Le 24-avr.-08, à 18:26, nichomachus a écrit :
>
> On Apr 22, 11:28 pm, "Brian Tenneson" <tenn....domain.name.hidden> wrote:
>> Perhaps Hilbert was right and Physics ought to have been axiomatized
>> when he
>> suggested it. ;) Then again, there might not have been a motivation
>> to
>> until recently with Tegmark's MUH paper and related material (like by
>> David
>> Wolpert of NASA).
>
> The logical positivists were motivated to axiomatize in the predicate
> calculus the laws of scientific theories in the early 20th century,
> first because they believed that it would guarantee the cognitive
> significance of theoretical terms in the theory (such as the
> unphysical ether of maxwell's electromagnetism), and then later
> because it had evolved into an attempt to specify the proper form of a
> scientific theory. In practice this had too many problems and was
> eventually abandoned. One of the consequences of this program was that
> axiomatizing the laws of a theory in first order predicate calculus
> with equality was that such a formulation of a theory always implied
> various unintended interpretations. The amount of effort needed to
> block these unintended interpretations was out of proportion with the
> benefit received by axiomatization.
It is a bit weird because it is just logically impossible to block
those unintended interpretations. And This should not be a problem.
The reason why physical theories are not axiomatize is more related to
the fact that axiomatization does not per se solve or even address the
kind of conceptual problem raised by physics.
>
>
>>
>> I was trying to answer Bruno's objections regarding set theory being
>> too
>> rich to be the 'ultimate math' the MUH needs to propose what the
>> universe is
>> and I quipped that that was because math is invented or discovered to
>> further its own end by logicians, for the most part, and that
>> metamathematicians such as Cantor had no apparent interest in physical
>> things or furthering the pursuit of Physics.
>>
>> Another question of Bruno's was my motivation. I started this quest
>> hoping
>> that three truth values were sufficient to develop a set theory with a
>> universal set that was in a classical logic sense consistent to ZFC
>> set
>> theory. Or, if not true, prove that and figure out why. Perhaps
>> more truth
>> values would solve that. My main motivation has definitely not been
>> to
>> "rescue" a major apparent shortcoming in the MUH as I started this
>> on-and-off quest in 2003 with no internet connection or resources
>> such as a
>> deluge of journals (ie, a good library). How it started was that
>> someone
>> online in a place such as this used Russell-like arguments to -prove-
>> that
>> the Physic's universe -does not exist- for essentially the same
>> reasons a
>> universal set can't seem to be non-antimonious.
>>
>> Suppose Everything is well defined along with its partner,
>> containment (such
>> as the earth is contained in the solar system by the definitions of
>> both).
>> Then Everything does not exist. Proof:
>> Consider the thing, call it "this something," that is the qualia of
>> all
>> things that do not contain themselves.
>> Then this something contains itself if and only if this something
>> does not
>> contain itself.
>
> I am suspect of the claim that a logical argument such as the above,
> which relies on a paradox of self-reference, could be used to
> demonstrate the non-existence of the so-called Everything.
Indeed. It will just prevent the "Everything" to be a thing (to belong
to Everything).
> Also, I
> personally remain unconvinced that there is anything problematic about
> the exitence of the universe of universes, or the ensemble of all
> possible mathematical structures, thought it may not be well defined
> at present. I don't believe that this is simply the union of all
> axiomatic systems. If trying to define the Everything as a set implies
> a contradiction, then fine -- it isn't a set, it's an ensemble, which
> doesn't carry any of the connotations that are implied by the use of
> "set" in the mathematical sense. Therefore each entity in the ensemble
> is a unique collection of n axioms that has no necessary relationship
> to any other axiom collection. What happens in an axiom system stays
> in that axiom system, and can't bleed over to the next one on the
> list. Some of these may be equivalent to each other.
>
> A = The collection of all finite axiom systems
> B = The collection of all consistent finite axiom systems
I guess you mean "recursively enumerable" instead of finite. You would
loose first order Peano Arithmetic (my favorite lobian machine :).
Note also that SAS occurs very quickly. SAS occur in theories which are
much weaker than the SAS themselves (ex: SAS occur in Robinson
Arithmetic, i.e. when you can define successor, addition and
multiplication. SAS themselves need induction.
>
> The "cardinality" of B is not greater than the "cardinality" of A.
> (Scare qutoes since cardinality is a property of sets and these may
> not be sets if that would imply contradiction.) It is B that is
> interesting from the point of this discussion since it is believed (I
> don't know of any proof of this) that only systems in B could produce
> the type of rational and orderly physical existence capable of
> containing observers who can think about their existence as we do
> (SASs, or Self-Aware Substructures). The collection of all those
> systems capable of containing SASs is the most interesting from the
> point of view of the present discussion, and must have a "cardinality"
> not greater than that of B, since many axiom systems are too simple to
> contain SAS, and the ones with them are expected to predominate.
>
> The idea of this ensemble so propounded does not seem to entail an ad
> absurdum paradox such as you gave above. Further, didn't I see you say
> somewhere that you don't even believe in sets? I apologize if I am
> mistaken, but if that is true, I can't see how that statement would
> reconcile with sincere belief in the validity of the agument you gave
> above.
>
> If there is some genuine logical inconsistency in the above, please
> point it out to me as to me this (which is Tegmark) seems like a good
> direction to go in trying to formulate a proper definition of the
> Everything.
I suggest you read my papers (published in French a long time ago, in
english a bit more recently) or my posts to this list. Those should
help you for making clear what is missing in Tegmark or Schmidhuber
(and many others): the fact that they do not take into account the
first person (plural or not) indeterminacy.
With comp (and its transfinitely numerous weakenings) we cannot know
which computational consistent histories (in case we bet there is one)
do support our experiences, and this put total constraints on the
nature of any observable and sharable realities.
This *changes* everything ... Like in Plato, the physical world can
only be the border of "our" ignorance or first person plenitude (as
George Levy called it). "our" is "us the (hopefully) sound machines".
(or alpha-machine, a constructive ordinal based weakening of the notion
of machine, but I use "omega-machine" (that is: machine) for reason of
simplicity).
>
>>
>> By a simple logical tautology (a variant of ad absurdum), this proves
>> that
>> "Everything is well defined" is a false statement. It also raises
>> doubts as
>> to the existence of this so called Everything. Maybe this google
>> group
>> should end?
>>
>> I don't think so.
>>
>> My quip was something along the lines of, "however, in any ternary
>> logic, ad
>> absurdum is not a tautology and therefore, can't be used here."
>>
>> That discussion got me going and while mostly off task, I've been
>> thinking
>> about this on and off since then. Basically, my motivation to
>> "rescue" a
>> universal set is so that Cantor's dream of formalizing in a
>> mathematical way
>> some type of deity could be realized. The analogy would be Abraham
>> Robinson
>> is to Issac Newton (on infinitesimals) as Quinne (et al) are to
>> Cantor (on a
>> universal set). Right idea, but never considered using fuzzy logic
>> not to
>> be delved into much until Lukaseiwicz, Zadeh, and others revitalized
>> FL. As
>> it took an army of giants to "rescue" Newton's intuition which was
>> criticized by another philosopher (Berkeley, akin to Russell) to
>> develop
>> enough tools (compactness theorem), it is taking an army of logicians
>> to
>> "rescue" Cantor's intuition about God which, and this may be
>> apocrypha, he
>> believed to be his maximally infinite set. He thought infinity must
>> be an
>> attribute of God and therefore delved into infinite sets, hoping, I
>> assume,
>> to reach some type of Omega set that contains all sets and would then
>> be
>> necessarily the "biggest" infinity. Cantor proved that the power set
>> of any
>> set is "larger," however, and settled his own quest in his own way
>> though
>> I'm guessing he -desired- the opposite conclusion to have been
>> reached.
>>
>> Others in the FL army are trying to reach that conclusion which
>> Cantor,
>> chronologically, would have to have re-discovered much mathematics to
>> realize in the way this army is doing.
>>
>> So the basic motivation is to find some type of thing with maximality
>> in
>> some important sense and study it. With the MUH, now I suspect that
>> Everything would be a likely candidate for a literal God and atheism
>> might
>> have to suddenly be the irrational side to be on.
>>
>> So on this note, the works of David Hawkins (a psychiatrist and
>> spiritualist) inspired me to ponder the following question, along with
>> Tegmark's articulation of the MUH, of course.
>> Which mathematical structure -is- the universe in Physics?
>>
>> I suspect it might already exist and has been studied.
>
> I agree. We could exist in the Mandlebrot set for all we know.
*assuming* comp, we do. This follows from comp and result by Blum, Shub
and Smale I have conjecture this, as Penrose did for a slightly weaker
statement.
> Determining which mathematical structure is our own universe is likely
> practically impossible, though determining which classes of
> mathematical structures are more likely candidates may be doable.
I think we have no choice in the matter (once we assume the
"unbelievable comp hyp."). The physical is not just a mathematical
structure among others. The physical emerged from a sort of sum
pertaining on the whole of the mathematical possible histories. If this
does not give the empirical physics, then comp will be refuted. But
preliminary results give already a sort of quantum topology. The one I
have more or less extracted from the comp hyp, at the modest
propositional level, has not yet been prove to be be equivalent to
universal quantum topology, but they are clues indicating that comp
could be the promising path. It is quasi obvious that comp entails many
consistent histories, and the math gives reasons why such histories
interferes statistically in a "quantum way", i.e. with a
perpendicularity relation on the possible incompatible states/stories.
Ah yes the "truly" parallel realities are perpendicular, but this is
already the case with quantum mechanics and its "scalar product".
What is hard, and on which I am stuck since years is to find the
(arithmetical) needed tensor product, or how does a first person plural
reality occur. Mathematically it is enough to assume at some place a
linearity condition. But this is cheating; we have to justify that
linearity from comp only, as comp justifies we have to do. Sorry if I
am a bit short.
bruno
>
>
> It's like finding
>> the correct non-Euclidean Geometry applicable to the universe we
>> perceive
>> gets us to a GR that coincides with observation (for the most part?).
>> I am
>> guessing that the universe must have an MV-algebra
>> structure.http://en.wikipedia.org/wiki/MV-algebra
>>
>> I was trying to rejoin Bruno's "too rich" -valid- (imho) objection to
>> Tegmark's approach in his MUH paper by concocting a theory that was
>> far less
>> rich. All I need are things and a notion of containment. I was
>> going to
>> call it container theory. Then there'd be no need to develop
>> something
>> strong enough to do numbers, infinite sets, and such, so with those
>> goals
>> gone, so much more is available to Physics without having to squeeze
>> any set
>> theory or logic into Physics. It's there, I suspect, in -classical
>> logic-
>> and recent -algebra- in the guise of MV-algebra. This area is
>> exactly what
>> I mean by thing and containment. Now if you look at the wiki article
>> above,
>> observe, firstly, how little there is reliance on sets or
>> non-classical
>> logic.
>>
>> Secondly, I could view all the letters that would normally be
>> variables as
>> things in the "container theory" I was trying to work on. In
>> MV-algebras,
>> the variables represent truth degrees and the carrier of the
>> MV-algebra is
>> the truth set, the set of all truth values which has cardinality two
>> in all
>> classical logics. But this seems promising for my 'container theory'
>> which
>> I was assuming someone had done that I just had to find somewhere.
>> Now if
>> each variable is now a worldline, one think of it that way. The
>> carrier of
>> the MV-algebra is the set of all worldlines in one parallel universe.
>> An
>> ideal could be a sub-universe that isn't parallel. The circle-plus
>> is the
>> notion of joining and the circle-times is the notion of intersecting
>> or
>> meeting (to use Boolean terminology which is much more compatible
>> with most
>> natural languages).
>>
>> The 0 in the MV-algebra could be intuitively compared to that which
>> contains
>> nothing or the empty container.
>>
>> The notion of containment is given by the ordering induced by the
>> circle-plus and negation operator, listed in detail in Siegfried
>> Gottwald's
>> "A Treatise on Many-Valued Logics" in section 9.2.1 on pages 215-234.
>>
>> So if each variable represents a world-line consistent with -some-
>> laws of
>> some Physics, which vary from parallel to parallel (a parallel would
>> be an
>> ideal of an MV-algebra), then maybe this way to view MV-algebras
>> would prove
>> interesting to a Physicist.
>>
>> To glue MV-algebras together into what the multiverse might be, not
>> much
>> more complex than a simple union would suffice, I think (not having
>> thought
>> along those lines yet)?
> >
>
http://iridia.ulb.ac.be/~marchal/
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Received on Fri Apr 25 2008 - 06:27:42 PDT