Subject: Reply to "Reply to Do Points Have Area

Reply to Jesse

Subject:      Reply to "Reply to Do Points Have Area?"

Author:       Candice Hebden <dreamy_aurora@hotmail.com>

Date:         21 Jan 98 08:54:05 -0500 (EST)

Hey Jesse,

On January 20, 1998

[You said]

>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 Euclid's arealess point, but

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

sense.

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 Euclid's Axiom One >(definition

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

yours needs!

Yours,

Candice Hebden

dreamy_aurora@hotmail.com

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