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The day Johann Galle invented Neptune...

#### ...and Eli Whitney discovered the cotton gin

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

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-david stanke