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Topic: the ugly contradiction at the heart of High School Geometry #35.3
Uni-text 8th ed.: TRUE CALCULUS

Replies: 24   Last Post: Nov 3, 2013 7:44 PM

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 plutonium.archimedes@gmail.com Posts: 18,572 Registered: 3/31/08
some changes in True Geometry, once "empty space" is factored in
#35.5 Uni-text 8th ed.: TRUE CALCULUS

Posted: Nov 2, 2013 3:48 PM

On Saturday, November 2, 2013 12:21:29 AM UTC-5, Archimedes Plutonium wrote:
(snipped)
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> The solid facts are that Geometry in High School and College is a sewer of contradiction where they are so lazy and dumb as to think that geometry is only about:
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> (1) points
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> (2) lines
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> When in truth, the rudiments of Geometry are about:
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> (1) points
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> (2) finite versus infinite causing a macroinfinity and microinfinity
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> (3) microinfinity causing Empty Space of infinite numbers that are not allowed
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> (4) lines, which are built from points, empty space, and successor points
>

Now I do not have much time to spend here on exploring what True Geometry looks like for High School and University. A lot will change since Geometry has holes and gaps between successive points. Of course the "continuum hypothesis" of Old Math is a pure slab of crackpot nonsense as is the Cantor crackpot nonsense.

As I write True Calculus, it is obvious that we can only have a Calculus if there are gaps and holes between successor points.

But let me mention a consequence of these gaps and holes on True Geometry.

In True Geometry we have gaps and holes in not only Euclidean geometry but also in Elliptic and Hyperbolic.

In Euclidean geometry we know that a line not parallel to another line, intersects at a point with the other line if both go to infinity. Well that is not necessarily true in True Geometry, for example in the 10 Grid if we extend it to 11 in both directions of the x-axis where 11 is an infinity number, we can picture many lines in that 11 by 11 and the lines not parallel, yet never intersecting.

So in Old Fake Geometry, parallel lines and intersection need a grand overhaul.

Another curiosity that arises in True Geometry but did not in Fake Geometry is the idea of a "solid line" versus a "dashed line".

In Old Fake Geometry, every line was solid and solid with points that had no gaps or holes between successive points. So in Old Fake Geometry, there was no such thing as a "Dashed Line".

In True Geometry there is a solid line which is a connecting of every finite point and extending through the empty space of infinite points. However there also exists a dashed line that connects two successive finite points, misses connecting the next successor finite point, but connects the 3rd with the 4th successive finite point and repeats the pattern. So in True Geometry we can have a solid line__________>

and we can have a dashed line ------------>. So every solid line in True Geometry has a subset dashed line. Now, if the Geometry extends to infinity in both directions as this <_______> then we have
2 dashed lines as subsets. Whereas a line ray can only have one dashed line.

Now a dashed line to infinity may have no significance to mathematics, but to physics, a dashed line would have huge significance such as in the Maxwell theory and how light or neutrinos travels in space as a longitudinal wave.

Now let me mention something else with the intersection of two lines in True Geometry. Some are going to complain that if you have holes and gaps in lines that who knows if the intersection is a finite point on both lines that intersect? Who knows if the intersection is two gaps or holes in both lines and thus no intersection at all? Well, the gaps and holes quantize the angles that can exist. Those angles allow only specific lines to exist. So that if two lines intersect in True Geometry, their intersection is indeed a finite point on both the lines that intersect, because the lines can only be of specific angles. You see, the borderline of finite with infinite has quantized the graph grid of geometry, so that only specific holes exist and correspondingly, specific angles exist.

AP