Early in this chapter, Prof. Banchoff discusses the notion of the ancient Greeks that through their study of Geometry, they were uncovering the natural laws of the world, where the gods built the universe with their Olympian compasses and straightedges. It follows, then, that their description of theorems and postulates is a process of uncovering what was already true. With the whole of B3d being devoted to legitimizing higher-dimensional geometry, it would seem fair within the context to say that people like Grassmann and Riemann were, likewise, discovering the rules of nature. And that is precisely what is implied on page 195, which mentions the discovery of the Klein bottle. However, on the preceding page, August Moebius is discussed, along with his invention of the Mobius strip. Why the distinction?
The difference between invention and discovery could be called a "supratheme" (forgive my butchering of ancient tongues, but I mean to say a theme which is not within a discipline but rather addresses that discipline from an external reference frame) of mathematics, and certainly of B3D. Do mathematicians really discover things? On a certain level, it is clear that they do not, for, as we all know, there are many possible means of representing mathematical concepts and operations. Many or all of these systems can, however, be mathematically associated and demonstrated to use essentially equivalent forms of logic; perhaps the mathematician invents a language to describe a discovery? With respect to higher dimensions, one is allowed to wonder if mathematicians are really discovering or merely inflating their inventions, and whether these processes are in fact different at all.
Prof. Banchoff's response.