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, 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/




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