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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Received on Tue Apr 21 2009 - 17:31:57 PDT