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"
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:
▪ straight line;
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
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
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
Google's New-Newsgroups censors AP posts but Drexel's Math Forum does
not and my posts in archive form is seen here:
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies