Re: Universal prior - Go FORTH and Run Backward

From: Marchal <marchal.domain.name.hidden>
Date: Fri Nov 12 03:36:16 1999

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.

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.

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!

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.

Bruno
Received on Fri Nov 12 1999 - 03:36:16 PST