Attempting an understanding of John Collier's posting given below --
He mapped systems with Atlan's "infinite sophistication" (which cannot be
modeled solely from the properties of their components) to computational
systems that do not halt given all relevant inputs, and so are not, in his
terminology, "globally mechanical", as well as to Rosen's systems that do
not have "synthetic models" (i.e., model and system do not 'commute').
Such systems (e.g., living systems) can, however, be "locally mechanical"
and support (Rosen's) 'analytical models'. That is, some of their
functions can be given limited logical representations, one or a few at a
time. Thus, there can be no global model of a cell or an organism -- and,
I would add, of any dissipative structure (i.e., complex, organized
material systems). These have infinite sophistication just from the fact
that they can be classified as having scalar hierarchical structure -- that
is, they are susceptible to being classified as compositional hierarchies,
as in [context [whole [part]]]. A whole is 'emergent' from cohesion of its
parts in a given context.
Biological functions and the traits entrained by them are the products of
biological inquiry. Gell-Mann's and J. Crutchfield's 'effective' or
'structural' complexity is estimated by the length of a list of such
functions, each of which could be represented by an analytical model which
is stepwise mechanical. The fact of infinite sophistication is reflected
as what I call (borrowing a term from Jerry Chandler) 'perplex complexity',
the aspect of complexity that makes it resistant to prediction.
STAN
------------------------------------------------
John said:
Pedro has pointed out a real problem, I think. I have a few words to say on
it that may be of some help in sorting out the issues. They derive partly
from my trying to make sense of Atlan's use of computational language along
with his claim that some biological (biochemical really) structures have
"infinite sophistication". A structure with infinite sophistication cannot
be computed from the properties of its components. Sophistication, as far
as I can tell, is a measure of computational depth, which depends on the
minimal number of computational steps to produce the surface structure from
the maximally compressed form (Charles Bennett). Atlan has made the
connection, but also noted it is not fully clear as yet, since Bennett's
measure is purely in terms of computational steps, and is relative to
maximal compression, not components. Cliff Hooker and I noted these
problems (before we knew of Atlan's work -- well, I did, but it was
presented poorly by one of his students -- see Complexly Organized
Dynamical Systems, Open Systems and Information Dynamics, 6 (1999):
241-302. You can find it at
http://www.Newcastle.edu.au/centre/casrg/publications/Cods.pdf).
The
question relevant to Pedro's post is why is computation relevant if common
biological systems have infinite sophistication, and thus are not
effectively computable, even if they have finite complexity?
Here is my stab at an answer: the notion of mechanical since Goedel and
Turing (I would say since Lowenheim-Skolem, since Turing's and Goedel's
results are implicit in their theorems) breaks up into two notions,
stepwise mechanical and globally mechanical.
A globally mechanical system can be represented by an algorithm that halts
on all relevant inputs (Knuth algorithm); these are computable globally.
The stepwise ones have no global solution that is effectively computable,
but are computable locally (to an arbitrarily high degree of accuracy).
The difference is similar to that between a Turing machine that halts on
all relevant inputs and one that does not. Both are machines, but only the
latter corresponds to Rosen's restricted notion of mechanical. So
computation theory can help us to understand the difference between things
that are stepwise mechanical, and things that are not. Things of infinite
sophistication are not globally mechanical. I will say without proving that
they correspond to Rosen's systems that have analytical models but no
synthetic models. They may still be mechanical in the weaker sense. In fact
I have not been able to see how they cannot be mechanical in this way.
Consequently, there are Turing machines that are mathematically equivalent
to systems of infinite sophistication, but they do not halt.
So you are probably wondering how processes of this sort can occur in
finite time. The answer is dissipation. I'll not give the solution here, as
my coauthor on another paper just came into the room and asked me how it
was going, and I said I was writing something else that was peripherally
relevant :-) A case in point is given in my commentary on Ross and Spurrett
in Behavioral and Brain Sciences titled Reduction, Supervenience, and
Physical Emergence, BBS, 27:5, pp 629-630. It is available at
http://www.nu.ac.za/undphil/collier/papers/Commentary%20on%20Don%20Ross.htm
as well as the BBS site.
All spontaneously self-organizing systems (see the Collier and Hooker CODS
piece) are only locally mechanical. I won't prove that here, but there is a
clue in the BBS commentary.
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Received on Fri Nov 24 11:27:47 2006
