Non-E geometries were discovered by Johann C.F. Gauss in the late 18th and early 19th centuries, Farkas Bolyai a little afterwards, and around the same time by Nikolai Ivanovich Lobachevsky (the subject of a famous song by Tom Lehrer ("Plagiarize! Don't shade your eyes! That's why got made your eyes!")), as well as G.F.B. Riemann, around mid-century.
The key thing is what happens when you deny Euclid's parallels postulate (parallel lines never meet), which had for thousands of years struck mathematicians as unobvious in theway that Euclid's other postulates are not. Turns out taht you can construct consitent geometries when you deny the p-posstulate, thus blowing everyone's minds and setting the statge for Einstein to posit that these geometries describe actual space(time).
--- Charles Brown <cbrown at michiganlegal.org> wrote:
>
>
>
> On the curved space-time, interesting that in his
> Ph.d thesis , Marx favored
> Epicurus over Democritus , and Epicurus held that
> there was a declination of
> a falling atom from a straight line, that atoms
> "swwerved" , which is a
> curve, in contrast with Democritus who held they
> fall in a straightline.
>
> Democritus speculatively thought of atoms.
>
> Did Epircurus speculatively discover curved
> space-time ?
>
> ( See below)
>
> Charles
>
> ^^^^^^
>
>
> andie nachgeborenen
>
>
>
> Gravity is not a force. That is the fundamental
> negative result of GTR -- the elimination of the
> idea
> of the "force" of gravitation posited by Newton,
> thus
> the elimination of the problem of action at a
> distance
> that so puzzled him.* Gravity is an effect of the
> curvature of spacetime near massy objects. It is an
> consequence of geometry. Neither is it correct to
> say
> that it in virtue of some occult force that
> mass-energy "causes" spacetime to curve. That
> presupposes tata spacetime is somehow naturally
> Euclidean and that something must be posited to
> expalin deviations from flatness. This is an error.
> S-T is locally Euclidean, which may be where the
> error
> comes from. It may be (but probbaly isn't)
> Euclidean
> in on the large scale, but once you give up the idea
> of Newtonian absolute space, fixed, immovable, the
> same everywhere, it is no special mystery why the
> presence of mass-energy would produce variations in
> the shape of spacetime, and why it would produce the
> precise highly preductable vatiations it does. No
> occult powers are called for to account for this
> unless you illegitimately import closet Newtonian
> assumptions. As noted by me and others, a
> unification
> of GTR with qwuantum theory, which hopefully will
> give
> us more ins ights into gravity, still eludes us.
>
> * Action at a distance creeps back in quantum theory
>
> -- in a context totally removed from gravitation --
> via Bell's Theorem, but quantum is so fundamentally
> bizarre and incomprehensible that there isn't much
> we
> can do about that
>
> ^^^^^^
>
> Karl Marx
> The Difference Between the Democritean and Epicurean
> Philosophy of Nature
> Part II: On the Difference between Democritean and
> Epicurean Physics In
> Detail
>
http://www.marxists.org/archive/marx/works/1841/dr-theses/ch04.htm
> ________________________________
>
>
> Chapter One: The Declination of the Atom from the
> Straight Line
>
> ________________________________
>
>
>
>
> Epicurus assumes a threefold motion of the atoms in
> the void.(1)
>
<http://www.marxists.org/archive/marx/works/1841/dr-theses/ch04.htm#1>
> One
> motion is the fall in a straight line, the second
> originates in the
> deviation of the atom from the straight line, and
> the third is established
> through the repulsion of the many atoms. Both
> Democritus and Epicurus accept
> the first and the third motion. The declination of
> the atom from the
> straight line differentiates the one from the
> other.(2)
>
<http://www.marxists.org/archive/marx/works/1841/dr-theses/ch04.htm#2>
>
>
> This motion of declination has often been made the
> subject of a joke. Cicero
> more than any other is inexhaustible when he touches
> on this theme. Thus we
> read in him, among other things:
>
> "Epicurus maintains that the atoms are thrust
> downwards in a straight line
> by their weight; this motion is said to he the
> natural motion of bodies. But
> then it occurred to him that if all atoms were
> thrust downwards, no atom
> could ever meet another one. Epicurus therefore
> resorted to a lie. He said
> that the atom makes a very tiny swerve, which is, of
> course, entirely
> impossible. From this arose complexities,
> combinations and adhesions of the
> atoms with one another, and out of this came the
> world, all parts of it and
> its contents. Besides all this being a puerile
> invention, he does not even
> achieve what he desires."(3)
>
<http://www.marxists.org/archive/marx/works/1841/dr-theses/ch04.htm#3>
>
>
> We find another version in the first book of
> Cicero's treatise On the Nature
> of the Gods:
>
> "Since Epicurus saw that, if the atoms travelled
> downwards by their own
> weight, nothing would be within our control, for
> their motion would be
> determined and necessary, he invented a means for
> escaping this necessity, a
> means which had escaped the notice of Democritus. He
> says that the atom,
> although thrust downwards by its weight and gravity,
> makes a very slight
> swerve. To assert this is more disgraceful than to
> he incapable of defending
> what he wants."(4)
>
<http://www.marxists.org/archive/marx/works/1841/dr-theses/ch04.htm#4>
>
>
>
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>
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>
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