Off-line Categories back On-line

[Marken_Richard_S1]

[From Rick Marken (931022.1000)]

I have been having a little off-line discussion with Martin
about categories. He said I could bring it back to the net
so I am posting Martin's most recent comments and my most
recent reply. I hope we can keep this on the net from now on
(if there is a "from now on" for this topic).

···

------------
Martin Taylor to Rick Marken (cc to Bill Powers):

I think our apparent disagreement has been largely over the use of
particular words, rather than about the underlying concepts (though
see my response to Bill's provocative posting of yesterday).

The way I've been using the term, "category" is something that occurs
within the hierarchy. Discreteness is the primary issue, though Bill's
long posting of yesterday (to which I responded publicly) makes me see
that I may have overemphasized the notion of a binary value of the perception.
The "category" perception is something I think we all agree, perhaps
with slightly different connotations, exists in the hierarchy.

As I see you using the term, categorization is equivalent to a reduction
in measure. If more than one set of input values to a function results
in the same output value, then that function categorizes. 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. In most of
the lower-level cases you have brought up, all that the perceptual function
does is to reduce to one the dimensionality of its input. It measures
(in my terms) the amount of its input along the single dimension that
the function defines. I (outside observer) can categorize that measure
by giving a number to it, a number that would be different if the measure
changed by some finite amount greater than epsilon, but not for lesser
changes in the measure.

I hope that describes the effects we are talking about. What remains, if so,
is the words we use. 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, and I like the term "measure" or "evaluation"
or simply "function" for the effect that happens at every perceptual level
(including category). But I'm not fixated on terminology. The important
thing is to have our discussions be about concepts, and not to wast net
bandwidth when it is only words that are at issue.
------
Marken to Martin Taylor (cc to Bill Powers):

As I see you using the term, categorization is equivalent to a reduction
in measure. If more than one set of input values to a function results
in the same output value, then that function categorizes.

Yep.

I would prefer
to say that an outside observer can categorize the inputs to the
function, using his/her category-level perception.

If that's your preference.

The perceptual function itself doesn't categorize unless it IS a
category-level function.

And how do we know that it is a category level function? What is the
difference between a category level function and other perceptual
functions?

In most of
the lower-level cases you have brought up, all that the perceptual function
does is to reduce to one the dimensionality of its input.

All perceptual functions reduce the dimensionality of the input --
to ONE dimension. This would also be true of category level
perceptual functions as well.

It measures
(in my terms) the amount of its input along the single dimension that
the function defines. I (outside observer) can categorize that measure
by giving a number to it, a number that would be different if the measure
changed by some finite amount greater than epsilon, but not for lesser
changes in the measure.

It seems to me that you are saying that categorization involves
discretizing a perceptual signal. And that such a discretized
signal IS a category perception. I think this view misses an
important aspect of the PCT model, viz. the notion that The
SEMANTICS OF PERCEPTION (whether what we experience is a sensation,
transition, configuration, principle OR category) are determined
by the nature of PERCEPTUAL FUNCTIONS, NOT BY THE NATURE OF
PERCEPTUAL SIGNALS.

Maybe you mean that the a category perception is always the result
of a perceptual function that produces a non analog output. This
is a little better, but, still, I think, inadequate. Other, non-
category type perceptual functions could produce discrete (categorical
in your sense) output. It's not the nature of the OUTPUT of the
perceptual function (discrete, analog, time coded, whatever) that
determines what is perceived -- it is the nature of the function
itself. The problem to be answered in this discussion (from the
perspective of the PCT model) is "what is the nature of the
function that produces the perceptual signal that corresponds to
the experience that we (well, Bill) calls "category". This
question can only be answered by research -- as I said, testing
to see what variables are controlled when people control for "category"
or "class membership" or whatever you want to call it -- and modelling.

So if you want to continue thinking of categories as discrete
perceptual signals, that's fine. The only problem with doing
so, I think, is that it keeps you from being able to formulate
any useful research questions (or models, for that matter).

I hope that describes the effects we are talking about.

Obviously, I don't think so.

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

That's fine; but this perception must be defined by the possibly
discrete results of a perceptual FUNCTION, not as the result of
simply discretizing a perceptual signal.

Why am I having a tremendous sense of deja vue. Didn't we already
go through this?

Best

Rick

[_Martin_Taylor]

[Martin Taylor 931022 18:40]
(Rick Marken 931022.1000)

I tried to keep this discussion off-line until there might be some
breakthrough that would interest the wider readership of CSG-L, but have
been outvoted. So, without apology for boring people (Rick and Bill can
apologize if they see fit), here is another episode in the saga.

Rick...

It seems to me that you are saying that categorization involves
discretizing a perceptual signal. And that such a discretized
signal IS a category perception.

No, but any category level perceptual function must (in my view)
generate a signal that changes abruptly when its input shifts by
an infinitesimal amount across some boundary in the multidimensional
input space.

I think this view misses an
important aspect of the PCT model, viz. the notion that The
SEMANTICS OF PERCEPTION (whether what we experience is a sensation,
transition, configuration, principle OR category) are determined
by the nature of PERCEPTUAL FUNCTIONS, NOT BY THE NATURE OF
PERCEPTUAL SIGNALS.

Yep.

Maybe you mean that the a category perception is always the result
of a perceptual function that produces a non analog output.

Not sure what you mean by "non-analogue." All neural signals are analogue.
Many kinds of perceptual function, at any (?) perceptual level MIGHT
produce outputs that have only discrete levels. A category-level
perceptual function MUST have at least one discrete transition boundary
(I started in this discussion by saying that it must have only a one-zero
pair of possible outputs, but Bill's long posting made me see that as an
unnecessary restriction).

This is a little better, but, still, I think, inadequate. Other, non-
category type perceptual functions could produce discrete (categorical
in your sense) output.

Yes, they could, but they would not necessarily be categorical in my sense.

It's not the nature of the OUTPUT of the
perceptual function (discrete, analog, time coded, whatever) that
determines what is perceived -- it is the nature of the function
itself. The problem to be answered in this discussion (from the
perspective of the PCT model) is "what is the nature of the
function that produces the perceptual signal that corresponds to
the experience that we (well, Bill) calls "category".

I agree with this. What I have tried to do is to describe at least some
of the characteristics that such a perceptual function must have.

This question can only be answered by research -- as I said, testing
to see what variables are controlled when people control for "category"
or "class membership" or whatever you want to call it -- and modelling.

In principle, you have to be right. But if you don't know what you are
looking for, you won't know when you see it. At the moment, I wouldn't
know at all how to begin such a study. I might take a picture of a chair
and ask "Is this a chair," and then construct different modifications to
it to see what modifications are corrected by the subject, but that would
only tell me that the label is used in a certain way. I might ask the
subject to keep this "the same." But then I would be open to control of
all the lower-level perceptions as well. I wouldn't know where to start.
Would you?

There has been, as Bill says, an enormous amount of conventional research
on the concept of "category." I doubt whether it has much relevance to
what we are discussing, which is the structure of the PCT hierarchy. In
that context, I might put my side of the discussion in the form of a
different question: "Is there a specific level of the hierarchy in which
the perceptual functions are such as to provide a discrete boundary in
the space of its inputs, so that the perceptual signal changes discontinuously
as the input crosses the boundary, and such that the boundary exhibits
hysteresis." I think those two characteristics probably specify the
category level, if it exists, though they may not be sufficient (and the
second one may not be necessary).

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

That's fine; but this perception must be defined by the possibly
discrete results of a perceptual FUNCTION, not as the result of
simply discretizing a perceptual signal.

I'm trying to see in what way you are not repeating back to me the point
I have been trying to get you to see. How is it different from the theme
of my last many postings?

···

============
(Bill Powers 931022.1230 MDT)

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.
...
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.
...
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.
...
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.

And so on.

It seems to me that you argue against yourself. Take the last point
first. If there exists a category-level PIF that answers yeah or nay
to "tying a bow knot", and another that answers to "eating an ice-cream
cone," neither should have any influence on the other, and the perceptual
output of one should not be expected to be related to the perceptual
output of the other. Only at some logic level to which these feed might
there be a relationship that enforced an exclusive-or. The facts of the
world might mean that both were never 1 at the same time, but nothing
at the category level would be expected to do so. In terms of the
linguistic discussion that started this thread, there is no "contrast"
between these two categories.

Now let's go backwards in your posting. It is true that anything that is
not within a category boundary is outside it. What does this mean, in
process terms? Assuming that a category perceptual function exists at all,
it has certain signals as inputs. In the space of all perceptions, it
matters not a whit to that PIF what the state of any signals might be
if they are not connected to its input. So, if the category is based only on
visual sensory signals that have passed through many lower-level functions,
no variations in taste, touch, spoken language, etc. will affect the
outputs of its PIF. Seeing an object as category "chair" is not mutually
exclusive with smelling something of category "cigarette smoke." And
category PIFs that use the same input signals can have overlapping reagions
in which they say "yeah, that's me." An object can be a chair and an
antique at the same time. Both are categories, both based on exactly
the same sensory signals.

If a category PIF does not get signals at its input that lie within its
boundary, it will say "nay." That does not influence any other category
PIF, except those that happen to use its signal as part of their input
set (I am allowing, here, for the flip-flop connection). There's no
logical relation among them. That lies higher in the hierarchy.

Only at this [logic] 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 no PIF for "ruler" category will be giving any output, and presumably
the logic level that is testing this relationship will be cut off in some
way (we haven't discussed the mechanisms of logic levels yet, so I don't
think I am begging the question).

I think you are confusing an outsider's view of "A vs not-A," which defines
the location of the boundary of the category's acceptance region in the
space of its inputs, with the category PIF's "view," which is that it
provides a significant output when it has appropriate input and not
otherwise.

To belabour the point:

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 the question is ill-posed, because it presumes that
there must always be a positive category (and seems to presume that
there is always exactly one). Another answer is "Anything, or nothing,
provided it isn't a fluge."

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.

And, I hope, not only the language of mathematics, but also other kinds
of language, including music.

Martin