[From Bill Powers (931022.1230 MDT)]
Martin Taylor(direct post, also relayed by Rick to CSG, also cc-d
to me, so I now have three copies. Keep this up and I'll bill
someone for a bigger hard disk)
I think you're getting close to a good expression of the problem
with category perceptions, and that Rick also is being helpful by
reminding us that it's the nature of the perceptual function, not
the signal, that we have to dope out.
I would prefer to say that an outside observer can categorize
the inputs to the function, using his/her category-level
perception. The perceptual function itself doesn't categorize
unless it IS a category-level function.
Good. This expresses in another way the general problem that I
bring up now and then: the use of a human level of perception to
characterize the functions of a level in a model or in someone
else that is actually lower (like trying to characterize the
stretch reflex as a program, as in the TOTE unit). If any level
of perception is seen as a categorizing level, then we can be
sure of one thing at least: the person doing the seeing is using
a category level of perception. Whether this is justified is
another question. This is sort of the opposite of reductionism;
inflationism?
My attempt to push the level of discretization up to the logical
level has some bugs, but it does take care of one of your
criteria for defining a category level.
I like the term "category" for the perceptual level that
divides the world into discrete aspects that can be used in
logical and linguistic operations ...
One aspect of this division has come up before, in your writings
and in certain cyberneticists' uses of the Spencer-Brown
"calculus of distinctions." A categorical distinction, it has
been said, divides the world into two parts: the category and its
complement. My claim is that this is a misattribution of a
logical perception to the category level. Logically, all that is
not A is not-A. But that is a logical or set-theoretic concept,
not a categorical concept. I don't think it belongs at the
category level, but at the logic/program/rule-driven level.
Suppose I construct a perceptual function that will respond with
a (discrete if you like) signal if any one of the following
signals is present at the input: a signal indicating presence of
a red circle, or one representing a green diamond, or one
representing a black vertical line. This is one way to create a
category: it is the category of things which are a red circle, a
green diamond, or a vertical black line. If any one or more of
these elements is present at the input, the output will be 1.
Otherwise it will be 0. I could also include an event-perception,
"fluge". Now the term "fluge" is equivalent to the other inputs,
in that it also results in the same category signal.
Now we have the category named "fluge." What is the complement of
this category? Presence of the category is indicated by a
perceptual signal with a value of 1. Presence of the complement
is indicated by the same perceptual signal with a value of 0. But
what are the perceptual items at the lower level that are
represented by a 0?
The answer is that there are none. Presence of the category is
indicated when one or more of the lower perceptual signals that
reach the input function is distinctly nonzero. But the
perceptual output signal with a value of 0 tells us only that
none of the four inputs is active. It doesn't tell us whether any
other set of inputs to any other category detector is present,
nor does it tell us what other category-perceivers exist. In no
way does the absence of a category signal indicate the presence
of any other category signal. To say "not-category-A" is
equivalent to saying "not fluge AND (NOT each of the other
figures)." And that is all. It is not equivalent to the assertion
of all other categories, or any other category.
The creation of the complementary category is an artifact of the
logic level. The logical treatment of class membership or set
membership carries a tacit assumption of a set of elements, all
of which exist but only some of which are included within a given
set, class, or boundary. It follows axomatically that all
elements which are not in one set are outside it -- i.e., in the
complement of the set. This is the playing field on which set
theory games and the like are conducted. But what if each element
could be present or not present, depending on what is going on at
some lower level? Now it is not true that if element A is not
inside set S, it must be outside it. Element A might not even
exist at the moment, although it might come into existence at
some other time.
Suppose we have two categories: "tying a bow knot" and "eating an
ice cream cone." Does it follow that if "eating an ice cream
cone" is not true, we must be "tying a bow knot?" In fact, both
category signals could be 0, both could be 1, or only one of them
could be 1.
It's only at the logic level where we have a logical variable
which always exists but can have either of two values, either of
which is meaningful in a logical expression. Only at this level
does _absence_ of a signal standing for a proposition indicate
_presence_ of some equally significant proposition: absence of a
signal meaning x <= y logically means x > y. That is not the sort
of computation that goes on at the category level as I understand
it. At the category level, absence of x <= y (a signal standing
for a relationship) might mean only that x and y, at the moment,
don't exist. In that case, the relationship x > y doesn't exist,
either. One ruler must be either the same or less in length than
another, or greater in length than the other. But what if there
are no rulers being perceived? Then neither relationship-
perceiver is generating any signal. That's a case with which
logic can't deal: it violates the implicit premise, "given two
rulers...".
I have a feeling than when we finally get all this straightened
out, the result is going to be something of value for our
understanding of the relationship of mathematics to the rest of
experience.
ยทยทยท
--------------------------------------------------------------
Hans Blom (931022) --
That was a nice straighforward analysis of one aspect of
conflict. It would have been better, however, if you had actually
set up two control systems WITH A DISTURBANCE of x, and looked at
the results. Optimizing for the case of no disturbance might give
you a result very different from optimizing for best resistance
to disturbance, particularly when you use nonlinear criteria of
what is "best."
Also, conflict is not a problem unless one system or the other,
or both, has physical limits on the possible outputs. If you
don't take that into consideration, and just look at the abstract
algebra, you could easily come up with optimum parameters that
oppose a disturbance of 1 unit by producing outputs of 1,000,000
units and -1,000,000.99 units.
Kent McClelland has done some conflict simulations in which
output limits are taken into account; they will be published in
due course, but if asked he might send you a preprint.
I agree that nonlinear sensitivity of output to error will
partially solve some conflict problems by de-emphasizing small
errors in favor of avoiding larger ones. But if achieving both
goals is considered very important, conflict will just drive both
systems to their maximum outputs and neither one will have any
control when disturbances appear.
By the way, control systems that use absolute values or squares
of errors will work only with errors of one sign. If the errors
switch to the other sign, the feedback will become positive. I
presume you weren't saying that the control systems themselves
used these functions of error.
---------------------------------------------------------------
Best to all,
Bill P.