On 16 Oct 2015, at 02:36, Bruce Kellett wrote:
It is the failure to clearly distinguish between these different
senses of the word 'exists' that cause most of your confusion.
The mathematical theorem is that when a machine looks inward, in the
sense made precise by Gödel, Kleene and others, the machine is forced
to distinguish 8 main different sort of existence, and in fact many
more.
The ontic basic existence can be based on any first order logical
specification of a Turing universal language or system. Once chosen,
the ExP(x) means it exists a basic element which has the property P.
To fix the thing I use as basic element the number+basic +/*laws, but
the theology of the machine, including physics, will not depend on
which universal basic system has been taken. The first basic Turing
universal system is like a sort of base in which we can describe and
study all the others.
I will also say that a relation R(x,y, ..) or a property P(x) exists
for a shorten of it is true that R(x, y, z) for some x, y, z.
Then we have the 8 nuances that no machine can miss when looking
inward deep enough, which is exactly what they can do when they
believe in enough induction axioms.
Then I define the set of beliefs of the ideally correct machine *in*
the language of the machine, which here will be elementary arithmetic,
given that we have fixed that one. The most typical machine/number/
theory/belief-set is Peano Arithmetic.
RA can prove that PA exists (trivially actually), and RA imitate all
machine, notably in proving all the details of the computations that
PA does when doing her "thinking".
I write []A for "PA proves A", and I think about it as translated in
the language of the machine. In our case this makes []A an
arithmetical proposition.
Then you get all the sort of existence by the quantified modal logics:
ExP(x)
[]ExP(x)
[]Ex []P(x)
With [] put for []p, and for []p & p, and for []p & <>t, and for []p &
<>t & p, and some infinity of graded variants, depending on the points
of view.
The apparent primary matter is given by the quantization:
[]<>ExP(x)
[]<>Ex []<>P(x)
(but here only on []p & p, []p & <>t, []p & <>t & p, with p obeying p -
> []p).
So there are indeed many sort of existence, and most error in
philosophy and theology can be reduced to a confusion between such
existence, or a confusion between the corresponding hypostases.
If you define [1]p = []p & p, [2]p = []p & <>t and [3]p = []p & <>t &
p, and [0]p = []p, you get the five hypostases, and even 8, as three
of them split between a provable and true part. The incompleteness
makes the true part extending properly the provable part, and that
split is inherited by [2]p and [3]p.
Lucas-Penrose invalid use of Gödel's incompleteness can be seen as a
confusion between [0] and [1]. The separation between science and
theology can be sees as a confusion between [0] ans [0]* for the logic
which split along proof and truth. Obviously some confusion entail
others.
I recall the epistemological the plotinian lexicon:
p
[]p
[]p & p
[]p & <>t
[]p & <>t & p
One
Intellect
Soul
Intelligible Matter
Sensible Matter
Truth
Provability
Knowledge
Observable
Sensible
Only One (God) and the Soul do not split along proof and truth.
Up to now it works. It is a pure mathematical theory, and physics is
determined by them, so just let us look if it works. Thanks the
quantum weird logic and possible interpretations, it works.
The neoplatonism of the universal (Turing) machine suggests that the
Heisenberg relations *are* consequence of the (mathematical) self-
reference limitations. Aristotelians cannot see that because they tend
to confuse the One with the Observable.
The theory gives the tools to test all this, and measure a possible
departure from the neopythagoreanism or neoplatonism canonically
associated to the universal machine.
If the quantum logic we got is good enough, we can apply Gleason
theorem and prove the unicity of the measure on the computations (when
seen from inside, observed), and if some quantum logicians are correct
(hard paper!), the whole standard model might follow.
UDA is for the human babies, AUDA, the translation, is for all
universal Löbian number, where a number is Löbian when it can prove p -
> []p for all Sigma_1 arithmetical sentences.
Note that the ontology is given by RA, for which p -> []p is true, but
RA don't know that. For PA, which exists in the mind of RA (so to
speak) p->[]p is not only true, but provable. She knows, like you,
that she is Turing universal, and Löbian.
Keep this post, as further conversation could help to make all this
clear and simple. You might put some good book on logic (Mendelson,
Boolos and Jeffrey, Epstein & Carnielli) near your bed.
Bruno
http://iridia.ulb.ac.be/~marchal/
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