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Do Points
Have Area? Reply to John Conway Subject: Re: Reply to Do Points Have Area? Author: Jesse Yoder < jesse@flowresearch.com> Date: 22 Jan 98 14:32:16 -0500 (EST) Hi John - On January 21, 1998, you wrote: >" [John Conway] > >"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." > > [Jesse] > "Response: You seem to understand pretty well what I mean. Here is how > a plane would look, with lots of points; >" It still surprises me that you didn't even notice the double use of the word "point" in the sentence I obliquely quoted from you! How can two points touch at a point?" RESPONSE: How can two Points touch at a Point? I don't know; perhaps they touch at a point. But I don't understand why my position is so much less understandable than the Euclidean one. On Euclid's account a line if made up of infinitely many dimensionless points. So the points are compactly packed, yet there is always room for one more between any two points! Does this mean there is empty space between two Euclidean points? I'm not even sure they touch -- what I'm claiming is that when two Points are next to each other, they have the same relation as when two physical objects are next to each other. But I still don't even know if two objects that are touching have a point in common, or if they are just "up against" each other the way a baseball would be in a glove. You then continue: >" Of course, you've now agreed to distinguish between "Points" and "points", but it really seems to me that in a fundamental sense this vitiates your system, because it bases it on the traditional notions. Surely you should be able to describe the structure and arrangement of your Points without using Euclid's points? If not, it can hardly be true that "a Point is the smallest allowable unit of area"." RESPONSE: I don't think that distinguishing between Points and points vitiates my system. If you say that I've reintroduced Euclidean points by talking about intersecting Points, then I would refer you to the above paragraph, where I say it is not yet clear how two Points next to each other relate to each other--it may not require introducing the idea of point. You then continue: >"I have difficulty in following your comments about switching to new frames of reference. Do you think this is legal, or were you really saying it was impossible? It seems to me that it's obviously impossible in your system. If a Point is really the smallest allowable unit of area, then no kind of changing frames of reference can possibly produce a smaller Point." RESPONSE: If my "frames of reference switching" comments are hard to follow, I apologize. Perhaps I haven't adequately explained the idea. But the idea is, I claim, not hard to understand, though perhaps the term "frame of reference" is too abstract. What I am saying is that when someone uses a coordinate system, they should specify their unit of measurement (which itself is embedded in a frame of reference). I believe there is a unit of measurement implicit every time a measurement is made. For example, if I'm measuring distance to the sun, it's miles. If it's gasoline, it's tenths of a gallon. Once the unit of measurement is known, this determines the size of the points in the line. If it's tenths of an inch, the Points are 1/10 of an inch in length (or diameter). If it's 1/100th of an inch, the Points are 1/100th of an inch in length (or diameter). This is how changing frames of reference (which is really just changing units of measurement) can produce a smaller Point. I prefer this to saying that the points are infinitely small. Jesse http://mathforum.org/kb/forum.jspa?forumID=130 Back to "Do Points Have Area?"
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