On Sat, 16 Oct 1999, Alastair Malcolm wrote:
> I would be interested to know if anyone can find any problems with the
> following new solution to the much discussed flying rabbit problem
> (discussed below as the dragon universe problem).
It's NOT new, at least not the idea. But it is in the direction
of what I have argued for.
> Here the measure is initially based on the symbols comprising the formal
> systems which include the mathematical structures specifying universes
> (as in Tegmark's paper).
> The starting point is all possible symbol strings containing all
> possible symbols. Now, bearing in mind that any symbol can have any
> interpretation, we can start to select those strings which contain
> symbols that can be interpreted in their entirety as well-formed
> formulae (wff's). Of these we can select those which comprise mutually
> consistent axioms (both directly, and after taking into account derived
> theorems via standard rules of inference), connected by an 'AND' (that
> is, a string of the form 'Axiom1 AND Axiom2 AND ...'). We can also
> strike out any cases of duplicate axioms.
The above I have a problem with. I do not see how the above is
supposed to be defined. The basic idea, however, is very similar to the
idea I have discussed in the past in which all Turing programs are
running. The main difference is that the latter is reasonably well
defined. Your symbols become tape symbols, interpretation of them
becomes the specification of the Turing machine, etc. I do agree that the
analog sector should be included somehow, but I think it would be (if
possible) necessary to generalize the Turing machine to deal with that
case.
> We now have a list of symbol strings that are interpretable as axiom
> sets of all possible consistent theories (theorem sets). Since all axioms
> are representable by an (anthropically unbiassed) subset of all possible
> combinations of symbols, then we should not lose generality by scaling up
> our measure from symbols to axioms. Now, since the axioms for a TOE
> completely specify our universe (and perhaps others), it is clear we
> cannot just add extra axioms to specify a dragon event, because that
> would lead to an inconsistency (theorems derivable from the TOE would
> specify a non-dragon event at that time and place). However it is
> possible to have a set of axioms that would specify the path of our
> visible universe, but with a visible dragon event - a dragon universe
> theory; this would have to be represented by a much longer axiom string
> than the TOE set.
> Now, as an extreme approximation (all that is necessary here), we can
> say that our TOE requires n axioms, the dragon universe 2n axioms. In
> comparing all possible theories containing a finite number of axioms,
> then it is reasonable to suppose that occurences of TOE's will vastly
> outnumber dragon universe theories, once functionless axioms are taken
> into consideration: consider all strings up to m axioms in length, for
> large finite m; there will be n more surplus axioms for the TOE than for
> the dragon universe theory (m-n as opposed to m-2n), and so bearing in
> mind that each axiom can contain an infinite variety and number of
> different symbols, and that each axiom (functionless or not,
> participating in the specification of another universe or not),
> contributes to the measure, then there will be far more combinations of m
> axioms that include our TOE than is the case for dragon universe
> theories, with the result that we are far more likely to be in a simpler
> (TOE-based) universe - dragon universes are not more probable.
Again, this is clearly the same argument that I made, and that Wei
Dai made for a different reason, that the set of all Turing programs
should lead to the appearance of simple physical laws. It helps to have,
as in the Turing case, an 'end of program' symbol; the rest of the string
after this is functionless. I think you misspoke about 'axioms containing
symbols' above. In fact things work best when the number of different
types of symbols is minimal (e.g. 0 or 1).
> (Two further points here: firstly, the outnumbering of dragon universes
> is maintained as m increases to infinity - simply take a finite random
> sample; secondly, if one wishes to take into consideration all possible
> paranormal events, one also needs to take into consideration the much
> larger number of equal-axiom-number theories that specify our universe,
> but with invisible paranormal events, events in other universes and so
> on - these will vastly outnumber visible-paranormal-event theories.)
Yes, these non-functional symbols are responsible for the desired
statistics. It is easy to see that the fraction of programs containing a
desired initial string of length n, followed by a loop or end of program,
then infinite 'junk' code, is proportional to s^-n, where s is the number
of different types of symbols (e.g. 2).
A way should be found to handle, precisely, the analog sector. It
must be shown that the physical laws we observe are actually typical of
those predicted to be observed by the model. Of course, before that, a
way must be found to relate math to conscious observations. E.g.
computationalism + an implementation criterion + ASSA.
- - - - - - -
Jacques Mallah (jqm1584.domain.name.hidden)
Graduate Student / Many Worlder / Devil's Advocate
"I know what no one else knows" - 'Runaway Train', Soul Asylum
My URL:
http://pages.nyu.edu/~jqm1584/
Received on Sat Oct 16 1999 - 16:03:09 PDT