You're quite right, "discovery" vs. "invention" is a theme that lies right at the heart of a course on mathematical ways of thinking, and there are indeed different ways. The "discovery" people subscribe to a Platonic view that says that mathematics has an independent existence, whether or not it resides "in the mind of God". Some more humanistic thinkers consider mathematics as one of the noblest of human creations, but no more than that. The word "invention" seems a bit contrived in these circumstances, since I for one think of gadgets when that word is mentioned. Although I am in the discovery camp, I still think that there is a great nobility to the human effort. I also think that there are some mathematical facts that inevitably would have been found and others that might never have seen the light of day unless some particular mathematician had become intrigued with a very usual relationship that he or she had spotted almost by accident. I myself am proudest not of the theorems I happened to get to first among a crowd of people with pretty much the same goal in mind, but rather the results that are more idiosyncratic, and not at all inevitable. It is tempting to think of metaphors for this creative enterprise, but maybe it's just as good to try to observe it without dwelling on it too much. Noble it is.
Whether or not working in higher dimensions has any more or less nobility than other parts of mathematical enterprise is surely a matter of taste, and I know where I come down on that one.