Le 21-mars-07, à 17:41, Brent Meeker a écrit :
> Bruno Marchal wrote:
>>
>> Le 20-mars-07, à 18:05, Brent Meeker a écrit :
>>
>>> What are those relations? Is it a matter of the provenance of the
>>> numbers, e.g. being computed by some subprocess of the UD? Or is an
>>> inherent relation like being relatively prime?
>>
>>
>> It is an inherent relation like being prime, or being the godel number
>> of a proof of f, etc.
>
> I didn't think godel numbering was unique? If I just cite a number,
> like 12345678987654321, is it either the godel number of a proof or
> not?
The whole point of Godel's proof is that, exactly like you can define
"prime" in the language of Peano Arithmetic PA (or weaker non lobian
entity like Robinson Arithmetic), you *can* define "being the godel
number of a proof by PA" *in* PA.
In all situation, when we are using a language we have to agree on some
convention. Even the question "is the number 12345678987654321 prime or
not" presuppose the conventional use of base ten position notation for
number and the arithmetical definition of primeness.
The same occurs for the notion of arithmetical provability.
Of course the finite string "12345678987654321" could be a godel number
of a proof or not depending of the way you represent the variable or
the quantifier or the connector in term of numbers, but the *string*
"12345678987654321" will represent a prime number or not according to
the way you represent decimal numbers in Pean Arithmetic. It is just
because we already agree, since high-school, on conventional notation
for number that you can directly talk about the *number*
123456787654321". If we would have learn in high school some canonical
way to represent proof (in PA) by numbers, like those used to explain
to PA what a proof is, I would have been able to say yes or no
directly.
Now, concerning Peano Arithmetic (PA), obviously (ask if not)
12345678987654321 is NOT the godel number of a proof of f
(independently of any choiçce of proof representation). This is because
PA is consistent, so no number can represent a proof of a falsity f.
And a lobian machine like ZF can prove that.
Note that, although Stathis defends many argument which are going in
the direction of the ultimate consequence of comp, I do agree with some
critics you are doing here and there. To believe that the observer
moment OM are disconnected is perhaps a remnant ASSA thought (Absolute
Self Sampling Assumption).
Let us try to be a little more specific. From a computational point of
view, a computation can be considered equivalent with a proof of an
arithmetical statement having the shape "it exists a number having this
or that decidable property", that is "En P(n)" (it exists a n such that
P(n)). So the universal dovetailer infinite execution is equivalent
with the set of true Sigma1 sentences together with their proofs. So we
can define easily a notion of atomical (primitive) third person
"Observer Moment" by them. I put OM in quotes, because strictly
speaking OM are conceived as first person experience and not as third
person computational states. The UD Argument does force us to take
those distinction into account. In particular it shows that, to
predict anything physical, among other things, we have to define a
notion of first person plural indeterminacy, bearing on the OMs as seen
by a machine itself embedded in UD*.
What happens is that each internal view, or hypostases, like the first
plural person points of view, are defining a non trivial (thanks to
Godel & Co.) mathematical and quasi geometrical structure on the OMs.
That is each hypostases relate the OMs. They single out arithmetical
relations in between the OMs. They remains disconnected, in a sense,
like the numbers themselves can be considered disconnected. But of
course, numbers are connected to each other by the usual arithmetical
relations (definable in term of sums and products), and when the
internal points of view are taken into account, we got for each of them
a collection of different connections. If you recall some of my older
post, the three primary hypostases rise Kripke multiverses, and the
secondary (material, physical, natuiral) hypostases rise Scot-Montague
sort of continuum topological multiverses, where the OMs appears to be
as connected as "points" on continuous manifolds.
Now if comp is correct, it is expected that the relations between the
OMs, from some first person point of view, restricted to the atomical
Sigma1 propositions (by the UDA) gives rise to the laws of physics,
defined by what is invariant for all observer point of views (arguably
given by the third or fourth or fifth "hypostases"). Those are indeed
the one which technically give purely arithmetical interpretation of
Quantum Logic, the real signification of which remains to be seen (to
be sure).
The lobian machine already gives some clues that a *physical*
computation has to be given by a unitary transformation (and thus a
solution of the SWE). I don't obtain quantum entanglement, because I am
still unable to recover arithmetical tensor products. This is a bit
ironical, giving that "non locality" (a typical feature of quantum
entanglement) is the easiest consequence about the physical world that
we can derive from UDA, but apparently, when interviewing the lobian
machine, it appears to be rather well hidden.
Obviously, the comp-physics is not yet as elaborated than current
modern physics. But thanks to the fundamental difference between G and
G*, inherited by most "material (first person plural) intensional
(modal) variant" of it, the lobian interview does not eliminate the
mind or the first person views, with all their un-communicable
sensations and qualia. here the comp physics is in advanced compared to
the physicists who, in majority, are still attaching their feet to the
primary Aristotelian matter bullet.
To be sure, I love Aristotle. His theology is enough precise to be
falsified. And, as I explained in my last (SIENA) paper, Plotinus
theory of Matter is really the theory of Aristotle reconsidered in the
Platonic theology.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Thu Mar 22 2007 - 07:33:12 PDT