|
welcome to |
|
Do Points
Have Area? Reply to John Conway Subject: Reply to "Do Points Have Area?" Author: Jesse Yoder < jesse@flowresearch.com> Date: 18 Dec 97 17:59:08 -0500 (EST) Hi John - You began by quoting some back and forth comments between me and Candice, then said: >" 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". Response: John, possibly you have not seen my effort to better define my terms, but in response to some of your earlier posts I have proposed nine axioms as a replacement for the first nine axioms of Euclid. I could reiterate these here, but I would simply refer you to this earlier post. I'm not sure of the exact title (I believe it was "Re: Pi", and it was around 11/21/97). The idea that circles don't really exist is Candice's so I won't comment on that (since I disagree with that anyhow). But on whether points have area, I believe that if we define a point as having area, we can avoid Zeno-like paradoxes (as I've said before). A point is the smallest allowable round unit of measure in a system. This definition is more coherent than Euclid's, which is "A point is that which has no part." You then continue: >"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." 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. You then continue: >"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." Response: I'm not keen on giving up the term 'point', since I am simply responding to what I see as a paradox in the traditional concept of 'point'. Also, I don't want to have to invent a new term 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. You then continue as follows: >" 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." 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). >" 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. You then continue, beginning with a quote from an earlier post of mind: >" 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"." 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. You tnen quote me as saying, and then comment: ">> Have a supergreat holiday season! > > Jesse and the same from me!" Response: Since you are displaying such good will, I will apologize if I seemed sardonic in some of my earlier responses. You are forcing me to state my case as clearly as possible. And thanks for your holiday wishes! I will be out till 12/23, but will return then. Best wishes, Jesse
http://mathforum.org/kb/forum.jspa?forumID=130
©1999-2007 |
|
|
