Do Points Have Area?
Subject: Re: Reply to "Do Points Have Area?
Author: John Conway <conway@math.Princeton.EDU>
Date: Tue, 16 Dec 1997 12:29:28 -0500 (EST)
On 16 Dec 1997, Jesse Yoder wrote:
> Hi Candice -
> It was very nice to hear from you again.
> You (Candice) wrote the following:
> >" Since circles (as I believe) do not exist (I believe they are just
> polygons with many many sides and angles) it makes sense to have the
> exist??? Jesse, I don't understand how a circle can exist...What
> exactlly is a point??? That is what geometry is really based
> assumption that makes no sense..."
 and Jesse replied:
> Since circles exist, there is a need to find their areas. And this can
> be done by means of the formula 4*r*r, where 'r' equals the radius,
> (or, alternatively, d*d, where d equals the diameter). And the result
> will be in round inches, instead of square inches.
 [Candice again]:
> ">What
> exactlly is a point??? That is what geometry is really based
> assumption that makes no sense..."
[Jesse again]:
> In reference to 2, in Euclidean geometry, a point has no area. In
> circular geometry, it is the smallest unit area--hence, a point has
> area in circular geometry. 
 These discussions all seem very confused to me. Neither of the
participants seems to "believe" in Euclidean geometry. That's fine,
but they don't say what they MEAN by such statements as "circles
don't really exist they are just polygons with many sides" or "points 
really have area". 
 What ARE these "circles", "polygons", and "points" being spoken of?
Are we talking about points in real physical space, or in some purely
conceptual one? All the statements are nonsense for real physical space,
which behaves very strangely indeed when dimensions get small, and is,
in particular, so unlike Euclidean 3-dimensional space that all these
terms are utterly meaningless. To learn the appropriate questions to
ask about real physical space, you first have to learn a lot of physics.
Euclidean 3-space is only an approximation that's valid when no dimensions
are two large or too small.
 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.
 But in this second use, the word "point" seems to be used in
something like its Euclidean sense! The double use is confusing.
Since "point" is well-established with its Euclidean meaning,
Jesse should use a new term, say "spot", for his new object.
 Now I want to know how these spots are arranged. I presume
they can't overlap (for otherwise the area of the overlap of two
would be smaller than either). Are they arranged hexagonally,
like this:
 o o o o o o
 o o o o o o o
 o o o o o o
 o o o o o o
(but magnified so as to touch each other)? More importantly
than any particular such question is the meta-question : where
do we get all this information from? How does Jesse know that
these spots touch each other, are circular, and all have the
same area? I presume this is not by examination of physical space, 
but somehow by pure thought. 
> It is tempting to view a point as the limiting case of circle (a
> circle with no area). Is it contradictory to say "A circle has no
> area, yet it is solid"? kirby has taken me to task for using the
> phrase "radius of a point", yet if a point has area, it should be
> possible to meaningfully use this phrase.
 It's an example of the same kind of confusion. "Radius" has
a well-defined meaning in Euclidean geometry, as the distance 
from the center of a circle to any point on its periphery. It
has no meaning in Jesse's geometry until he gives it one. What's
your definition, Jesse?
> I agree with you that the key to unlocking the mysteries of geometry
> lie in a correct understanding of the concept of a point.
 What does "CORRECT" understanding MEAN? Just what kind of
system are you talking about? We know what "point" means in
Euclidean geometry, but you seem to think that this word has a
life of its own, and also means something outside of Euclidean
geometry. Well, I don't know what meaning you intend, and so
have no idea what it could possibly mean for a statement about
your new kind of "point" to be correct.
 It's as if you started to deny the truth of Lewis Carroll's
poem by saying that no snark is a boojum. Until you've given
meanings to the terms involved, it's silly to say that this
statement is either "correct" or "incorrect". 
> Have a supergreat holiday season!
> Jesse
and the same from me!
 John Conway



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