myself, but my greatest confusion was in putting the pieces together, and it's generous to assume that someone could ask 20 questions about similar topics and expect them all to be answered to the same level of detail or accuracy.

Is it controversial because people feel it is a "soft question" (I would actually disagree with that characterization to some extent) or because it is a "big list"? (again, it seems like most of these are closely related and should be possible to fold into each other easily, plus I specifically state I'm not concerned with extensive justification of the answers, just what the answer is).

The reason why I haven't made the question community wiki yet is (1) I don't know how to, (2) I don't know when it is appropriate to do so.

To the extent that I felt it necessary/appropriate to answer my own question, this does seem to suggest that community wiki would be appropriate, but then again, I don't quite understand the purpose of community wiki.


1 Answer 1


Short answer: I don't see how making the question on main "community wiki" would help.

The purpose of Community Wiki is not to answer your own question. You are welcome to provide excellent answers to your own questions without it being made CW.

I hear some conflicting strains in the meta question posed above. On one hand you say you really want a unified answer to what might well be twenty ordinary questions, yet you feel "it necessary/appropriate to answer" your own question. Are you perhaps more interested in attention than in learning?

Another tension I sense in your framing of the Question is between tackling a very broad topic and trying to characterize it narrowly as a matter of "taxonomy" or "hierarchy", as if a canonical answer can be given.

Is there a single hierarchy or taxonomy of "algebraic objects" used to "capture geometric intuition"? Certainly there are axiomatic systems that capture the "intuition" of geometry in algebraic terms, but even the very successful approaches are attractive for reasons of concision and adequacy, not because there are no alternatives.

I believe there are aspects of mathematics that we can commonly acknowledge as "geometric intuition", but I expect this becomes personalized, if not substantially subjective, as researchers apply it in varied and deeper arenas.

For these reasons I suspect that twenty separate questions would make a better, more lasting contribution to Math.SE. This finer granularity could more closely match the ordinary size and detail of Answers. While it is likely that not all twenty Questions would initially get equally good treatment in this divided format, they would all have opportunity for improvement (curation) over time. I would find it daunting to attempt to improve an Answer combining all twenty parts.

  • $\begingroup$ I thought the question might be supposed to be community wiki because an answer that addresses all points would be long (although not necessarily; see the excellent answer provided by another user), not because I answered it. I answered it because I was (1) finding some of the answers based based on the comments given to the question and wanted to leave them there for future reference, (2) thought that any answer might be too long and that would dissuade people from trying to answer. I know how to make my answer community wiki but I don't know how to make the question community wiki. $\endgroup$ Jun 30, 2016 at 1:23
  • $\begingroup$ Also in my question I gave a specific list of algebraic objects which can be used to capture geometric intuition; I just left the title vague so anyone could add whatever else they felt also fit in well. $\endgroup$ Jun 30, 2016 at 1:24
  • $\begingroup$ There was a checkbox to make my answer community wiki (which I just did) but I don't think I can make the question itself community wiki (there was no checkbox when I clicked to edit it) $\endgroup$ Jun 30, 2016 at 1:27
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    $\begingroup$ I suspect your ability to post a CW Question is limited to the original posting, because the implication of a CW Question is that all Answers are automatically CW. Perhaps it is something a Moderator can accomplish after the fact, because in the past when posts were auto-converted to CW, a Moderator could be asked to undo that. $\endgroup$
    – hardmath
    Jun 30, 2016 at 1:28
  • $\begingroup$ At least once I have made a self-answer to a Question community wiki, for the purpose of providing some background material that could expedite responses by other Readers. Perhaps that fits your situation to some degree. $\endgroup$
    – hardmath
    Jun 30, 2016 at 13:45

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