On 19 Dec 2013, at 22:46, Jason Resch wrote:



> 8. There is no need to build the computer in step 7, since the executions of all programs exist within the relations between large numbers.

That would only be true if everything that could exist does exist, and maybe that's the way things are but it is not obviously true.

It doesn't require that everything to exist, it requires only one particular program to exist: the universal dovetailer. This program and its execution exist within mathematics.

Yes, even in arithmetic, and under different important forms. Its many descriptions exist, and the computation are "truly" emulated in the truth referred by the theorems concerning those description. That is a point which met some difficulties for non-logician, as it is impossible to ever point a computation, without mentioning a description of it. The computation itself is captured by the truth of certain arithmetical statements, not by the existence of a description of those computations. The nuance is subtle, because we infer the existence of the computation by looking at the existence of some description of them, and to show that this is equivalent is by no means a trivial affair, linking the syntax of the theory and its intended meaning (and that is why we need AR). There is a need to really study how simple theories (like RA) can represent in some strong sense the partial recursive function. It is well done in Boolos and Jeffrey, or in Epstein & Carnielli. The whole difficulty of step 8 is in this paragraph. Those who believe that a filmed boolean graph can be thinking commit a confusion between use and mention (like I have just described).


For example, it is a true statement that the state of this program after the 10^100th step of its computation has some particular value X, and it is also a true statement that the 10^100 + 1 step has some other particular value Y. It is also a true statement that the program corresponding to the emulation of the wave function for the Milky Way Galaxy contains John Clark and this particular John Clark believes he is conscious and alive and sitting in front of a computer in a physical universe.

OK.




> Hence, arithmetical realism is a candidate TOE.

A candidate certainly, but is it the real deal? Maybe but it's not obvious.

Right, but it is a scientific question. It will not be easy but we can refute or confirm the theory by seeing what the UD implies for the physics that observers see. Everett's theory was a great confirmation, for without it, conventional QM with collapse (and a single universe) would have ruled it out. As it stands, there are several physical concepts that provide support for the UD being a valid TOE:

Quantum uncertainty
Non clonability of matter
Determinism in physical laws

Actually, this one is the problem. There might still be a too big FPI, like with the "white rabbits".




Information as a fundamental "physical" quantity

Yes, and even obeying different "information laws" above and below the substitution level. And this is confirmed by the difference between quantum information and classical (Shannon) information theory.


(I think there is something I am forgetting, but Bruno can fill in the gaps)

May be after more coffee :)

What we need to do, or at least what mathematicians should do is to compare the empiric quantum logics with the quantum logics provided (by Goldblatt's result) on the (three) arithmetical quantum logic provided by the arithmetical quantizations (S4Grz1, Z1*, X1*). It fits up to now, but the program I wrote to test this should be optimized.
We can come back on this.

Bruno




http://iridia.ulb.ac.be/~marchal/



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