Assume, just for the sake of argument, that space /isn't/ quantized at
the Planck length. Push the glass forward on the table about an inch.
Based on the mathematical assumption that the distance traveled by the
glass can be viewed as a segment of a number line with an infinite number
of rational numbers contained therein, you've just made the glass assume
an infinite number of positions on the table in a finite amount of time.
If the transition from one infinitesimal point to the other takes some
finite amount of time, however minute, the glass's journey would take
forever, which is obviously not the case. If, on the other hand, the
transitions happen instantaneously, you have a seeming paradox, where
one infinity tries to cancel out the other, leaving you with however
long it took you to move the glass, say, 0.765 seconds. Obviously,
something is wrong here as well. However, if your primary assumptions
are that space is quantized (the Planck length) /and/ time is quantized
(the chronon), then your calculations describe what actually happened,
and also predict what will happen. This leads me to describe one of the
filters I use when I think about things like this:
A paradox is nature's way of telling us one of three things:
1) Our initial assumptions were erroneous
or
2) Our assumptions were sound but out analysis was faulty
or
3) Both.
Dan
> a possibly "less than infinite" approach to the concept of infinity ..
>
> 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 ? Any spot you pick, I (for example) relocate the glass
>to a new spot 1/sqrt(3) from the left side if the original distance
>was "1" . Starts to look like an infinite number of locations on the
>table top .. even using only one dimension. And then there's the
>"front-back" dimension ... and the rotation of the logo on the glass ...
>each of these "variables" seem to have an infinite number of states.
> As for the amount of water in the glass ... if composed of something
>like quarks, sure, there's a finite number of ways to partition this
>quantity, but the spatial relationships between these quarks exhibit
>something like "uncountably infinite" number of states.
> The language of mathematics shapes the concepts easily framed.
>The language of an aborigine may not contain a word for "snow", but
>this does not mean that snow does not exist.
>
>
>... the home for non-verbal memes ...