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

 

 

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