On 12 Mar 2011, at 19:17, Brent Meeker wrote:

On 3/12/2011 1:48 AM, Bruno Marchal wrote:

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.

OK.  That's what I thought, but I wanted to be sure.


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.

They are all integers, aren't they? So am I to understand Z, U, and Y as parameters which can be chosen and each choice will pick out a set of numbers, such as the primes, the evens,..., Goldbach pairs?

Yes. Even non negative integers in that particular example.





Then X belongs to W_Nu is equivalent with

So the solutions, X, will be a set of integers which, in our chosen coding scheme, express some recursively enumerable set of numbers?

Yes.
The set of numbers X will constitute such a W_Nu, and is a recursively enumerable set of all solutions of the system of diophantine polynomials for some choice of the parameters Z, U, Y. And all W_i can be retrieved by chosing well Z, U, Y.

A number Nu can be said to be a universal numbers if Nu = ((ZUY)^2 + U)^2 + Y, + the other polynoms, and W_nu is a universal recursively enumerable set (a creative set, a m-complete set or a sigma_1 complete set, those are equivalent notion). There will be an infinity of them, but few numbers, in proportion, will be universal. Like there is an infinity of codes for a LISP interpreter, but few programs are LISP interpreter.

It is hard to believe that a super growing function like y = x^(x^(x^(x^ ...x) with x occurrence of x (which grows quicker than all exponential functions!) can be represented by a polynomial. But this has been proved and is a direct consequence, of course, of the existence of universal diophantine polynomial. There is also parameter Z, U, Y coding the current state of our galaxy, or coding the quantum vacuum ... Those are (sophisticate) example of universal numbers.

Note that the degree of the universal polynomial below is 5^60. We can reduce the degree to four, but then we will need 58 variables.

Note that if the variables are real numbers, we get a decidable, even finite, and thus non universal, set of solutions. The universality comes from the digitality, or from the natural numbers. The natural numbers behave in an infinitely more complex way than the real numbers, and reminds us of the non triviality of what is a computation. That can perhaps be used to mitigate the idea that a rock can think, because I can imagine that point particles (in quantum non relativistic physics for example) can simulate the real-number version of that polynomial, but I am less sure for the diophantine (integers) version.

Note also that, although the first order theory of rational numbers is Turing universal, it is an open problem if there is a universal polynomial on the rational numbers. Another open problem is the existence of a cubic universal diophantine polynomial. We know that there is no quadratic universal polynomial, though.

Bruno



Brent


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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