On Mon, Sep 17, 2007 at 12:36:51PM +0200, Bruno Marchal wrote:
> >
> > 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.
>
>
>
> I have no clue of what you are saying here. Perhaps you could elaborate
> or give a reference where you say more.
>
Is there a problem with the notation? Perhaps you are reading too much
into it?
>
>
>
>
>
> >>> 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.
>
>
> Which kind of map? This is already problematic once CT is assumed: it
> should be at least a map between descriptions and set of
> interpretations (or you assume a form of operational interpretation,
Yes.
> but then you are implicitly assuming some universal machine behind the
> curtains ...
>
No. It is just a map. Not all maps correspond to machines.
>
>
>
>
> > 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),
>
>
> I guess you mean: a machine can be interpreted as a very special sort
> of subset of such a mapping (which one?).
>
Sorry my fingers are slipping. Machines (computable functions) are a
type of map, but not all maps are machines (or perhaps you prefer the
word function to map).
>
> > 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.
>
>
>
> Yes, ok. I was just alluding to the 1-3 distinction. With comp you can
> associate a mind to machine, but you have to associate an (uncountable)
> infinity of machine to a mind, and all the problem consists in making
> this clear enough so as to be able to measure the amount of white
> rabbits. This has been done for important subcases in my work, like the
> case of probability/measure/credibility *one*, which does indeed obey
> to (purely arithmetical) "quantum law". This makes the quantum feature
> of the observable realities a case of "digitality" as seen from inside.
>
>
>
> >>>> 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.
>
>
> Completely so indeed. But then his proposal for a constructive (and
> apparently deterministic) physics appears to be in contradiction with
> the comp consequences about the 1-3 relations.
>
>
>
> > However I take the ensemble as simply
> > existing, not requiring an further justification.
>
>
> ?
>
>
>
> > It has equivalent
> > status to your "arithmetical realism".
>
> How could I know? You assume the existence of a (very big set) without
> making clear what are your assumptions in general. A priori, accepting
> the (ontic) existence of such big sets means that you presuppose a part
> of set theory (and thus with infinity). This is a far stronger
> assumption than arithmetical realism (accepted by most intuitionists
> and finitists). That cannot be equivalent.
Not equivalent. Equivalent status. Assumption of the set of all
infinite strings plays the same role as your assumption of
arithmetical realism, and that is of the ontological background.
> I make clear (well I try)
> that uncountable sets and informal set theories (and many continua)
> appears in the *first person* plenitude, or at the metalevel. Ontically
> we need only numbers with addition and multiplication (the ontic
> existence of more than that is undecidable by any sound machine, and
> provably useless by lobian machine).
>
> About reals or infinite strings, a big difference is that the set of
> reals is uncountable (not enumerable), but recursive or comp-reals are
> countable although not recursively countable. So there does not exist a
> universal dovetailer operating only on the constructive reals. The set
> of constructive reals is equivalent with the set of total computable
> functions from N to N. There is no universal dovetailer for them. I
> call that the "Graal" in "Conscience et Mécanisme", because it gives a
> picture of the first person plenitude, and machines can approximate
> this by going into the constructive transfinite (which I have described
> in the list with the growing computable functions).
>
It might seem like such uncountable sets are too much to assume, but
in fact it is the simplest possible object. It has precisely zero
information. No countable set has this property. I put your objection
into the same category as those who claim the multiverse is
ontologically profligate. Apologies to intuistionists out there.
>
>
> > 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
>
>
> (which you confuse some time with the UD, I think).
>
He does actually dovetail, so it is a universal dovetailer in all but
name perhaps. But the ontological basis of the "Great Programmer"
differs very much from COMP.
>
>
> > 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.
>
>
> I agree. The two papers are not obviously related. I have also
> different versions of his second paper. This is not a reason to
> attribute to Schmidhuber things *you* introduce. I would prefer to call
> the set of all infinite strings the "Russell ensemble", even if that is
> mean to criticize the idea. Hope you are not worried by my frankness.
>
Of course, although you'd better say the Standish ensemble so as not
to misattribute it to Bertie. Also, it is quite clearly a set (I think
you've read enough English papers to know the difference between and
ensemble and a set in English, I hope?)
>
>
>
>
> >>>> 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.
>
>
>
> But as sets, they are isomorphic, and if you don't have extra-structure
> on your "ensemble", the relation between your "observers" and your
> "ensemble" is even more obscure, it seems to me.
>
You've lost me here.
>
>
>
>
> >
> >>
> >> 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] -
>
>
> So here you do explicitly accept extra-structure, a measure, on your
> ensemble, making them again quite close to the reals.
> You cannot derive the existence of a measure from just a definition of
> a set. (There are *many* possible measures on any set).
>
Yes of course. The uniform measure has always been part of the definition.
>
>
>
> > see Li & Vitanyi example 4.2.1 for a detailed
> > discussion. The idea is simple enough, however.
>
>
> ... where they describe how to put a measure on some set of *subsets*
> of an uncountable sets. You have to define a Borel structure on it,
> etc.
> It is indeed explained in Li & Vitanyi (page 214q).
>
We must have different editions. On mine its page 243 :). So there
must be some non-measurable subsets. But I fail to see how these can
be inverse images of an observers interpretation (O^{-1}(n)) must be
measurable). But then I admit I am acting like a physicist in glossing
over these sorts of details.
>
>
>
> >>> 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.
>
>
>
> (OK, but only since 525 after J.C., and just because scientists have
> been forced to let the fundamental questioning to authorities mixing
> political and spiritual power ....).
>
What happened in 525 CE again?
>
>
>
> > 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.
>
>
> OK. But again it could be misleading to call that "the semantic level",
> because a relation between "semantic" and first person would be a very
> interesting things to dig on, but nobody has done that yet.
> All hypostases (first person, third person, first person plural, etc.)
> have syntax and semantics.
Yes but again we're mixing terminologies. When I refer to syntactic
level, I'm refer to what stuff is, and when I refer to semantic level,
I mean how it is interpreted. This can be applied to all situations
where emergence is occurring. So in the case of an ideal gas, the
molecular description is syntactic, and its thermodynamic description
is semantic. In the Game of Life, the update rule is syntactic, the
description in terms of gliders, puffers and guns is semantic. It may
not be the best terminology, but it is the best I've come across to date.
> I have given in this list and in all my papers on the subject two main
> definitions of the first person. In UDA it is the memory content of a
> diary that a candidate for self-multiplication keep with him, and in
> AUDA I define the first person by the "knower" (and thus the knower
> modal logic S4(*)) by using the more abstract theaetetical notion of
> knowledge given in the Theaetetus by Plato (they are related through
> the usual platonist "dream argument").
>
>
> Bruno
>
> (*) knowing p -> p (incorrigeability)
> knowing p -> knowing(knowing p) (introspection)
> knowing (p -> q) -> [(knowing p) -> (knowing q)]
> (rationality or weak omniscience).
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
--
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A/Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 hpcoder.domain.name.hidden
Australia http://www.hpcoders.com.au
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Received on Mon Sep 17 2007 - 08:22:45 PDT