Subject: Reply to Do Points Have Area

Reply to Candice

Subject:      Reply to Do Points Have Area?

Author:       Jesse Yoder jesse@flowresearch.com

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."

[You]

"Response: You seem to understand pretty well what I mean. Here is how

a plane would look, with lots of points;

oooooooooooooooooooooooooooooooooooooooooooooooooo

oooooooooooooooooooooooooooooooooooooooooooooooooo

oooooooooooooooooooooooooooooooooooooooooooooooooo

oooooooooooooooooooooooooooooooooooooooooooooooooo

oooooooooooooooooooooooooooooooooooooooooooooooooo

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.

Yours,

Jesse

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