```Date: Mar 7, 2006 5:42 PM
Author: RCA
Subject: Euclids postulates and non-Euclidean geometry

Hi,I am trying to understand the motivation behind non-Euclidean geometry.1. I do not understand why Euclid's fifth postulate is any differentfrom the other postulates. For instance, it seems as intuitive to me toaccept the fifth postulate as to accept the first one in one viewpoint.2. If I assume that a Euclidean geometry refers to an infinite planesurface which closely matches our intuition at small scales, I findboth the first and the fifth postulate to be equally believable (purelyby intuition in both cases).3.  If I focus on the errors arising due to the approximation of aplane of intuitive scales actually being a part of the curved surfaceof the earth, then, of course, I begin to see the Euclidean rulesfailing, since we are on a different surface. I notice 'curvedtriangles' actually having 'curved sides' on the earth's surface. Inthis case, I completely redefine the angle between the 'curved sides'of the 'curved triangles' on the surface to be the angle betwen thetangents of the curves at that point. Now we are no longer talkingabout the angle between strictly straight lines - we are referring tothree angles between tangents to 'bulging' curves, which, quitintuitively would add to more than 180 degrees.4. If we set aside intuition for a moment and focus completely on theabstract platonic world of ideal forms, then every postulate and axiomcan be questioned, and we can have very many amusing platonic worldsbased on different rules. I could have one platonic world whereEuclid's first postulate is wrong and the fifth is valid and see whatinteresting behaviors I can find in that world. I could create severalsuch worlds and derive interesting properties in all of them. To thatextent, a non-Euclidean geometry may be accepted as existing in oneworld among many worlds, where other worlds had other forms ofnon-Euclidean geometries formed by questioning each postulate and axiomof Euclid in different combinations.5. If the justification for singling out the fifth postulate to focuson is merely driven by the utility of the resulting worlds, then I amalso willing to accept a statement such as " there are several platonicworlds formed by negating each axiom and postulate of Euclid, but theonly ones using practically useful results and map to our intuition arethe Euclidean geometry (at small scales) and some hyperbolic geometries(at the large scale of the universe)". However, this seems more of aconvenience argument than a scientific one.I am very new to this entire field, so could someone help me get pastthese fundamental confusions?Thanks
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