Hyperbolic Geometry

Date: 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/