Hi,
I take the opportunity that the list is calm to send a first
approximation of a possibly extendable post which addresses the
beginning of the background needed for the interview of the universal
machine on the physical laws.
It also addresses some point relevant for discussing the link "formal
system" <======> "Computation" as in Tegmark diagram page 18 of his
paper "the mathematical universe" (cf a post by Mark Geddes).
Although schematic, it could already help me if you can list the points
for which you would like examples or more technical details, or just
explanations. You can also ask for *less* technical details like an
explanation in (pure) english, perhaps.
I am building again from Robinson Arithmetic, but I could have use the
combinators or any logical description of what a universal machine (a
computer) can do. The adavantage of using Robinson Arithmetic (or its
"little" variant) is that provability in Robinson Arithmetic
corresponds to universal computability, but not of universal
provability (which does not exist).
I am perhaps on the verge of not being able to explain the sequel in
informal term, but I keep hope that non expert, but computer-open
minded people, can learn and help me to be clearer or more pedagogical,
without having us to study thoroughly mathematical logic.
Tell me perhaps if you don't understand what I call "Searle's error" in
the comp setting below.
***
0) historical background
ARISTOTLE: reality = what you see
PLATO: what you see = shadows of shadows of shadows of shadows of ....
what perhaps could be.
And would that be? nobody can say, but everybody can get
glimpses by looking inward, even (universal) machines.
Twentieth century: two creative bombs:
- The Universal Machine (talks bits): UM (Babbage, Post, Turing,
Church, Suze, von Neumann, ...)
- The other universal machine (talks qubit): QUM (Feynman,
Deutsch, Kitaev, Freedman, ...)
Comp = Milinda-Descartes Mechanism in a digital version. = (also) "YES
DOCTOR" + CHURCH'S THESIS. (I suppress the arithmetical realism,
because it is implicit in CHURCH'S THESIS).
UDA: a reasoning which shows that if comp is correct then the physical
laws have to be derived by a measure on states (the measure being made
up through their computational histories).
Subgoal: extract QUM from UM's self-observation.
Link with everything-list: search for the "observer moments" and the
relevant structure operating on them (not yet solved).
***
1) The ontic theory of everything: LRA (Little Robinson Arithmetic),
CLASSICAL LOGIC (first order predicate logic axioms and inference
rules)
AXIOMS OF SUCCESSION
AXIOMS OF ADDITION
AXIOMS OF MULTIPLICATION
That's all. It is the "Schroedinger" equation of the comp-everything!
The reason is that LRA is already as powerful as a universal machine.
LRA proves all verifiable sentences with the shape ExP(x), with P(x)
decidable. It is equivalent with the universal dovetailer.
Now we have to do with LRA what Everett has done with QM. Embed the
observer in the ontic reality.
For this we have to "modelize" the observer/knower/thinker.
***
2) The epistemic theory, or the generic observer theory: PA (the lobian
machine I will interview).
CLASSICAL LOGIC (first order predicate logic axioms and inference
rules)
AXIOMS OF SUCCESSION
AXIOMS OF ADDITION
AXIOMS OF MULTIPLICATION
AXIOMS OF INDUCTION
Note: the observer extends the ontic reality! It extends it by its
beliefs in the induction axioms. They are as many as they are formula
F(x), and they have the shape:
[F(0) & Ax(F(x) -> F(x+1))] -> AxF(x)
***
OBVIOUS IMPORTANT QUESTION: How to interview PA when we dispose
ontologically only of LRA?
NOT OBVIOUS SOLUTION: just try to obviate the fundamental SEARLE ERROR
(cf Mind's I, Hofstadter -Dennett describe it well) in front of the LRA
theorems.
I explain: Searle's goal consisted in arguing against mechanism, that
is arguing we are not machine, and in particular that a simulator is
not the real thing. He accepts the idea that in principle a program can
simulate a chinese speaker. Knowing the program, Searles accepts he can
simulate it, and this without understanding chinese. He concludes that
we have to distinguish between speaking chinese and simulating speaking
chinese.
True! but with comp you have to distinguish between the simulated
chinese speaker and the simulator of the chinese speaker! By being able
to simulate the chinese speaker, Searle can have a conversation with
the chinese speaker (well assuming that the chinese speaker can talk
english, or that Searle knows chinese).
This is particularly important in our setting. LRA has the power of a
universal turing machine, so it has universal computability power, and
can act as a universal simulator. In particular LRA can simulate PA,
and any recursively enumerable theory/machine. But compute or simulate
are not similar to believing or proving (or talking in some genuine
personal way).
LRA provides a view of computability as a very particular and quasi
debilitating case of provability. LRA can prove almost only the true
Sigma1 sentences (which is enough to run the UD). For example, LRA
cannot prove (about its ontic reality or intended interpretation) that
Ax (x + y = y + x). PA can. PA is already a sort of Ramananujan, a
total genius compared to LRA, despite the fact that LRA is already
universal for computation. For the notion of provability there is no
universal notion. There are as many notion of provability than there
are machines (human included).
If you want, LRA is gifted in proving the existence of number having
verifiable property, and giving that PA can be defined, by Godel
arithmetizability, in the language of LRA , although LRA has no opinion
on the induction axioms (believed by PA), LRA can prove that PA proves
this or that. And with strong-AI (that is Searle's weak Turing notion)
LRA will prove "and Bruno says this or that". In this setting, Searle
would confused LRA and PA, or LRA and Bruno.
Now, I will not interview the brunos, but the PAs, as they have much
less prejudices, if only that. By the "PAs" I mean the many recurring
proofs by LRA that PA proves something (you can see that as the many
simulations of PA by LRA). Well, the main advantage of interviewing PA
is that anyone just a bit reasonable, believes in PA's beliefs.
The PAs, are Löbian. In a nutshell, if LRA is indeed universal, PA is
not only universal, but knows, in some weak sense, that she is
universal.
LRA has universal "existential provability" power about the verifiable
propositions, but LRA has almost no universal "universal provability"
power. LRA's talks go like this: oh a prime number! oh a proof by ZF
that PA is consistent, oh ...", but LRA cannot prove the infinity of
the primes, nor can LRA prove any reasonable result in number theory
begining by a "Ax" (for all x) quantification. The induction axioms, in
which PA believes, provide her with, not only a tremedous
generalization power, but confers to PA the (modal-logically-viewed)
maximal indepassable self-referential power (the one which are
axiomatisable by the modal logics G, G*).
Being universal LRA is under the joug of the incompleteness phenomenon.
But this is not saying a lot given that we already know LRA's
provability power are so weak. PA, being universal too, is obviously
under that joug too, and that is more astonishing a priori, because, as
I said, PA is already quite a genius with tremendous provability power.
Now, by its powerful self-referential knowledge inherited by that
provability power, knowing her universality, PA *knows* that she is
under the joug of the incompleteness phenomenon.
This makes PA, and all her correct (with respect to the usual model
(N,+,x)) recursively enumerable extensions, incredibly modest! (if not
a bit naive). But it makes also PA "aware" of intensional distinctions,
already described in the Theatetus, which provide then the arithmetical
interpretations of Plotinus Hypostases. Including his "Matter"
hypostases, making them comparable with quantum logic/quantum
computations.
OK, that last paragraph was quick. I have to explain more on logic
(godelian, modal, but also the weak logics like intuitionistic logic or
(mainly!) quantum logic, ...). Asap.
Does this help a bit?
Bruno
http://iridia.ulb.ac.be/~marchal/
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