Re: Euclids postulates and non-Euclidean geometry


Gregory L. Hansen wrote:
In article <1141769783.409836.114650@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
RCA <rcanand@xxxxxxxxx> wrote:
Hi,

I am trying to understand the motivation behind non-Euclidean geometry.

The geometry of surfaces. Having grown bored with drawing straight lines,
geometers started to ask questions like what is the shortest path from A
to B if we must stay on a surface? What is the sum of angles of a
triangle that is drawn on a curved surface? What do we compare to a
"straight line" if we must stay on a surface?

Riemannian geometry, a generalization of Euclidean,

False statement. The two geometries employ different postulates.
Furthermore, as Klein proved, the Euclidean sphere serves as a model
for Re. geometry and the latter is consistent if, and only if, the
former is consistent.

has some set of
postulates like the distance from AB equals the distance from BA, the
distance between two points is zero if and only if they're the same point,
distances are positive, and the distances AB+BC>=AC. And there's some
condition that small enough regions must become increasingly close to
Euclidean geometry, e.g. parallel line segments remain parallel over short
enough distances (or no sharp corners on your manifold).

But the moment any postulate is discarded, geometers will wonder what
happens if the others are discarded. Let us not insist that the distance
between A and B is zero if and only if they're the same point. Then we
have a pseudo-Riemannian geometry, which is the geometry of Einstein's
theories of relativity. And a few new dimensions elevated that from the
geometry of a surface. Semi-Riemannian and quasi-Riemannian
describes those geometries that discard one of the other postulates.

Pseudo-Riemannian geometry has received a lot of attention because of its
physical applications.

What physical applications you are talking about? There is none. It has
been used in a model of gravitation that is not widely accepted as
valid and at the same time resists quantization and conformation with
the widely accepted and experimentally confirmed QM theory.

Besides that, attention tends to be focused on the
fifth postulates because it's natural to violate it by drawing things on
curved surfaces, and again because of its physical applications like
navigating a spherical Earth. The other geometries are too esoteric and
weird to generate much public attention, but that doesn't mean they've
been ignored.

Yes, I agree, there are so many nuts around who do not know what they
are talking about...


Mike



--
"The hardest conviction to get into the mind of the beginner is that the
education he is receiving in college is not a medical course but a life
course for which the work of a few years under teachers is but a
preparation." -- Sir William Osler

.

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