>
> I propose comments on the last posts by
> GSlevy and Russell Standish, which will also summarize
> my critics of Tegmark and Schmidhuber.
>
> GSLevy wrote:
>
> >Pushing beyond the world of discrete programming, the question arise, is it
> >possible to create languages for the control of:
> >
> >1) a unidimensional unidirectional continuous computer.
>
>
> The problem is that until now nobody has found a definition of
> what could be a continuous computer. There is no corresponding
> Church's Thesis in the continuous domain. Worst: there is no
> unanimity about what the continuum really is. The intuitionistic
> or constructive approachs toward the continuum seems to me the
> most interesting, but there are numerous and not equivalent.
> The non-constructive approach is more standart, but still will
> depend on the model choose. Most of these models are mathematical
> structures which are impossible to describe using a finite or even
> recursively enumerable theory.
>
> This is linked to something said by Russell Standish in his paper
> and in some posts. Russell Standish pretends that (I quote him):
>
> Each self-consistent mathematical structure is completely described
> by a finite set of symbols, axioms and rule.
>
> This is totaly incorrect, I'm afraid. It is known that you cannot
> find a recursively enumerable (RE) set of axioms to specify the
> (N,+,x) structure (N) of the natural numbers with addition and
> multiplication. Each RE description of N admit plenty of non
> isomorphic models which belongs to any reasonable set of
> mathematical structures. And things are worst with sets,
> categories, etc.
> Of course you can guess this from the fact that the set of RE
> theories are countable, and the set of possible mathematical
> structure is ... much much bigger, to say the least.
This is very interesting. I will need to think some more on
this. Clearly, when I studied pure mathematics, such problems were
glossed over. We did touch on Goedel's theorem (more as
extra-curricular interest), but clearly these issues go beyond Goedel.
>
> People should realize that notion like "definability",
> "provability" etc. are highly relative mathematical concept, by
> which I mean that they are formalism-dependent.
> "computability" is the first and the last (until now) purely
> absolute concept, which doesn't depend on the formalism choosed.
> Godel said that with the computability concept there is some
> kind of miracle. The miracle is the apparent truth of Church's
> thesis.
>
> That is why Tegmark or GSlevy approach, although very interesting,
> is still ill-defined. Tegmark will need some powerfull constructive
> axiom to give a sufficiently precise sense to the "whole set
> of mathematical structures" so that he can make the white
> rabbit disappearing, by isolating the right prior.
>
> Schmidhuber has chosen the "right" (absolute) objective realm:
> the universe of computations, but still doesn't realise that
> with comp, the prior cannot be defined on the computation,
> but only on the relative continuation, and this in a "trans-universe"
> manner (cf the UDA).
> The Universal Dovetailer Argument (UDA as I
> now call it, hoping Chris appreciates) shows also that we must
> take into account the continuum.
> It seems that the UDA makes some links between Tegmark and
> Schmidhuber approach, indeed. Because it shows that with comp
> some prior needs non RE mathematical structures.
I think Schmidhuber is the best starting point also. With the
"Occam/Wigner" argument I'm advancing, clearly this narrows the range
on interesting universes to the "intersection" of Tegmark and Schmidhuber.
>
> BTW, Russell, I publish the "white rabbit" problem in my 1991
> paper: Marchal B, 1991, Mechanism and Personal Identity, Proceedings
> of WOCFAI, De Glass & Gabbay (Eds), pp. 335-345, Angkor, Paris.
> Those who want that paper just send me out-line an address. I do NOT
> put them on my WEB page because it is in horrible WORD format, and
> I have not the time to change all the format of my older publications!
>
Thanks for the ref. Word files can be converted easily into postscript
(or some people can also convert it into PDF - same diff). However, it
would be better for you to convert the file and ship the postscript,
rather than shipping the Word file as different versions of Word seem
to produce incosistent results.
Nevertheless, I'd be interested in the paper (whether Word or ps).
> I have still a question for Russell, what is the meaning of
> giving a "physical existence" to a mathematical structure ? This is
> not at all a clear statement for me.
I am using it in the sense of Tegmark - essentially saying that some
structure really _does_ exist. A bit like saying Platonic forms really
_do_ exist.
Cheers
>
> Bruno
>
>
----------------------------------------------------------------------------
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
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Received on Fri Nov 12 1999 - 16:45:30 PST