All the other axioms seemed so obvious as to be unquestionable, but this one did not. Yet you couldn't get rid of it without destroying huge portions of the mathematics, and no one seemed able to reduce it to anything more elementary. What vast effort had been wasted in that chimeric hope was truly unimaginable, Poincaré said.
Finally, in the first quarter of the nineteenth century, and almost at the same time, a Hungarian and a Russian...Bolyai and Lobachevski...established irrefutably that a proof of Euclid's fifth postulate is impossible. They did this by reasoning that if there were any way to reduce Euclid's postulate to other, surer axioms, another effect would also be noticeable: a reversal of Euclid's postulate would create logical contradictions in the geometry. So they reversed Euclid's postulate.
Lobachevski assumes at the start that through a given point can be drawn two parallels to a given straight. And he retains besides all Euclid's other
axioms. From these hypotheses he deduces a series of theorems among which it's impossible to find any contradiction, and he constructs a geometry
whose faultless logic is inferior in nothing to that of the Euclidian geometry.
Thus by his failure to find any contradictions he proves that the fifth postulate is irreducible to simpler axioms. It wasn't the proof that was alarming. It was its rational byproduct that soon
overshadowed it and almost everything else in the field of mathematics.
Mathematics, the cornerstone of scientific certainty, was suddenly uncertain. We now had two contradictory visions of unshakable scientific truth, true for all men of all ages, regardless of their individual preferences.
This was the basis of the profound crisis that shattered the scientific complacency of the Gilded Age. How do we know which one of these geometries is right? If there is no basis for distinguishing between them, then you have a total mathematics which admits logical contradictions. But a mathematics that admits internal logical contradictions is no mathematics at all. The ultimate effect of the non-Euclidian geometries becomes nothing more than a magician's mumbo jumbo in which belief is sustained purely by faith!
And of course once that door was opened one could hardly expect the number of contradictory systems of unshakable scientific truth to be limited
to two. A German named Riemann appeared with another unshakable system of geometry which throws overboard not only Euclid's postulate, but
also the first axiom, which states that only one straight line can pass through two points. Again there is no internal contradiction, only an inconsistency with both Lobachevskian and Euclidian geometries.
According to the Theory of Relativity, Riemann geometry best describes the world we live in. Continua...