Dear Hal,
Consider the last two paragraphs from one of Stephen Wolfram's papers:
http://www.stephenwolfram.com/publications/articles/physics/85-undecidability/2/text.html
***
"Quantum and statistical mechanics involve sums over possibly infinite sets
of configurations in systems. To derive finite formulas one must use finite
specifications for these sets. But it may be undecidable whether two finite
specifications yield equivalent configurations. So, for example, it is
undecidable whether two finitely specified four-manifolds or solutions to
the Einstein equations are equivalent (under coordinate
reparametrization).[24] A theoretical model may be considered as a finite
specification of the possible behavior of a system. One may ask for example
whether the consequences of two models are identical in all circumstances,
so that the models are equivalent. If the models involve computations more
complicated than those that can be carried out by a computer with a fixed
finite number of states (regular language), this question is in general
undecidable. Similarly, it is undecidable what is the simplest such model
that describes a given set of empirical data.[25]
This paper has suggested that many physical systems are computationally
irreducible, so that their own evolution is effectively the most efficient
procedure for determining their future. As a consequence, many questions
about these systems can be answered only by very lengthy or potentially
infinite computations. But some questions answerable by simpler computations
may still be formulated."
***
It has been pointed out, by Roger Penrose himself (!), that the
decidability problem for Einstein's equations is equivalent to Halting
Problem of Turing Machines (pg. 337 of "Shadows of the Mind"). When we put
these two arguments together, what do we get?
See:
http://arxiv.org/abs/quant-ph/0304128 ;-)
Stephen
----- Original Message -----
From: "Hal Finney" <hal.domain.name.hidden>
To: <everything-list.domain.name.hidden>
Sent: Tuesday, January 20, 2004 7:18 PM
Subject: Re: Is the universe computable
> CMR writes:
> > Then question then becomes, I suppose, if in fact our universe is a
digital
> > one (if not strictly a CA) havng self-consistent emergent physics, then
> > might it not follow that it is "implemented" (run?) via some
extra-universal
> > physical processes that only indirectly correspond to ours?
>
> This is a good point, and in fact we could sharpen the situation as
> follows.
>
> Suppose multiverse theory is bunk and none of Tegmark's four levels work.
> The universe isn't infinite in size; there is no inflation; the MWI is
> false; and all that stuff about Platonic existence is so much hot air.
> There is, in fact, only one universe.
>
> However, that universe isn't ours. It's a specific version of Conway's
> 2D Life universe, large but finite in size, with periodic edge conditions.
>
> Against all odds, life has evolved in Life and produced Self
> Aware Subsystems, i.e. observers. These beings have developed a
> civilization and built computers. See the link I supplied earlier,
> http://rendell.server.org.uk/gol/tm.htm for a sample of such a computer.
>
> On their computers they run simulations of other universes, and one
> of the universes they have simulated is our own. Due to a triumph
> of advanced mathematics, they have invented a set of physical laws of
> tremendous complexity compared to their own, and these laws allow for
> atoms, chemistry, biology and life of a form very different from theirs.
> They follow our universe's evolution from Big Bang to Heat Death with
> fascination.
>
> Unbeknown to us, this is the basis for our existence. We are a simulation
> being run in a 2D CA universe with Conway's Life rules.
>
> Now, is this story inconceivable? Logically contradictory? I don't
> see how. The idea that only one "real" universe might exist, but that it
> could create any number of "simulated" ones, is pretty common. Of course
> it's more common to suppose that it's our universe which is the "real"
> one, but that's just parochialism.
>
> And what does it say about the physical properties which are necessary
> for computation? We have energy; Life has "blinkiness" (the degree to
> which cells are blinking on and off within a structure); neither property
> has a good analog in the other universe. Does the "real" universe win,
> in terms of deciding what properties are really needed for computation?
> I don't think so, because we could reverse the roles of the two universes
> and it wouldn't make any fundamental difference.
>
> Hal
>
>
Received on Tue Jan 20 2004 - 20:04:04 PST