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Re: Euclids postulates and nonEuclidean geometry
Posted:
Jun 25, 2007 7:23 AM


The crucial points: > 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. Yes, you can "show" that Euclidean geometry holds on a flat surface, nonEuclidean geometry holds on a nonflat surface such as a sphere. The problem is "How do you define 'flat'?". There are formulas from differential geometry for "curvature" but that begs the question since they are based on calculus which is in turn based on Euclidean geometry. In basis, a "flat" surface is one on which Euclidean geometry it true! AS far as mathematicians are concerned, there are in fact many different geometries depending upon which axioms you accept. Your argument that "this seems more of a convenience argument than a scientific one" applies to the scientist choosing which mathematical model to use and that's always a matter of "convenience".
Indeed, your insistence that the fifth postulate is "believable purely by intuition" indicates that you are looking at it as a matter of "convenience" that model that most closely matches your experience.



