for pumpkin 

when pumpkin, buttercup, and i were traveling in africa we would drink. heavily. just to pass the time. and dull the pain. we would then start discussing noah's ark. as normal drunk people do. buttercup would declare an unshakable faith in the story. something he can't explain. just faith. pumpkin would say the story is ridiculous. illogical. scientifically impossible. we would then start talking about the highest truth. and parallel lines. we had loads of discussions about the highest truth. and parallel lines. is faith any less of a truth than logic (science or mathematics)? 

poincare lived from 1854 to 1912, a professor at the university of paris. this man was an international celebrity at thirty-five, a living legend at fifty-eight...

... during poincare's lifetime, an alarmingly deep crises in the foundations of the exact sciences had begun. for years scientific truth had been beyond the possibility of a doubt; the logic of science was infallible, and if the scientists were sometimes mistaken, this was assumed to be only their mistaking of it's rules. the great questions had all been answered. the mission of science was now simply to refine these answers to greater and greater accuracy... it was hardly guessed by anyone that within a few decades there would be no more absolute space, absolute time, absolute substance or even absolute magnitude; that classical physics, the scientific rock of ages, would become "approximate"...

... in his foundations of science poincare explained that the antecedents of the crisis in the foundations of science were very old. it had long been sought in vain, he said, to demonstrate the axiom known as euclid's fifth postulate and this search was the start of the crisis. euclid's fifth postulate of parallels, which states that through a given point there's not more than one parallel line to a given straight line,  we usually learn in tenth-grade geometry. it is one of the basic building blocks out of which the entire mathematics of geometry is constructed...

... finally, in the first quater if the nineteeth century, and almost at the same time, a hungrian 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 hypothesis 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 the euclidian geometry. thus by failure find any contradictions he proves that the fifth postulate is irreducible to simpler axioms.

it wasn't proof but it was alarming. it was it's 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 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.