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.

>>> 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.


> 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.

> 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 ?

> 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).

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 ...



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