On 20 Apr 2009, at 17:41, Brent Meeker wrote:

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>> >> A computation is a sequence of numbers (or of strings, or of >> combinators, etc.) as resulting by an interpretation. For such an >> interpretation, you don't need a "world", only an "interpreter" that >> is a universal system, like elementary arithmetic for example. > > You put scare quotes around "interpreter". Just because it is not a human interpreter, but a programming language interpreter. I use the term in the computer science sense. > I don't see how arithmetic > is an interpreter - isn't it an interpretation (of Peano's axioms)? Usually I use "Arithmetic" for the (usual) standard interpretation (in the human sense) of arithmetic. By arithmetic I was thinking of a formal system such as the formal system Robinson Arithmetic (or Peano Arithmetic depending on the context). It is not so easy to show that Robinson Arithmetic is a Turing Universal interpreter, but it is standardly done in most good textbook in mathematical logic(*). It is no more extraordinary that the Turing universality of the SK combinators, or the universality of the Conway's game of life, or the universality of any little universal system. > And > how does arithmetic avoid the problem of arbitrarily many mappings, as > raised by Stathis? Once you accept the computationalist hypothesis, not only that problem is not avoided but the problems of the existence of both physical laws and consciousness is entirely reduced to it, or to the digital version (UD) version of that problem. The collection of all computations is a well defined computational object, already existing or defined by a tiny part of Arithmetical Truth, and not depending on the choice of the initial basic formal system. The mapping are well defined though. The way Putnam, Mallah, Chalmers and others put that problem just makes no sense with comp, given that they postulated some primitively material or substantial universe which does not makes any sense (as I have argued already). Then they confuse a computation with a description of a computation. Sometimes they use also the idea that real numbers occurs actually in nature which just add confusion. Now I usually don't insist on that, because, even if such mapping would make sense, it just add computational histories in the universal dovetailing, or in Arithmetic, and this does not change the measure problem. The only important fact here is that with comp, the digitalness makes the measure problem well defined: none mappings are arbitrary: either there is a computation or there is no computation. For example, with numbers and succession (but without addition and multiplication) there is no universal computation, even if there is a sense to say there is all description of computations there. A counting algorithm does not constitute a universal dovetailing. Now, numbers + addition + multiplication, gives universal computations and thus all computations with its typical super-redundancy, and the measure problem makes sense. Ontologically we need no more. Epistemologically we need *much* more, we need something so big that even with the whole "Cantor Paradise" or the whole "Plato Heaven" at our disposition we will not even been able to name what we need (and that is how comp prevents first person reductionism or eliminativism, and how it makes theology needing a scientific endeavor). (with science = hypothetical axiomatics). I agree with Kelly that we don't need a notion of causality, but we need computations (Shannon information measures only a degree of surprise, and consciousness is more general than being surprised, and I agree with you that information is a statical notion). But the notion of computations needs the logical relations existing among numbers, although other basic finite entities can be used in the place of numbers. In all case, the computations exists through the logical relations among those finite entities. We could say that a state A access to a state B if there is a universal machine (a universal number relation) transforming A into B. This works at the ontological level, or for the third person point of view. But if A is a consciousness related state, then to evaluate the probability of personal access to B, you have to take into account *all* computations going from A to B, and thus you have to take into account the infinitely many universal number relations transforming A into B. Most of them are indiscernible by "you" because they differ below "your" substitution level. (*) - Richard Epstein and Walter Carnielli, Computability, computable Functions, Logic, and the Foundations of Mathematics, Wadsworth & Brooks/Cole Mathematics series, Pacific Grove, California, 1989. - Boolos, Burgess and Jeffrey, Computability and Logic, Cambridge University Press, Fourth edition, 2002. 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 -~----------~----~----~----~------~----~------~--~---