On 11 Mar 2011, at 19:53, Brent Meeker wrote:

## Advertising

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. Acomputation (of phi_4(5) is just a sequencephi_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 isimplicit in any textbook on theoretical computer science, or ofrecursion theory (like books by Cutland, or Rogers, or Boolos andJeffrey, ...).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 ofnumbers <x,y> are code by numbers, and you can mechanically extractx and y from <x,y>Then u is a universal number if for all x and y you have thatphi_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 RobinsonArithmetic prover. I will say that a number u is universal if RAcan prove the (purely arithmetical) relation phi_u(<x,y>) = phi_x(y).The notion is not entirely intrinsic (so to be universal is notlike to be prime), but this is not important because from themachine's point of view, all universal numbers have to be takeninto account.By "not intrinsic" I assume you refer to the fact that the Godelnumbering scheme that idenitfies a number u with a program is notsomething 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 totaken into account"? If I understand correctly *every* naturalnumber 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/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.