[From Bill Powers (930409.0930 MDT)]
Allen Randall (930408) --
You would like Rick Marken or me to admit that at least a teeny
tiny smidgen of information gets from the disturbance into the
perceptual signal. You probably wonder why there is such massive
resistance to making this harmless-seeming admission. After all,
what does it matter if 0.001% of the information in the
disturbance is transmitted into the perceptual signal? Who would
ever notice such an insignificant amount? Couldn't we just go
ahead like nice guys and give you this one little crumb?
The reason we resist is the same as the reason for your
insistence. You recognize full well that no matter how tiny the
amount, a principle is at stake -- and so do we. You are trying
to establish the principle of explaining the operation of a
system in terms of a measure called information, which involves a
metaphor in which information is like a substance that can be
transmitted from one place to another, and which can cause things
to happen or put constraints on physical interactions. We are
rejecting that metaphor, saying that this kind of principle can
lead only to pseudo-explanations.
Let me draw an analogy. Before Newton, it was believed that
objects persisted in their motion after being hurled because the
act of hurling them transmitted into them a Principle or Quality
called "impetus." The impetus in the object carried it forward,
being used up by the motion. When the impetus ran out, the object
simply fell to the ground. Newton, on the other hand, proposed
that objects move only according to the sum of forces acting on
them, and that in the absence of such forces their motion
continued indefinitely in a straight line, unchanged.
Newton's explanation of the flight of hurled objects is totally
different from that of the Scholastic who believed in the
metaphor of impetus. What do you think that Newton would have
said if a Scholastic had tried to wheedle him into admitting that
at least a teeny tiny smidgen of impetus must get from the hurler
into the hurled object? The Scholastic, who has the image of
impetus firmly in mind, can't imagine how the motion could
persist without any impetus at all -- after all, it's impetus
that carries the motion forward, isn't it? To imagine motion
taking place without any impetus whatsover would be, to the
Scholastic, a contradiction, an absurdity.
I think that you are in the position of the Scholastic, insisting
on the metaphor of information being transmitted from one place
to another and accounting for what happens at the destination.
This is a totally different principle of explanation from the one
used in PCT.
The PCT explanation says that physical variables outside the
nervous system (or the control system) affect sensors, which
produce signals that continuously represent the state of the
variables; that the perceptual signal which results is subtracted
quantitatively from a reference signal; that the reference signal
actuates an output function which creates an output that is some
continuous time-function of the error signal, and that finally
the output acts physically on the local environment, the state of
which contributes to the state of the sensors at the same time
the independent disturbing variables are making their
contribution. By representing all these relationships and
interactions as quantitative mathematical functions, we can solve
the system of equations and predict the states of all the
variables as time-functions of the two independent variables :
the reference signal and the disturbing quantity. And we have, in
many simple cases, made just such predictions and shown that they
match what we can measure about behavior. In our simulations, we
can show that every variable is accounted for at every moment.
Now, where are we going to fit the concept of "information" into
this picture? What could it possibly do that the model does not
already explain? The only answer I can see is that the
methodology of information theory can explain the uncertainty in
our observations -- the unpredictable and apparently unsystematic
deviations of real behavior from the smooth behavior of the
model. In all of the situations we have investigated so far,
these deviations amount to something like 5 to 10 percent of the
range of the variables. As we improve the models, the unexplained
deviations become smaller; we have yet to determine what part of
the remaining unexplained variance can be explained by a
systematic model of improved design. When we do reach an apparent
limit, any further treatments of the data will have to be done in
terms of probabilities and the mathematics of uncertainty. But it
will never be necessary to invoke the metaphor of information.
I have been trying to point out that the measure you call
information, however you calculate it, is simply a way of
characterizing a variable that shows a behavior over a finite
time. You can apply this calculation to the disturbance, and to
the perceptual signal, and to any other signal or quantity you
please. You may even find that there are certain global
relationships among these measures or characterizations. But the
existence of such relationships in no way implies that the
measure of one variable somehow gets through the intervening
space and turns up as part of the measure of the second variable.
This is a naive and almost anti-physical view. It's like saying
that the position of one end of a seesaw somehow gets through the
wood and metal and turns up as the position of the other end. It
substitutes the transmission of some hypothetical thing for the
physical relationships that actually explain what is going on.
If the concept of information were related to the physical
processes in such a way that the results were exactly equivalent
in terms of either kind of representation, matters would be
entirely different. The law of conservation of potential +
kinetic energy is exactly equivalent to saying that a force
exerted through a distance, plus a mass moving with a given
velocity (squared), yield a constant sum. This generalization
exposes a relationship between physical quantities that is not
evident in any single measurement, but which can be converted
into terms of physical measurements.
But this is not how the concept of information works. Saying that
a certain amount of information is transmitted from one variable
to another in no way can be used to calculate how the second
variable depends on the first. There are no equivalences running
from the informational view to the functional view. However,
there are equivalences running the other way. Given the physical
parameters of a channel, one can calculate how signals with
different frequency distributions will be affected as they pass
through the channel, and what the resulting waveforms at the
output will look like. From such calculations we can arrive at a
concept of channel capacity, and from channel capacity we can
pass to the equations of information theory. We can show why
different amounts of "information" would be calculated from
measures of the input and output variables, and we can predict
exactly how they will differ.
In the same way, Newton could explain the flight of a hurled
object, and therefore explain how it might appear to the
Scholastic that a certain amount of "impetus" is present in the
object at all times. But Newton could explain far more about the
flight of the object, aspects of its flight that the Scholastic
never even noticed. The Scholastic, of course, would insist that
Newton has simply verified the existence of impetus, and might
even claim that impetus is the underlying explanation of WHY
Newton's laws work.
I suppose that at some time in his life, Newton might have
believed in impetus. But he also had other knowledge about the
way things work, knowledge that suggested a different view to
him. At some point he must have found himself torn between the
old interpretation and the dawning new one. Perhaps at some point
he, too, felt that impetus was the true explanation for his laws
of motion, both of which concepts made sense to him. But there
must have come a time when he finally realized that the laws of
motion stand by themselves and are fundamental, while impetus is
merely a metaphor with no explanatory power. If Newton were an
information theorist today, and were beginning to grasp the
relationships of PCT, he might once again be torn between two
concepts. But I think I know where he would end up.
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Best,
Bill P.