Re: Question - Embedded Spaces - Properties versus Relations

It is time to draw the attention to the concepts of =property= and =relation= and the difference between them. I will highlight just one part of your latest post:

(.....)

Agreed, but calling them 2-dimensional is just a matter of conventions. A
plane in R3 has normals everywhere, and R2 has no such normals.

They are different.

You can specify R2, or any plane comtained in R2 completely with 2-tuples.
Cant do that with a plane in R3. Any plane in R3, or any representation of
any such plane either is, or is equivalent to a representation via 3-tuples.
(.....)

Being 2D is a property of planes. 

Having a normal line at every point is a relation of planes to ordinary 3D space.

Having a normal plane at every point is a relation of planes to 4D space.

About representation by 2-tuples: once you chosen a plane in space, you will always need precisely two parameters to represent a particular point in that plane.

Take for instance an arbitrary plane in space that is not parallel to the Z axis. Then you only need x and y to specify an arbitrary point in that plane. The z coordinate follows by z = Ax + By + C for some constants A, B, C, and so z ==depends on x, y== and is =not= a third parameter with its own life.

The z coordinate as a function of x and y stands here for the relation of the plane to the entire space.


In formal logic properties and relations are easily distinguished: properties are boolean functions of one independent argument, whereas relations are boolean functions of two or more independent arguments.

IHTH: Johan E.Mebius



Lefty wrote:

"Jean-Claude Arbaut" <jean-claude.arbaut@xxxxxxxxxxx> wrote in message
news:BEE5191F.7B77%jean-claude.arbaut@xxxxxxxxxxxxxx



On 27/06/2005 04:28, Lefty wrote:



2D, because a plane is a geometric object of dimension 2:


Well, yes. But it is also described by 3-vectors, hence 3D.



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