Re: Which mathematical structure -is- the universe in Physics?

From: Brian Tenneson <tennesb.domain.name.hidden>
Date: Thu, 24 Apr 2008 10:08:16 -0700

I was attempting to -invalidate- that argument against the existence of the
universe, actually, by saying that in three truth values, which the
Physicists can't rule out as being the more accurate logic of their
universe, the argument "reductio ad absurdum" is not a tautology and,
therefore, can't necessarily be applied.

However, in binary logic, the Physicist's universe (or whatever Everything
means) can't exist.

I doubt self-reference is inherently the problem in light of things like
Tarski's fixed point theorems which provide concrete examples of wffs that
are self-referencing, in terms of Godel numbers, if I recall. That proof I
was exposed to was not an existence proof of self-referencing wffs merely by
"logical flamboyancy" but by the providing an example of an actual -class-
of self-referencing wffs. Obviously, the above argument does not explicitly
involve wffs (it does, however, implicitly), and I am -only- making a case
for plausibility at this particular moment.

I see no problems with the argument given that in binary logic, their
universe can't exist; this, to me, convinces me that the Physicist's
universe can't operate on binary logic by Occam's Razor as -none- of the
data in any experiment would fit the result that confirms their speculation
that their universe exists.

Ergo, the Physicist's universe must operate on at least three truth values.
(Consequently, it exists.) This to me is a more elegant solution to the
argument than citing self-referencing issues as automatically damning. If
natural language can be used to prove the Heine-Borel theorem, without the
need for wffs, then why must a statement about Everything be formalized in
machine-level code with wffs?

If there is further objection to my line of thinking, -please- point it out
to Everyone (which I hope is well-defined or else no one would know what I
mean, right?) ;)

Thank you for your remarks; I find all input extremely productive!!

On Apr 24, 9:26 am, nichomachus <Steven.Payne.L....domain.name.hidden> wrote:
> 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
>
> ...
>
> read more

On Thu, Apr 24, 2008 at 9:26 AM, nichomachus <Steven.Payne.Long.domain.name.hidden>
wrote:

>
> 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 - 13:08:31 PDT

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