`Do Points Have Area?`
` `
`Subject:      Re: Reply to "Do Points Have Area?`
`Author:       John Conway <conway@math.Princeton.EDU>`
`Date:         Tue, 16 Dec 1997 12:29:28 -0500 (EST)`
` `
` `
` `
`On 16 Dec 1997, Jesse Yoder wrote:`
` `
`> Hi Candice -`
`> `
`> It was very nice to hear from you again.`
`> `
`> You (Candice) wrote the following:`
`> `
`> >" Since circles (as I believe) do not exist (I believe they are just`
`> polygons with many many sides and angles) it makes sense to have the`
` `
`............................`
` `
`> exist???  Jesse, I don't understand how a circle can exist...What`
`> exactlly is a point???  That is what geometry is really based on...an`
`> assumption that makes no sense..."`
` `
`............................`
` and Jesse replied:`
` `
`> Since circles exist, there is a need to find their areas. And this can`
`> be done by means of the formula 4*r*r, where 'r' equals the radius,`
`> (or, alternatively, d*d, where d equals the diameter). And the result`
`> will be in round inches, instead of square inches.`
` `
` [Candice again]:`
` `
`> ">What`
`> exactlly is a point???  That is what geometry is really based on...an`
`> assumption that makes no sense..."`
` `
`[Jesse again]:`
` `
`> In reference to 2, in Euclidean geometry, a point has no area. In`
`> circular geometry, it is the smallest unit area--hence, a point has`
`> area in circular geometry. `
` `
`  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".  `
` `
`   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.`
` `
`   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.`
` `
`  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.`
` `
`   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.  `
` `
`> 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?`
` `
`> 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".  `
` `
`> Have a supergreat holiday season!`
`> `
`> Jesse`
` `
`and the same from me!`
` `
`         John Conway    `

http://forum.swarthmore.edu/epigone/geometry-research/swenkhartil/Pine.3.07.9712161228.D1444-d100000@okra.princeton.edu