On 10 Nov 2009, at 19:29, Brent Meeker wrote:

>>
> But this seems like creating a problem where none existed.  The
> factorial is a certain function, the brain performs a certain  
> function.
> Now you say we will formalize the concept of function in order to  
> study
> what the brain does and perhaps understand what is consciousness.  But
> there is nothing that requires that you start over with all possible
> computations.  That is like explaining the factorial function by
> considering all possible computations that produce it (like the  
> above).
> It's not wrong, but it doesn't follow from saying that the factorial  
> is
> a function.  That's why I say I take it as an ansatz - "Let's consider
> all possible computations and see if we can pick out physics and the
> brain and consciousness from them."


Hmm... The seventh step comes after six other steps. I think you  
confuse UDA and Tegmark or Egan speculation on the mathematical nature  
of physics. But when we assume comp, the physical appearance cannot be  
describe by *any* computation a priori: it *has* to be retrieve from  
all computation. Roughly speaking, if we are universal machine, we do  
belong to an infinity of computation, and matter, or anything below  
our substitution level, has to be described by "all computations". It  
is an open problem if that sum converge toward something we could  
describe by "one" computation. It is the whole point of the reasoning.  
It is "theorem" in the comp theory that matter emerges from all  
computations. From this you can prove that comp implies the non- 
cloning of any piece of matter, like it proves the existence of a  
strong form of indeterminacy, etc.


>>>>
>>> Could you remind me what the phi_i are?  The enumerated partial
>>> functions?
>>
>>
>> The enumerated so called "partial /computable/ functions".
>>
>> To get them, take your favorite universal system (fortran, lisp, c++,
>> java, whatever), write down the grammatically correct description of
>> function (of one argument, say, that is, from N to N). Put those  
>> codes
>> in lexicographical order, and you get the corresponding phi_i: phi_1,
>> phi_2, ..., and their domain W_1, W_2, W_3, ...
>>
>> With Church thesis, all the computable functions (having the domain  
>> N)
>> will belong to that list, but there will be no algorithm capable of
>> telling in advance for any phi_i if it compute partial computable
>> function or a computable functions.
>>
>> Given that this is a key point for everything which will follow, I
>> have to be sure that most people understand exactly why this has to  
>> be so.
>
> Ok, I think I understand.  It's probably not relevant here, but
> physicist usually think of functions which can be computed  
> approximately
> by a uniformly convergent algorithm as computable (e.g. compute pi)  
> but
> I think in the above you mean computable in the Turing sense that the
> computation stops with the answer (e.g. compute pi to 100 decimal
> places).  Right?


Right. There is a vocabulary problem about what is a function, and  
unfortunately english speaker and french speaker have different  
conventions, and sometimes I slip from one to other, and this does not  
help. Usually a function from N to N is supposed to be defined on all  
element of N. And thus a computable function will have an algorithm  
which does stop on all of its input.
But the Kleene diagonalization shows that there is no computable list  
of all computable functions, so if a language is universal, it means  
that the computable functions can only belong to a list of something  
else. That something else are called partial computable function: they  
are allowed to be not necessarily define on some natural number. So to  
get ALL functions, in some computable way, we have to take something  
larger: all partial functions, and to get all execution of all  
algorithm, we will have to dovetail, and from the first person point  
of view, there is an emerging continuum of computations, and it plays  
the role of the background roots of the physical laws, below our  
substitution level.

The physical world is not just a mathematical space among mathematical  
spaces, it is really a sort of summary of the whole border of the  
whole set of "mathematical spaces" as seen by mean universal machines.  
That the laws of physics seems computable is a mystery now.

I am just showing that the comp theory reduces the mind-body problem  
to a pure arithmetical (or combinatorial, ...) body problem.

Bruno


http://iridia.ulb.ac.be/~marchal/




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