`Points & points`
` `
`Subject:      Re: Reply to "Do Points Have Area?"`
`Author:       John Conway <conway@math.Princeton.EDU>`
`Date:         Thu, 18 Dec 1997 18:16:28 -0500 (EST)`
` `
` `
` `
`On 18 Dec 1997, Jesse Yoder wrote:`
` `
`> Hi John -`
`> `
`> 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.`
` `
`   OK, so what does "touch" mean, for your "points"?  I'm not`
`going back to your old postings, but I definitely recall`
`that in one of them, it was said that adjacent ones touched `
`at a point!  `
` `
`   Again, what does "circular" mean?  Euclid's definition`
`is that a circle consists of all points at a given distance`
` from another point, called the center of the circle.  What's`
`your definition?`
` `
`   Your figure above suggests that your "points" in a given`
`plane are arranged in a square array.  Is this true?  `
` `
`   In Euclidean geometry, when circles are arranged like this,`
`there are some spaces in between.  Is this also the case in`
`your new geometry?  If so, what are these spaces "made of"? Points???`
`.................................`
`> 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.`
` `
`    It certainly would.  A very big problem with all your descriptions`
`is that they presuppose an underlying Euclidean geometry.  For instance,`
`you say you want to use the term "circle", a concept that, to the rest`
`of the world, is defined in terms of Euclid's notion of "point" rather`
`that your new one.  They are perfectly comprehensible if you do allow`
`yourself to use Euclid's notions, but in that case it's confusing,`
`and also to my mind improper, to use some of the Euclidean terminology`
`with different meanings to his.`
` `
` `
`> 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).`
` `
`    Aha!  So your geometry fails to be isotropic, and has a preferred`
`system of coordinate-axes!  But I thought the point of your system`
`was that it gave in some sense a better fit to our native ideas`
`about the world, or at least about geometry.  Was this my misapprehension?`
`Was it really supposed just to be a better fit to the pixels on`
`a computer screen?`
` `
`> >" 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.`
` `
`    And in what sense is the word "point" being used in that last`
`clause?  Yours, or Euclid's?  Also, what's "periphery" mean in`
`your new system?  As far as I can see, there are NONE of your`
`"points" that are actually ON the periphery of a given one,`
`and exactly FOUR "points" that are adjacent to it.  But the `
`distance from any of these to the given one is what Euclid`
`and I would call its DIAMETER, rather than its RADIUS  (and I`
`might remark, that since you seem to be adopting Euclid's `
`definition of a circle, that this circle seems to consist`
`just of four of your "points"!)`
` `
` `
` `
`> You then continue, beginning with a quote from an earlier post of`
`> mind:`
`> `
`>     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.  `
` `
`    I read your nine axioms when you posted them, and, then as now,`
`found your language replete with tacit assumptions from the very`
`Euclidean geometry you were trying to replace. I've since deleted`
`them, but will happily respond to them if you'll send me another`
`copy.     `
` `
`   Some time ago, you were critical of the logic of the calculus,`
`and now you have some similar criticisms of Euclidean geometry.`
`But those who live in glass houses should at least be careful`
`when they throw stones!  In particular, you really shouldn't`
`give a word two meanings in the same sentence (as I believe I`
`caught you doing with "point").  If you do so, then you are`
`clearly the one to blame if other people misunderstand you as`
`a result.  If you intend to reject some of Euclid's ideas`
`and definitions while accepting others, then you must be just`
`as careful to say what you accept as well as what you reject.`
` `
`   Also, you cannot allow yourself to make tacit assumptions`
` from classical geometry in the way that you repeatedly have;`
`for it's improper to do so if your reader may not; but if you`
`allow your reader to make such assumptions from classical geometry`
`in the way that you do, then he might well make so many of them that`
`in effect he assumes ALL of Euclidean geometry.  [To tell you`
`the truth, I think that you are effectively doing this, while`
`appearing to deny it.]`
` `
`        I hope you read this before you get out of touch for the`
`season, because you may need quite some time to think out your`
`position!  `
` `
`             Regards,  John Conway`

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