I have mentioned this before, but it keeps haunting me.
If geometry did not exist.
Could you invent it with mathematics alone?
And if you could do that...
Why would you?
For instance: A triangle can be defined mathematically in different ways,
but without the inherently geometric presentations of lines and angles, it
seems that all you could generate is a description of a set of values which
have the same relation as the values which would be present if a geometric
shape were measured or sampled from optical or tactile detections.
That is not to say that the list of mathematical definitions which satisfy
triangularity (a^2 + b^2 = c^2 for example), even an exhaustive list, would
suggest anything like the visible presence of a shape. All of the
mathematics can be done completely in the dark, and no realism of points,
plots, displays, manifolds, topologies, etc, ever need to literally appear
to anything. So why do they?
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