`Reply to Candice`
` `
`Subject:      Reply to Do Points Have Area?`
`Author:       Jesse Yoder < jesse@flowresearch.com>`
`Date:         22 Jan 98 13:57:31 -0500 (EST)`
` `
`Hi Candice -`
` `
`Referring to the idea I suggested that the number of Points in a line`
`might vary with the frame of reference, you said:`
` `
`>"So,  if this is all true, then there would be much "empty" space in`
`certain frames of reference and less "empty" space in others.  Not`
`everyone measures the amount of gas in their car by tenths of a`
`gallon.  In fact, most of the world doesn't even know what a gallon`
`is!  Every frame of reference you make will have to be stated before`
`any work is done on the problem.  Still then, many people might not`
`understand your frame of reference!"`
` `
`RESPONSE: Perhaps you're right about the empty space. But I don't`
`really understand why this is so different from the Cartesian`
`coordinate system, where you have an x and y axis serving as a frame`
`of reference for the points on the plane. The area in the plane is`
`empty, until you put in some additional lines or points. If you draw a`
`curve on a Cartesian Coordinate system, what is the rest of the area`
`if not "empty space"?`
` `
`Let me address the issue about shifting frames of reference. I realize`
`that more people use the metric system of liters than the American`
`system of gallons, so I don't expect everyone to measure gas in tenths`
`of gallons. The point of saying you have to specify a refernce system`
`is that it provides a way to avoid the paradox of arealess points. If`
`you specify a frame of reference, by which I mean saying what your`
`unit of measurement is, then you can picture the x and y axes as`
`composed of a corresponding number of Points. For example, if you are`
`measuring in tenths, then a Point can be 1/10 of an inch (I realize`
`I'm switching from gallons to inches here). And if you are measuring`
`to the 1/100 position, then the Points are 1/100th of an inch. This`
`avoids the paradoxes that arise from arealess points. And if this`
`seems awkward or unduly complex, keep in mind that every time you make`
`a measurement, some unit of reference is explicitly or implicitly`
`implied. If you say "It's 93 million miles to the sun," you're using`
`miles as your unit. I'm just saying "Let's make this assumption`
`explicit, and we avoid the paradoxes that arise form arealess points."`
` `
`Candice, you then continue as follows:`
` `
`>" Sometimes the simplier theory is more "correct" because it makes`
`sense.  I certainly am not a believer of Euclid's arealess point, but`
`it does have it's merits.  People once thought that the Earth was the`
`center of the Universe.  Aristole made all kinds of rules to support`
`his theory in respect to the "strange" orbits of Jupiter's satilites`
`and moons.  But Copernicus's idea of the Heliocentric gallaxy`
`(although not widley accepted at first) was simplier and makes more`
`sense."`
` `
`RESPONSE: Agreed about simpler theories sometimes being true, but`
`simple theories can also contain hidden paradoxes that aren't`
`immediately obvious.  It sounds easy to say "OK, I'm standing here on`
`a dimensionless point. Now if I move to the other side of the room,`
`I'll be located at another dimensionless point." It's not until you`
`bring out Zeno's paradox, which seems ot show you can never reach the`
`other side of the room, that you realize that it may not be such a`
`great idea to say that a 3-dimensional object can be located at an`
`arealess or dimensionless point. `
` `
`In general, I'm not a big fan of the "Simpler, and therefore more`
`likely to be true" theory. If the world is complex, it may take a`
`complex theory to adequately explain it!`
` `
`You then say:`
` `
`>"I am not doubting the accuracy for you circular geometry.  It seems`
`it will make sense once certain things are worked out."`
` `
`RESPONSE: Thanks for the vote of confidence. I realize there are`
`problems with the theory I'm presenting too, but I think that`
`sometimes  certain people lose track of the problems in their own`
`positions as they get carried away criticizing the positions of others`
`(I realize this applies equally well to me).`
` `
`Best wishes,`
` `
`Jesse`

http://forum.swarthmore.edu/epigone/geometry-research/swenkhartil/79jyd7fzl79y@forum.swarthmore.edu