On 11 Mar 2011, at 19:53, Brent Meeker wrote:
On 3/9/2011 9:14 AM, Bruno Marchal wrote:
On 08 Mar 2011, at 20:11, Brent Meeker wrote:
On 3/8/2011 10:41 AM, Bruno Marchal wrote:
We could start with lambda terms, or combinators instead. A
computation (of phi_4(5) is just a sequence
phi_4^0 (5) phi_4^1 (5) phi_4^2 (5) phi_4^3 (5) phi_4^4 (5)
phi_4^5 (5) phi_4^6 (5) ...
_4 is the program. 5 is the input. ^i is the ith step (i = 0, 1,
2, ...), and phi refers implicitly to a universal number.
Bruno, I don't think I understand what a universal number is.
Could you point me to an explication.
The expression "universal numbers" is mine, but the idea is
implicit in any textbook on theoretical computer science, or of
recursion theory (like books by Cutland, or Rogers, or Boolos and
Jeffrey, ...).
Fix any universal system, for example numbers+addition
+multiplication, or LISP programs.
You can enumerate the programs:
P_0, P_1, P_2, ...
So that you can enumerate the corresponding phi_i
phi_0, phi_1, phi_2, ...
Take a computable bijection between NXN and N, so that couples of
numbers <x,y> are code by numbers, and you can mechanically extract
x and y from <x,y>
Then u is a universal number if for all x and y you have that
phi_u(<x,y>) = phi_x(y).
In practice x is called program, and y is called the input.
Now, I use, as fixed initial universal system, a Robinson
Arithmetic prover. I will say that a number u is universal if RA
can prove the (purely arithmetical) relation phi_u(<x,y>) = phi_x(y).
The notion is not entirely intrinsic (so to be universal is not
like to be prime), but this is not important because from the
machine's point of view, all universal numbers have to be taken
into account.
By "not intrinsic" I assume you refer to the fact that the Godel
numbering scheme that idenitfies a number u with a program is not
something intrinsic to arithmetic; it is a mapping we create. Right?
Partially right. If you start from Robinson arithmetic, each universal
numbers will have many syntactical variants due to the provable
existence of different coding scheme. This does play a role in the
measure problem a priori. But this is true for your brain too (clearly
so with comp).
But then what does it mean to say "all universal numbers have to
taken into account"? If I understand correctly *every* natural
number is a universal number, in some numbering scheme or another.
So what does "taking into account" mean?
Not every natural number will be universal. Once you have choose your
initial system, only a sparse set of numbers will be universal, and
the computations will rely on the relations between the numbers. You
are not supposed to change the initial system. The laws of mind and
the laws of matter does not depend on the choice of the theory
(initial system), but you cannot change the theory, once working in
the theory. To say that all universal numbers have to be taken into
account means that you have to take into account all the relations
which constitute computations in the initial theory. The observer
cannot distinguish them from its first person point of view. The
measure does not depend on the codings, but does depend on the
existence of infinitely many coding schemes.
Let me be completely specific, and instead of using RA, let us use a
universal system of diophantine polynomials:
The phi_i are the partial computable functions. The W_i are their
domain, and constitutes the recursively enumerable sets.
The Turing universal number relation (X is in W_Nu) can be coded as a
system of diophantine polynomials(*). I will say that a number Nu is
universal if W_Nu is the graph of a universal computable function.
The detail of the polynomials is not important. I want just to show
that such numbers are very specific. Then you can represent
computations by their recursively enumerable sequence of steps, etc.
Löbian machine can likewise be represented by their set of beliefs,
themselves represented by RE set W_i, etc. The same is true for
"interaction" between Löbian machine and environments (computable or
computable with oracle), etc.
For any choice of representation, the number relation organizes itself
into a well defined mathematical structure with the same internal
hypostases.
Here is a universal system of diophantine polynomial, taken in a paper
by James Jones
James P. Jones, Universal Diophantine Equation, The Journal of
Symbolic Logic, Vol. 47, No. 3. (Sept. 1982), pp. 549-571.
To define universal numbers here is very short. To do the same with RA
would be much longer and tedious, but the principle is the same. Once
the initial universal system is chosen, there is no more arbitrariness
in the notion of universal numbers, computations, etc. Except for
different internal codings, but this is the price for comp. I can
change your brain in many ways, but I have to keep the relations
between your constituents intact, making your brain a non relative
notion. A brain is not intrinsic, but is not arbitrary either.
In the universal system of diophantine polynomials below, there is an
infinity of number Nu coding the primes numbers, or the even numbers,
or the universal numbers, etc. If all numbers code precise things, it
is false that precise things can be coded by arbitrary numbers.
(*) (Numbers are named by single capital letters, or by single capital
letters + one non capital letter. In the first line Nu, Z, U, Y, are
integers. Then X belongs to W_Nu is equivalent with
Nu = ((ZUY)^2 + U)^2 + Y
ELG^2 + Al = (B - XY)Q^2
Qu = B^(5^60)
La + Qu^4 = 1 + LaB^5
Th + 2Z = B^5
L = U + TTh
E = Y + MTh
N = Q^16
R = [G + EQ^3 + LQ^5 + (2(E - ZLa)(1 + XB^5 + G)^4 + LaB^5 + +
LaB^5Q^4)Q^4](N^2 -N)
+ [Q^3 -BL + L + ThLaQ^3 + (B^5 - 2)Q^5] (N^2 - 1)
P = 2W(S^2)(R^2)N^2
(P^2)K^2 - K^2 + 1 = Ta^2
4(c - KSN^2)^2 + Et = K^2
K = R + 1 + HP - H
A = (WN^2 + 1)RSN^2
C = 2R + 1 Ph
D = BW + CA -2C + 4AGa -5Ga
D^2 = (A^2 - 1)C^2 + 1
F^2 = (A^2 - 1)(I^2)C^4 + 1
(D + OF)^2 = ((A + F^2(D^2 - A^2))^2 - 1)(2R + 1 + JC)^2 + 1
Z, U, Y codes just the parameter Nu. The other "letters" are the
variables.
We could transform this into one giant universal polynomial, and we
could manage its degree to be equal to 4. But that would take a lot of
place and variables. The solutions of that equations provide a
universal dovetailing, notably because computations can be seen as RE
relations. For further reference, I will call that system JJ (for
James Jones). JJ, like RA, is an acceptable theory of everything, at
the ontological level.
Only the left (terrestrial) hypostases can be named in that system,
the right hypostases, the divine one, are, in the general case purely
epistemological and cannot be named in the system, like the universal
soul itself S4Grz. Machines can point on it only indirectly, but can
study, nevertheless, the complete theory for simpler Löbian machines
than themselves. So JJ provides the ontology, but the internal
epistemology necessarily escapes the ontological TOE (whatever the
initial theory choice). This explain that you can *see* consciousness
there, no more than you can see consciousness by dissecting a brain.
I recall the 8 hypostases (three of them splits according to the G/G*
splitting)
p
Bp
Bp & p
Bp & Dp
Bp a Dp & p
The "left one" are the one ruled by G, and the right one are the one
ruled by G*. Which corresponds the terrestrial and the divine
respectively in the arithmetical interpretation of Plotinus. Those
with (& p) are not solution of the JJ equations, nor any of them when
extended in first order modal logic. Philosophy of mind, with comp, is
intrinsically very complex mathematics. The contrary would have been
astonishing, to be sure.
To sum up: don't confuse the flexibility of the codings with the non
arbitrariness of the relations we obtain between those numbers in any
initial theory chosen.
Note that JJ is simpler than RA. It has fewer assumption. It does not
need logics, but only equality axioms, and a the existential
quantifier (or the distinction variable/parameter). It is the
equivalent of the other "everthing ontological theory" based on the
combinators:
((K x) y) = x
(((S x) y) z) = ((x z) (y z))
Which is quite much readable and shorter! In that theory you might
understand that there will be an infinity of universal combinators
(which will be very long!), but to be universal will not be an
arbitrary notion. Non-intrinsic does not mean arbitrary.
Bruno
http://iridia.ulb.ac.be/~marchal/
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