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).
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.
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. 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 Wed Apr 23 2008 - 00:30:34 PDT