Calculus, si, "dynamical systems", no!

CSGnet Archive CSG1993
1

[From Rick Marken (931214.0800)]

Me:

What would make you think that
dynamical mathematics would no longer apply?

Martin Taylor (931213 12:50) --

I've been arguing that they [control systems] ARE physically
realizable, and that the mathematics of dynamical systems DOES apply.
Every time I do, I get shot down, or at least shot at.

I think of calculus as the "mathematics of dynamical systems".
Are you saying that there has been a gross misunderstanding --
that all you are saying is that calculus applies to control
system behavior?

Me:

Aren't the variables
involved in perceptual control changing over time? Who disagrees with
the idea that a lot (all?) of dynamical mathematics is useful to to
PCT; how else would you mathematically describe a dynamic phenomenon?

Martin:

Do you, perhaps, own a mirror?

So you think I have been disagreeing with your proposal that calculus
applies to the behavior of control systems? I don't disagree with
this. Let me be very clear: calculus applies to the analysis of
control systems.

In your posts on this subject you mentioned "attractors" and
"attractor basins" which led me to believe that you were talking
about an approach to understanding purposeful systems based on
the idea that their "apparent" goals can be viewed as the "equilibrium
states" of "dynamical", "self- organizing" "complex" systems.
This "complex" or "dynamical" systems approach to understand-
ing living systems has become quite popular in some circles (the
Artificial Life Institute in Santa Fe, for example, is dedicated
to understanding living systems from this point of view). This
"dynamical" approach to living systems is nearly the opposite of
the approach taken by perceptual control theory; this is what I
thought you were referring to when you talked about the potential value
of applying dynamics to behavior: I though you were talking about these
kinds of "dynamical" models -- not calculus. Of course, these
"dynamical" models are completely useless and misleading (as far as
purposeful behavior is concerned) so it was probably silly of me to
think that you were claiming that such models could have much value
for the study of behavior. I should have known that you meant that
"calculus can be usefully applied to the study of behavior" -- a
statement with which I heartily agree.

Best

Rick

2

[Martin Taylor 931214 14:20]
(Rick Marken 931214.0800)

So you think I have been disagreeing with your proposal that calculus
applies to the behavior of control systems? I don't disagree with
this. Let me be very clear: calculus applies to the analysis of
control systems.

So far, so good. Nice to know.

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

Yep. Follows from the above.

which led me to believe that you were talking
about an approach to understanding purposeful systems based on
the idea that their "apparent" goals can be viewed as the "equilibrium
states" of "dynamical", "self- organizing" "complex" systems.

Which doesn't follow from the above. It follows from your having read
the kind of thing Tom was quoting, from which I dissociated myself.

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

I would be happier with saying that it is nearly orthogonal to the
approach taken by PCT. I don't see an opposition. The attractor systems
they discover may very well be there in the dynamical interactions among
different control systems. That says nothing about why the attractor
systems might exist, which is what PCT is about. My argument in Durango
was that interacting complexes of control system will reorganize to
avoid positive feedback being experienced by any of them, so far as
each can do. The dynamical system that consists of these many individual
control systems will manifest attractor basins. We call the attractors
themselves "the language" in which the individual control systems
communicate.

Read between the lines of the above. What I'm saying is that the calculus
of the control systems may not be explicitly soluble in such complex
systems, but it is normal for dynamical systems to settle toward attractors
if such exist (that's the definition of an attractor, after all). PCT
has a central construct called "reorganization." Reorganization serves
the control system by allowing the control of arbitrary percepts to effect
the control of intrinsic variables. It is a stabilizing mechanism for
the individual--a "feel good" system. The individual is one in a crowd,
and the actions of the others who perceive the one all are perceptions
that the one may control. If control fails, the individual is likely
to reorganize until it succeeds. Hence an attractor in the social
dynamic. And the attractor is an artifact that can be studied in its
own right, independently of the control systems whose behaviour induces it.

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. If there are independent
and non-orthogonal high-level ECSs, then there is a potential for conflict
among them. If that is so, they can develop their own attractor dynamics,
settling into systems of belief that are hard to disturb, even though
the individual elements of the structure might not be controlled with
any high gain. Be clear: I'm not saying this is so. I'm saying that
I don't reject the possibility it is so. And it would be a proper
application of dynamical thinking WITHIN the HPCT structure.

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

Martin