On Sat, Sep 15, 2007 at 03:13:09PM +0200, Bruno Marchal wrote:
>
>
> Le 14-sept.-07, à 01:02, Russell Standish a écrit :
>
> >
> > On Thu, Sep 13, 2007 at 03:04:34PM +0200, Bruno Marchal wrote:
> >>
> >>
> >> Le 13-sept.-07, à 00:48, Russell Standish a écrit :
> >>
> >>> These sorts of discussions "No-justification", "Zero-information
> >>> principle", "All of mathematics" and Hal Ruhl's dualling All and
> >>> Nothing (or should that be "duelling") are really just motivators for
> >>> getting at the ensemble, which turns out remarkably to be the same in
> >>> each case - the set of 2^\aleph_0 infinite strings or histories.
> >>
> >>
> >> Once you fix a programming language or a universal machine, then I can
> >
> > You don't even need a universal machine. All you need is a mapping
> > from infinite strings to integers.
>
> Which one?
>
>
It doesn't matter. The most interesting ones, however, have inverse
images of non-zero measure. ie \forall n \in N, the set
O^{-1}(n) = {x: O(x)=n}
is of nonzero measure.
>
> > And that can be given by the
> > observer,
>
>
> But what is the observer? Is the observer an infinite string itself, a
> machine, ?
>
The only thing assumed about the observer is that there is a map
between descriptions and interpretations. The additional assumption
about inverse images having nonzero measure is needed to solve the
White Rabbit problem.
An observer can be a machine (which is a subset of such mapping), but
needn't be a machine in general.
Some strings, _under the interpretation of the observer_, are mapped
to observers, including erself. Without the interpretation, though,
they are just infinite strings, inert and meaningless.
>
>
> > where the integers are an enumeration of the oberver's
> > possible interpretations.
>
>
> I still don't understand what you accept at the ontic level, and what
> is epistemological, and how those things are related.
>
I'm not sure these terms are even meaningful. Perhaps one can say the
strings are ontic, and the interpretations are epistemological.
>
>
>
> >
> >> imagine how to *represent* an history by an infinite string. But then
> >> you are using comp and you know the consequences. Unless like some
> >> people (including Schmidhuber) you don't believe in the difference
> >> between first and third person points of view.
> >>
> >>
> >> (Youness Ayaita wrote:
> >>
> >>> When I first wanted to capture mathematically the Everything, I tried
> >>> several mathematicalist approaches. But later, I prefered the
> >>> Everything ensemble that is also known here as the Schmidhuber
> >>> ensemble.
> >>
> >>
> >> Could you Youness, or Russell, give a definition of "Schmidhuber
> >> ensemble", please.
> >
> > The set of all infinite length strings in some chosen alphabet.
>
>
> Is not Shmidhuber a computationalist? I thought he tries to build a
> constructive physics, by searching (through CT) priors on a program
> generating or 'outputting" a physical universe. Is not the ensemble an
> ensemble of computations, and is not Schmidhuber interested in the
> finite one or the limiting one? Gosh, you will force me to take again a
> look at his papers :)
>
Schmidhuber has his ensemble generated by a machine, and perhaps this
makes him computationalist. However I take the ensemble as simply
existing, not requiring an further justification. It has equivalent
status to your "arithmetical realism". Obviously I'm departing from
Schmidhuber at that point, and whilst in "Why Occam's Razor" I use the
term Schmidhuber ensemble to refer to this, in my book I distinguish
between Schmidhuber's Great Programmer idea and my "All infinite
strings exist prima facie" idea. This is mostly because Schmidhuber's
second paper (on the speed prior) makes it quite clear he is talking
about something quite different.
>
>
>
> >
> >> Also I still don't know if the "physical universe" is considered as an
> >> ouptut of a program, or if it is associated to the running of a
> >> program.)
> >
> > No, it is considered to be the stable, sharable dream, as you
> > sometimes put it.
>
>
>
> It is the case, by and through the idea that the observer is a lobian
> machine for which the notion of dream is well defined (roughly
> speaking: computations as seen through the spectacles of the
> hypostases/point-of-vies).
>
> The set of all infinite strings, according to the structure you allow
> on it, could give the real line, the set of subset of natural numbers,
> the functions from N to N, etc. It is not enough precise I think.
All of these concepts are more precise and have additional properties
to the set of all infinite strings. For instance, the reals have
group properties of addition and multiplication that the strings
don't.
>
> I don't understand either how you put an uniform measure on those
> infinite strings, I also guess you mean a (non-uniform) measure on the
> subsets of the set of infinite strings. Interesting things can come
> there.
>
>
About the only important property the strings have is the uniform
measure. This is basically the same as the uniform or Lebesgue measure
on the interval [0,1] - see Li & Vitanyi example 4.2.1 for a detailed
discussion. The idea is simple enough, however.
>
>
> > It is the interpretation of the observer, but it
> > isn't arbitrary.
>
>
> Certainly not in Schmidhuber, as I remember (cf our discussions in this
> list). OK, with comp, but in some RSSA way, and not in any ASSA way
> based on an ensemble.
>
Schmidhuber downplayed the role of the observer, as is typical of a
scientist. Since this appears to be the point of departure between you
and he, I'll state that I've always followed you in this point, that
the 1st person pov (what I call the semantic level) is important.
>
>
> Bruno
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
--
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A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 [EMAIL PROTECTED]
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