Re: Uniqueness of physical objects in the universe.

From: robert j. kolker (nowhere_at_nowhere.net)
Date: 10/31/04 

Date: Sun, 31 Oct 2004 00:06:26 -0400

Lefty wrote:
>
>
> I am trying to approach the problem of proving more important things about
> the universe. I would like to see a proof that space is continuous or not. I
> dont know if it's possible, but that is where I am trying to go. I know that
> this is a difficult problem, but I also know that there is an answer out
> there somewhere and I would like to see it demonstrated once and for all -
> regardless of who completes it.

Let us assume, arguendo, that spacetime is granular and very
fine-grained. In order to do any useful calculations you will have to
use averages and continuous approximations. We do not deal with gases by
trying to figure out the trajectory of every last molecule. It is an
practical impossibility to do so. A mole weight of gase has someghint
like 10^24 molecules. We have neither the time nor the computer power to
deal with molecules as individual entities. That is why we do
statistical mechanics so we can average things out.

Matter consits of atoms and is discrete, but we do elasticity theory by
assuming an elastic body is continuous with a variable modulus of
elasticity and density. We idealize a such a body by treating it as a
continuum.

If we tried to deal with fine grained space in a discrete fashion we
would run into intractable mathematical difficulties. So, in a way, it
does not matter if spacetime is discrete. We will treat it continously
and our math will be based on compact spaces.

Even in quantum theory we end up using averages to deal with large
aggregates or large bodies. And the math of quantum theory is real and
complex analysis with operators on Hilbert spaces. So the math is going
to be based on continuity and compactness regardless of the structure of
spacetime.

Bob Kolker

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