>

`Foundations`
` `
`Subject:      RE: Coordinate systems`
`Author:       John Conway <conway@math.Princeton.EDU>`
`Date:         Tue, 10 Feb 1998 14:11:15 -0500 (EST)`
` `
` `
` `
`On Tue, 10 Feb 1998, Jesse Yoder wrote:`
`> RESPONSE: As I have pointed out in previous discussion, there is an`
`> anti-Eucldean and an anti-Cartesian element to what I call Circular`
`> Geometry:`
`> `
`> The anti-Euclidean element consists of saying that Points have area`
`> (unlike points, which have no area), and that Lines have width and`
`> length (unlike lines, which have only length. `
` `
`   Yes, this is the sort of thing I thought you were trying to say.`
`But with "Points" being defined in Euclidean terms as certain discs it's`
`a tautology - obviously discs have area.  Also, what purpose is there in `
`defining your anti-Euclidean ideas in Euclidean terms?  You don't seem`
`to realise how silly it sounds to say that "Points touch at points"`
`when half of your purpose is to abolish the use of points.  An anti-Euclidean`
`geometry that can only be built on a foundation of Euclidean geometry`
`doesn't sound to me to be a very successful opponent!   Aren't you capable`
`of developing it in its own terms?`
` `
`> The anti-Cartesian element consists of analyzing circular area in terms`
`> of round inches rather than square inches.`
` `
`    I don't know why you want to do this, and why it isn't any more`
`than a triviality.  In Euclidean terms we can define "a circular inch"`
`(I deliberately use a term other than "round inch" because I'm not`
`quite sure what you want that to mean) to be the area of a circle`
`of radius 1.  Then it follows from Euclid's theorems that the area`
`of a circle of radius R is  R^2 round inches.  `
` `
`    I think that this means that Euclid provides a foundation that`
`can do what you want about round inches (and, of course, can also`
`do much more, that you don't want).  So again we come to the `
`question you haven't really faced: it seems to be trivial to define`
`your kind of geometry on the foundation of Euclidean geometry - can`
`you do it WITHOUT assuming this foundation?`
` `
`    I may remark that even in Euclidean terms I still haven't got`
`much of a clue about the meaning of your proposed terms.  This may`
`be just because I've forgotten answers you may have given to some`
`of my questions, so I'll repeat them. `
` `
`   At one time we agreed that Points could be taken to be discs of`
`diameter one.  Is every such disc a Point, or only those centered `
`at points (x,y) with integer coordinates; or maybe some other set?`
`In particular, can Points overlap without being equal?`
` `
`  Also, can you remind me what Circles and straight Lines are; and`
`if they are not made up of Points, what it means for a Point to`
`be "on" a Line or Circle?`
` `
`     John Conway`

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