Hi,
Roadmap:
There are two things I can hardly separate: UDA, that is the Universal
Dovetailer Argument, which is an argument showing that if you take
seriously enough the hypothesis that we are digitalizable machine then
it follows that *necessarily* the physical laws, among more things (see
below), emerge from the relation between numbers, in a verifiable way.
This includes relative relations between particular numbers and what I
will call from now on Universal Numbers.
And UDA points on a way to explicitly extract the physical laws: just
interview one of those universal numbers.
I say above that the physical laws emerges, among more things
Among more? Not only UDA forces the quanta to emerge from numbers, but
the interview shows that they emerge in the company of many non
communicable truth, even many sort of non communicable truth, some of
which are good candidates for playing the role of qualia. In that case
quanta appears to be sharable qualia.
It is in the UDA that I have introduce the first and third person
discussion (the expression itself coming from philosophy of mind). In
the UDA the definition of first person *discourse* is quite specific:
it is the content of a diary, or of any local memorization of a result
of self-localisation (like Washington and Moscow) after
self-duplicating experiments. Note that here some "universe" or
"universal story" is presupposed in the thought experiment. Third
person person discourse are then defined by the content of a diary of
some external observer, where external refer to the
reading-annihilating (cut) and reading reconstituting (copy) pair of
boxes.
The last version of UDA has 8 steps: (CT is for Church's thesis)
1) classical teleportation (Here the 1-discourse is equivalent with the
3-discourse, up to the use of the indexical "I" by the candidate)
2) Classical teleportation with delay (Here already the 1 and 3
discourses diverge)
3) Self-duplication (Here the first person indeterminacy appears)
4) Self-duplication with asymmetric delay.
5) Teleportation without annihilation of the "original"
6) Virtual 1+2+3+4+5 (the preceding steps with virtual reconstitutions)
7) Universal 1+2+3+4+5 (Here appears the universal Machine and the
Universal Dovetailer. CT is used)
8) Arithmetical 1+2+3+4+5 (Here the UD appears to be "just" Sigma1
truth).
It follows from that that "ontologically" (perhaps in a weaker sense
than usual, or at least in a weaker sense than Peter D. Jones' use of
the term) the following theory is enough. It is often named Robinson
Arithmetic and denoted by Q. It is a tiny segment of elementary
arithmetic. As a theory, it is a very poor theory, completely incapable
of making the slightest generalization. But yet able to prove all the
sigma1 truth, which by Church thesis, makes it capable of representing
all computable functions, that is the Fi and the Wi, etc. Put in
another way, although Q has few provability abilities, it has the full
universal power of a universal turing machine or of any (relative)
universal number.
So:
TOE 1 (Ontology, formally presented)
Classical first order logic with equality, together with the symbol
0, s, +, *
The intended meanings of 0, s, + and * are respectively the number
zero, the successor function (which sends a number n on its successor
n+1), and addition and multiplication.
The axioms are: (where "Ax" should be read: "for any number x", and
"Ex" should be read "it exists x such that ", and "x y" abbreviates
~(x = y))
Q1) Ax 0 s(x)
Q2) AxAy x y -> s(x) s(y)
Q3) Ax (x 0 -> Ey(x = s(y))
Together with the definition of addition:
Q4) Ax x + 0 = x
Q5) AxAy x + s(y) = s(x + y)
and the definition of multiplication:
Q6) Ax x * 0 = 0
Q7) AxAy x * s(y) = (x * y) + x
Just remember that the roadmap point: UDA shows that "ontologically"
Q1+Q2+Q3+Q4+Q5+Q6+Q7 is enough for a (comp) TOE. As a machine, which
dynamics is governed by the inference rules of logic (modus ponens,
rule for the quantifiers), Q is recursively isomorphic to a UD. By its
Church-Turing completeness, it is universal for computability.
KEY REMARK: although computability-complete, the theory Q is very weak
in term of provability power. It cannot even prove that any numbers has
a successor (Ax Ey (y = s(x)), although Q can prove all individual
statements Ey(y = s(0)), Ey(y=s(s(0))), Ey(y=s(s(s(0))), etc. Q can
prove (modulo the notation) that 1+3 = 2*2, that 1+3+5 = 3*3, that
1+3+5+7 = 4*4, etc., but Q cannot, like the greeks, generalizes and
concludes that the sum of any sum of the first odd positive integers
are perfect squares. Q can prove, for each number n and m, that n+m =
m+n, but cannot infer that (n + m = m + n) is true for all n. Still, Q
has the full power of being able to prove ExP(x) in case P is decidable
and true, and this makes Q sufficiently powerful to prove
metamathematical statements like the theory "ZF can prove the
consistency of me (Q)". Here we are invited to NOT DOING Searle's
error, consisting in confusing computation (simulation, emulation) and
provability (or any higher belief system notion). In a sense Q can
imititate ZF proving Q's consistency, but Q can hardly believe (prove)
ZF's consistency, and Q can certainly not infer her consistency from
ZF's proof of it. Give me time and Einstein's brain, and I can emulate
Einstein's opinion about the loop-gravity/string theory debate, but
this will not entail I will understand it. It is capital to understand
this, because it will explain eventually how from a little ontology (Q
theory), we will get an infinitely rich and unbounded epistemology
"from inside Q" or from inside the work of a UD.
TOE 2 (Epistemology, will permit the "universal interview" which will
include "physics" and in fact all Plotinus' hypostases (n-person point
of view).
An observer will be any consistent extensions of Q. Peano Arithmetic is
probably the most celebrate one, like ZF, or axiomatizable second order
arithmetic (Analysis), same+omega-rule (as example of angels), or
("gods") the set of all statement of arithmetic (written in ZF) which
are true in all transitive models of ZF, etc.
I will fix the idea by choosing Peano Arithmetic (PA) for the begining
of the universal interview. Q, as the UD, generate (in a precise
arithmetical sense) all the proofs, made by all the consistent
extensions of itself, and we will keep an eye on what PA says about
itself and its consistent extensions. This will be equivalent with the
study of what a self-referentially correct machine can prove about
itself.
Formally PA is just Q1+Q2+Q3+Q4+Q5+Q6+Q7 together with an infinity (but
mechanically generable) of induction axioms:
For any predicate P definable in Q (or PA, they have the same
language), PA believes that (think about an infinite range of
dominoes):
[ if P(0) and Ax (P(x) -> P(s(x)) ] -> AxP(x)
This gives to PA a tremendous ability to generalize. It is not even
clear if not all current mathematic (except logician's set theories and
categories) can be proved in PA. And of course PA is lobian, by which I
mean PA's provability are captured by the modal logics G and G*.
Suddenly grand-mother stops me: Bruno, you get too much technical ...
keep the hypostases for another posts, it will be too much ...
OK, it was an attempt. Please keep this post for letter references. But
the extraction of physics (and the other hypostases/persons) is
entirely based on the arithmetical self-reference logics, and I can
hardly explain this without introducing the basic notions. Recall that
once we have a modal logic we can use some canonical multiverse
attached to it.
But don't hesitate to comment. Nevertheless in the next one we get the
n-person arithmetical notions.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Sat Aug 12 2006 - 10:25:16 PDT