`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 aways 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 (whcih is really just changing units of`
`measurement) can produce a smaller Point. I prefer this to saying that`
`the points are infinitely small.`
` `
`Jesse`

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