Plamen, Loet, Pedro,
> On 2 Mar 2018, at 10:36, Dr. Plamen L. Simeonov
> wrote:
>
> I know him: his name is God, the meta-observer + meta-actor at the same time.
> Correct, Bruno?
God has no name that can be invoked … in the antic greek scientific approach of
theology. So it is only a subject of inquiry and never an answer. The God of
plato was arguably the notion of truth, with the understanding that it
transcend us, or is “beyond” us, or bigger than us. But then who are “us”?
The use of “God” was as a form of pointer to the question of what is real, with
the doubt about the natural criterion: what we see is what is real, that
Aristotle will yet come back on, and which please our sense and intuition.
Now, if we start from some theological assumption like Mechanism (the believe
that we can survive some digital transformation), then, the constraints of
digitalness are enough big and counterintuitive to be able to refute Aristotle
theology (where God is the physical reality) and to force the rationalist to
envisage a coming back of the God of the Pythagoreans: the Numbers, or the
arithmetical reality.
Indeed, it is a proven fact that the elementary arithmetic reality emulates
(executes, run, …, in the precise mathematical sense of Church, Turing, Kleene,
…) *all* computations, and it is a fact that a universal machine cannot
distinguish by introspection if it is run by an arithmetical relation or any
Turing universal machinery. It is also a fact that such computations are
implemented in arithmetic in a highly distributed way, and that observation
provides information coming from a self-localization in an infinite
distribution, and highly structured, complex net of computations. The structure
is imposed by the mathematics of computability versus provability versus
knowability versus observability, all modes of the universal machine ability to
refers to itself.
So when Pedro asks “The impending agenda is on something general universal as
an object, and yet concrete particular enough in process. The richness resides
within the concreteness down to the bottom.”, I would suggest the concept of
universal machine, or universal word, number, digital program, etc. It is
something very general, and admitting many very particular instances, yet all
mimicking each other in arithmetic. But this leads to the reversal between
physics and number’s psychology/theology. We are distributed in infinitely many
computations, making any attempt to predict anything into a statistics on all
computations, again structured by the universal machine ability to refer to
itself. That makes mechanism testable, and indeed, this leads to quantum logic
for the logic of the observable of (any) universal machine/number. Yet, that
means that there is no physical bottom, or that the physical bottom is not
really a bottom, but a statistical sum on infinities of computations, something
rather confirmed by quantum mechanics or quantum filed theory.
And that put even constraints on what “God” can be. Unlike a common idea about
God, there will be a trade-off between science and potence. Quasi-omniscience
leads to quasi-impotence, and the price of potence (ability to act on the
reality) leads to loss of science: it looks we cannot have both at once. The
finite creature, being participating to the building of the realities, can act
by lacking knowledge, and can awaken in the infinite by loosing acting powers.
If Mechanism is true, from inside, the arithmetical truth is made equivalent
(yet in a necessarily non provable way) with the semi-computable universality,
and god is the universal subject associated with the universal machine. It is a
not a creator, more like a terrible child, and rarely if ever satisfied despite
the range of its distribution.
The “correct” machine avoids the contradictory blasphemy by adding an
interrogation mark for the propositions corresponding to their true but
unjustifiable, and the logic of Gödel-Löb-Solovay, accessible to the machine
itself provided a very small amount of inductive abilities, provides the way
to handle them with the needed caution.
On the propositions which are semi-computable truth and proof meets and join: p
<-> []p, but only at the truth level: G* proves []p -> p, but G does not even
for p restricted at sigma_1 (semi-computable). Note that G, for p restricted to
sigma_1 proves p -> []p, which is what makes the machines Löbian. It directly
implies a form of self-referential awareness ([]p is itself sigma_1 so this
implies []p -> [][]p).
A nice recreative introduction to the key tool here, the modal G, is given by
Smullyan’s book “Forever Undecided”. It makes it look like a fairy tale,
because the K4 reasoner needs to visit a very special Knight-Knaves Island, but
that is the case for all self-referential relatively finite entities by Gödel
Diagonal Lemma (or by Kleene’s second recursion theorem).
With the number there are two sort of