On 06 Jan 2009, at 20:18, Brent Meeker wrote:

>
> Stathis Papaioannou wrote:
>> 2009/1/6 Abram Demski <abramdem...@gmail.com>:
>>> Thomas,
>>>
>>> If time is merely an additional space dimension, why do we  
>>> experience
>>> "moving" in it always and only in one direction? Why do we remember
>>> the past and not the future? Could a being move in some spatial
>>> dimension in the same way we move through time, and in doing so  
>>> treat
>>> time more like we treat space? Et cetera.
>>
>> You could model a block universe as a big stack of Life boards, where
>> the time dimension is represented by the spatial displacement between
>> the boards. There's no way the observers in such an arrangement can
>> step out of one board onto another, backwards or out of sequence.  
>> Some
>> would say that the stack of boards does not count as a computation,
>> and others that even if it counts as a computation it doesn't count  
>> as
>> a conscious computation; that to reach such states you need causality
>> and for causality you need fundamentally real time, not block
>> pseudo-time. I don't see any justification for such claims beyond a
>> desire to preserve the magic in the world.
>
> If you don't require causality or something else that provides a  
> continuum
> topology then the boards can be infinitesimally thin and without any  
> intrinsic
> order.  That would mean that a single "board", by itself (a "state"  
> in machine
> terminology) would have to count as a computation.  That's why Bruno  
> insists on
> a digital structure, but even in his model there is the UD running  
> in the
> background and providing an order.


OK.

What remains to be (re)explained, to grasp completely the steps 7 and  
8 of the UDA,  is that the background for a "running UD" does not need  
more than a tiny part of arithmetic (or combinator, ...).

The computation steps of the UD can be defined entirely in arithmetic,  
and this determines the topology and the measure on the "boards" in a  
way where it makes no sense to "change" anything, like you cannot  
change the property of the numbers at will, once you have accepted the  
(usual) definitions.

This is not easy to explain. It is implicit in the proof that the  
recursive or computable functions are representable in (Robinson)  
Arithmetic, or that the recursively enumerable sets are representable  
in (Robinson) Arithmetic, or combinators(*).

Actually the AUDA shows that there will be as many topologies and  
measures than there are types of points of view. Starting from the  
usual, and best known, Gödelian, notion of effective self-reference  
for defining the third point of view, and accepting the Theaetetus'  
definitions of knowledge for defining the other person's point's of  
view, we can extract the logics corresponding to those views, (they  
correspond to the arithmetical "hypostases" in the plotinus paper).  
They determine the corresponding topology and the corresponding  
measure. Then it should be "just" math. I see indeed this as mainly an  
attempt to formulate the mind body problem in math, as it can be done  
when assuming digital mechanism.

AUDA protects explicitly machines against any reductionist conception  
of what a machine, or number (or combinator, ...) can be. This is due  
to the fact that correct universal machines can prove their own  
incompleteness and can distinguish "truth" from their own provability  
predicate. They can known they cannot define or give a name to an  
ultimate notion of "truth", yet they can be guided by it and to it, in  
a non enumerable varieties of ways.


Bruno

(*)This is rather well done in the two following (quite good) books:

George S Boolos, John P. Burgess,  and Richard C. Jeffrey.  
Computability and Logic, fourth edition, 2002, Cambridge University  
Press.

Richard L. Epstein and Walter A. Carnielli. Computability, Computable  
Functions, Logic, and the Foundations of Mathematics.  1989, Wadswoth  
& Brooks/Cole Mathematics Series, Pacific Grove, California.

Those who appreciates the combinators, (or those who dislikes the  
numbers),  can read "How to Mock a Mocking Bird" by Raymond Smullyan,  
who gives a big part of the ideas and technics for representing the  
recursive functions with the combinators S and K.




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




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