Re: Flying rabbits and dragons

From: Alastair Malcolm <amalcolm.domain.name.hidden>
Date: Tue, 9 Nov 1999 18:42:56 -0000

----- Original Message -----
From: Russell Standish <R.Standish.domain.name.hidden>
> 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.

This appears to me to be some kind of conflation of (at least) the two
schemes discussed earlier (my axiom-based one, and our earlier bit-based
one) - it is certainly distinct from the basic axiom-based one that you have
been criticising. Since this (axiom-based) scheme is not bit-based as it
stands, the 'Schmidhuber plenitude' assumption does not apply to it, and so
I can legitimately dismiss non-wffs as irrelevant to the dragon analysis as
far as my scheme is concerned.

However, let me follow your route (as best I can) for a little. I presume
you must be choosing a particular interpretation of bits corresponding
simply to symbols. Some such strings lead to wff's, theories, and, for a
minority of these, (the specification of) universes. Others are meaningless
strings (non-wffs). I think you would need to clarify:
1) How any *additional* interpretation of bits (over and above that leading
to symbols) could lead to actual universes.
2) What actual role (if any) TM's play.
3) Whether bitstrings could be immediately interpreted as anything other
than symbol strings.
4) How K-complexity (if relevant) could be compatible with nonsense bits.
(I must say I also have a severe problem with a bitstring corresponding to a
non-wff being interpretable as anything other than the non-wff itself.)

> 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.

Note that many other systems (for example pre-programmed microchips, brains,
other mathematical mappings) can implement transitional rules - there is
nothing special about UTM sequential processors.

> > 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.

I can only think that we have fundamentally differing conceptions of what
constitutes a logically inconsistent universe. For me, a logically
inconsistent universe would be one where (P AND (NOT-P)) is true somewhere
within it (and from which *any* statement is derivable); this would
presumably have to fall out from some logically inconsistent theory. Unless
we are straying beyond the realms of science here, dreams are quite
comfortably compatible with physics (apart conceivably from the 'qualia'
problem - which would apply at least as much to the waking state and
shouldn't be relevant to logical consistency anyway). If physical laws can
accommodate dreams there is not even physical inconsistency, let alone
logical inconsistency. (If by chance you are referring to dream *content*,
these can hardly be considered as universes, but dream images anyway are
logically consistent, though dreamed flying rabbits obviously don't have to
conform to our physical laws.)

> > > 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.

If you are referring to our earlier derived explanation for the paucity of
dragons, that was what I was referring to as well. If not, please elucidate.

Thanks

Alastair
Received on Tue Nov 09 1999 - 10:54:33 PST