[From Bill Powers (930128.2200)]
Avery Andrews (930129.1000) --
You've made a start on a formidable paper. I hope that others
will join with you in bringing it to completion. I think it will
be known in the future as a fundamental work in the field of PCT.
I would like to clear up one point that is still unclear, which
is control of fast movements, or rather the obvious lack of it:
From this it seems clear that feedback will not be very useful
in controlling certain high-speed movements, such as the left
jab of Muhammad Ali, clocked at 40ms.
Imagine a control system that has a time-constant of 40
milliseconds (we don't actually have to imagine it; the Little
Man models it). This control system can control the position of
an arm, so that one or more perceived joint angles follows a
smoothly varying reference signal, moving the arm under complete
control at all times.
But now take this very same control system, and instead of giving
it a reference signal that passes smoothly from one magnitude to
another, make the reference signal jump instantaneously -- in,
say ten milliseconds -- from a frequency of 200 impulses per
second to a frequency of 500 impulses per second, with no passage
through intermediate frequencies.
The arm will move from the initial position corresponding to 200
impulses per second to the final position corresponding to 500
impulses per second. If the control system is critically damped,
it will do so in a single swift move that goes to the new
position in an exponential approach. With a time constant of 40
milliseconds, it will go 63% of the way to the new position in 40
milliseconds, 86% of the way in 80 milliseconds, and 95% of the
way in 120 milliseconds. This represents the fastest speed at
which the control system can correct a suddenly-appearing error
of any magnitude, large or small.
During this maximum-speed movement, there is naturally no
control. The control system is already producing the largest
output it can consistent with stopping in the new position. Any
disturbance that came and went during the approach would simply
cause a deviation from the nominal path. So, paradoxically it may
seem, the control system's control action is uncontrolled after a
step-change in the reference signal -- although the final
position is as controlled as ever.
This is simply the nature of control. As the speed of movement
increases, due to more and more abrupt changes in the reference
signal, the error signal becomes larger and larger (producing
faster and faster movement) until the change becomes a true step-
change, at which point the movement will reach its maximum speed.
The resistance to disturbances that appear during the movement
will be, at low speeds, essentially the same as for static
reference signals. As the rate of change of the reference signal
increases, the resistance to disturbance during the transition
becomes smaller and smaller, until in the limit it vanishes.
There is therefore no inconsistency between saying that slow
movements are controlled while the fastest ones are not. There is
no need to posit one organization for slow movements and another
for fast ones. The same control system, with the same parameters,
explains the behavior we see over the whole range.
There is another side to this story. One common idea about the
hierarchy of control is that within it, goals are set and then
the control systems alter perceptions to match them. But this
appearance, I think, is misleading. On the time scale appropriate
to any level of control, behavior is not a process of error
correction. On that time scale, errors never become large.
Instead, as reference signals vary, perceptions simply follow
them. The changes we see reflect changes in reference signals,
not the process of error correction. On the appropriate time
scale, which has been called the "specious present," error
correction takes no time.
Ray Allis (930128) --
Reading your post on language was an eerie experience. It was as
though I could see your meanings, and they were the same as mine.
I say with some hestitation that we have the same perception of
what language is. The words point to the perception, provided
that those involved have the same perception. If they don't, the
words just evoke irrelevancies.
Years ago a mathematician named Richards said to me, apropos of
similar concepts, that while he could describe a mathematical
proof to me, he couldn't show it to me. To a mathematician, a
proof is a perception; the description is only an attempt to
evoke a similar perception in someone else.
The real machinery of the brain operates without benefit of
words, I think. The words are just perceptions that we manipulate
to evoke experiences in others. This evocation is as near as we
can come to knowing that someone else exists in this universe,
someone who resembles us and perhaps experiences the world in
similar terms. I know you agree with that. I wonder if anyone
Avery Andrews (930129) --
The reason the nonword perceptions drop out of the picture so
quickly is that their connection to the word-ish ones seems to
be indirect, mediated by the lexicon.
To "mediate" means to me to stand between. What is "the lexicon"
that it can stand between nonword perceptions and the words that
we use to indicate nonword perceptions? I don't find any such
mediating process in my own experiences. Perhaps there are
lexicons in linguistics, in the sense of formal lists of words in
conjunction with other words. But such lexicons, taken by
themselves, are only dictionaries; they tell you what words to
use in place of other words, or what words to use with other
words. The link I speak of, and that Ray Allis described so
eloquently, lies between words and the experiences we indicate by
passing words among us. This applies to the words in the lexicon
as well as to those in ordinary sentences.
For example the verb `eat' has one version (the active voice)
whereby the preverbal NP is the eater and the postverbal one
the thing eaten, and another, the `passive', wherby the
preverbal NP is the thing eaten, and the eater is either left
out or expressed postverbally, by a prepositional phrase with
the preposition `by'.
And why is this so? I don't deny the truth of what you say, but
if what you say is all there is to the matter, the fact that you
present is meaningless. Why should there be two such usages of
"eat"? Just what is it that these two usages convey, and what
differences of experience are conveyed by their differences? Why
is it that we refer to one usage as "active" and the other as
"passive?" To what do those two terms refer? What is this thing
you call the "eater?" What distinguishes the eater from the
eaten? None of these questions can be answered by further
examination of the words and their customary usages. The very
terms in which you describe the word, the verb, themselves must
be defined in terms of experiences, not words. The words are just
ripples on the surface, enlarging whatever they pass over. When
the light is wrong, all you can see are the ripples. From another
point of view, you see through them to something underneath, the
meanings, which are nothing more than the world you experience
when the ripples die out.
Best to all,