Re: Examining Mathematical Approaches

From: Ahmed Ouahi, Architect (ahmed.ouahi_at_welho.com)
Date: 10/19/04 

Date: Tue, 19 Oct 2004 16:34:49 +0300

................ ...That, if a straight line falling on two
straight lines makes the interior angles on the same side less than two
right angles,
the two straight lines, if produced indefinitely, meet on that side on which
the angles are less than two right angles!!!!!!!!!!!!!!............ ...

-- Euclid, 

--
Ahmed Ouahi, Architect
Best Regards!
"Gnivilq" <gnivilq@aol.com> wrote in message
news:20041019091817.26518.00002549@mb-m03.aol.com...
> Examining Mathematical Approaches
>
>      I have received communications from a mathematician who asserted that
my
> statements about the validity of non-Euclidean Geometry were erroneous. He
> apparently objected to the assertion that the definition of a straight
line was
> faulty for all geometries and that any non-Euclidean Geometry of N
dimensions
> could be contained in a Euclidean geometry of N+1 dimensions. (The latter
> objection might be considered important because it implies that GR has a
> responsibility to describe the Euclidean geometry of 4 spatial dimensions
> implied by its assertion that our space of three dimensions is curved. The
> failure to include a description of such a space does General Relativists
> little credit.)
>
>      One of the arguments presented is that the geometry represented by a
> circle, a mobius strip or a klein bottle cannot be embedded in a three
> dimensional Euclidean geometry but can only described isometrically. If I
were
> to take his arguments literally I, as someone who IS embedded in a three
> dimensional Euclidean space, could not draw a circle on a piece of paper,
and
> could not construct a klein bottle or a mobius strip! Obviously I, and
anyone
> else, can do so. It is apparent then that the correspondent's mathematical
> approach is faulty. What he should be doing is making sure that his
mathematics
> is sufficient to do the job before asserting that someone else's
understanding
> is deficient.
>
>      To illustrate the second conclusion, that the true definition of a
> straight line for any geometry is the shortest distance between two points
> WHICH REMAINS WITHIN THE GEOMETRY, consider the distance between New York
and
> Los Angeles. In the two dimensional geometry which represents the surface
of
> the Earth, the shortest distance is along a great circle. It will be noted
that
> this meets the revised definition of the straight line for that geometry.
In
> terms of the three
> dimensional geoetry represented by the Earth as a whole, the definition
also is
> valid. The shortest distance which remains within the three dimensional
> geometry is though a tunnel which passes about 200 miles below the
Mississippi
> River. The revised definition works for all geometries.
>
>      It is asserted that the force which we sense as gravity results from
the
> geometry of space. If one would take the trouble to examine this
conclusion he
> would recognize that there is no way that this would occur. Imagine a
perfectly
> smooth spherical planet sufficiently isolated so that external
gravitational
> effects could be ignored. On this planet are scribed two great circles at
right
> angle to each other. At the intersection of the two lines rests a
perfectly
> smooth steel ball and along the first line, perhaps 30 degrees away, rests
an
> identical ball. To simplify the argument let us consider that the balls
can
> roll over the planet's surface with zero friction. If the balls are
started
> rolling in a direction at right angles to the first line, they will follow
> great circle paths and will approach each other. A GR geometer would
assert
> that, due to the
> curved space represented by the two dimensional geometry, the balls
experienced
> a force which would cause them to approach each other since they followed
a
> curved path. There is a difficulty with this idea however. The balls could
be
> stopped anywhere along the path and they would remain where they were
showing
> that an attractive force between them did not exist! The force that caused
them
> to approach each other is a vector component of the force of gravity which
is
> at
> all times normal to the surface and therefore cannot be sensed in the two
> dimensional geometry of the surface. To make the statement more
forcefully,
> THERE IS NO WAY IN @#$% THAT GEOMETRY CAN PRODUCE THE FORCE OF GRAVITY!
>
>      The source material for this posting may be found in "Gravity"
(1987),
> "The Einstein Hoax" (1997), and "Corrections to Residual Errors in Special
> Relativity (1999) located at
http://www.members.aol.com/einsteinhoax/site.htm.
> EVERYTHING WHICH WE ACCEPT AS TRUE MUST BE CONSISTENT WITH EVERYTHING ELSE
WE
> HAVE ACCEPTED AS TRUE, IT MUST BE CONSISTENT WITH ALL OBSERVATIONS, AND IT
MUST
> BE MATHEMATICALLY VIABLE. PRESENT TEACHINGS DO NOT ALWAYS MEET THIS
> REQUIREMENT. THE WORLD IS ENTITLED TO A HIGHER STANDARD OF WORKMANSHIP
FROM
> THOSE IT HAS
> GRANTED WORLD CLASS STATUS.
>
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please
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read at
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>
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>
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have
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