Let me try to answer some of your not-unreasonable questions. This might be of some general interest.
1. The importance of Euclid's fifth postulate is mostly historically. Do not consider it to be the main reason for studying non-Euclidean geometry. The most important reason is that the geometry of a sphere--where straight lines are great circle arcs (the bulging we view from an external position is considered irrelevant)--is vitally important to us for navigation, astronomy and these days a great variety of industrial and scientific applications (GPS, satellites, geodesic surveying etc). The geometry on the surface of a sphere is analogous to Euclidean geometry and approximates it in the small. Some things are directly the same--the altitudes of a spherical triangle still intersect at a point. Some things are similiar but different--there is now not one but two medians from a vertex to the opposite side, and it turns out there are four centroids, not just one. The trigonometric laws are similiar, perhaps somewhat more complicated. Even rational trigonometry works in the spherical case (and of course it greatly simplifies things there, as it does in the Euclidean case).
2. The second good reason is hyperbolic geometry, which is the `other' non-Euclidean geometry. This is to spherical geometry as the hyperboloid of two sheets (x^2+y2-z2 = -1) is to the sphere (x^2+y^2+z^2 = 1). This theory turns out to be deeper in some sense than the spherical one, largely due to the possibilities for a greater variety of symmetries. The most obvious manifestation is that on the sphere there are 5 Platonic solids--on the hyperboloid there are an infinite number. These turn out to have beautiful connections to surfaces (like two-holed tori), to complex analysis, number theory and modern physics. Hyperbolic geometry plays a big role in the understanding of 3-manifolds and other esoteric stuff from mathematical physics. There are also some industrial applications, but these are not as important as the spherical ones.
3. Of course you can try to create theories by mixing and matching definitions as you like, but mathematics is not such an ad-hoc subject. Most randomly chosen definitions turn out to worthless--for the simple reason that unless there are good and interesting examples, you can't get any one else interested.
Cheers, Norman Wildberger (Assoc Prof, School of Maths, UNSW)
>Hi, > >I am trying to understand the motivation behind non-Euclidean 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. >2. If I assume that a Euclidean geometry refers to an infinite plane >surface which closely matches our intuition at small scales, I find >both the first and the fifth postulate to be equally believable (purely >by intuition in both cases). >3. If I focus on the errors arising due to the approximation of a >plane of intuitive scales actually being a part of the curved surface >of the earth, then, of course, I begin to see the Euclidean rules >failing, since we are on a different surface. I notice 'curved >triangles' actually having 'curved sides' on the earth's surface. In >this case, I completely redefine the angle between the 'curved sides' >of the 'curved triangles' on the surface to be the angle betwen the >tangents of the curves at that point. Now we are no longer talking >about the angle between strictly straight lines - we are referring to >three angles between tangents to 'bulging' curves, which, quit >intuitively would add to more than 180 degrees. >4. If we set aside intuition for a moment and focus completely on the >abstract platonic world of ideal forms, then every postulate and axiom >can be questioned, and we can have very many amusing platonic worlds >based on different rules. I could have one platonic world where >Euclid's first postulate is wrong and the fifth is valid and see what >interesting behaviors I can find in that world. I could create several >such worlds and derive interesting properties in all of them. To that >extent, a non-Euclidean geometry may be accepted as existing in one >world among many worlds, where other worlds had other forms of >non-Euclidean geometries formed by questioning each postulate and axiom >of Euclid in different combinations. >5. If the justification for singling out the fifth postulate to focus >on is merely driven by the utility of the resulting worlds, then I am >also willing to accept a statement such as " there are several platonic >worlds formed by negating each axiom and postulate of Euclid, but the >only ones using practically useful results and map to our intuition are >the Euclidean geometry (at small scales) and some hyperbolic geometries >(at the large scale of the universe)". However, this seems more of a >convenience argument than a scientific one. > >I am very new to this entire field, so could someone help me get past >these fundamental confusions? > >Thanks > > >