On Mon, 18 Oct 1999, Alastair Malcolm wrote:
> From: Jacques M. Mallah <jqm1584.domain.name.hidden>
> [AM:]
> > > 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.
>
> Depends on how you define 'define' I suppose - I thought that I had just
> defined it! But seriously, my aim was to formalise my ideas sufficiently to
> rebut one of the main criticisms of all-possible-universes hypotheses,
> namely that there would appear to be far more lawless paths for our universe
> to take than that (or those if QM is included) prescribed by the known
> physical laws.
For a definition to be acceptable, it must be shown that the
proceedure you propose is both possible and unique. By way of example let
me point out that even the Turing machine case has problems. This is
similar to the issues involved in defining an absolute Algorithmic
Information Content of a string, which I have discussed on a previous
occasion. The problem there is that each Turing machine ("interpretation
of symbols") gives different results, and to average over all of them it
would first be necessary to have a natural measure over the machines. In
the Turing case though I suspect it is possible to define such a measure.
> > > 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 believe I was too generous above, because in the Turing case
using only a few symbols the combinatorical argument works.
If the number of symbols is large, however, there is a problem.
If the amount of information to select even one symbol (out of N, say) is
large compared to the amount of information needed to describe the
universe, it still takes just 1 symbol to describe either our universe or
a more complicated universe. (The symbol may imply all the needed
formulas.) Combinations of symbols have the same problem. You may say
here that even the 1 symbol will imply more 'copies' of the simpler
systems, but that is exactly what we would need to prove, and how have we
gotten any closer to doing so? Again it is a question of establishing a
natural measure distribution on the interpretations, which seems much
easier if the symbols can only take on 2 values and therefore must be
combined to produce the description.
> The scheme I have described is different in that it does not necessitate
> TM's - all the computational-based
> explanations for simple physical laws that I have seen have tended to rely
> on the specific mechanism of a sequential TM processor ('program stop'
> codes, loop-backs and so on).
I did not realize last time that you had intentionally tried to
generalize the TM-based argument.
> As you stated in our last discussion (11/7),
> only *some* mathematical structures are TM's, and one would need first to
> show that TM's were the most prolific way of generating universes out of all
> possible such structures before any of these ideas could be given good
> credance. (From what you have said before it doesn't seem to me that you
> hold that *only* TM's are suited to generating consciousnesses - but no
> doubt you will tell me if I am wrong in this surmisal.) All this however is
> not to say that TM's may not provide a useful analytical tool in this area.
Yes. It is also possible that the measure problem might have a
solution for TMs but not in general, in which case one could argue that
TMs are really special. (Probably quantum TMs have any relevant
characteristics of TMs as well.) It is also possible that it can't be
solved even for TMs, in which case one might forget the whole
zero-information idea. I have sometimes argued that a typical TM based
consciousness is likely to see a simpler world than we do.
- - - - - - -
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 Tue Oct 19 1999 - 13:39:51 PDT