Bruno Marchal wrote:
> Le 02-mai-06, à 00:18, Tom Caylor a écrit :
> >
> >
> > Bruno Marchal wrote:
> >> Le 25-avr.-06, à 17:37, Tom Caylor a écrit :
> >>
> >>>
> >>> In fact, "closed system" and "meta element" seem to be contradictory.
> >>
> >> Not necessarily. It could depend of what you mean exactly by "closed".
> >> Closure for the diagonalization procedure is the key. Diagonalization
> >> is the key of the "heart of the matter". I will come back on this
> >> later.
> >>
> >
> > Closed system (Principia Cybernetica): An isolated system having no
> > interaction with an environment.  A system whose behavior is entirely
> > explainable from within, a system without input...
> >
> > Mathematically, a closed system contains its boundary, or it contains
> > its limit points.  In other words, anything expressable with the given
> > axioms/language is itself a member the system.
> All right. Topologically they are "closure" systems, and they provide
> "natural" models for both first person and S4 type of modal knowledge
> theory. A set is included in its closure, the closure of a closure is a
> closure, etc. Example: a theory (set of formula closed for the
> application of the inference rules), a closed subset of a topological
> space (not necessarily Hausdorff), closed subpace of Hilbert spaces.

Bruno, thanks for understanding what I mean by closed.
John, I shouldn't have mentioned the "closed" definition that allows
output from the system, because this is not what I mean by closed.  My
meaning of closed does not allow knowledge from the outside.

> >
> >>
> >>> And, back to the original question, "closed system" and "erasing
> >>> information" seem to be contradictory.
> >>
> >> Why?
> >>
> >
> > I'm at an impasse with myself in trying to explain my intuition
> > further.
> OK.
> > Meanwhile I'm studying up on diagonalization, waiting for
> > your "heart of the matter" (which I take as just a pun and not
> > referring to physical matter, heaven forbid).
> Heaven forbid? Comp forbids!  ;-)
> About the heart of the matter I have begun a post but I realize it will
> be far too long and technical, and I am still searching a way to
> present that "heart of the matter"  in some swallowable way ....
> Of course "heart of the matter" is an allusion to a section in
> Smullyan's "Forever Undecided" which got that name.

I am beside myself  ;)  Perhaps the interactive step-by-step approach
that you've used in the past would be easier for you and more
profitable for us.

> >
> > Speaking of "impasse with myself" and diagonalization, a thought
> > occurred to me that an instruction that "erases information", like a
> > Turing machine "goto" statement (e.g. Wei Dai's "go to the beginning of
> > the tape" instruction)
> ? Why a goto should erase anything ?

Actually, in reviewing the definition of Turing machine (it's been over
2 decades since I studied it) I agree with you.  The Turing machine
leaves behind a memory of its past through its "writes" to the tape.
Maybe I don't understand what Wei Dai was saying with his setting of
the head back to the start of the tape.  In order to get back to the
exact beginning *state* the Turing Machine would have to be instructed
to do an inverse of all of the writes it has done and then go back to
the start of the tape.

> > seems to be a *self-referential* instruction.
> > Maybe this has something to do with the original question and (I
> > maintain) the need for a meta viewpoint, or an open system, to
> > understand it.
> But then how will you explain how that "meta-open" system understand
> anything. You take a risk of being lead to infinite regress (but then
> see for a case below).
> The heart of the matter, which will really be the "closure of some set
> for the diagonalization procedure" will make it possible to find some
> fixed point for the "meta" operation itself, so that it will be
> possible for a system belonging to a closed system to refer to itself
> in a relatively correct way, with some probability (normally determined
> from inside).

I believe that in the meta system, being "open" requires a paradigm
shift in the meaning of understanding.  If we just stick to our
reductionist meaning of understanding, then we are still closed and we
haven't really gone out of the system.  This new sense of understanding
is what allows us to not go into an infinite or circular regress.  It
is what allows us to assign *true* meaning in the first place.

> But now, I must confess (!) that I am discovering that if the Riemann
> critical zeros really describe a spectrum related to a quasi (?)
> classical chaotic regime---as it can be suspected from experimental
> (but still purely mathematical!) evidences---then I could imagine that
> the prime numbers could eventually describe not only a Universal Wave
> Function (even if only by pieces but the first person doesn't care as
> far as those pieces have a positive density) but would also describe a
> sort of universal wave reduction like if an absolutely external
> observer was included freely in the number's gift !
> So, recursion theory (computer science) allows internal "metas", but
> primes, by their so much irregular behavior could still provide an
> apparent reduction justifying some external metas. Weird. I tend not to
> believe in it, though.
> Who did invite the primes to the banquet?
> Just thinking aloud. Perhaps my Spring Riemann fever ...
> Bruno

Wishful, but good, thinking in my view.  I take your "I tend not to
believe in it, though" as saying that you don't think it's worth
investing a lot of your resources in pursuing it.  I tend to think that
pursuing anything is worth it if it allows us to see in a new way why
it is closer or further away from reality.  The theory is that we can
use these experiences to formulate a viewpoint of reality that is
closer and closer to reality.  Believing in this theory is actually an
act of faith in the goodness of reality that goes beyond what evolution
can explain.  I use the word "goodness" over and above
"understandability".  If reality is understandable *by us* in any way
close to the aspirations of the Everything List, then I feel pulled to
express this as, "Someone out there is truly being good to us."


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