
Re: Euclids postulates and nonEuclidean geometry
Posted:
Aug 11, 2007 2:18 PM


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 "NonEuclidean" 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 online, explores in this direction.

