Reply to Candice
Subject: Reply to Do Points Have Area?
Author: Jesse Yoder
Date: 20 Jan 98 17:39:32 -0500 (EST)
Hi Candice -
Good to hear from you again! You recently asked a couple of questions
about my geometry, as follows:
I have two questions about your circular geometry. On December 18,
you posted
[John Conway]
>"If we're just talking about some purely conceptual space then the
assertions are meaningless until that space is somehow defined. 
Jesse speaks of "circular geometry", in which a "point" is the
smallest unit area, and in other statements he's made it clear
that he thinks of these "points" as little circles and lines
as like strings of beads: oooooooooooooooooo, in which 
any two adjacent ones touch each other at a point."
"Response: You seem to understand pretty well what I mean. Here is how
a plane would look, with lots of points;
The above points are circular, solid, and touching horizontally as
well as vertically. I can't draw a solid circle with this email
system. A point, as you say, is the smallest, allowable round unit
area in a system."
 What do you call the area between the points? Isn't there always a
smaller sized point?
Response: First off, let me take the second question. John Conway has
suggested I adopt a convention for indicating when I am using 'point'
in my sense, so I am capitalizing Point and Line. The answer is No,
there isn't always a smaller sized Point, since when a measurement is
made, you have to specify a frame of reference that says how small the
points are allowed to go. Thsi is often implicitly understood. For
example, if I'm measuring miles from work to home, I measure in tenths
of a mile. When I measure the amount of gas put in my car, I measure
in tenths of a gallon. The distance from here to the sun is measured
in miles. The positions of computer chips on a board might be measured
to the ten thousandth of an inch. Deciding what your frame of
reference is determines the size of your Points. Of course, there is
always ROOM FOR another point, but all that means is that you are
shifting to a different frame of reference, in which case again there
will be no smaller sized Points within this new frame of reference.
As for the space between points, the answer is that this is
mathematical space that can be referenced in relation to Points on the
coordinate system.
I hope this helps. I just read an account of the Euclidean idea that
points have no area, yet somehow make up a line in a book called The
Non-Euclidean Revolution by Richard Trudeau. This convinces me once
again that it is simply paradoxical to say, on the one hand, that
points have no dimension, and, on the other hand, that a line, which
has length, is made up of infinitely many of these dimensionless
points. Mutiplying 0 by infinity still equals 0. As far as I can see,
this remains an unresolved problem for Euclid's Axiom One (definition
of point), and I believe that ascribing area to points is the only way
around it.




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