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.
>
> 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. 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
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.
>
> 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.
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.
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)?
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Received on Thu Apr 24 2008 - 12:27:01 PDT