> From: "Joel Dobrzelewski" <[EMAIL PROTECTED]>
> Subject: Re: Countable vs Continuous
> Date: Thu, 21 Jun 2001 08:41:13 -0400
> > There is the rather harmless kind: the countable one. And some say
> > there is another kind, a strange one, the one associated with the
> > uncountable continuum, the one whose very existence many deny.
> > Do not lump them together.
> Yes, I can see how this distinction might be useful in some esoteric
> discussions. But it seems (to me) to have little relevance to achieving a
> successful Theory of Everything. And it might even distract us..
> Both kinds of infinities are inaccessible to humans. So they can play no
> part in our theories. In this sense, I think it's fair to lump them into
> the pile of things that are not helpful.
I think we may not ignore infinities for quite pragmatic, non-esoteric
reasons. Many believe the history of our own universe will be
infinite - certainly there is no evidence against this possibility. Also,
any finite never-halting program for a virtual reality corresponds to
an infinite history. TOEs ignoring this seem unnecessarily restrictive.
> We cannot build the universe out of pieces themselves we cannot construct.
> (e.g. Pi)
What you cannot construct in finite time is just a particular
representation of Pi, namely, the one consisting of infinitely
many digits. But this is not a problem of Pi, it is a problem of
this particular representation. There are better, finite, unambiguous
representations of Pi: its programs. You can manipulate them, copy them,
insert into other finite programs and theorem provers, etc. Proofs of
Pi's properties are essentially proofs of properties of Pi's programs.
Really bad are those things that do not have ANY finite representation.
> Juergen, what do you think about the minimal cellular automaton? Is it a
> good candidate ATOE (algorithmic theory of everything)?
it depends - minimal for what purpose?