On 10/6/2012 8:54 AM, Platonist Guitar Cowboy wrote:
Yes, but it is also in its infancy. With Aczel's work not 30 years
old, and this admittedly weak analogy to consciousness only a few
years old, which Aczel does not seem to be following up on himself:
My point is, this is very young, what's young is always messy and will
hopefully tidy itself up, and I can feel some funky aspect,
specifically the observer aspect of a non-well-founded set defining a
Russell operator, hinting at quantum physics perhaps in the future.
It should be called the Cantor operator, but as Zuckerman notes: "the
importance of PR and publishing makes the difference." So, knowing
this, why doesn't he call it the Cantor operator...
I suspect that he named it after Russell because Russell's
canonical (?) definition of the paradoxical set. I don't know that
Cantor drew any attention to that set, thus he doesn't get credit for it.
On Sat, Oct 6, 2012 at 9:12 AM, Bruno Marchal <marc...@ulb.ac.be
On 06 Oct 2012, at 02:37, Stephen P. King wrote:
Mathematical Foundations of Consciousness
Willard L. Miranker
(Submitted on 23 Oct 2008)
We employ the Zermelo-Fraenkel Axioms that characterize sets
as mathematical primitives. The Anti-foundation Axiom plays a
significant role in our development, since among other of its
features, its replacement for the Axiom of Foundation in the
Zermelo-Fraenkel Axioms motivates Platonic interpretations.
These interpretations also depend on such allied notions for
sets as pictures, graphs, decorations, labelings and various
mappings that we use. A syntax and semantics of operators
acting on sets is developed. Such features enable
construction of a theory of non-well-founded sets that we use
to frame mathematical foundations of consciousness. To do
this we introduce a supplementary axiomatic system that
characterizes experience and consciousness as primitives. The
new axioms proceed through characterization of so- called
consciousness operators. The Russell operator plays a central
role and is shown to be one example of a consciousness
operator. Neural networks supply striking examples of
non-well-founded graphs the decorations of which generate
associated sets, each with a Platonic aspect. Employing our
foundations, we show how the supervening of consciousness on
its neural correlates in the brain enables the framing of a
theory of consciousness by applying appropriate consciousness
operators to the generated sets in question.
This is part of what I have been assuming form the beginning
of my conversation with Bruno so many moons ago. Its nice to see
its independent discovery.
As the cow-boy guessed right this is assuming too much, both for
the formalism used (which is OK), and the ontology, so it uses
implicitly non-comp hypothesis, which is less OK, as comp is also
assumed implicitly. IT is not uninteresting for possible progress,
but it is unaware that matter as to be explained by statistics on
computations "seen from inside". The role of "Russell operator" is
played by the Kleene second recursion theorem, which encapsulates
the "non foundation" well enough.
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