The first thing I'd like to add, is that I don't believe we're
philosophically opposed to each other. There must be some kind of
misunderstanding on one of our parts - either of what each other is
saying, or of the problem we're trying to solve. It'd be great for
someone else with an external perspective to butt in and tell us where
we might be going wrong.
1) As you say yourself, a non-wff is just a meaningless string of
symbols, i.e. one of the bitstrings making up the Schmidhuber
plenitude. As pointed out, the Schmidhuber plenitude does seem to have
a natural measure defined on it through K-complexity and Turing
interpretability. The non-wff strings, as simple the K-incompressible,
or random ones, however, they are still formal entities, and so
contribute to the discussion of issues related to measure. I also
belief that they are interpretable as well, just as Rorschach plots
are interpretable, even when they have no information content at all.
The reason that they're interpretable, is that they are close to a wff
that has information.
2) There is a curious duality here relating the Tegmark plenitude with
the Schmidhuber one, that I've not yet figured out. On one hand, all
consistent mathematical systems are based on a finite set of axioms,
and a finite set of transition rules for deriving theorems. These can
be encoded as a bitstring and a UTM, so all members of the Tegmark
ensemble must be contained within the Schmidhuber one. Contrariwise,
the UTM structure, interpreting bitstrings is but one such
self-consitent mathematical structure, so the Schmidhuber ensemble is
contained with the Tegmark one. So are the two plenitudes equivalent?
If not, then why not? Is it something to do with noncomputable
mathematical objects, such as noncomputable real numbers?
>
> ----- Original Message -----
> From: Russell Standish <R.Standish.domain.name.hidden>
> > I agree, that if you restrict your attention to wffs only, then there
> > is no problem. It is also no surprise. However, I don't believe one
> > can solve the White Rabbit problem by defining it away.
>
> It is most certainly NOT just being 'defined away'.
>
> The whole point is that we are confining ourselves to *logically consistent*
> universes (and perhaps other entities), that is, universes with some kind of
> logical structure. The only assumption in my analysis is that these must be
> definable with some kind of formal system (I don't even assume that it would
> be contained within fig 1 of Tegmark's table). A non-wff is just a
> meaningless string of symbols - we can't even *consider* any logically
> consistent universe without at least a wff-string. Once we have the
> wff-string, then we can start to build towards defining logically consistent
> universes (via axiom sets, *consistent* theories, metric spaces (probably),
> and so on). It is *then* that the analysis of axioms comes into play (see my
> original post), and flying rabbits are banished to a very small minority of
> universes.
>
> There is no 'defining away' of the white rabbit problem. It sounds to me as
> if you are confusing white rabbit universes with logically inconsistent
> universes (the latter should not be able to exist in anyone's
> book).
I guess this is the contentious point. Dreams appear to be examples of
logically inconsistent universes. I would explain them in terms of the
brain's capacity to interpret meaningless data streams generated in
the brain during sleep. I guess you would need to explain either that
dreams are in fact logically consistent, or that somehow this example
doesn't count.
Perhaps
> if I add some comments to an earlier post of yours, it may help to clarify
> the wider issue, but remember that what follows below is mainly based on our
> earlier scheme and not the same as one described at the start of this
> thread.
>
> > I don't think anyone said they don't exist. Its the measure of the
> > worlds that contain that is interesting. There are two possibilities:
> >
> > a) The dragon universe is the outcome of a mathematical
> > description. In the sense we use dragon here of being non-lawlike, we
> > may suppose that they are the outcome of a very complex mathematical
> > description. As we well know, the measure of such universes is much
> > smaller than the very lawlike universe we inhabit.
>
> We don't know this unless we can find a proof. In fact the analysis that we
> developed earlier can provide a strong indication of the small measure of
> dragon universes. (See my web pages.)
>
This result follows directly from the universal measure or universal
prior employed in the Schmidhuber plenitude. It may be more
problematic if your ensemble of mathematical systems is not a proper
subset of the Schmidhuber plenitude.
> > b) The dragon universe corresponds to one of the "nonsense
> > bitstrings", which we know to vastly outnumber the lawlike ones. In
> > this case, we must apply the argument I worked out with Alistair
> > Malcolm to show that indeed these dragon universes are of small
> > measure compared with lawlike universes, and that the remaining
> > "nonsense" universes are like noise - unobservable.
>
> If the bits are interpretable as anything, they would normally be
> interpretable mathematically (or at least as part of some formal system,
> otherwise we would be ranging beyond the formality of information theory
> itself), in which case what they represent will be coherent, but not in
> general visible to us (usually because what is represented will be in
> another universe) - again the general argument we derived holds as required
> for their small measure. For totally nonsense bitstings, or for any parts of
> bitstrings which are non-interpretable - these are ontologically
> problematical (as are non-wffs); I think that we have to remember that we
> are just trying to find generic schemes that can subsume all logically
> possible universes - the particular set of building blocks used to ground
> each scheme ought not to affect the relative measures of various types of
> universes.
No problems with the above.
Neither non-wffs nor non-interpretable bits can represent
> anything, and to include them in any analysis is giving them unwarranted
> ontological credibility. Fortunately neither are required to ascribe low
> measure to dragon universes.
However, they exist in the Schmidhuber "all bitstrings" version of the
plenitude.
>
> Alastair
>
>
>
>
----------------------------------------------------------------------------
Dr. Russell Standish Director
High Performance Computing Support Unit,
University of NSW Phone 9385 6967
Sydney 2052 Fax 9385 6965
Australia R.Standish.domain.name.hidden
Room 2075, Red Centre
http://parallel.hpc.unsw.edu.au/rks
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Received on Sun Nov 07 1999 - 18:49:30 PST