On 10 Jan 2011, at 00:16, Brian Tenneson wrote:

Those are some very interesting results, I must say.

Thanks.



Why the letter choice of B?

B is for Gödel's arithmetical provability predicate 'Beweisbar'. It is applied to a number which is coding or expressing a statement.



Does it "secretly" stand for "believes" or something else?

It is believability, by any (correct) machine believing in the recursive definition of addition and multiplication (and in not too much more), and is such that if she believes soon or later A, and believes soon or later A -> B, then she will believe B, soon or later.

Solovay theorem shows that the modal logic G and G* solves the provability issue, of those machines/theories. At the propositional level. G and G* asserts which statements are provable (and true) or true but unprovable by the machine. G is mainly axiomatized by Lôb formula B(Bp -> p) -> Bp which explains my Löbian machine terminology. Löb formula expresses a form of extreme modesty. If you replace p by the constant false, f, you get Gödel incompleteness theorem: ~Bf (consistency) -> ~B(~Bf). The consistency of the machine entails the unprovability of the self-consistency by the machine. I hope you see that B(Bf -> f) -> Bf is equivalent with Dt -> ~BDt, or that Dt -> DBf. D is ~B ~. D is for Diamond (he modal dual of B, for the modal box. The modal box B of G (and G*) captures completely the logic of Beweisbar B. G capture the part provable by the machine. And G* captures the truth, like Dt, DDt, DBf, etc.




If x is true then ~Bx seems to imply that if a hypothesis is true then science will never prove it no matter how much time you give them (barring any sort of infinite time). Is that right?

Science never proves. It asserts hypothesis, like

x + 0 = x
x + s(y) = s(x + y)

and common inference rules.

Science proves things in theories. My theory is the two axioms above +
x * 0 = 0
x*s(y) = x*y + x

and the inside epistemology is given by those four same axioms together with the induction principle on expressible formula (leading to being Löbian).





I've had some exposure to Alan Watts and all I have seen is both profound and simple.

Wow. Well said.



DDDDDt is a bit hard for me to understand. Would you elaborate for me?

D is ~B ~
Dt is for ~Beweisbar('~t'), that is ~Beweisbar('f'), that is the non provability of the (Gödel number representing the) false, that is (self) consistency, as expressed by the machine. It is third person self-reference. DDDDDt is the consistency of the consistency of the consistency of the consistency of the consistency of t. It is far stronger than just consistency. It is a typical truth (provable by G*), but unprovable by the machine. DDt <-> Dt, but the machine cannot see (rationaly believe) that Dt -> DDt.

By Gödel's completeness (not: incompleteness!) we have that consistency (Dt) is equivalent with the existence of a model. Dt is "invention of reality" for a machine. The machine bets instinctively "she does not believe in BS (f, the vulgar name of falsities, like 0 = 1.)".

You get the arithmetical hypostases by defining new "predicate" connecting believability with truth (first person move) and consistency (material move), eventually iterated. The logic of Bp & p, with p arithmetical gives a non arithmetical knowability or first person view of the machine. You bypass "Tarski theorem" (the non definability of arithmetical truth in arithmetic) by "modeling" the non arithmetical "true('p') by using p itself. That's the very essence of the idea of Theaetetus, and somehow a consequence of the dream argument in metaphysics. Tarski's theory of truth can be relied to this. ##Y*%° is true if it happens that ##Y*%°, or if it is the case that ##Y*%°.

The machine cannot prove p <-> Bp, but the machine can prove Bp -> p each and at only time she can prove p, (Löb's theorem); and she can prove p -> Bp for all sigma_1 formula. 'Bp' itself is sigma_1 so that she can prove Bp -> BBp, or dually DDp -> Dp. That is what provides the needed introspection to know that she is universal and that she is incomplete and 'uncompletable', as far as consistent, and that she can crash anytime, and she has already many questions. She can even make money (prove new true theorems) with consistent, but false, proposition (Shit (Bf) can happen: G* proves DBf).

I identify a theory with a (chatty) machine, or even a set of beliefs (closed for the inference rules). I study what ideally self- referentially correct universal machine can believe about themselves (Bp) and reality (Dt) and other realities (DBf, DDt, etc.).

The logics of Bp & Dt, and Bp & Dt & p, the 'material hypostases', give clues on the way universal machines can succeed in dream sharing and building/filtering realities.

Good books are by Boolos (1979, 1993) and Smorynski (1985). Smullyan wrote a recreative introduction to the logic G (Forever Undecided, 1987), where 'B' is for believe (by rational believers) (and 'know' is Bp & p, informally).

In that 'theology', somehow GOD is already overwhelmed by the DIVINE INTELLECT which is completely overwhelmed by the UNIVERSAL SOUL which is completely overwhelmed by the material (intelligible and sensible) realities. This appears in the first order modal extension of G and G*. Arithmetical truth viewed from inside by universal machines is FAR bigger than Arithmetical truth.

Bruno




Bruno Marchal wrote:


On 05 Jan 2011, at 21:45, Brian Tenneson wrote:

"The Tao that can be described is not the ultimate Tao"

I <3 Lao tseu, and all the taoists. There is a full chapter on Lao- tseu in the long version of my PhD(*). They were aware of the dream argument. In fact, I call the modal formula:

x -> ~B x (or its contrapositive Bx -> ~x) with B intended for the scientific communication or proof: the Lao-Tseu Watts Valadier principle, there.

Alan Watts describes indeed something similar in his book "the wisdom of insecurity", and Valadier, a french jesuit, wrote a remarkable book where it shows that making moral is immoral.

Gödel's theorem (and Löb, Solovay) provides many solutions, having an arithmetical content, for such an equation. Indeed all x belonging to G* \ G obeys to that equation.

With x = Dt (= ~Bf = consistency), you get Gödel's second incompleteness theorem: Dt -> ~BDt. But DDDDDt is also a solution. Most formula beginning by D (= ~B ~) are solutions. Correct machines cannot prove that they cannot prove something.

Tarski's theorem provides even more insightful solutions, which are analytical, and on which the correct machine can only be mute.

It led me also to a very simple theory of intelligence. A machine is intelligent if she is not stupid, and a machine is stupid if either she believes that she is intelligent, or she believes she is stupid. Aagain incompleteness provides solution. From that I showed that intelligence has a positive feedback on competence, but that competence has a negative feedback on intelligence.




Interesting. I wonder if it's so. Whether or not the ultimate Tao can be described has been the object of all my research-related thinking for a while now. I finally made a breakthrough this year on the problem. I still have to manipulate what I think on it and massage the document about it. At least I can say that I'm not trying to describe the Tao. I'm trying to describe a description of the Tao. The reduced product of all structures is my candidate for my description for a description of the Tao.

Perhaps Lao Tzu already put in his two cents regarding this kind of TOE.

In "conscience and mechanism" I argue in detail that most of the writing of Lao-Tseu, Tchouang-Tseu, and especially (my favorite) Lie-Tseu can be interpreted by the discourse of the self- referentially correct machine. But Plotinus is closer to us. I have studied classical chinese and modern chinese, for years, to discuss on Lao-Tseu with scholars. It is difficult.

Mechanism makes a bridge between Smullyan's "Tao is silent" and Smullyan's "Forever undecided". I still don't know if Smullyan would agree on this. Some remark by him makes me think he is not aware of that connection, or that mechanism favors that connection.

If you like Lao-Tseu, you might appreciate Smullyan's book "tao is silent".

Bruno

(*) http://iridia.ulb.ac.be/~marchal/bxlthesis/consciencemecanisme.html


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