Four Geometries
Subject: RE: Coordinate systems
Author: Jesse Yoder <>
Date: Tue, 10 Feb 1998 13:54:31 -0500
On Feb. 8, John Conway wrote:
> I think that what you call "selecting a Point size to suit a
> particular measurement" is obviously very sensible, but that it's
> silly to use "Point" and "point" with different meanings. Have you
> any strong reason for doing this rather than using the same language 
> as everyone else - for instance saying something like "working to
> within a tolerance of 1/100 of an inch"? I thought you had, namely
> that you felt Euclid's "points" didn't exist, but perhaps some
> absolute "Points" did. 
> I'm surprised that you regard changing a coordinate-system as
> "extreme": to me it seems a rather trivial and practical matter.
> I have a suggestion to make. If it's really the case that
> your distinguishing between "points" and "Points" is practical
> rather than theoretical, why not just drop it and use a more
> traditional language in the interests of better communication 
> even if (like many other people) you'd really prefer it if language 
> hadn't developed in the way it has? I say this because talking
> about the use of words is much less valuable than talking about
> the things they represent, and your present terminology has made
> it very difficult (at least for me) to understand what you're
> really trying to say.
RESPONSE: As I have pointed out in previous discussion, there is an
anti-Eucldean and an anti-Cartesian element to what I call Circular
The anti-Euclidean element consists of saying that Points have area
(unlike points, which have no area), and that Lines have width and
length (unlike lines, which have only length. 
The anti-Cartesian element consists of analyzing circular area in terms
of round inches rather than square inches.
These makes four possible geometries:
1.Euclidean and Cartesian (our current geometry)
2. NonEuclidean and Cartesian (Using Points and Lines, but sticking with
square inches instead of round inches)
3. Euclidean and Non-Cartesian (Sticking with Euclidean points and
lines, but using round inches instead of square inches)
4. Non-Euclidean and Non-Cartesian (Circular Geometry, i.e., using
Points and Lines AND starting with round inches instead of square
I have been arguing for 4, partly because I think it's the most
interesting of the four possibilities. But I could drop the terminology
of Points and Lines altogether and even accept Euclidean concepts of
point and line, and still have an interesting alternative geometry (#3).
I am willing to consider your suggestions, though, since they usually
turn out to be good ones. As for your suggestion of dropping the
terminology of Points for points, Lines for lines, and Circles for
circles, I have no problem with this (as I've said) except that since I
claim that Points have area and Lines have width, it still seems like a
useful technique (one inspired by your suggestion, as I've noted). 
Does "Points have area" just mean "When any measurement is made, it's
made within a certain tolerance", the answer is basically yes, EXCEPT
THAT I want to block the possibility of saying there are infinitely many
points on a line (I claim there are finitely many, and how many there
are varies with the measurement being made).
Jesse Yoder
P.S. I think that Circular Geometry is a lot more compelling if you can
view the graphic of the Coordinate Systems, which unfortunately didn't
come across on the Geometry-Research post, so I will repeat my offer to
fax this to anyone who sends me their fax number. Eventually, I expect
to be able to post this on a website. I also have a Word for Windows
document I can email that does have the graphic--again, this didn't make
it through the forum posting.





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