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From: Russell Standish <R.Standish.domain.name.hidden>

Date: Thu, 21 Oct 1999 10:08:45 +1000 (EST)

*> >
*

*> > 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

Date: Thu, 21 Oct 1999 10:08:45 +1000 (EST)

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).

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.

----------------------------------------------------------------------------

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

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