Reply to John Conway
Subject: Re: Reply to Do Points Have Area?
Author: Jesse Yoder < email@example.com>
Date: 22 Jan 98 14:32:16 -0500 (EST)
Hi John -
On January 21, 1998, you wrote:
>" [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
> a plane would look, with lots of points;
>" It still surprises me that you didn't even notice the double
use of the word "point" in the sentence I obliquely quoted from you!
How can two points touch at a point?"
RESPONSE: How can two Points touch at a Point? I don't know; perhaps
they touch at a point. But I don't understand why my position is so
much less understandable than the Euclidean one. On
's account a Euclid
line if made up of infinitely many dimensionless points. So the points
are compactly packed, yet there is aways room for one more between any
two points! Does this mean there is empty space between two Euclidean
points? I'm not even sure they touch -- what I'm claiming is that when
two Points are next to each other, they have the same relation as when
two physical objects are next to each other. But I still don't even
know if two objects that are touching have a point in common, or if
they are just "up against" each other the way a baseball would be in a
You then continue:
>" Of course, you've now agreed to distinguish between "Points"
and "points", but it really seems to me that in a fundamental
sense this vitiates your system, because it bases it on the
traditional notions. Surely you should be able to describe
the structure and arrangement of your Points without using
Euclid's points? If not, it can hardly be true that "a Point
is the smallest allowable unit of area"."
RESPONSE: I don't think that distinguishing between Points and points
vitiates my system. If you say that I've reintroduced Euclidean points
by talking about intersecting Points, then I would refer you to the
above paragraph, where I say it is not yet clear how two Points next
to each other relate to each other--it may not require introducing the
idea of point.
You then continue:
>"I have difficulty in following your comments about switching to
new frames of reference. Do you think this is legal, or were you
really saying it was impossible? It seems to me that it's obviously
impossible in your system. If a Point is really the smallest
allowable unit of area, then no kind of changing frames of
reference can possibly produce a smaller Point."
RESPONSE: If my "frames of reference switching" comments are hard to
follow, I apologize. Perhaps I haven't adequately explained the idea.
But the idea is, I claim, not hard to understand, though perhaps the
term "frame of reference" is too abstract. What I am saying is that
when someone uses a coordinate system, they should specify their unit
of measurement (which itself is embedded in a frame of reference). I
believe there is a unit of measurement implicit every time a
measurement is made. For example, if I'm measuring distance to the
sun, it's miles. If it's gasoline, it's tenths of a gallon. Once the
unit of measurement is known, this determines the size of the points
in the line. If it's tenths of an inch, the Points are 1/10 of an inch
in length (or diameter).If it's 1/100th of an inch, the Points are
1/100th of an inch in length (or diameter). This is how changing
frames of reference (whcih is really just changing units of
measurement) can produce a smaller Point. I prefer this to saying that
the points are infinitely small.
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