Subject: RE: Reply to "Do Points Have Area?"
Author: Jesse Yoder < email@example.com>
Date: Mon, 2 Feb 1998 09:01:27 -0500
Bill - On the face of it, I agree. I think it is very difficult to come
up with a noncircular definition of a circle. Notice, however, that the
definition you quote is not viciously circular, since it does not
contain the word 'circle' - instead, it contains the word 'circular.' On
the other hand, as I have argued elsewhere (in a post to Candice), the
Euclidean definition of the circle as a set of points equidistant from a
fixed point is BANKRUPT, since it allows for the possibility that four
points form a circle.
In any system, some concepts or terms must be accepted as undefined. Not
every concept can be defined in terms of previously defined concepts.
Garry (Mr K K G Yau) has urged me in the past to accept 'point' as
undefined. I am unwilling to do this, because the difference between
points and Points (which are points with area) is fundamental to the
Circular Geometry I have developed.
On the other hand, it may be necessary to accept certain operations as
fundamental and undefined. In particular, it may be necessary to accept
as undefined the concept of circular motion (as in a Circle is generated
by moving a Point in a circular motion). Likewise, if we are to suffer
with the concept of a straight line (which I have it appears been forced
to accept against my wishes, since I need the concept of radius and
diameter), perhaps the concept of "straight" should also be undefined,
as in a straight Line is generated by moving a Point in a single uniform
direction--the path so generated is a straight Line. (Note that a curve
is generated by moving a Point in a nonuniform direction).
Possibly the idea of circular motion could be specified without using
the concept of circle or circularity, e.g., by moving a Point in such a
way as to maintain a constant distance from a fixed point until the
Point intersects with itself--the path so generated is a Circle. If you
accept this, then this is a noncircular definition of the circle.
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