[From Bill Powers (931104.0830 MST)]
Norm Holland (931103.2150)--
Welcome aboard, Norm. I admit that I have not met what you would
call an excessive number of people who combine literary criticism
with electrical engineering and psychoanalysis. Your example
supports my contention that there is really only one science of
life and that people simply approach it from different angles
(and with varying success). This isn't an eclectic or
microtheoretic approach, is it? I think it's almost the opposite,
being based on the idea that human behavior should make sense
when you get the right idea about it: there is really something
there to learn, and some ideas are better than others.
I'll have a copy of "I" soon -- the interlibrary loan card just
arrived in yesterday's mail.
The "theme and variations" concept of individual organization
fits with my ideas about system concepts and principles, and of
course the psychoanalytic emphasis on conflicts is perfectly
consistent with PCT. I think that Freud leaned too heavily on
_particular_ conflicts, just as Bill Glasser leans too heavily on
_particular_ "basic needs." The urge to divide people into a
small number of categories is nearly irresistible (for instance,
those who divide people into categories and those who don't).
Life seems simpler when you can say that a person falls into a
category which you have already encountered, so you have a ready-
made way to treat him or her. But I think that with PCT in mind,
it's possible to deal with people far more as individuals, even
while recognizing that in some ways they are all alike. I have a
vision of psychotherapy that's utterly simple in principle: you
just try to find out what the person has difficulty controlling,
and why, and see what can be done to deal with the difficulty.
With PCT in the background I think we have ways of defining
difficulties and ways of resolving them that ought to be very
powerful in the hands of an experienced therapist. And all
without thinking in terms of conditions, complexes, or other such
narrow and arbitrary categories.
I know Maturana and Varela, and there are some deep disagreements
between them and me despite areas where we seem to be working in
parallel. Both of them reject the concept of purposive behavior,
Maturana by saying that outcomes are simply the chance result of
converging inner and outer forces, and Varela by saying that a
purpose is nothing but the use to which something can be put, or
the function it can fulfill. Such ideas, I'm sure you can see,
represent a serious difference between the basic ideas of PCT and
the ideas of these two biologists. It's always been a frustration
for me that the pleasant personal relationships between them and
me have not led to any clear resolution of these differences and
thus to some sense that we are working toward the same ends.
Someone explained to me once that the academic style in many
South American countries is to make flat statements about how
things are and then to keep repeating them in the same words
until the opposition is worn down, without justifications or
explanations. I don't know how useful such a generalization is,
but it does seem to explain some of the failures of communication
between these biologists and me. I've never been able to get an
answer to a simple question like "What is your basis for making
that statement?" The nearest that Maturana has come to
acknowledging a relationship between PCT and his theories was at
a public meeting where I badgered him into the grudging admission
that control theory might amount to an explanation of how
autopoiesis works. He didn't seem to think it very important to
explain how autopoiesis works.
Looking forward to your contributions.
···
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Oded Maler (931104) --
All mathematics is a discrete activity, expressed and
communicated in finite symbol strings (with some "hand-waiving"
to convince informally). The magic of "continuous" mathematics,
is that you take a continuous physical process, translate it to
a sequence of discrete symbols (e.g., an algebraic or a
differential equation), perform discrete symbolic operations on
these symbols, and finally take the result to imply something
on the continuous phenomenon you started with. This is exactly
what you do when you prove that controllers stabilize - by
algebraic manipulation rules and not by your private continuous
intuitions.
I agree that this is by far the most common way of dealing with
mathematical processes, but it's not my preferred way. When I say
that I'm a poor mathematician, it's exactly this style of
mathematics to which I refer. For me, discrete symbol-
manipulation is just too far removed from the phenomena I'm
trying to handle mathematically. Looking at a Laplace transform,
I get only the vaguest picture of how the system in question
actually works. I realize that others have no such difficulties
and are perfectly comfortable with abstractions like Bode plots
and Nyquist stability criteria; I'm not claiming superiority for
my approach. I'm just saying that a different way of doing
mathematics is far more comfortable for me.
This way is simulation. When I set up a model as a simulation, I
don't have to use most of the rules relating to the manipulation
of discrete symbols (although they do play a part -- I'm not
totally ignorant of that kind of mathematics). I don't ever
really sit down and solve any differential equations. For me
that's a terribly tedious process in which I can easily get lost,
and I am never sure that the answer I got is right. Instead, I
just set them up as simulations and watch what they do. After a
while I get a feel for the way the equations behave and for the
effects of parameters and variables on each other.
One disadvantage of my approach is that I don't derive general
theorems or discover optimal conditions in this way. So I very
likely miss concepts of interest or importance. But my demands
are as modest as my symbol-handling capabilities; I don't really
care about optimal control, for example, as long as my models
faithfully reproduce the actual behavior of subjects. If a
nonoptimal model fits their behavior, then they aren't
controlling optimally, either, so optimal control doesn't bear on
the question I'm trying to answer.
Similarly, I'm not much concerned about kinds of control that
might take place under conditions different from the ones that
actually obtain in this environment with real organisms, so
general theorems of controllability and so forth cover a lot of
territory that doesn't pertain to my work. For instance, lately
I've realized that roboticists are trying to solve problems that
are far more difficult to solve than the problem of how an actual
human arm is controlled. Robot manipulators often have more or
different degrees of freedom than the human arm, and they allow
ranges of joint angles that bring in ambiguities missing in the
human arm, which has much narrower limits of motion at the
joints. It's much easier to model a real human arm than it is to
model a robot whose joints can swivel through 360 degrees! I like
to think that perhaps nature is telling us to pick a simpler
design with a more limited arm, and accomplish the rest of what
needs to be done in a different way. Maybe the 6+ d.f. problem of
fast efficient real-time arm control with unlimited range of
joint angles is really intractable, and evolution did not solve
it for a good reason!
Anyhow, I find that setting up simulations and running them gets
me to usable results much more easily than trying to solve formal
equations using rules of symbol-manipulation. Also, I have the
great advantage using my method that I can handle the actual
relationships found in nature where the formal methods of symbol-
manipulation fail. As most engineers discover when they get
outside the range of textbook problems, nature hardly ever
cooperates by making the best describing equations solvable.
Before most real problems can be put in a form that symbol-
manipulation can handle, the actually-observed relationships must
be converted to the nearest mathematical representation for which
solutions are known or discoverable. This means that you're never
actually solving the presented problem, but only some problem
similar to it.
In a simulation, if you find that a muscle responds to a driving
signal s as s to the power 1.8, you just put that relationship
into the model and run it. It doesn't matter than you can't solve
the system of equations analytically. It doesn't even matter if
you can't find any exact analytic form that fits the data; you
just tabulate the data and use it in a table lookup, with or
without interpolation as needed for the precision you want. So
the simulation approach is much better suited to the messy
relationships we find in natural system; it requires far less
idealization.
Analog computing was displaced almost totally by digital
computing some 30 years ago. One of the great advantages of
digital computers, put forth by its proponents, was its great
accuracy. The parameters of an optimized control system could be
computed to 10 or more decimal digits! But those who had to build
examples of such systems were not as impressed by this accuracy
as the accountants who bought the same computers. How could you
make the parameters adjustable to that degree of accuracy? The
real devices worked on analog, not digital, principles, and an
analog computer could derive the same parameters 100 times as
fast, providing results that were as precise as would make any
practical difference in a real device.
The analog approach is far more suited to the real world of
organisms than is the digital or analytic approach. Analytically,
you can compute the inverse kinematics and dynamics of a system
and show that in principle the proper driving waveform would then
be integrated to the exact form of motion wanted. But those who
derive such solutions forget that real integrators and computers
such as exist in a nervous system do NOT arrive at exact
solutions of anything. The computations vary, drift, and generate
noise; if the same computations that come out so nicely in
analytical form were embodied in a real organism, the outcome
would be wildly different from what is observed. In the real
world, the double integral of sin(omega*t) is not the expected -
(omega^2)*sin(omega*t), but a waveform that drifts farther and
farther from that form in phase, amplitude, and offset as time
passes. And not much time at that.
So I can make a case for a kind of computation that is not the
customary manipulation of discrete symbols, and which arrives at
results that are just as useful in terms of the behavior of real
organisms in a real world. This is not to downplay the usefulness
of the analytical approach; it's only to say that that approach
isn't the only game in town, or even the best one to bet on for
handling certain problems of modeling behavior.
Obviously, if I and hundreds of others can use a simulation
method of representing the behavior of systems, then the ability
to do this must be granted to any model of the brain's
organization. This is why I suggest broadening the definition of
what I've called the "program" level by renaming it the
"computation" level, where now "computation" is understood to
include all sorts of symbol-manipulation, including analog
manipulations of continuously-variable signals that are symbols
standing for continuous variables at lower levels, as in an
analog computer.
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Best to all,
Bill P.