The reason for seeking alternate formulations of the parallel postulate isOriginally Posted by ThePerfectHacker
that for more than 2000 years Mathematicians have not found the existing
This even lead Gauss to go out and attempt to measure the angle sum
of a large triangle to see if it was measurably different from 180 degrees.
The resolution of this problem is that the parallel postulate is what
differentiates Euclidean geometry from other similar geometries which
differ only in terms of how many parallels exist through a given point to a
Elliptic geometry - no parallels
Hyperbolic geometry - more than one parallel.
These are homogeneous geometries, if we lift the homogeneity constraint
we allow inhomogeneous geometries, which include our best current
models of the geometry of space time.