Sometimes you find out things, for which were under your nose all
along. This happens to me sometimes with finding the keys. But in this
case, I was looking for what in geometry demands there to be holes
between successive Reals. I often mentioned that the Calculus could
not exist without holes between points and that the Cartesian
Coordinate System is more like a Graph Paper system with empty space
between points to plot rather than the system of absolute continuity.
So, I was struggling to find where Geometry demands these holes. And
just recently I found it serendipity.
All three of the geometries begin with a primitive notion or axiom of
"what is a point" and "what is a line". So that if there is any
contradictory conflict between point and line, and the resolution of
that conflict is tiny holes in between every point, then, I found the
demand of holes on geometry.
Now let me just refer the reader to the history of the axioms of
Geometry. I know them mostly from Euclid and from Hilbert.
--- quoting Wikipedia on point and line in geometry ---
Points are most often considered within the framework of Euclidean
geometry, where they are one of the fundamental objects. Euclid
originally defined the point as "that which has no part".
The notion of line or straight line was introduced by ancient
mathematicians to represent straight objects with negligible width and
depth. Lines are an idealization of such objects. Thus, until
seventeenth century, lines were defined like this: "The line is the
first species of quantity, which has only one dimension, namely
length, without any width nor depth, and is nothing else than the flow
or run of the point which [...] will leave from its imaginary moving
some vestige in length, exempt of any width. [...] The straight line
is that which is equally extended between its points"[1]
Euclid described a line as "breadthless length", and introduced
several postulates as basic unprovable properties from which he
constructed the geometry, which is now called Euclidean geometry
Hilbert's axiom system is constructed with nine primitive notions:
three primitive terms:
▪ point;
▪ straight line;
▪ plane;
and six primitive relations:
--- end quoting Wikipedia ---
I need the above reference to show the contradiction.
The point is a number-point and has no length, no width and no depth.
The line is a union of points but has length but no width nor depth.
Can you sense the contradiction? Probably not, for if it was that
easy, someone would have discovered this a long time ago rather than
me in 2012.
The contradiction is in many folds. One of them is that if a point has
no length, then no matter how many of them you stack or union together
there is still no length.
Now if you say that length is not a measure of length of points but of
a distance from one point to another, well that is just sidestepping
the issue by redefining length as distance as if they are different
things.
If a point has no length and a line is a union of points with length
then you have the hidden assumption that 2 points make a length
whereas 1 point does not make a length. And if 2 points make a length,
then you have the hidden assumption that points are not absolute-
continuity in that between any 2 points lies a 3rd new point.
So in other words, the Euclidean and Hilbert axiom program of Geometry
have hidden assumptions about points and lines that makes the entire
geometry program self contradictory.
The only way out of this contradiction axioms, is to make the first
two axioms or primitive notions to be that a point has length, width
and depth and that the line is a union of such points that has a width
and depth be the smallest increment (the Planck's constant of width
and depth).
The point and line is defined by the borderline of finite to infinite.
So the Calculus is the major breakdown of when you use absolute
continuity and you keep points and lines as what Euclid and Hilbert
had used. But the entire program of Geometry is a gaggle of
contradiction, unless the first two axioms or primitive notions are
mended.
Google's New-Newsgroups censors AP posts but Drexel's Math Forum does
not and my posts in archive form is seen here:
<plutonium.archime...@gmail.com> wrote:
> Sometimes you find out things, for which were under your nose all
> along. This happens to me sometimes with finding the keys. But in this
> case, I was looking for what in geometry demands there to be holes
> between successive Reals. I often mentioned that the Calculus could
> not exist without holes between points and that the Cartesian
> Coordinate System is more like a Graph Paper system with empty space
> between points to plot rather than the system of absolute continuity.
> So, I was struggling to find where Geometry demands these holes. And
> just recently I found it serendipity.
> All three of the geometries begin with a primitive notion or axiom of
> "what is a point" and "what is a line". So that if there is any
> contradictory conflict between point and line, and the resolution of
> that conflict is tiny holes in between every point, then, I found the
> demand of holes on geometry.
> Now let me just refer the reader to the history of the axioms of
> Geometry. I know them mostly from Euclid and from Hilbert.
> --- quoting Wikipedia on point and line in geometry ---
> Points are most often considered within the framework of Euclidean
> geometry, where they are one of the fundamental objects. Euclid
> originally defined the point as "that which has no part".
> The notion of line or straight line was introduced by ancient
> mathematicians to represent straight objects with negligible width and
> depth. Lines are an idealization of such objects. Thus, until
> seventeenth century, lines were defined like this: "The line is the
> first species of quantity, which has only one dimension, namely
> length, without any width nor depth, and is nothing else than the flow
> or run of the point which [...] will leave from its imaginary moving
> some vestige in length, exempt of any width. [...] The straight line
> is that which is equally extended between its points"[1]
> Euclid described a line as "breadthless length", and introduced
> several postulates as basic unprovable properties from which he
> constructed the geometry, which is now called Euclidean geometry
> Hilbert's axiom system is constructed with nine primitive notions:
> three primitive terms:
> ▪ point;
> ▪ straight line;
> ▪ plane;
> and six primitive relations:
> --- end quoting Wikipedia ---
> I need the above reference to show the contradiction.
> The point is a number-point and has no length, no width and no depth.
> The line is a union of points but has length but no width nor depth.
> Can you sense the contradiction? Probably not, for if it was that
> easy, someone would have discovered this a long time ago rather than
> me in 2012.
> The contradiction is in many folds. One of them is that if a point has
> no length, then no matter how many of them you stack or union together
> there is still no length.
> Now if you say that length is not a measure of length of points but of
> a distance from one point to another, well that is just sidestepping
> the issue by redefining length as distance as if they are different
> things.
> If a point has no length and a line is a union of points with length
> then you have the hidden assumption that 2 points make a length
> whereas 1 point does not make a length. And if 2 points make a length,
> then you have the hidden assumption that points are not absolute-
> continuity in that between any 2 points lies a 3rd new point.
> So in other words, the Euclidean and Hilbert axiom program of Geometry
> have hidden assumptions about points and lines that makes the entire
> geometry program self contradictory.
> The only way out of this contradiction axioms, is to make the first
> two axioms or primitive notions to be that a point has length, width
> and depth and that the line is a union of such points that has a width
> and depth be the smallest increment (the Planck's constant of width
> and depth).
> The point and line is defined by the borderline of finite to infinite.
> So the Calculus is the major breakdown of when you use absolute
> continuity and you keep points and lines as what Euclid and Hilbert
> had used. But the entire program of Geometry is a gaggle of
> contradiction, unless the first two axioms or primitive notions are
> mended.
Now keeping in tenure with the perspective above we see that a line in
Euclidean Geometry involves more of an angle than anything about "more
points". So that we start at a point and move at an angle from that
point to traverse over more points of that angle.
When the Infinity borderline is 10^603 then the holes or gaps between
successive numbers are of a metric 10^-603 and from the ellipsoid
surface area formula the smallest ellipsoid is of length 10^603 but
depth and width of 10^-603
leaving us with a metric for the size of a point which has a 10^-604
length and the same for width and depth.
Now the three geometries are different and have their own axioms which
implies that they have their own special points and lines.
In Elliptic the lines are great circles and we expect the points in
Elliptic to be tiny circles of 10^-604 diameter. For Euclidean
geometry the points are likely to be tiny squares of sides 10^-604.
Now those reading these posts after being brainwashed about geometry
all their lives, well, they will reject everything said here. But then
that is the usual case of mathematicians, when they learn by doing
what others do and never learn by reasoning and using logic. Imitation
learning is far more widespread than a person who uses logic, and
reasoning to find out what is true and what is false.
It has taken mathematics, the science of precision, taken it more than
2,000 years to realize that if you have finite transitioning or moving
into the infinite, that there must be a number which is the borderline
between finite and infinite and not some pitiful ghost number like
Omega. If geography can teach students that there are two countries in
Europe of France and Germany and they are side by side, that the
student immediately recognizes there must be borderlines to tell one
what is Europe and then borderlines to tell one what is the difference
between France and Germany. But not mathematicians, who for over 2,000
years knows that finite moves into infinite, yet not a single one of
them interested in finding that borderline. And why is that? Easy
answer is that most every mathematician learns not from logic and
reason but learns from copying or imitating another mathematician
(parrot learning). They are unable to think for themselves or by
themselves, but have to copy what others print or say.
Google's New-Newsgroups censors AP posts but Drexel's Math Forum does
not and my posts in archive form is seen here:
Alright, this is good, really good because the bulk of the
length of a line is derived from the empty space between two
successive Reals. A point itself has a tiny bit of length of
10^-604, but the bulk of the length is the empty space of 10^-603 that
lies between two successive Reals.
Now a line itself in Euclidean Geometry is constructed from the
ellipsoid in 3rd dimension Elliptic Geometry given its formula which
relates pi with phi. And it tells us that the length of a line to
infinity in Euclidean geometry is 10^603 while its other axes are
10^-603 respectively. The total surface area of that line, since it is
a ellipsoid is 12 square units and that provides us with the fact that
a Number point or a point in Geometry has some size, albeit a very
tiny size of 10^-604.
In Old Math, they started the entire program of Geometry with an
unassailable big glaring Contradiction. They started geometry with two
axioms, some like to call them primitive notions (Hilbert) that a
point has no length, width, depth and that a line has length but no
width and no depth.
If a point has no parts, then just stringing them together will not
provide length, especially if you have ever more of them between any
two given points (absolute continuity). What gives the line a length,
is when the points are separated by a distance of empty space. So that
when we say a line has length, we are summing up the distances of all
those empty spaces between successive Real points.
So the program of Euclid's axioms on geometry and the program of
Hilbert's axioms of geometry fall dead due to contradiction before
they even get off the ground to fly.
So I never really needed to show that Calculus cannot exist with
absolute continuity for there is no picket fence summation since the
picket fences have 0 width, nor that a slope or tangent are impossible
to construct in many graphs because of the density of points between
any two Real points.
So why has humanity never taken this alternate route in the history of
mathematics and questioned absolute continuity? Well, so much of early
mathematics believed in **continuity** for our eyes cannot see small
things such as the size of atoms and subatomic particles. If our eyes
could see on the level of atoms and subatomic particles, we would
recognize that continuity is a false-perception and that empty space
of forces of electricity and magnetism is what the world is made out
of. Absolute continuity is a false perception for eyes that are big
and never small and we plastered our mathematics with this false
perception. Physics learned that true physics is quantum physics, the
small phenomenon and replaced the Newtonian physics with its absolute
continuity, absolute time and absolute space with quantum mechanics of
a discrete space and empty space.
It is about time that mathematics catches up to the advances of
Physics as mathematics is at least a century behind physics in
understanding and wisdom.
Google's New-Newsgroups censors AP posts but Drexel's Math Forum does
not and my posts in archive form is seen here:
you are evincing the self-same problem
of definition, which you gave with your miscuing
of the the four color conjecture,
about "the line is a seperate color!"
