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

The question was whether information was enough, or whether something 
else is needed for consciousness.  I think that sequence is needed, 
which we experience as the passage of time.  When you speak of 
computations "going from A to B" do you suppose that this provides the 
sequence?  In other words are the states of consciousness necessarily 
computed in the same order  as they are experienced or is the order 
something intrinsic to the information in the states (i.e. like 
Stathis'es observer moments which can be shuffled into any order without 
changing the experience they instantiate).

A related question in my mind has to do with reversibility.  
Computations in general are not reversible: Turing machines erase 
symbols. You can't infer the factors from the product.  But QM (without 
collapse) is unitary and reversible in principle (though not in practice 
because of statistical and light-speed reasons).  So my question is, are 
the computations of the UD reversible? 

> 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.

Does the UD have to complete the infinitely many computations from A to 
B, i.e. we must think of these computations as being complete in Plationia?


> (*)
> - 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
> >

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