🚀 go-pugleaf

Ken Presting writes:

>What makes the multi-body problem a counter-example is not just the fact
>that the problem has no closed-form solution, but the chaotic nature of
>the mechanical system.

Chaos is only a problem if we need to model the behaviour of a particular
system over a particular period of time exactly -- i.e., not just capture
how it might go, but how it *does* go.  This isn't what we're trying to do
in cognitive science, so it's not a problem.  We can model the system to
a finite level of precision, and be confident that what we're missing is
only random "noise."  So while we won't capture the exact behaviour of System X
at 3 p.m. on 12/22/89, we'll generate equally plausible behaviour -- in other
words, how the system *might* have gone, if a few unimportant random 
parameters had been different.

This leads to a point which is a central tenet of functionalism -- you don't
need to capture a system's causal dynamics exactly, but only at a certain level
of abstraction.  Which level of abstraction?  Well, this is usually
specified teleologically, depending on what you're trying to capture.  Usually,
it's a level of abstraction that captures plausible input/output relationships.
Anything below this, we can consider either implementational detail, or noise.
Just what this level of abstraction is, of course, is a matter of some debate.
The most traditional functionalists, including the practitioners of
"symbolic" AI, believe that you may go to a very high level of abstraction
before missing anything important.  The move these days seems to be towards
a much less abstract modelling of causal dynamics, in the belief that what
goes on at a low level (e.g. the neural level) makes a fundamental difference.
(This view is sometimes associated with the name "eliminative materialism",
but it's really just another variety of functionalism.  Even at the neural
level, what we're trying to capture are causal patterns, not substance.)

>What makes the analog causal system different from the algorithm is that
>each state of the analog system encodes an infinite amount of information.

Arguable.  My favourite "definition" of information is due to Bateson, I
think (no endorsement of Bateson's other views implied): "Information is a
difference that makes a difference."  An infinite number of bits may be
required to descibe the state of a system, but in any real-world system, all
of these after a certain point will not make any difference at all, except as
random parameter settings.  (The beauty of Bateson's definition is that the
final "difference" depends on our purposes.  If we wanted a precise simulation
of the universe, these bits would indeed be "information".  If we want a
cognitive model, they're not.)

Incidentally, you can concoct hypothetical analog systems which contain
an infinite amount of information, even in this sense -- by coding up
Chaitin's Omega for instance (and thus being able to solve the Halting
Problem, and be better than any algorithm).  In the real world, quantum
mechanics makes all of this irrelevant, destroying all information beyond
N bits or so.  

Happy Solstice.
--
Dave Chalmers     (dave@cogsci.indiana.edu)      
Concepts and Cognition, Indiana University.
"It is not the least charm of a theory that it is refutable"