Reply to Jesse
Subject: Reply to "Reply to Do Points Have Area?"
Author: Candice Hebden <firstname.lastname@example.org>
Date: 21 Jan 98 08:54:05 -0500 (EST)
On January 20, 1998
>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. This 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.
So, if this is all true, then there would be much "empty" space in
certain frames of reference and less "empty" space in others. Not
everyone measures the amount of gas in their car by tenths of a
gallon. In fact, most of the world doesn't even know what a gallon
is! Every frame of reference you make will have to be stated before
any work is done on the problem. Still then, many people might not
understand your frame of reference!
Sometimes the simpler theory is more "correct" because it makes
sense. I certainly am not a believer of
's arealess point, but Euclid
it does have it's merits. People once thought that the Earth was the
center of the Universe. Aristotle made all kinds of rules to support
his theory in respect to the "strange" orbits of Jupiter's satellites
and moons. But Copernicus's idea of the Heliocentric galaxy
(although not widely accepted at first) was simpler and makes more
I am not doubting the accuracy for your circular geometry. It seems it
will make sense once certain things are worked out.
[You then continue, answering my first question]
>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 still don't understand. All space must contain points right?
Aren't points supposed to define the space of something? If this is
true then there is space unaccounted for... making an infinite amount
of non-space! If it isn't true then state it. And then explain to
me, please, how space could go from empty to not-empty with a change of
frame of reference.
John Conway earlier posed the sujestion that you lay your points on a
hexagonal frame. As the frame of reference decreases, there is always
a model for the arrangement of the points so that there is less empty
space. Would you want to use something like that? Or would that
further confuse the issue?
[You continue again]
>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
's Axiom One >(definition Euclid
of point), and I believe that ascribing area to points is the >only
way around it.
I am in complete agreement with the failure of Euclidean's arealess
points. I also am in agreement with you that a new type of geometry
should be devised. However, I do not believe it has been completed
yet! Good luck. I really hope you can "fix" what the new geometry of