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April 9th, 2017, 09:19 PM   #21
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Changed my mind about expanding points. Back to Euclid and Deascartes.

The line is the fundamental geometric figure of Euclidean geometry.

Points are not geometric figures, they are locations.

(2) is a line from an assumed location of length 2.
(2,3) are two "different" lines from an assumed location.
(1,3,..an) are n different "lines" from an assumed location.
The lines above also identify locations.

You can't create geometry from points, but you can discuss and analyze sets of points. That's why Point Set Geometry is misleading, it is ambiguous- it implies you start with points and create geometry, making point sets foundational- they aren't.



You can abstract infinite sets of "things" and relate them with "rules" using geometric names, which is fine until you derive things using your abstract things and rules and give the result geometric names- Banach-Tarski paradox for example; or claim you have established the foundation of geometry.

EDIT
This should be generalized to: You can't create a geometric dimension from a zero dimension.
Example, you can't create a square by laying lines (0 width) next to each other.
Example, Banach Tarski Paradox.

Last edited by zylo; April 9th, 2017 at 10:14 PM.
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April 10th, 2017, 06:32 AM   #22
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Beer soaked query follows.
Quote:
Originally Posted by zylo View Post
Changed my mind about expanding points. Back to Euclid and Deascartes.

The line is the fundamental geometric figure of Euclidean geometry.

Points are not geometric figures, they are locations.

(2) is a line from an assumed location of length 2.
(2,3) are two "different" lines from an assumed location.
(1,3,..an) are n different "lines" from an assumed location.
The lines above also identify locations.

You can't create geometry from points, but you can discuss and analyze sets of points. That's why Point Set Geometry is misleading, it is ambiguous- it implies you start with points and create geometry, making point sets foundational- they aren't.



You can abstract infinite sets of "things" and relate them with "rules" using geometric names, which is fine until you derive things using your abstract things and rules and give the result geometric names- Banach-Tarski paradox for example; or claim you have established the foundation of geometry.

EDIT
This should be generalized to: You can't create a geometric dimension from a zero dimension.
Example, you can't create a square by laying lines (0 width) next to each other.
Example, Banach Tarski Paradox.
This is like a belated April Fools thing, right?
Thanks from Denis
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April 10th, 2017, 08:03 AM   #23
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Quote:
Originally Posted by jonah View Post
This is like a belated April Fools thing, right?
Right. ZyLOO apparently tried to backdate to Apr.1st:
denied by moderator as a POINTless post
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