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.

All Newsposts by this Website are available at
http://www.members.aol.com/postinglog/newspostings.com

Please make any response via E-mail as Newsgroups are not monitored on a
regular basis. Objective responses will be treated with the same courtesy as
they are presented. To prevent the wastage of time on both of our parts, please
do not raise objections that are not related to material that you have read at
the Website. This posting is merely a summary.

E-mail:-

The material at the Website has been posted continuously for over 5 years.
In that time THERE HAVE BEEN NO OBJECTIVE REBUTTALS OF ANY OF THE MATERIAL
PRESENTED. There have only been hand waving arguments by individuals who have
mindlessly accepted the prevailing wisdom without questioning it. If anyone
provides a significant rebuttal that cannot be objectively answered, the
material at the Website will be withdrawn.

................. ...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" wrote in message
...
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.

All Newsposts by this Website are available at
http://www.members.aol.com/postinglog/newspostings.com

Please make any response via E-mail as Newsgroups are not monitored
on a
regular basis. Objective responses will be treated with the same courtesy
as
they are presented. To prevent the wastage of time on both of our parts,
please
do not raise objections that are not related to material that you have
read at
the Website. This posting is merely a summary.

E-mail:-

The material at the Website has been posted continuously for over 5
years.
In that time THERE HAVE BEEN NO OBJECTIVE REBUTTALS OF ANY OF THE MATERIAL
PRESENTED. There have only been hand waving arguments by individuals who
have
mindlessly accepted the prevailing wisdom without questioning it. If
anyone
provides a significant rebuttal that cannot be objectively answered, the
material at the Website will be withdrawn.

Gnivilq wrote:

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.

That is not true. The two dimensional Klein Bottle requires a four
dismensional space for an embedding. Hassler Whitney proved that an
n-dimensional manifold can be embedded in a 2n dimensionsal euclidean
space and no smaller number is generally sufficient, although in some
cases a smaller dimension might suffice. What this means is there exist
n dimensional manifolds which -require- 2n dimensions for an embedding.

Once again you exhibit your ignorance and even your stupidity. We are
all born ignorant but that can be remedied. Stupidity is the refusal to
cure one's ignorance when the means to do so exist.

Bob Kolker

Ahmed Ouahi, Architect wrote:

................ ...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,

Do not top-post. Top-posting is sure evidence of sexual inadequacy and
too frequent masturbation.

Bob Kolker

Not at all!
.................. ...However, what you do claim, does only clarifies as
explains, that you are a deeply suffering along a sexual problem during your
existence. Therefore, undercover of your mental masturbation, definitely as
a matter a fact!!!!!!!!!!!!!!!!!................... ...

--
Ahmed Ouahi, Architect
Simply As that!

"robert j. kolker" wrote in message
...


Ahmed Ouahi, Architect wrote:

................ ...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,

Do not top-post. Top-posting is sure evidence of sexual inadequacy and
too frequent masturbation.

Bob Kolker

and arrogance is a guarantee of what?



"robert j. kolker" wrote:

Gnivilq wrote:

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.

That is not true. The two dimensional Klein Bottle requires a four
dismensional space for an embedding. Hassler Whitney proved that an
n-dimensional manifold can be embedded in a 2n dimensionsal euclidean
space and no smaller number is generally sufficient, although in some
cases a smaller dimension might suffice. What this means is there exist
n dimensional manifolds which -require- 2n dimensions for an embedding.

Once again you exhibit your ignorance and even your stupidity. We are
all born ignorant but that can be remedied. Stupidity is the refusal to
cure one's ignorance when the means to do so exist.

Bob Kolker

Gnivilq:
Examining Mathematical Approaches

I have received communications from a mathematician who asserted that my
statements about the validity of non-Euclidean Geometry were erroneous.

Why, I'm shocked! Not to worry, though. Your policy of diligently
disregarding such communications has kept all of your erroneous statements
intact. If you get greedy and try to misconstrue that information into
another misstatement in one of your spams without thoroughly checking to
make sure it's wrong, you could accidently include a true statement. While
you might legitimately argue that doing so still constitutes a mistake,
your intended audience is unlikely to appreciate that subtlety.

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.

There exists a thing known as a ``math book''. Have you entertained the
idea of purchasing one or even taking the plunge with google and trying to
figure any of this out for yourself? Or, is this a trial balloon for a new
article full of misconceptions and falsehoods?

(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.

Yes, this paragraph will be perfect as a maximally misconceived
non-sequiter with no connection to the words it contains in the
way anyone else normally uses them. How _do_ you do it?