RCA
Posts:
11
Registered:
11/26/05
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Re: Euclids postulates and non-Euclidean geometry
Posted:
Mar 7, 2006 8:03 PM
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Thanks all for your great responses. My key takeaway from all your
responses are:
My interest is logical and not historical - most of the literature
seems to imply that the fifth postulate is more questionable in an
absolute and obvious sense, which is what confused me. This discussion
has clarified that it is not obvious.
I found an interesting relevant discussion about this issue at
http://www.cut-the-knot.org/triangle/pythpar/Geometries.shtml
1. There are several nonscientific (historical - Euclid didnt introduce
it for a while, aesthetic - it "looks" uglier) reasons why it turned
out that the fifth postulate was debated more than the others.
2. The negation of the fifth postulate led to practically useful
geometries and thus came to the forefront.
3. There is active work in logic that investigates negating the other
postulates as well, based on (Bolyai et al's) further abstraction of
denying the common meanings of line and the plane.
Thanks very much
RCA
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