Points & points
Subject: Re: Reply to "Do Points Have Area?"
Author: John Conway <conway@math.Princeton.EDU>
Date: Thu, 18 Dec 1997 18:16:28 -0500 (EST)
On 18 Dec 1997, Jesse Yoder wrote:
> Hi John -
> Response: I think it's pretty clear here that we're talking about
> mathematical conceptual space, not just physical space which
> apparently is more Riemannian than Euclidean. Even in Euclidean
> geometry when I measure a trianglular object, I bestow 180 degrees on
> it even though the physical object may not be perfectly triangular.
> Likewise, "straight" lines like ropes are not perfectly straight, but
> we treat them as straight when we measure (even a physical ruler isn't
> perfectly straight).
> You then continue:
> >"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 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.
 OK, so what does "touch" mean, for your "points"? I'm not
going back to your old postings, but I definitely recall
that in one of them, it was said that adjacent ones touched 
at a point! 
 Again, what does "circular" mean? Euclid's definition
is that a circle consists of all points at a given distance
 from another point, called the center of the circle. What's
your definition?
 Your figure above suggests that your "points" in a given
plane are arranged in a square array. Is this true? 
 In Euclidean geometry, when circles are arranged like this,
there are some spaces in between. Is this also the case in
your new geometry? If so, what are these spaces "made of"? Points???
> for 'circle', etc. But I will accept your idea for now (though perhaps
> I would prefer the term 'ball' to 'spot.'), if it would help clarify
> the discussion and avoid ambiguity.
 It certainly would. A very big problem with all your descriptions
is that they presuppose an underlying Euclidean geometry. For instance,
you say you want to use the term "circle", a concept that, to the rest
of the world, is defined in terms of Euclid's notion of "point" rather
that your new one. They are perfectly comprehensible if you do allow
yourself to use Euclid's notions, but in that case it's confusing,
and also to my mind improper, to use some of the Euclidean terminology
with different meanings to his.
> Response: See above -- the hexagonal idea is interesting, but what I
> have in mind is simply a bunch of "spots" or "balls" that touch each
> other above, below, and on the sides (also, there is an x - y
> coordinate system, with one of these rows serving as an x axis and one
> row serving as a y axis).
 Aha! So your geometry fails to be isotropic, and has a preferred
system of coordinate-axes! But I thought the point of your system
was that it gave in some sense a better fit to our native ideas
about the world, or at least about geometry. Was this my misapprehension?
Was it really supposed just to be a better fit to the pixels on
a computer screen?
> >" You then contine, quoting from an earlier post of mine, then
> commenting:
> > 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?"
> Response: I suppose this is a fair question. I would give the same
> definition as in Euclidean geometry -- a raidus is the distance from
> the center of a circle to any point on its periphery.
 And in what sense is the word "point" being used in that last
clause? Yours, or Euclid's? Also, what's "periphery" mean in
your new system? As far as I can see, there are NONE of your
"points" that are actually ON the periphery of a given one,
and exactly FOUR "points" that are adjacent to it. But the 
distance from any of these to the given one is what Euclid
and I would call its DIAMETER, rather than its RADIUS (and I
might remark, that since you seem to be adopting Euclid's 
definition of a circle, that this circle seems to consist
just of four of your "points"!)
> You then continue, beginning with a quote from an earlier post of
> mind:
> 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"."
> Response: Again, John, I would refer you to my nine axioms. But as for
> a point, I will stick with this definition: "A point (or spot, or
> ball) is the smallest round unit area allowable in a system." This
> seems to be more informative than the Euclidean "area with no part,"
> which you feel is imbued with so much meaning. I am not resorting to
> uttering meaningless phrases, as in Carroll's poem. 
 I read your nine axioms when you posted them, and, then as now,
found your language replete with tacit assumptions from the very
Euclidean geometry you were trying to replace. I've since deleted
them, but will happily respond to them if you'll send me another
 Some time ago, you were critical of the logic of the calculus,
and now you have some similar criticisms of Euclidean geometry.
But those who live in glass houses should at least be careful
when they throw stones! In particular, you really shouldn't
give a word two meanings in the same sentence (as I believe I
caught you doing with "point"). If you do so, then you are
clearly the one to blame if other people misunderstand you as
a result. If you intend to reject some of Euclid's ideas
and definitions while accepting others, then you must be just
as careful to say what you accept as well as what you reject.
 Also, you cannot allow yourself to make tacit assumptions
 from classical geometry in the way that you repeatedly have;
for it's improper to do so if your reader may not; but if you
allow your reader to make such assumptions from classical geometry
in the way that you do, then he might well make so many of them that
in effect he assumes ALL of Euclidean geometry. [To tell you
the truth, I think that you are effectively doing this, while
appearing to deny it.]
 I hope you read this before you get out of touch for the
season, because you may need quite some time to think out your
 Regards, John Conway




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