Le 14-mai-05, ā 16:04, Lee Corbin a écrit :
If they are furthermore enough rich in complexity to have "abstract inhabitants", it is reasonable or plausible (at least) that for those inhabitants their abstract universe will look as it is real.
This rests on the surprising conclusion that the inhabitants actually compute later states from earlier ones. Of course, it can always be made to *look* as though that is what "happened". Eternal Truth #6: seek and ye shall find.
It is precisely this latter "surprising conclusion" that is resisted by so many (including me). Just as it would seem that anyone should resist making conclusions about the relationships between the books in Borges' Library of Babel.
Borges' Library is different from the UD in this very respect. The UD does not simply generate all programs, but it executes all program. In the platonic sense that it does relate the computational states to each other. Its existence is far from from obvious. It seems easy for a mathematician to refute its existence by simple cantor-like diagonalisation. It takes the genius of Kleene or Post to unravel the paradoxes, to show that the set of programmable functions is close for the diagonal procedure, to show this entails the absoluteness of the computability relation, and the relativeness of any provability relation. It is a whole new mathematical world which has been discovered.
And this will make sense if, furthermore again, their relative abstract
computational continuations have the right measure. And theoretical
computer science can justify the existence of such relative measure.
We may take the books in the Borges library---admittedly in his scenario none of which is infinite---and begin willy-nilly assigning a greater measure to some than to others. It is extremely tempting to assign greater measure to short ones. But in the infinite-string version of Borges' library, Russell Standish (for one) begins by assigning equal measure to each bit string.
I do not assign measure to short strings. Nor to any strings. I just use the Godel-Lob Logic to define the particular case of "measure one" in the language of a (Lobian) Universal Machine. Lobian is equivalent with having enough provability ability. I use a result from Goldblatt which translates Quantum logic in modal logic.
I compute the inverse of Goldblatt transformation on the lobian machine description of that measure one, and I compare with quantum logic. Looks simple but those transformation are not easy to handle and lead to intractable problems and open question.
Compare to QM or even Newton, I agree, the comp-physics is still too young (to say the least), but this only with respect to pure 3-person prediction, let us say the quanta.
But thanks to incompleteness, which in the modal setting is captured by the gap between two amazing modal logics G and G*, I can say that with respect, to the existence of 1 and 3 person, an to the explanation of why quanta and qualia, and why they are different, the comp-physics is already in advance.
Modal logic is just a tool for adding possible nuance to classical logic. My experience is that modal logic is more easy to understand than to understand the difference between atheist and agnostic. Indeed, I am used to explain the difference between atheist and agnostic by using the belief modality. I'm sure you know the difference between atheist and agnostic, but let me explain it with an explicit modality. Let d be the proposition according to which God exists. Let B be the belief modality as applied to someone. And let - represent negation. Then, by definition I would say:
-a religious believer is someone who believes in the existence of God, i.e. about who 'Bd' is true.
-an atheist is someone who believes God does not exist, i.e. about who 'B-d' is true.
-an agnostic is someone who does not believe in the existence of God, and he does not believe in the non existence of God. Both '-Bd' and '-B-d" are true for him (or her, it, ...).
People tends to confuse, and natural language does not help, propositions like -Bd and B-d. Most acts of putting the mind-body under the rug, or using Godel theorem against comp, can be seen at some level of description of the argumentations as error of that kind. That's why the discovery of G and G* by Solovay is a formidable event for simplifying the life of those who want to study the counter-intuitive lesson of the universal machine which introspect itself: what machine can prove and cannot prove but can correctly guess about their consistent extensions, and their geometry. The modal logic G is the study of the Beweisbar Gödel probability predicate Bew(x) which is true if there is a number coding a proof of x.
You should'nt reduce the whole of computer science in the Library of Babel. Machine have dynamics, even if they are on the type discrete, digital, and, when seen in Platonia, looks like static abstractions. Here the movie-graph or Olympia+Klara (Klara = the counterfactual block) makes understand that comp makes it impossible for a machine to distinguish the real and the virtual, but also the virtual and the arithmetical. With comp, machine lives in Platonia, but with Godel, platonia admits internal view which looks quite dynamical. At least that what the machine says, according to the arithmetical translation.