Re: Euclids postulates and non-Euclidean geometry


mmeron@xxxxxxxxxxxxxxxxxx wrote:
In article <1141776535.402811.96760@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "Eric Gisse" <jowr.pi@xxxxxxxxx> writes:

RCA wrote:
Hi,

I am trying to understand the motivation behind non-Euclidean geometry.

You would be better off asking this in sci.math.

Simply put, the motivation was to prove that Euclid's 5th postulate is
provable from the others. What came out of that are things such as
Hyperbolic, Elliptic, and Spherical geometry. It was nearly 2,000 years
before there was a self-consistant non-Euclidian geometry.



1. I do not understand why Euclid's fifth postulate is any different
from the other postulates. For instance, it seems as intuitive to me to
accept the fifth postulate as to accept the first one in one viewpoint.

The fifth postulate is independant from the first four. In other words,
you cannot prove the fifth postulate from the first four.

Well, to be precise, the sam is true about any of the first four as
well. None of the five can be proven from the remaining four (else,
you wouldn't need all five).

To be precise, it is likely you are correct but it may turn out in the
future you are not. Do you think all those thousands of Jesuit monks
who spent their whole life trying to derive the 5th from the other four
knew less geometry than we do today?

It makes sense to many people that the 5th should be derivable from the
other four. What maybe missing is a higher principle, a meta-axiom.

It is clear that the 5th axiom is more than a postulate of a formal
system. It defines the type of world we live in. If we knew with
certainty what type of world is that, then we would know either to
derive it and or to assume its true.

The issue is still open since Klein's theorem was only capable of
proving relative consistency only.

Mike







Mati Meron | "When you argue with a fool,
meron@xxxxxxxxxxxxxxxxx | chances are he is doing just the same"

.

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