Dynamics and calculus

[From Rick Marken (931214.1530)]

Martin Taylor (931214 14:20)--

Me:

In your posts on this subject you mentioned "attractors" and
"attractor basins"

Martin:

Yep. Follows from the above.[discussion of the calculus]

Attractors follow from calculus? Do the orbits of the planets
follow from the calculus?

Me:

This
"dynamical" approach to living systems is nearly the opposite of
the approach taken by perceptual control theory;

Martin:

I would be happier with saying that it is nearly orthogonal to the
approach taken by PCT. I don't see an opposition.

There is certainly no opposition if "dynamical" means "calculus".
But I don't understand how you could say that there is no opposition
between the "non-calculus" dynamical approach that I was referring to
above and PCT. The "non-calculus" dynamical approach with which I
am familiar (and that Tom Bourbon [931214.1217] gave some examples
of) is a MODEL of behavior -- its not the calculus.

I'm not going to reject out of hand the notion that the brain has a
somewhat modular hierarchy, involving high-level control systems that
are in a less-than-fully-connected structure.

There is a difference between "rejecting out of hand" and "not
including in the model because there is nothing to motivate its
inclusion". Just because we don't include something in PCT doesn't
mean we reject it. We put things in the model to make it behave right;
not because doing so is trendy and cool. I think that when you stay
close to the phenomena you are modelling you tend to be a LOT more
parsimonious in your thinking about what's really REQUIRED in a
model.

Accept that "calculus can be usefully applied to the study of behavior"
and you can't reject the dynamics that follows.

Of course; but its just a tautology if dynamics is calculus; you're
just saying "accept that calculus applies to behavior and you
can't reject that calculus applies to behavior". I have a suspicion
that you are not clearly making a distinction between the logical
system called "calculus" and the functional assumptions that
constitute a model of a system (functional assumptions -- like
f = ma -- that can be represented and manipulated using the
"calculus" but which are just assumptions[it might have been better
to assume f = m+a]).

Tom Bourbon (931214.1217) --

On my side, I am still trying to learn how Martin, and others, conceive of
the relationships among PCT, information theory and the study of nonlinear
dynamical systems.

Martin Taylor (931214 16:00) --

I hope my reply to Rick this afternoon has cleared up the
PCT<->dynamics link.

Not for me, no. I think what might help me is an example of a
dynamic analysis of behavior. If the result of that analysis
is to show that, say, behavior descibed as x^2 is equivalent
to the integral of behavior that is described as x then I have
no problem -- "dynamics" is just calculus. But if the analysis
includes assumptions about the functional relations that resulted
in the behavior described as x^2 then I will be able to test whether
those assumptions produce real behavior (control) and that model
just might be wrong (unlike calculus).

How about showing us what you consider to be a useful dynamical
analysis of some aspect of human behavior, equations and all.
Or, better, write a little program for a dynamical system of the
sort you think might be useful. Of course, this isn't necessary
if all you mean by the dynamical approach is "calculus" (I think
there are lots of texts on this kind of dynamical approach in our
library).

Best

Rick