> Juergen wrote:
> >Your vague answers to questions I did not ask keep evading the issue of
> >continuum vs computability in the limit. I give up. JS
>
> Let us try to be "very precise", then. I propose you the iterated
> self-duplication experience.
>
> Assuming computationalism, we survive. (I guess you agree).
>
> Here is the question. Do you expect the (infinite) sequence in {W,M}*
> appearing on your t-shirt to be
>
> computable or uncomputable ?
>
> In case you want to restrict to the finite sequences appearing at each
> step,
> I propose then we stop the experience after 1000 steps. In that experience
> do you expect the sequence of W and M (lenght = 1000) to be
>
> compressible or not compressible ?
> Bruno
Very precise? T-shirt? Never mind, I think I now completely understand
your reasoning. Contradicting my announcement to give up I'll try once
more to point out your unspoken assumptions and what is unsatisfactory
about them.
You expect finite future observation sequences to be incompressible,
because most binary strings are incompressible.
The unspoken assumption is a uniform distribution on equal-sized prefixes
of possible futures.
When you say indeterminacy you just mean probability 2^-n for each prefix
of size n. There is nothing "shocking" or "weird" about it (your words);
you just assume one particular computable probability distribution on
finite prefixes of futures.
That's ok. Although it does represent a limitation, because there are many
other computable probability distributions on possible universe histories,
and theories of everything (TOEs) should take all of them into account.
Then, however, you make a bold step and generalize from computable things
to noncomputable things: you expect infinite future observation sequences
to have no finite description, because most infinite binary strings do
not have one.
The unspoken assumption is a uniform distribution on all infinite strings.
The problem was pointed out repeatedly - one cannot generalize from
finite prefixes to a continuum of infinite objects without leaving the
realm of computation and dovetailers. Any dovetailer producing a growing
binary tree can output only countably many nodes, never a continuum.
The issue of 3rd person vs 1st person is irrelevant here. So is the issue
of enumerating the reals in a "list" - it is already enough to observe
that in countable time you cannot produce uncountably many things, only
their prefixes.
For a long time your claim that "there is no computable universe to which
we belong" has not made any sense to me, but now I realize that what you
really mean is just that if our universe is uniform randomly sampled from
the interval of real numbers then with probability 1 it is not computable.
This is an ancient result.
Your underlying assumption is the existence of a noncomputable selection
process selecting one of uncountably many objects. This has nothing to do
with your computable 2^-n prefix probabilities above, or with computation
in the limit, or with dovetailing, or with truly algorithmic TOEs. You
have a NONalgorithmic approach here.
To summarize: There are many possible computable distributions on
equal-sized prefixes of futures. You pick a uniform one, perhaps
because unbiased coin tosses seem natural to you. But as soon as
it comes to infinite objects the uniform distribution is LESS natural
than many others, because infinite sequences of coin tosses do not yield
describable results, and thus do not even exist from any descriptive point
of view. You seem to think you can save the situation by "dovetailing"
over ALL infinite sequences constituting a continuum. But the continuum
is beyond dovetailing. This forces you to leave the countable and
constructive realm and invoke your nonconstructive "arithmetic realism",
whatever that may be. Your "proof" just restates that the reals are not
countable, or that one cannot compute universes that are not computable
in the limit. On the other hand you do assume the very existence of
things noncomputable in the limit. This existence does not derive from
your dovetailer argument, since the dovetailer always will stay in the
computable realm. So you make a major assumption besides those rooted
in computability theory.
---
I guess Occam wouldn't like this additional noncomputability assumption,
this assumption of the existence of nondescribable things, because it is
unnecessary, given what we know. There are describable ways of assigning
nonvanishing probabilities to certain infinite futures characterized by
finite descriptions. Of course, the corresponding describable selection
processes will never select any future that is not computable in the
limit, but we can discard those anyway, because they don't even exist, in
the sense that we cannot even describe them, not even through a dovetailer
which at best can output finite representations of all futures computable
in the limit, but never the full continuum!
Such reasoning leads to algorithmic TOEs as opposed to nonalgorithmic
approaches like yours (or Tegmark's, who does mention Kolmogorov
complexity at some point but avoids details).
As long as the distribution is approximable or computable
in the limit, one can show that only histories with finite
descriptions can have nonvanishing probability, and that only
histories with short descriptions can have large probability:
http://www.idsia.ch/~juergen/toesv2/node34.html
That's why flying rabbits etc. are unlikely.
Received on Wed Feb 07 2001 - 05:34:06 PST