> >
> > All axiomatical mathematical systems can be encoded as bit strings,
> > with the rules of symbol manipulation encoded as the rules of the
> > Turing machine, and the all the output strings are theorems.
>
> I think it is important to clarify here the difference between the two
> potential applications of TM's (there are others of course).
>
> Case 1) In my interpretation of Jacques Mallah's ideas from some of his
> posts, TM's form a subset of all possible mathematical structures. Assuming
> that TM's could somehow generate universes, then program loops and so on
> might be able to explain the simple physical laws in most of these kinds of
> universes. However it would have to be shown that the measure of these
> universes was higher than those from any other possible subset of
> mathematical structures.
>
> Case 2) A TM generates theorems from axioms, as you describe above. Note we
> are talking about one particular TM here (or perhaps a subset of possible
> ones, dependent on the precise definition of the TM) - an inference engine.
> This is possibly useful as an analytical tool, but to play any (and the
> only) actual role
> in the production of universes it would have to counter the challenge of
> 'why that particular TM with that particular set of machine instructions
> (Modus Ponens/substitutivity or whatever) and no other generative entity?'
> (I come back to the case of all possible TM's below). Note also that 'loop
> back' arguments will not apply to a program with fixed (inference)
> instructions.
>
Fundamentally, we are considering the universes as bitstrings. The TMs
provide interpretations of those bit strings. Now a TM can be
specified as a bit string also (as say a transition table), so one
could say that the bitstring plus TM specification is a candidate
universe. (I don't actually know how to do this consistently, I was
hoping that Li and Vitanyi might have an answer, when I'm finally able
to borrow that book from the library).
Now, since all mathematical structures can be encoded as a series of
axioms, and the laws of symbol manipulation as a transition table that
is prepended to the bitstring, it would seem that the "all
mathematical structures" plenitude is a subset of the "all bitstrings"
plenitude. However, not all bitstrings correspond to well formed
formulae, so it seems to me that the "all maths plenitude" is a strict
subset of the "all bitstring" plenitude. (I'm prepared to admit doubt
here - it could be that equlity applies, but I don't think it is a
superset) However, by arguments, which you will no doubt be well
versed with now, the "all maths plenitude" dominates the "all
bitstrings plenitude" (at least in terms of generating self aware subsystems).
>
> > However,
> > not all valid TM and initial bitstrings can be represented by a consistent
> > mathematical system (a computer scientist might like to comment on
> > this assertion?)
>
> This seems clearly wrong to me (any tape sequence is in principle physically
> (and so mathematically) realisable - unless you are asserting the
> mathematical un-modellability of our universe, which takes us into a whole
> new ball game) - I am guessing that what you really mean here is that in
> general TM's can do more than just generate theorems (and so mathematical
> structures) from axioms, and so (to continue to what you say below) are more
> 'basic'.
>
> > Therefore, the all TM bitstring model is more general than the all
> > mathematical structure model.
>
> (If my guess above is correct, then this conclusion hardly follows,
> otherwise brains, analogue computers and pre-programmed micro-chips
> (which could also churn out theorems from axioms given simple inference
> rules, as well as outputting other things) would also be more 'basic'.)
> We are back here with what I think you are advocating - a *basic*
> all-possible TM scheme. An incidental point to be made here is that the
> scheme I have introduced can also be deemed to be more basic than
> mathematical structures - it starts with symbol strings (which could be bit
> strings), only some of which are wff's suitable for formal systems. But a
> more important point is that any set of sequential TM processes (necessary
> to invoke the advantages of loops etc) is itself a logical structure
> (definable by a formal system), and for it to be considered to have any (and
> the only) ontological/platonic role in the generation of universes (as
> opposed to specific TM subsets being used as modelling tools), it must
> be able to counter the charge 'why is the (wider) universe endowed with
> that particular logical structure (and no other)?'
>
> > Your argument would appear to work if
> > you asserted the essential mathematical nature of the universe, but
> > fails miserably in the TM bitstring case.
>
> If you are referring to the argument I gave in response to Jacques Mallah, I
> hope I have responded sufficiently, but if you are referring to the argument
> in my original post in this thread, please say why you think it fails.
>
> > However, we have already
> > worked out the solution to the TM bitstring case (see the conversation
> > between myself and Chris Maloney earlier in the archives).
>
> What thread/date is this? If by chance you are mistaking me for Chris
> (heaven forbid :), the scheme we worked on a few months back necessitated
> neither TM's nor loops; however a more careful explication of it (see my web
> pages) turned out to be rather long-winded, hence the new (conciser) scheme,
> certainly bearing similarities to our one, but not dependent upon bits or
> K-specifications.
>
Culpa mea - yes it was you. I had a look at your web page, and it the
prose you gave seems to descriobve the gist of our argument quite
well.
More later - I just has some visitors come to my office.
> Alastair
>
> Many worlds discussions begin at:
> http://www.physica.freeserve.co.uk/p105.htm
>
>
>
>
>
>
>
>
>
>
>
>
>
----------------------------------------------------------------------------
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
----------------------------------------------------------------------------
Received on Wed Oct 20 1999 - 17:07:53 PDT