Le 19-sept.-07, à 11:56, Russell Standish a écrit :
>
> On Tue, Sep 18, 2007 at 04:23:58PM +0200, Bruno Marchal wrote:
>>
>
>> OK. You know I like your little book as an introduction to the field,
>> but, as you have already acknowledge, there is some lack in rigor in
>> it, and it is not even clear if eventually you are of the ASSA type or
>> RSSA type, or if you accept comp or not. Use of Bayes and Prior, for
>
> I am clearly on the record, both in the book and also in the list
> archives as an "RSSA type".
I do not pretend the contrary. Only that it is not clear. (We have a
problem of communcation I think, not more)
>
> As far as comp is concerned, I do not assume it, but accept it as a
> model of what's going on. See page 79 of my book.
This does not help, unless you take some conseq of comp as granted,
like the reversal physics/number-theology, or a form of
mathematicalism, etc.
I do consider that the discovery (by Babbage, Post, Church, Turing,
...) of the Universal Machine is a major discovery of our time which
changes almost all what has been thought about machine up to then. This
is reflected in the computability theory, and I exploit those
theoretical consequences.
>
>> example, is a symptom of ASSA type reasoning. Distinction between 1
>> and
>> 3 person points of view is symptom of the RSSA type of reasoning, (and
>> favored with comp).
>
> Not if the prior were actually given by the observer erself.
?
> This is
> the main point of departure between Schmidhuber's and my approach.
>
>>>
>>> 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 don't know. Let us fix a simple alphabet: {0, 1}. Then an infinite
>> string like
>> 111101000100111000000010100101100001101001 ..... (infinite on the
>> right) can be seen as the chracteristic function of a subset of N (the
>> first 1 in the string means then that 0 is in the set,, the second one
>> that 1 is in the set etc. The resulting set is
>> {0, 1, 2, 3, 5, 9, 12, 13, 14, 22, 24, 27, 29, 34, 35, 37, 40, ...}
>> So there is a bijection between the set of infinite strings on the
>> {0,1} alphabet, and the subset of N. So without putting any
>> extra-stcruture on the set of infinite strings, you could as well have
>> taken as basic in your ontology the set of subset of N, written P(N).
>> Now, such a set is not even nameable in any first order theory. In a
>> first order theory of those strings you will get something equivalent
>> to Tarski theory of Real: very nice but below the turing world: the
>> theory is complete and decidable and cannot be used for a theory of
>> everything (there is no natural numbers definable in such theories).
>> From this I can deduce that your intuition relies on second order
>> arithmetic or analysis (and this is confirmed by the way you introduce
>> time). But then this again is really a strong assumption, far stronger
>> than arithmetical realism.
>
> Stronger in what sense?
In the syntactical, or proof-theoretical sense. A theory A is stronger
than a theory B if A proves all theorems of B.
The set of theorems of B is included in the set of theorems of A. For
example PA is stronger (in that sense) than ROBINSON. (ROBINSON is PA
without the induction axioms).
Another example: QM + physical collapse is stronger than pure QM.
Caution: if a theory A is syntactically stronger than B, then B is
semantically stronger than A. Given that A has more axioms, it will
have less models. It is like in algebra: a big set of equations has
less solution than a little one. Syntax (theory+proof) and Semantics
(mathematical models) are in a sort of Galois correspondence: the more
you have axioms (the richer in theorems your theory is), the less you
have models.
> I have only assumed just enough to make sense
> of the notion of complexity.
I still don't know if you take "all the strings" in some first order
logical setting (in which case it will be not enough for defining a
notion of complexity) or if you take "all the strings" in some larger
(second order, mathematical instead of logical, etc.) sense, in which
case you take far too much.
Given the relation between "all the strings" and the set of subsets of
N, sometimes it seems to me you are just formulating (in some awkward
way, with all my respect) some acceptation of classical logic (boolean
algbra) pertaining on the natural numbers. In that case, your
assumption would be arithmetical realism.
>
>> To be sure, I still don't know if your ontic base is just "nothing"
>> (but then in which theory?) or the infinite strings (again, in which
>> theory and as I said you will to use rich mathematics for that), etc.
>> As you know, I am trying to go a little beyond the UDA result so as to
>> give a little smell of the real thing. The trouble is that the basic
>> tools of logic and axiomatic are not very well known by anybody but
>> the
>> professional logicians.
>>
>
> Its not so much that, but in how you interpret the logical
> results. Calling G*/G an angel for instance might be colourful
> rhetoric, but it doesn't really mean much to me.
Rhaaaa.... You talk like John, now!
OK I say a bit more. First you are an old participant in the list, so
you remember surely that I have first presented G and G* as "the
machine-itself" and "the guardian angel of the machine". This was
mainly in honor of Judson Webb who is the first to clearly realised
that Godel's incompleteness results were protecting mechanist
philosophy against all possible diagonalization à-la LUcas (-Penrose):
and Judson call Godel's result the guardian angel of mechanism.
But more recently, perhaps because I have introduced the term
"theology", I have introduced the term "angels, Gods, Supergods ...",
but then I did define them. By angel I really mean any
self-refrentially correct entity which is NOT turing emulable. The
simple angels corresponds to the arithmetical (and analytical)
hierarchy in computability theory. They are the main object of study of
computability theory (alias "recursion theory"); Gods, supergods, are
just higher in such or similar hierachies. They corresponds often (but
not always) with machine in company of the so-called (by Turing)
Oracles.
Note: the term "angel" is never used by Plotinus, nor the term
"theology". Proclus uses the term "theology".
>
>>
>>
>>
>>> 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.
>>
>> Zero information. Zero justification. Occam razor ... I do agree with
>> these major motivations for the everything idea, but I disagree with
>> the proposition saying that the the set of strings needs
>> zero-information. Why not the infinite strings on both right and left
>> (coding the integers), or infinite many-dimensional lattices fit with
>> zero and one on the vertex, or etc. ?
>
> Information theory is defined on one-sided strings. It would be
> possible to redefine complexity to use two-sided strings, or subsets
> of N, or real numbers, but you just end up with an isometric theory,
> it wouldn't be saying anything different/
?
>
>> There is just a lack of enough precise definition so as to verify your
>> statements that strings needs zero-information, and as I say above,
>> from some standard and traditional view points, infinite strings needs
>> a lot of information to be define.
>>
>>
>>> No countable set has this property.
>>
>> Why?
>>
>
> For finite sets, one has the objection - why that finite number? For
> infinite countable sets, can one even define a measure?
We can put many measures on all sets, finite, infinite countable,
infinite uncountable, and beyond.
Only when there is some extra-structure, making the set a vectorial
topological space for example, can we find in the lucky case some
unique measure theorems.
>>>>> 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,
>>
>>
>> We have discuss this. In the first paper the "great programmer" is not
>> a dovetailer, and indeed there is nothing in the ASSA approach for
>> which dovetailing could play a role.
>>
>
> No the great programmer writes and runs a dovetailer on er great
> computer.
? (what is a great computer? A Macintosh, a PC?, a concrete machine, an
abstract one ??)
>
> I missed the dovetailer on my first reading, as indeed it doesn't seem
> too important. I suspect the Scvmidhuber intended the dovetailer to
> "breathe fire" into the ensemble.
?
> I'm not sure that Schmidhuber accepts arithmetical realism.
? (I have no doubt he accepts it; for example he studies computation in
the limit, he accepts Church thesis, etc.)
> His
> ontology is that the great programmer exists, whatever e may be.
> The
> GP is not necessarily a machine, or an arithmetical formula.
That's correct, and that is why the notion of dovetailing does not
apply to his way of working (and this is a major departure between us).
>> Well, I hope you are not referring to the notion of "ensemble" as it
>> occurs in physical statistics.
>
> Not at all! Ensemble is usually used in mathematical terms to refer to
> a collection that need not satisfy set axioms.
You mean a set from naïve set theory? Given that we have never
mentionned even just one axiom of set theory, distinguishing set and
ensemble would be a 1004 fallacy.
>> I'm the one saying that I'm lost here. I am just asking: how do you
>> define "observer" in the infinite strings setting. (Actually with or
>> without extra-strcuture, like what you need to have a measure space).
>>
>
> I don't define the observer at all! If I knew enough to define this,
> I'd probably be very rich. I merely postulate what I need, and
> hopefully the postulates are reasonable.
>
> For the Occam's razor (and White rabbit argument), the observer
> defines a map O from the set of strings (descriptions, things which
> are observed) to concepts, or interpretations (results of
> observations). This latter set is assume discrete, and can therefore be
> indexed by N, so we can consider O: {0,1}^\infty->N.
>
> The requirement that the measure of O^{-1}(n) be non-zero is needed
> for the dewabbiting. It also ensures a positive complexity value for
> concepts.
>
>>
>>
>>>> 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.
>>
>>
>> This was not clear, sorry. Now you definitely need analysis or second
>> order arithmetic. This is everything but nothing!
>>
>
> It sounds like we have a different idea of "nothingness". I call the
> all-strings ensemble "nothing" as a rhetorical device to stress how
> wrong the "metaphysical profligates" are.
Is it a good idea? Should you not say something like "Nothing
Physical"? Saying that "all the strings = nothing" looks like a play
with words.
There are as many theories of nothing than there are theories of
things. The quantum nothing, to take a famous example, is everything
but nothing, but can be called a (quantum) nothingness because it has
an average energy value of zero (well, this is typically false but I
guess you see my point).
>>>> ... 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.
>>
>> This is not even provable in formal set theory like ZF (Zermelo
>> Fraenkel). You need the axiom of choice, if I remember well. This
>> means
>> you presuppose set theory (a vastly bigger and richer ontology than
>> arithmetic).
>>
>
> It strikes me as unimportant if these sets exist or not. But I could
> be wrong - sometimes corner cases are very important.
I agree! The devil is in the details .... I mean if we want to
progress. Looking too closely to the details at the start leads to 1004
type fallacies.
>
>>
>>
>>> 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.
>>
>>
>> I would not have dare to call you a physicist, but now that you admit
>> you are acting a bit like them, I can understand better :)
>>
>
> OK - I thought you'd appreciate the reference :)
Yes thanks. I took it as a moment of lucidity :)
>
>>
>>
>>
>>>>> 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,
>>
>>
>> You can do that. I see the point. But it is not standard at all and
>> has
>> to be explained in all detail, especially if you are not clear if you
>> follow comp or not. Even with comp this is highly ambiguous.
>>
>>
>>> and when I refer to semantic level,
>>> I mean how it is interpreted.
>>
>> This is even less standard, and although I could put sense on it, this
>> is only because I have tools for doing that. A term like
>> "interpretation" can be seen as syntactical at some level and
>> semantical at some other level.
>>
>>
>>
>>> This can be applied to all situations
>>> where emergence is occurring.
>>
>>
>> I knwo a lot of people who are searching their gun when they hear the
>> word "emergence". I mean this is a word which can be used once you
>> take many precautions. Also, if comp is correct, as you know, it is
>> matter (stuff) which emerges from consciousness/meaning (cf the
>> reversal result).
>>
>
> Sure there is a lot of confusion on the subject of emergence, so it is
> not surprising if there are people "reaching for the gun" when hearing
> the term. However, the term actually is quite well defined, and the
> way I use it is pretty consistent with what other recent thinkers have
> written, such as Bedau, or Ronald, or Fromm.
Yes. And personally I prefer when you use "emergence" instead of
semantiics, which is more involved and has more pecise connatation in
mathematical logic or computer science.
>
>>
>>
>>> So in the case of an ideal gas, the
>>> molecular description is syntactic, and its thermodynamic description
>>> is semantic.
>>
>> Hmmm.... I cannot really accept this. Here is a case where emergence
>> is far better than semantics.
>>
>
> I'm not sure what this has to do with "semantics", but everything to
> do with emergence. Semantic means "meaning". The "semantic layer" is
> the descriptive layer that means something to the observer.
In logic, semantics is related to the attribution of truth value to
(closed) formula. In classical prpositional logic, a semantics is a
function from the alphabet of propositional variables in the set {0, 1}
or {true, false}, and a way to extend those values to more complex
formula (like in truth table). For first order logic it is still rather
simple to define such semantics, but it is longer (and a bit boring).
In non classical logic, semantics are in general more subtle.
"meaning" is related to this, but is a vastly more general and informal
terms.
> The
> "syntactic layer" is typically what the system is specified on
> (although it need not be). The terms are my invention, but really of
> the other terms I've across, microscopic/macroscopic will not do, as
> it often doesn't relate to size, and L_1/L_2 (This is Ronald, Sipper
> and Capcarre's term) - I mean WTF?
?
>
> I have actually read quite a lot of literature on the subject, and
> there aren't any accepted terms for describing the levels of
> description in emergence. I'm trying to propose the semantic/syntactic
> level labels, but you know its a market of ideas out there.
Yes. Instead of using already known terms, in front of new ideas,
please just create your own new terms. Or just use periphrase like
"the levels of description of emergence". In fine, if you define
clearly your terms, then you can use any terms you want, in principle.
It helps the outside reader if those terms have some known
connotations.
If your theory is 100% scientifically clean, it should work with *any*
term (using "coffee" instead of ensemble, "tea" instead of "infinite"
etc.
I can do that for string theory (oops, not your strings, but those of
superstring theory by the physicists), or "my theory" (I mean the
theory of the lobian machine, etc. OK this is hard to get, so take this
remark with a grain of salt .... for a while .... (you can ask me to
explain how to do that in the case of "my theory" or in the case of a
mathematical physical theory (like strings).
Now that I am thinking about that the term "string" in the search of a
TOE is also a quasi copyright type of problem given that you string
have nothing to do (a priori) with the "bosonic string" or the
superstring physical theories ....
>
>>
>> Another point where your fuzziness does not help is that you are not
>> clear on the comp hyp.
>
> I never assume the comp hyp. I believe comp is consistent with what
> I've developed, and note that you can pretty much get the consequences
> of the UDA from a more general functionalist position.
This can be made clear with the UDA and after with the AUDA, where the
reasoning extends to a vast class of non turing emulable (non machine)
entities. The real key here is the notion of self-referential
correctness. To derive physics from functionalism is an interesting
idea, but this is different from deriving the necessity of
immaterialism from an hypothesis in the cognitive science. I do suspect
you have something interesting to say (which makes more frustrating
your incautious use of vocabulary).
>
>> Schmidhuber is clear on comp, even if he has
>> disagreed when discussing online in the list with the consequences I
>> derive from it. But the disagreement comes not from what comp means,
>> but from the 1/3 distinction, which Schmidhuber does not consider.
>> Tegmark does introduce an embryo of 1/3 distinction (but quite
>> different from mines (see below)), but Tegmark still uses some
>> identity
>> thesis implicitly for making observer belonging to universe, and minds
>> to brain. Such identities have just no meaning at all once you
>> postulate the comp hyp (or even strong weakening of it btw). OK? (by
>> UDA). Yes?
>
> Yes, but there is still some reason tying the observer to the
> universe.
In this context "universe" has no non ambiguous meaning at all.
> Otherwise you get the Occam catastrophe. (again my term, and
> others have called it differently) I do not know what that reason is,
> but suspect self-awareness has something to do with it.
You really should elaborate.
Russell, I am just mentioning "vocabulary problems" which could prevent
progress for those who want to take computer science /mathematical
logic as a tools, like we have to do once we postulate explicitly comp.
But I have realised in Siena that even such a thing as the mundane
classical "Church thesis" is not so easy even for logicians, so I know
that I have to explain that (and probably to write some more papers
...).
I am just telling that your use of vocabulary will not help to bridge
the gap between the physicist (not the "physician" 'course (thanks to
Brent)) and the logicians. I hope you manage my short and frank but
undiplomatical way of being direct.
Bruno
http://iridia.ulb.ac.be/~marchal/
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