`Definition of Point`
` `
`Subject:      Reply to "Do Points Have Area?"`
`Author:       Jesse Yoder < jesse@flowresearch.com>`
`Date:         8 Jan 98 07:25:57 -0500 (EST)`
` `
`Hi John -`
` `
`On December 23, 1997, John Conway wrote, in response to some comments`
`I made on circular geometry:`
` `
`>" 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?"`
` `
`Response: This is a good question, but I don't know why the idea of`
`two mathematical points touching is any more obscure in my geometry`
`than it is anywhere else, e.g. when two physical objects touch. And`
`I'm not sure what it means for two physical objects to touch -- do`
`they have a point in common, or do they share any molecules or`
`electrons? For two mathematical points to touch is for them to stand`
`in the same relation two spheres do when they touch (e.g., two`
`baseballs touching each other), whatever that is. `
` `
`Thinking along these lines, do the points on a Euclidean number line`
`ever touch? I assume the answer is "No", since you can always put`
`another point between two points on a Euclidean number line. Could`
`they ever touch, and how much space is between them?`
` `
`I don't have a problem with the standard definition of a circle, as`
`all points equidistant from a center point, except that this`
`definition doesn't insure that the circle is continuous. Four points`
`at north, south, east, and west from the center don't make a circle,`
`even though they are equidistant from the center of the circle. Also,`
`the points must somehow form a circular pattern, and I'm not sure how`
`to say that without smuggling the concept of "circular" in the`
`definition, e.g., "a closed, continuous, circular line whose points`
`are all equidistant from a fixed point."`
` `
`You then said:`
` `
` 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???`
` `
`Response: The points are conceived as existing in an array, somewhat`
`like the Cartesian Coordinate system. The plane they are on extends`
`indefinitely in all directions, so I don't see why it has to been seen`
`as "square".`
` `
`There are spaces in the this coordinate system, since you can't "tile`
`a plane" with circles i.e., fill up an entire plane with circles,`
`leaving no area uncovered. No, this empty space is not made up of`
`points -- it is simply empty mathematical space. Again, you seem to be`
`implying that there is some unexplained phenomenon in circular`
`geometry -- what is space 'made up of' -- that doesn't exist in`
`Euclidean geometry. But this question can also be asked of Euclidean`
`geometry.`
` `
`A better way to view this is as 3-dimensional, in which case the`
`points or "spots" as you prefer to call them become 3-dimenensional`
`balls with physical space between them.`
` `
`You then continue:`
` `
`>"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?"`
` `
`Response: Please explain what you mean by "isotropic" and by saying`
`that my geometry is not isotropic. I'm not sure what direction of`
`measurement has to do with this, unless you're using the term in some`
`special sense. Yes, my geometry does give a better fit to our native`
`ideas about the world and about geometry.`
` `
`You then continue, beginning with a quote from an earlier post: `
` `
`> "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"!)"`
` `
`Response: 'Point' is being used in my sense, not in Euclid's. Let me`
`try to give a better definition without using the term 'periphery': "A`
`circle is a continuous, closed line, all of whose points are`
`equidistant from a fixed point. This fixed point is called the center`
`of the circle, and the radius of the circle is the distance from the`
`center to any point on the continuous, closed line. This continuous,`
`closed line is called the circumference. The fixed point or 'spot' is`
`the smallest allowable unit area."`
` `
`You then wrote:`
` `
`">    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."`
` `
`Response: OK, I will resend them in a separate post.`
` `
`You continue:`
` `
`>"  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.]"`
` `
`Response: I understand more or less what you are saying here. But the`
`problem is that the terms we use like 'point' and 'line' are deeply`
`embedded in our conceptual framework, complete with Euclidean`
`interpretations and definitions. To see how completely Euclidean`
`geometry has penetrated our consciousness, notice how completely`
`architecture, furniture, and just about every other physical object`
`constructed by man conforms to straight-line geometry. This is not`
`"natural" -- it is the result of the extent to which Euclidean`
`assumptions have been adopted as if they were "common sense". `
` `
`Now if I come along and say "But there are paradoxes hidden in these`
`assumptions -- here are the assumptions I want to propose instead" --`
`then I can't even say this without using the terms I am trying to`
`propose better definitions for--or in some cases, the definitions may`
`remain the same, but the terms may be interpreted differently.`
`Nonetheless, I agree that the result can be confusing, so I will try`
`to be more aware of when I am using words in a Euclidean vs.`
`concentric geometry sense -- and specify the difference when this is`
`necessary.`
` `
`Happy New Year!`
` `
`Jesse `

http://forum.swarthmore.edu/epigone/geometry-research/thyspenddwox/48r9qp1x40im@forum.swarthmore.edu