Hyperbolic GeometryDate: 03/24/2003 at 20:31:21 From: Lis Subject: Hyperbolic Geometry Can you explain to me this assumption: Assuming that Euclidean geometry is consistent, had any of the failed attempts to prove Euclid's 5th Postulate from the other axioms succeeded, they would have actually completely destroyed Euclidean geometry as a consistent body of thought. Date: 03/25/2003 at 04:20:59 From: Doctor Jacques Subject: Re: Hyperbolic Geometry Hi Lis, There is some special logic involved here. Hyperbolic geometry states that you can draw more than one line parallel to a given line through an external point. This implies that the "5th postulate" is false. You can read more on Non-Euclidean geometry in the Dr. Math archives: Euclid's Parallel Postulate http://mathforum.org/library/drmath/view/54750.html Hyperbolic Geometry and the Euclidean Parallel Postulate http://mathforum.org/library/drmath/view/52832.html or other pages that you can find by using the Dr. Math searcher: http://mathforum.org/library/drmath/mathgrepform.html It has been proven that, IF Euclidean geometry is consistent, then there is a geometry that negates the 5th postulate and is also consistent. The idea of the proof is to build a model of non-Euclidean geometry within Euclidean geometry itself, and to prove that that model satisifies all the other axioms of Euclidean geometry, and therefore all conclusions that can be drawn from them. There are a few different such models. In two dimensions, one of the most widely known is due to Poincare. In that model the "plane" is the interior of a disk in the Euclidean plane, and the "straight lines" are circular arcs orthogonal to the perimeter of the disk. You can read more about that model on: World of Mathematics - Eric Weisstein http://mathworld.wolfram.com/PoincareHyperbolicDisk.html The important thing is that the model is built within the Euclidean plane itself, using properties of Euclidean geometry. The fact that the "plane" and "straight lines" (mind the quotes) satisfy the postulates of Euclidean geometry (except the 5th) can be proven *using Euclidean geometry itself*. The conclusion is that, IF Euclidean geometry is valid with the 5th postulate, it is also valid with its negation. Now, if we were able to prove the 5th postulate from the other axioms, then, as a consequence, we could also prove that it is false, and this would be a contradiction. In such a case, the conclusion would be that Euclidean geometry implies a contradiction, i.e. it is inconsistent. That is, in my opinion, the meaning of the sentence you refer to. I would like to add a few additional remarks: Euclidean geometry - as exposed is Euclid's Elements - is far from a rigorous discipline. Euclid assumes implicitly many more things than his explicit postulates. However, this can be remedied with a more rigorous approach, and the conclusion remains the same. The problem encountered here is rather common in mathematics, for example in set theory. It is usually not possible to prove that a system of axioms is consistent, but, in some cases, it can be proven that, IF you assume that a system of axioms is consistent, then it remains consistent if you add another axiom to it. Does this help? Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/ |
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