Subject: Re: gabriel's horn (long and goofy)
From: Paul Hertzel (hertzpau@niacc.cc.ia.us)
Date: Thu Feb 24 2000 - 17:03:45 EST
>anyone know of a (sort-of) intuitive explanation of the features of
>gabriel's horn?
>
>David J Etzwiler
Anne C. Flanigan Said,
The explanation I've seen is to think of a paint can that holds a
finite amount of paint, but whose surface can never be completely
covered by the paint because it's infinite.
and Bret Taylor said,
You can fill this up with paint, but you can't paint it. How can this be?
I offer 5 points on the next test to any student who can give
me a rationale explanation of this paradox.
This analogy makes the paradox clear, but it doesn't offer to me any
intuition about what's going on. The question still remains, How are we to
understand this apparent impossibility? Shouldn't it be a contradiction,
since
the liquid is everywhere in contact with the inner surface?!?
In Flatland, Edwin Abbott speculates about what would be the important
differences in the perceptions of two-dimensional creatures and us, three-
dimensional creatures. They would perceive Abbott's finger penetrating
their
world as a circle; "Oh, you're a circle", they would say.
The paradox of the horn brought up by David E always makes me think about
Flatland, not so much in terms of the differences in *perceptions* of the
respective
creatures, but rather in terms of the differences in the *fabrics* of the
respective
realities. In search of some intuition I find myself wondering about the
comparability of the yardsticks by which the fabrics of the different
dimensional
worlds are measured, that is, as George Yanos' remarks allude to, the worth
of a
square unit in 2-space is more than the worth of a cubic unit in 3-space.
Recall Zeno could not understand how the fox ever caught the hare. If each
leap
cuts the remaining distance in half, then infinitely many leaps are still
needed to
find one's lunch. How can you cover infinitely many leaps every time you walk
to the refrigerator? It's that the measure of the leaps is not a good
predictor of the
measure of their sum.
I think the measure of area is not a good predictor of the measure of
volume. You
really CAN paint the outside with a 3-d substance (in theory). You just
can't paint
it with a 2-d substance (whatever that is).
Paul Hertzel
NIACC
Mason City, IA
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