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Thank you for your explanation, Physicist! I won't need to ask you about proofs. However, I found this description which might be helpful to lurkers following the Euclidian v. Riemannian Geometry discussion:

The Origins of Proof

Both the Greeks of Euclid's time, and later Arabic mathematicians, had an intuition that the fifth postulate could actually be proven using the definitions and common notions and the first four postulates…

In fact, the fifth postulate is not derivable from the other postulates and notions, and nor is it universally true. Mathematicians continued to be fascinated by the fifth postulate throughout the centuries, but it was not until the nineteenth and twentieth centuries (through the efforts of a number of famous mathematicians including Legendre, Gauss, Bolyai, Lobachevsky, Riemann, Beltrami and Klein), that we came to know about geometries (called non-Euclidian geometries) where the fifth postulate is not true.

The fifth postulate can be shown to be true in a plane (or Euclidian) geometry. However, there are many other geometries where it is not true. Surprisingly enough, this is easy to illustrate! Consider the simple case of a sphere's surface.

It is impossible to draw a true straight line on a sphere without leaving the surface, So in spherical geometry the Euclidean idea of a line becomes a great circle. Thinking of the Earth, any line of longitude is a great circle - as is the equator. In fact the shortest path between any two points on a sphere is a great circle. (More generally, a minimal path on any surface is known as a geodesic.)

One of the consequences of Euclid's first four postulates is that if two different lines cross, they meet at a single point. This presents a small problem on the sphere, since distinct great circles always cross at two antipodal points! Two lines of longitude always cross at both the North and the South Pole!

But remember, we haven't yet said what the spherical analogue of a Euclidean point is! All we have to do is define a point in spherical geometry to be a pair of antipodal points and the problem promptly disappears. According to Euclid's definition number 23, "Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction". Given these definitions, it is easy to see that Euclid's first four postulates still make good sense. The fifth postulate, however, fails because it is impossible to draw two different lines that do not meet. In spherical geometry there are no parallel lines!

One of the consequences of the failure of the 5th postulate is that it is no longer true that the sum of the angles of a triangle is always 180 degrees.

Regarding your ongoing debate with Dr.Frank, I thought y’all might get a kick out of this: Hubert Yockey on a discussion board

Pose this proposition to your enemies (not your friends): Given any two theories, an experiment will decide between them and prove one true and one false. This is the philosophy of Sir Karl Popper. When a physicist does an experiment to prove that an electron is a particle, it behaves as a particle. When another physicist does an experiment to show an electron is a wave, it behaves as a wave. In some diffraction experiments ray tracing shows the electron or neutron was in two places at once. Thus these experiments prove the wave-particle dualism. Einstein was extremely annoyed by this and suggested experiments to explain what he regarded as a dilemma. He exclaimed: Der lieber Gott wuerfelt nicht mit der Welt! Bohr's reply was: "Einstein, stop telling God what to do!"

Faced with what physicstis and chemists have had to accept from relativity and quantum mechanics, taking the origin of life as an axiom seems rather tame.