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