Tracy Yucikas <tyucikas@cts.com> wrote:
> Consider a glass of water in your hand and a table to place
> it on ... How many locations are there to choose from ? Is
> two-dimensional space quantized ? Are there less than an
> infinite number of "places" on the tabletop?
While "space" itself may not be quantized, MATTER is. Assuming that we
could differentiate a movement so small as only 1 atom wide, there will
still be only a finite number of places in any 1 (or n) dimension(al)
length of table. Even if we could control the placement so smoothly as to
place an object in spaces between atoms, there is still only a finite
number of places on that table -- and I expect that in practice you'd find
yourself unable to place objects with that kind of precision anyway. Of
course, from a practical point of view, what is the difference between
6.022 x 10^23 (a mole of atoms -- about 12 grams of carbon)
and
infinity?
However, from a mathematical prespective, there is clearly an infinte
number of "places" -- this property of the real numbers is called
"compactness", meaning that between any two real numbers, I can *always*
find another real number. The fact that mathematics allows us to talk
about infinity in such precise terms still doesn't mean that infinity
exists... merely that it is conveniant to suppose that it does.
On a historical note, anybody interested in the history of "infinity"
should do some research into Georg Cantor. Some say his madness was a
result of the apparent contradictory results of his research into the
transfinite realm... it is these contradictions which Russell, amoung
others, set out to fix in this century with axiomatic set theory.
ERiC