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Re: Euclids postulates and non-Euclidean geometry
Posted:
Aug 11, 2007 2:18 PM
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Note that you're allowed to vary more than just the 5th
postulate.
There're also the definitions you might want to play
with.
Consider points, lines and planes all having the
same dimensions as any regular solid, but playing
the same roles, as vertices, edges and polygons --
a "geometry of lumps."[1]
Polyhedra provide the context, with any lower dimensions
conceptually impossible in some Kantian sense (not to
worry -- we still have flat surfaces and singularities,
are just less fixated on 'infinitely' as a attractive
attribute of anything).[2]
The above namespace'd be "Non-Euclidean" by today's
standards as surely as variants on the 5th Postulate.
Kirby
[1] such a "geometry of lumps" is proposed in Karl Menger's
"The Theory of Relativity and Geometry" anthologized in
"Albert Einstein: Philosopher ? Scientist" by Paul Schilpp, 1949.
[2] RBF's Synergetics, a metaphysics available on-line,
explores in this direction.
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