[From Bill Powers (930407.0830 MDT)]
Bruce Nevin (930407.0845) --
What's it like to wake up one morning and discover that you've
been transformed into a giant brain? Wow. Your summation of the
IT/PCT discussion cuts through all the confusion and brings
everything into focus:
So the information content of a perception controlled over time
is not a factor in the instant-by-instant control of that
perception, and the information content of o is in close
correspondence with the information content of d over time as a
byproduct of instant-by-instant control of p(t)=o(t)+d(t).
Let me tidy up after you a bit.
As Rick and Martin have both pointed out here and there, the
equation above holds true only when the kind of effect that o has
on p is the same as the kind of effect that d has on p. When d
and o both are linked linearly to p, the equation holds strictly
true and with good control we have mirroring of d and o. The
better the control, the more precise the mirroring.
When the physical links connecting o and d to p are differently
nonlinear or have different dynamical properties, we have to
express the "mirroring" relationship a little differently:
p = g(o) + h(d), where now g and h represent the partial
derivatives of p with respect to o or d, as functions of time.
In an environment where there are interactions between o and d
independently of the effects on p, we have to back up even
further:
p = f(o,d), where f is now a function with interaction terms
making the effect of d on p partly dependent on o, and the effect
of o on p partly dependent on d.
In all these refinements, time is a parameter. So the most
general form is
p(t) = f(o(t),d(t))
···
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Your discussion of the meaning of "information" suggests that we
ought to use a different term when speaking of the relation
between external and internal variables in the PCT context. I
suggest that "analog representation" will do the trick.
In an analog representation, one physical variable is used as the
analog of another one, often with a change of physical units as
in a transducer. An analog is not an analogy, for the
representation is strictly in terms of magnitude. Some one
dimension of an external situation is represented as a one-
dimensional analog signal. It's the one-dimensionality that makes
the analog representation possible and removes all ambiguity as
to what is being represented.
If a signal is present in the brain that covaries with the
horizontal angular position of an object on the retina, that
signal is an analog of the position. The intervening perceptual
function converts angular deviation into the magnitude of a
neural signal. The magnitude of the neural signal is then an
analog representation of the angular position. Whatever the
angular position is, however it changes, the neural signal
maintains a corresponding magnitude or changing magnitude. At
every moment, the magnitude of the neural signal represents
angular position.
Now suppose we simply alter the perceptual function so it
produces a signal proportional not to the angular position, but
to the first time derivative of the angular position. Now the
same signal is an analog of angular velocity. The same situation
as before holds on the retina -- the same object going through
the same pattern of positions. But now position is no longer
represented; only the moment-by-moment velocity corresponds to
the magnitude of the perceptual signal.
This concept of analog representation can be used to reduce
confusion about what perceptual signals "really" represent. One
of the problems that continually arises in discussing a hierarchy
of perceptions is that of using an inappropriate level of
perception while imagining it. If we consider a perceptual signal
that is the analog of a position, we can, unfortunately, also
notice that this signal can _vary._ In other words, we form the
velocity analog in our own minds while observing or imagining a
signal that is actually the analog of a position.
The position signal naturally varies if there is variation in the
lower-order situation that alters position. But the position
signal itself does not represent the variation; it represents, is
the analog of, position, not velocity or change. In order to
represent the velocity or change, we must form a new analog of
that signal: in other words, a higher-order perception of it. We
must use a transformation that makes a constant perceptual signal
correspond to a continuously-increasing or continuously-
decreasing quantity at the input to the perceptual function. Then
the perceptual signal will be the analog of rate of change, not
position. If the perceptual function involves taking two
successive time derivatives, the perceptual signal will be the
analog of acceleration.
One easily confused pair of perceptual classes consists of
configurations and relationships. When we look at a static field
of objects, we detect a set of configurations, familiar
arrangements of sensations. Each configuration is represented by
a configuration-analog signal. If a configuration changes shape,
it simply becomes more or less of that same analog signal.
However, when we look at a configuration, we can see that some
parts of it are shaded and some are light, some are one color and
some another, some are surface and some are edges. And we can see
that there are _relationships_ among these elements. Not only
that, we can see that there are relationships among
configurations. If the configurations are changing, we can see
relationships among the changes. So it begins to look as through
the configuration level is composed of relationships among
elements.
This is entirely a result of the observer or imaginer forming a
relationship-analog of the actual or imagined elements of lower
order. There is no signal at the configuration level that is the
analog of a relationship. At the configuration level, one signal
simply represents the state of a configuration, a one-dimensional
state that corresponds to more or less of that perceptual signal.
A similar confusion exists between the event and sequence levels.
At the event level, presence of a signal indicates a specific
event in progress. The magnitude of the signal indicates the
state of that event in a single dimension of change. But looking
from a higher level, we can see that events occur one after
another, in specific sequences. This is a sequence-level analog
being formed out of perceptions of events. The event-signals
themselves do not analogize sequence. A single event-signal
indicates one event now in progress. Only at the sequence level
do we notice that an event has a beginning, a middle, and an end,
and that they always occur in that order. Only at the sequence
level are there signals that are analogs of ordering.
You said, Bruce,
A measurement is not `about' the thing measured in the same
sense that uncertainty is or knowledge is `about' the thing.
If you just change "measurement" to "analog representation", you
will be saying exactly what I have said above. A measurement IS
an analog representation, a one-dimensional signal representing
just one aspect of a situation.
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The quantity of information is not the information quantified.
An analog signal can be analyzed as to the quantity of
information it contains, or the rate of information generation
involved in its production. This makes sense, however, only if we
have two ways of detecting the thing analogized; if we have only
the perceptual representation, we can't say how its information
content relates to the information content of the world from
which the perception was drawn. To speak of information
transmission is to compare two ways of analogizing what is
presumed to be "the same thing." Generally, one of the ways is
not really an analog signal, but a model, a hypothetical picture
of something in reality that corresponds to the perception: a CEV
or a disturbance acting on a CEV.
So there is really no transmission of information from an
external object to an internal perception of it. There are two
analogs, and we can compute the information content of each one
by log(R/r). Each analog is a moment-by-moment representation of
the state of something, the nature of the something being
determined by the intervening perceptual functions. If the two
analogs vary in corresponding ways, we can treat this as if
information had been transferred from one to the other (much as
Martin Taylor treats the transfer of "virtual messages" in his
Layered Protocol scheme). Thus "transmission of information" is a
metaphor.
An image that springs to mind is of Ed McMahon saying "Heeeeer's
Johnny!" while the audience simultaneously and in unison says the
same thing the same way. The appearance is that the information
in Ed's announcement has been transferred into the audience, or
vice versa, while in fact it has been simultaneously generated by
independent means.
You say:
The instantaneous comparison of the instantaneous values r(t)
and p(t) produces instantaneous error/output o(t) at moment t,
and in the environment the instantaneous value o(t) and other
environmental factors abstracted to a single value d(t) are
combined as p(t), moment by moment continuously--and this
suffices.
Yes. The mechanism by which o(t) comes to be nearly equal and
opposite to d(t) (in the mirroring case) results in creating
nearly the same amount of information in o as is found (by a
different means of observation) in d. But no information has
actually been transferred from d, through the control system,
into o. The transfer is "virtual." log(R/r) is a measurement, not
a substance. The actual process involved is one of analogizing
and then manipulating the signals that result.
As you say, to sum up:
The correspondence of variation over time in d(t) with
variation over time in o(t) is a result of continuous
entrainment, not of the transmission of information content
(whatever that is).
Or, d is outside of the control system, and p is inside it.
This is the meaning of Chuck's summation.
Which I would now amend to read: perception of d is inside the
observer, and so is the presumed state of p. The two perceptions
are obtained by different means, and do not communicate directly
with each other.
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Best,
Bill P.