Definition of Point
Subject: Reply to "Do Points Have Area?"
Author: Jesse Yoder < firstname.lastname@example.org>
Date: 8 Jan 98 07:25:57 -0500 (EST)
Hi John -
On December 23, 1997, John Conway wrote, in response to some comments
I made on circular geometry:
>" 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?
's definition Euclid
is that a circle consists of all points at a given distance
from another point, called the center of the circle. What's
Response: This is a good question, but I don't know why the idea of
two mathematical points touching is any more obscure in my geometry
than it is anywhere else, e.g. when two physical objects touch. And
I'm not sure what it means for two physical objects to touch -- do
they have a point in common, or do they share any molecules or
electrons? For two mathematical points to touch is for them to stand
in the same relation two spheres do when they touch (e.g., two
baseballs touching each other), whatever that is.
Thinking along these lines, do the points on a Euclidean number line
ever touch? I assume the answer is "No", since you can always put
another point between two points on a Euclidean number line. Could
they ever touch, and how much space is between them?
I don't have a problem with the standard definition of a circle, as
all points equidistant from a center point, except that this
definition doesn't insure that the circle is continuous. Four points
at north, south, east, and west from the center don't make a circle,
even though they are equidistant from the center of the circle. Also,
the points must somehow form a circular pattern, and I'm not sure how
to say that without smuggling the concept of "circular" in the
definition, e.g., "a closed, continuous, circular line whose points
are all equidistant from a fixed point."
You then said:
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???
Response: The points are conceived as existing in an array, somewhat
like the Cartesian Coordinate system. The plane they are on extends
indefinitely in all directions, so I don't see why it has to been seen
There are spaces in the this coordinate system, since you can't "tile
a plane" with circles i.e., fill up an entire plane with circles,
leaving no area uncovered. No, this empty space is not made up of
points -- it is simply empty mathematical space. Again, you seem to be
implying that there is some unexplained phenomenon in circular
geometry -- what is space 'made up of' -- that doesn't exist in
Euclidean geometry. But this question can also be asked of Euclidean
A better way to view this is as 3-dimensional, in which case the
points or "spots" as you prefer to call them become 3-dimenensional
balls with physical space between them.
You then continue:
>"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
Was it really supposed just to be a better fit to the pixels on
a computer screen?"
Response: Please explain what you mean by "isotropic" and by saying
that my geometry is not isotropic. I'm not sure what direction of
measurement has to do with this, unless you're using the term in some
special sense. Yes, my geometry does give a better fit to our native
ideas about the world and about geometry.
You then continue, beginning with a quote from an earlier post:
> "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"!)"
Response: 'Point' is being used in my sense, not in Euclid's. Let me
try to give a better definition without using the term 'periphery': "A
circle is a continuous, closed line, all of whose points are
equidistant from a fixed point. This fixed point is called the center
of the circle, and the radius of the circle is the distance from the
center to any point on the continuous, closed line. This continuous,
closed line is called the circumference. The fixed point or 'spot' is
the smallest allowable unit area."
You then wrote:
"> 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
Response: OK, I will resend them in a separate post.
>" 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.]"
Response: I understand more or less what you are saying here. But the
problem is that the terms we use like 'point' and 'line' are deeply
embedded in our conceptual framework, complete with Euclidean
interpretations and definitions. To see how completely Euclidean
geometry has penetrated our consciousness, notice how completely
architecture, furniture, and just about every other physical object
constructed by man conforms to straight-line geometry. This is not
"natural" -- it is the result of the extent to which Euclidean
assumptions have been adopted as if they were "common sense".
Now if I come along and say "But there are paradoxes hidden in these
assumptions -- here are the assumptions I want to propose instead" --
then I can't even say this without using the terms I am trying to
propose better definitions for--or in some cases, the definitions may
remain the same, but the terms may be interpreted differently.
Nonetheless, I agree that the result can be confusing, so I will try
to be more aware of when I am using words in a Euclidean vs.
concentric geometry sense -- and specify the difference when this is
Happy New Year!
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Wakefield, MA 01880