Le 17-sept.-07, à 08:51, Russell Standish a écrit :
>
> 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.
I have no clue of what you are saying here. Perhaps you could elaborate
or give a reference where you say more.
>>> 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,
but then you are implicitly assuming some universal machine behind the
curtains ...
> 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?).
> 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. 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).
> 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).
> 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.
>>>> 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.
>
>>
>> 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).
> 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).
>>> 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 ....).
> 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.
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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