Paper
presented at UQaM Summer Institute in Cognitive Sciences on Categorisation 2003
http://www.unites.uqam.ca/sccog/liens/program.html
To
appear in Lefebvre C., & H.
Cohen (Eds.) (2005) Handbook
on Categorization. Elsevier
To Cognize is to
Categorize:
Cognition is Categorization
Stevan Harnad
Chaire de recherche du Canada
Centre de neuroscience de la
cognition
Université du Québec à
Montréal
ABSTRACT: We organisms are
sensorimotor systems. The
things in the world come in contact with our sensory surfaces, and we
interact
with them based on what that sensorimotor contact "affords". All of our
categories consist in ways we behave differently toward different kinds of things --
things we do or
don't eat, mate-with, or flee-from, or the things that we describe,
through our
language, as prime numbers, affordances, absolute discriminables, or
truths.
That is all that cognition is for, and about.
KEYWORDS: abstraction,
affordances, categorical
perception, categorization, cognition, discrimination, explcit
learning,
grounding, implicit learning, invariants, language, reinforcement
learning,
sensorimotor systems, supervised learning, unsupervised learning
Pensar
es olvidar
diferencias, es generalizar, abstraer.
En el abarrotado mundo de Funes
no hab’a sino detalles,
casi inmediatos. Borges ("Funes el memorioso")
1.
Sensorimotor Systems. Organisms
are sensorimotor systems. The things in the world come in contact with
our
sensory surfaces, and we interact with them based on what that
sensorimotor
contact "affords" (Gibson 1979).
2. Invariant Sensorimotor Features ("Affordances"). To say this is
not to declare oneself a "Gibsonian" (whatever that means). It is
merely to
point out that what a sensorimotor system can do is determined
by what
can be extracted from its motor interactions with its sensory input. If
you
lack sonar sensors, then your sensorimotor system cannot do what a
bat's can
do, at least not without the help of instruments. Light stimulation
affords
color vision for those of us with the right sensory apparatus, but not
for
those of us who are color-blind. The geometric fact that, when we move,
the "shadows" cast on our retina by nearby objects move faster than the
shadows of
further objects means that, for those of us with normal vision, our
visual
input affords depth perception.
From
more complicated facts of projective and solid geometry it follows that
a
3-dimensional shape, such as, say, a boomerang, can be recognized as
being the
same shape Ð and the same size Ð even though the size and shape
of its shadow
on our retinas changes as we move in relation to it or it moves in
relation to
us. Its shape is said to be invariant under these sensorimotor
transformations, and our visual systems can detect and extract that
invariance,
and translate it into a visual constancy. So we keep seeing a boomerang
of the
same shape and size even though the shape and size of its retinal
shadows keep
changing.
3. Categorization. So far, the affordances I've mentioned have depended
on having either the right sensors, as in the case of sonar and color,
or the
right invariance-detectors, as in the case of depth perception and
shape/size
constancy. Having the ability to detect the stimulation or to detect
the
invariants in the stimulation is not trivial; this is confirmed by the
fact
that sensorimotor robotics and sensorimotor physiology have so far
managed to
duplicate and explain only a small portion of this subset of our
sensorimotor
capacity. But we are already squarely in the territory of
categorization here,
for, to put it most simply and generally: categorization is any systematic
differential interaction between an autonomous, adaptive
sensorimotor system and its world:
Systematic, because we don't want arbitrary interactions like the
effects of
the wind blowing on the sand in the desert to be counted as
categorization
(though perhaps there are still some inherent similarities there worth
noting).
Neither the wind nor the sand is an autonomous sensorimotor system;
they are,
jointly, simply dynamical systems, systems that interact and change
according
to the laws of physics.
Everything
in nature is a dynamical system, of course, but some things are not only
dynamical systems, and categorization refers to a special kind of
dynamical
system. Sand also interacts "differentially" with wind: Blow it this
way and it
goes this way; blow it that way and it goes that way. But that is
neither the
right kind of systematicity nor the right kind of differentiality. It
also
isn't the right kind of adaptivity (though again, categorization theory
probably has a lot to learn from ordinary dynamical interactions too,
even
though they do not count as categorization).
Dynamical
systems are systems that change in time. So it is already clear that
categorization too will have to have something to do with changes
across time.
But adaptive changes in autonomous systems are those in which internal
states
within the autonomous system systematically change with time, so that,
to put
it simply, the exact same input will not produce the exact same output
across
time, every time, the way it does in the interaction between wind and
sand
(whenever the wind blows in exactly the same direction and the sand is
in
exactly the same configuration). Categorization is accordingly not
about
exactly the same output occurring whenever there is exactly the same
input.
Categories are kinds, and categorization occurs when the same
output
occurs with the same kind of input, rather than the exact same
input.
And a different output occurs with a different kind of input. So that's
where
the "differential" comes from.
4. Learning. The adaptiveness comes in with the real-time history.
Autonomous, adaptive sensorimotor systems categorize when they respond
differentially to different kinds of input, but the way to show that
they are
indeed adaptive systems -- rather than just akin to very peculiar and
complex
configurations of sand that merely respond (and have always responded)
differentially to different kinds of input in the way ordinary sand
responds
(and has always responded) to wind from different directions -- is to
show that
at one time it was not so: that it did not always respond
differentially as it
does now. In other words (although it is easy to see it as exactly the
opposite): categorization is intimately tied to learning.
Why
might we have seen it as the opposite? Because if instead of being
designers
and explainers of sensorimotor systems and their capacities we had
simply been
concerned with what kinds of things there are in the world, we might
have
mistaken the categorization problem as merely being the problem of
identifying
what it is that exists (that sensorimotor systems can then go on to
categorize). But that is the ontic side of categories,
concerned with
what does and does not exist, and that's probably best left to the
respective
specialists in the various kinds of things there are (specialists in
animals,
vegetables, or minerals, to put it simply). The kinds of things there
in the
world are, if you like, the sum total of the world's potential
affordances to
sensorimotor systems like ourselves. But the categorization problem is
not
determining what kinds of things there are, but how it
is that
sensorimotor systems like ourselves manage to detect those kinds that
they can
and do detect: how they manage to respond differentially to them.
5. Innate Categories. Now it might have turned out that we were
all born with the capacity to respond differentially to all the kinds
of things
that we do respond to differentially, without ever having to learn to
do so
(and there are some, like Jerry Fodor (1975, 1981, 1998), who sometimes
write
as if they believe this is actually the case). Learning might all be
trivial;
perhaps all the invariances we can detect, we could already detect
innately,
without the need of any internal changes that depend on time or any
more
complicated differential interaction of the sort we call learning.
This
kind of extreme nativism about categories is usually not far away from
something even more extreme than nativism, which is the view that our
categories were not even "learned" through evolutionary adaptation: The
capacity to categorize comes somehow prestructured in our brains in the
same
way that the structure of the carbon atom came prestructured from the
Big Bang,
without needing anything like "learning" to shape it.
(Fodor's
might well be dubbed a "Big Bang" theory of the origin of our
categorization
capacity.)
Chomsky
(e.g., 1976) has made a similar conjecture Ð about a very special
subset of our
categorization capacity, namely, the capacity to generate and detect
all and
only those strings of words that are grammatical according to the
Universal
Grammar (UG) underlying all possible natural languages: UG-compliance
is the
underlying invariant in question, and, according to Chomsky, our
capacity to
detect and generate UG-compliant strings of words is shaped neither by
learning
nor by evolution; it is instead somehow inherent in the structure of
our brains
as a matter of structural inevitability, directly from the Big Bang.
This very
specific theory, about UG in particular, is not to be confused with
Fodor's far
more general theory that all categories are unlearnt and
unevolved; in
the case of UG there is considerable "poverty-of-the-stimulus" evidence
to
suggest that UG is not learnable by children on the basis of the data
they hear
and produce within the time they take to learn their first language; in
the
case of most of the rest of our categories, however, there is no such
evidence.
6. Learned Categories. All evidence suggests that most of our
categories are learned. To get a sense of this, open a dictionary at
random and
pick out a half dozen "content" words (skipping function words such as
"if," "not" or "the"). What you will find is nouns, verbs, adjectives
and adverbs all
designating categories (kinds of objects, events, states,
features,
actions). The question to ask yourself is: Was I born knowing what are
and are
not in these categories, or did I have to learn it?
You
can also ask the same question about proper names, even though they
don't
appear in dictionaries: Proper names name individuals (e.g., people,
places)
rather than kinds, but for a sensorimotor system, an individual is
effectively
just as much of a kind as the thing a content word designates: Whether
it is
Jerry Fodor or a boomerang, my visual system still has to be able to
sort out
which of its shadows are shadows of Jerry Fodor and which are shadows
of a
boomerang. How?
7. Supervised Learning. Nor is it all as easy as that case.
Consider the more famous and challenging problem of sorting newborn
chicks into
males and females. I'm not sure whether Fodor thinks this capacity
could be
innate, but the grandmaster, 8th-degree black-belt
chicken-sexers on
this planet Ð of which there are few, most of them in Japan Ð
say that it takes
years and years of trial and error training under the supervision of
masters to
reach black-belt level; there are no short-cuts, and most aspirants
never get
past brown-belt level. (We will return to this.) Categorization, it
seems, is a
sensorimotor skill, though most of the weight is on the sensory part
(and the
output is usually categorical, i.e., discrete, rather than continuous);
and,
like all skills, it must be learned.
So
what is learning? It is easier to say what a system does when
it learns
than to say how it does it: Learning occurs when a system
samples inputs
and generates outputs in response to them on the basis of trial and
error, its
performance guided by corrective feedback. Things happen, we do
something in
response; if what we did was the right thing, there is one sort of
consequence;
if it was the wrong thing there is another sort of consequence. If our
performance shows no improvement with time, then we are like the sand
in the
wind. If our performance improves Ð more correct outputs, fewer
errors Ð then
we are learning. (Note that this presupposes that there is such a thing
as an error,
or miscategorization: No such thing comes up in the case of the wind,
blowing
the sand.)
This
sketch of learning should remind us of BF Skinner, behaviorism; and
schedules
of reward and punishment (Catania & Harnad 1988). For it was
Skinner who
pointed out that we learn on the basis of feedback from the
consequences of
our behavior. But what Skinner did not provide was the internal
mechanism
for this sensorimotor capacity that we and so many of our
fellow-creatures
have, just as Gibson did not provide the mechanism for picking up
affordances.
Both these thinkers thought that providing internal mechanisms was
either not
necessary or not the responsibility of their discipline. They were
concerned
only with describing the input and the sensorimotor interactions, not
how a
sensorimotor system could actually do those things. So whereas they
were
already beginning to scratch the surface of the "what" of our
categorization
capacity, in input/output terms, neither was interested in the "how."
8. Instrumental (Operant, Reinforcement) Learning. Let us, too, set
aside the "how" question for the moment, and note that so-called
operant or
instrumental learning -- in which, for example, a pigeon is trained to
peck at
one key whenever it sees a black circle and at another key whenever it
sees a
white circle (with food as the feedback for doing the right thing and
no-food
as the feedback for doing the wrong thing) -- is already a primitive
case of
categorization. It is a systematic differential response to different
kinds of
input, performed by an autonomous adaptive system that responded
randomly at
first, but learned to adapt its responses under the guidance of
error-correcting feedback (thanks, presumably, to some sort of adaptive
change
in its internal state).
The
case of black vs. white is relatively trivial, because the animal's
sensory
apparatus already has those two kinds of inputs well-segregated in
advance --
although if, after training on just black and white, we began to
"morph" them
gradually into one another as shades of gray, and tested those
intermediate
shades without feedback, the pigeon would show a smooth "generalization
gradient," pecking more on the "black" key the closer the input was to
black,
more on the white key the closer the input was to white, and
approaching a
level of chance performance midway between the two. The same would be
true for
a human being in this situation.
9. Color Categories. But if the animal had color vision, and
we used blue and green as our inputs, the pattern would be different.
There
would still be maximal confusion at the blue-green midpoint, but on
either side
of that boundary the correct choice of key and the amount of pressing
would
increase much more abruptly Ð one might even say "categorically" --
than with
shades of gray. The reason is that between black and white there is no
innate
category boundary, whereas between green and blue there is (in animals
with
normal green/blue color vision). The situation is rather similar to hot
and
cold, where there is a neutral point midway between the two poles,
feeling
neither cold nor hot, and then a relatively abrupt qualitative
difference
between the "warm" range and the "cool" range in either direction.
10. Categorical Perception. This relatively abrupt perceptual
change at the boundary is called "categorical perception" (CP) and in
the case
of color perception, the effect is innate. Light waves vary in
frequency. We
are blind to frequencies above red (infrared, wavelength about 800 nm)
or below
violet (ultraviolet, wavelength about 400 nm), but if we did not have
color CP
then the continuum from red to violet would look very much like shades
of gray,
with none of those qualitative "bands" separated by neutral mixtures in
between
that we all see in the rainbow or the spectrum.
Our
color categories are detected by a complicated sensory receptor
mechanism, not
yet fully understood, whose components include not just light
frequency, but
other properties of light, such as brightness and saturation, and an
internal
mechanism of three specialized detectors selectively tuned to certain
regions
of the frequency spectrum (red, green, and blue), with an mutually
inhibitory "opponent-process"relation between their activities (red
being opposed to green
and blue being opposed to yellow). The outcome of this innate
invariance
extracting mechanism is that some frequency ranges are automatically
"compressed": we see them all as just varying shades of the same
qualitative
color. These compressed ranges are then separated from adjacent
qualitative
regions, also compressed, by small, boundary regions that look like
indefinite
mixtures, neutral between the two adjacent categories. And just as
there is compression
within each color range, there is expansion between them: Equal-sized frequency differences look
much smaller and are harder to detect when they are within one color
category
than when they cross the boundary from one category to the other
(Berlin &
Kay 1969; Harnad 2003).
Although
basic color CP is inborn rather than a result of learning, it still
meets our
definition of categorization because the real-time trial-and-error
process that "shaped" CP through error-corrective feedback from
adaptive consequences was
Darwinian evolution. Those of our ancestors who could make rapid,
accurate
distinctions based on color out-survived and out-reproduced those who
could
not. That natural selection served as the "error-correcting" feedback
on the
genetic trial-and-error variation. There are probably more lessons to
be
learned from the analogy between categories acquired through learning
and
through evolution as well as from the specific features of the
mechanism
underlying color CP -- but this brings us back to the "how" question
raised
earlier, to which we promised to return.
11. Learning Algorithms. Machine learning algorithms from
artificial intelligence research, genetic algorithms from artificial
life
research and connectionist algorithms from neural network research have
all
been providing candidate mechanisms for performing the "how" of
categorization.
There
are in general two kinds of learning models: so-called "supervised" and
"unsupervised" ones. The unsupervised models are generally designed on
the
assumption that the input "affordances" are already quite salient, so
that the
right categorization mechanism will be able to pick them up on the
basis of the
shape of the input from repeated exposure and internal analysis alone,
with no
need of any external error-correcting feedback.
By
way of an exaggerated example, if the world of shapes consisted of
nothing but
boomerangs and Jerry-Fodor shapes, an unsupervised learning mechanism
could
easily sort out their retinal shadows on the basis of their intrinsic
structure
alone (including their projective geometric invariants). But with the
shadows
of newborn chick abdomens, sorting them out as males and females would
probably
need the help of error-corrective feedback. Not only would the attempt
to sort
them on the basis of their intrinsic structural landscape alone be like
looking
for a needle in a haystack, but there is also the much more general
problem
that the very same things can often be categorized in many different
ways. It
would be impossible, without error-correcting supervision, to determine
which
way was correct in a given context . For the right categorization can
vary with
the context: sometimes we may want to sort baby chicks by gender,
sometimes by
species, sometimes by something else (Harnad 1987).
In
general, a nontrivial categorization problem will be "underdetermined."
Even if
there is only one correct solution, and even if it can be found by an
unsupervised mechanism, it will first require a lot of repeated
exposure and
processing. The figure/ground distinction might be something like this:
How, in
general, does our visual system manage to process the retinal shadows
of
real-world scenes in such a way as to sort out what is figure and what
is
ground? In the case of ambiguous figures such as Escher drawings there
may be
more than one way to do this, but in general, there is a default way to
do it
that works, and our visual systems usually manage to find it quickly
and
reliably for most scenes. It is unlikely that they learned to do this
on the
basis of having had error-corrective feedback resulting from
sensorimotor
interactions with samples of the endless possible combinations of
scenes and
their shadows.
12. Unsupervised Learning. There are both morphological and
geometric invariants in the sensory shadows of objects, highlighted
especially
when we move relative to them or vice versa; these can be extracted by
unsupervised learning mechanisms that sample the structure and the
correlations
(including covariance and invariance under dynamic sensorimotor
transformations). Such mechanisms cluster things according to their
sturctural
similarities and dissimilarities, enhancing both the similarities and
the
contrasts. An example of an unsupervised contrast-enhancing and
boundary-finding mechanism is "reciprocal inhibition," in which
activity from
one point in visual space inhibits activity from surrounding points and
vice-versa. This internal competition tends to bring into focus the
structure
inherent in and afforded by the input (Hinton & Sejnowsky 1999).
13. Supervised Learning. This kind of unsupervised clustering
based on enhancing structural similarities and correlations will not
work,
however, if different ways of clustering the very same sensory shadows
are
correct, depending on other circumstances (context-dependent
categorization).
To sort this out, supervision by error-corrective feedback is needed
too; the
sensorimotor structure and its affordances alone are not enough. We
might say
that supervised categories are even more underdetermined than
unsupervised
ones. Both kinds of category are underdetermined, because the sensory
shadows
of their members are made up of a high number of dimensions and
features, their
possible combinations yielding an infinity of potential shadows, making
the
subset of them that will afford correct categorization hard to find.
But
supervised categories have the further difficulty that there are many
correct
categorizations (sometimes an infinite number) for the very same set of
shadows.
If
you doubt this, open a dictionary again, pick any content word, say,
"table,"
then think of an actual table, and think of all the other things you
could have
called it (thing, object, vegetable, handiwork, furniture, hardwood,
Biedermeyer, even "Charlie"). The other names you could have given it
correspond to other ways you could have categorized it. Every category
has both
an "extension" (the set of things
that are members of that category) and an "intension" (the features
that make
things members of that category rather than another). Not only are all
things
the members of an infinite number of different categories, but each of
their
features, and combinations of features is a potential basis
(affordance) for
assigning them to still more categories. So far, this is again just
ontology.
But if we return to sensory inputs, and the problem facing the theorist
trying
to explain how sensorimotor systems can do what they do, then sensory
inputs
are the shadows of a potentially infinite number of different kinds of
things.
Categorization is the problem of sorting them correctly, depending on
the
demands of the situation.
Supervised
learning can help; if unsupervised learning ("mere exposure") cannot
find the
winning features, perhaps feedback-guided trial and error training will
do it,
as with the pigeon's black/white sorting and the chicken-sexing. There
are some
supervised learning algorithms so powerful that they are guaranteed to
find the
needle in the haystack, no matter how undetermined it is Ð as long
as it is
just underdetermined, not indeterminate (like the exact midpoint
between black
and white) or NP-complete -- and as long there is enough data and
feedback and
time (as, for the language-learning child, there is not, hence
the "poverty of the stimulus"; Wexler 1991). Our categorization
algorithms have to
be able to do what we can do; so if we can categorize a set of inputs
correctly, then those inputs must not only have the features that can
afford
correct categorization, but there must also be a way to find and use
those
affordances. (Figure 1 shows how a supervised neural net learns to sort
a set
of forms into 3 categories by compressing and separating their internal
representations in hidden unit space; Tijsseling & Harnad 1997.)
Figure 1. Upper: 3 sets of
stimuli presented to neural net: vertical arm of L much longer,
vertical and
horizantal about equal, horizontal much longer. Lower left: Position of the hidden-unit representations
of each of the three categories after auto-association but before
learning
(cubes represent Ls with long vertical arms, pyramids Ls with
near-equal arms,
spheres Ls with long horizontal arms). Lower right: Within-category compression and
between-category separation when the net has learned to separate the
three
kinds of input. (From Tijsseling & Harnad 1997.)
14. Vanishing Intersections? Fodor and others have sometimes
suggested otherwise: They have suggested that one of the reasons most
categories
can be neither learned nor evolved (and hence must be "innate" in some
deeper
sense than merely being a Darwinian adaptation) is the "vanishing
intersections" problem: If you go back to the dictionary again, pick
some
content words, and then look for the "invariance" shared by all the
sensory
shadows of just about any of the things designated by those words, you
will
find there is none: their "intersection" is empty. What do all the
shadows of
boomerangs or tables Ð let alone Jerry Fodors or chicken-bottoms
Ð have in
common (even allowing dynamic sensorimotor interactions with them)? And
if that
doesn't convince you, then what is the sensory shadow of categories
like "goodness," "truth," or "beauty"?
15. Direct Sensorimotor Invariants. There is no reason for invariance
theorists to back down from this challenge. First, it has to be pointed
out
that since we do manage to categorize correctly all those
things
designated by our dictionaries, there is indeed a capacity of ours that
needs
to be accounted for (see Appendix 1). To say that these categories are
"innate"
in a Cartesian, Platonic, or cosmogonic sense rather than just a
Darwinian
sense is simply to say that they are an unexplained, unexplainable
mystery. So
let us reject that. Let us assume that if organisms can
categorize, then
there must be a sensorimotor basis for that skill of theirs, and its
source
must be either evolution, learning, or both. Which means that there
must be
enough in those shadows to afford all of our categorization capacity.
16. Abstraction and Hearsay. Does it all have to be a
matter of direct sensorimotor invariants, always? No, but the path to
goodness,
truth and beauty requires us to trace the chain of abstraction
that
takes us from categories acquired through direct sensory experience to
those
acquired through linguistic "hearsay":
Consider
the five sensorimotor ways we can interact differentially with things,
the five
kinds of things we can do with things: We can see them, recognize
them, manipulate
them, name them or describe them. "Manipulate" in a
sense already
covers all five, because manipulating is something we do with
things;
but let us reserve the word "manipulate" for our more direct physical
interactions with objects, such as touching, lifting, pushing,
building,
destroying, eating, mating with, and fleeing from them. Naming them and
describing them is also a thing we do with them, but let us not subsume
those
two acts under manipulation. Seeing and recognizing are likewise things
we do
with things, but these too are better treated separately, rather than
as forms
of manipulation. And "seeing" is meant to stand in for all modes of
sensory
contact with things (hearing, smelling, tasting, touching), not just
vision.
Recognizing
is special, because it is not just a passive sensory event. When we
recognize
something, we see it as a kind of thing (or an individual) that
we have
seen before. And it is a small step from recognizing a thing as a kind
or an
individual to giving it a name. Seeing requires sensorimotor equipment,
but
recognizing requires more. It requires the capacity to abstract.
To
abstract is to single out some subset of the sensory input, and ignore
the
rest. For example, we may see many flowers in a scene, but we must
abstract to
recognize some of them as being primroses. Of course, seeing them as
flowers is
itself abstraction. Even distinguishing figure from ground is
abstraction. Is
any sensorimotor event not abstraction?
17.
Abstraction and Amnesia. To
answer, we have to turn to fiction. Borges, in his 1944 short story,
"Funes the
Memorious," describes a person who cannot abstract. One day Funes fell
off his
horse, and from then onward he could no longer forget anything. He had
an
infinite rote memory. Every successive instant of his experience was
stored
forever; he could mentally replay the "tapes" of his daily experience
afterwards, and it would take even longer to keep re-experiencing them
than it
had to experience them in the first place. His memory was so good that
he gave
proper names or descriptions to all the numbers -- "Luis Meli‡n
Lafinur,
Olimar, azufre, los bastos, la ballena, el gas, la caldera, NapolŽon,
Agust’n
de Ved’a" -- from 1 all the way up to enormous numbers (see Appendix
2). Each
was a unique individual for him. But, as a consequence, he could not do
arithmetic; could not even grasp the concepts counting and number. The
same
puzzlement accompanied his everyday perception. He could not understand
why we
people with ordinary, frail memories insist on calling a particular
dog, at a particular
moment, in a particular place, in a particular position, by the same
name that
we call it at another moment, a different time, place, position. For
Funes,
every instant was infinitely unique, and different instants were
incomparable,
incommensurable.
Funes's
infinite rote memory was hence a handicap, not an advantage. He was
unable to
forget -- yet selective forgetting, or at least selective ignoring, is
what is
required in order to recognize and name things. Strictly speaking, a
true Funes
could not even exist, or if he did, he could only be a passive
sensorimotor
system, buffeted about by its surroundings (like the sand by the wind).
Borges
portrayed Funes as having difficulties in grasping abstractions, yet if
he had
really had the infinite memory and incapacity for selective forgetting
that
Borges ascribed to him, Funes should have been unable to speak at all,
for our
words all pick out categories bases on abstraction. He should not have
been
able to grasp the concept of a dog, let alone any particular dog, or
anything
else, whether an individual or a kind. He should have been unable to
name
numbers, even with proper names, for a numerosity (or a numeral shape)
is
itself an abstraction. There should be the same problem of recognizing
either a
numerosity or numeral as being the same numerosity (numeral) on another
occasion as there was in recognizing a dog as the same dog, or as a dog
at all.
18.
Invariance and Recurrence. Funes was a
fiction, but Luria described a real person who had
handicaps that went in the same direction, though not all the way to an
infinite rote memory. In "The Mind of a Mnemonist" (1968) Luria
describes a
stage memory-artist, "S," whom he had noticed when S was a journalist
because
he never took notes. S did not have an infinite rote memory like
Funes's, but a
far more powerful and persistent rote memory than a normal person. When
he
performed as a memory artist he would memorize long strings of numbers
heard
only once, or all of the objects in the purse of an audience member. He
could
remember the exact details of scenes, or long sequences. He also had
synaesthesia, which means that sensory events for him were richer,
polysensory
experiences: sounds and numbers had colors and smells; these would help
him
remember. But his powerful rote memory was a handicap too. He had
trouble
reading novels, because when a scene was described, he would visualize
a
corresponding scene he had once actually seen, and soon he was lost in
reliving
his vivid eidetic memory, unable to follow the content of the novel.
And he had
trouble with abstract concepts, such as numbers, or even ordinary
generalizations that we all make with no difficulty.
What
the stories of Funes and S show is that living in the world requires
the
capacity to detect recurrences, and that that in turn requires the
capacity to
forget or at least ignore what makes every instant infinitely unique,
and hence
incapable of exactly recurring. As noted earlier, Gibson's (1979)
concept of an "affordance" captures the requisite capacity nicely:
Objects afford
certain sensorimotor interactions with them: A chair affords
sitting-upon;
flowers afford sorting by color, or by species. These affordances are
all
invariant features of the sensory input, or of the sensorimotor interaction with the input, and the
organism has to be capable of detecting these invariants selectively --
of
abstracting them, ignoring the rest of the variantion. If all
sensorimotor
features are somehow on a par, and every variation is infinitely
unique, then
there can be no abstraction of the invariants that allow us to
recognize
sameness, or similarity, or identity, whether of kinds or of
individuals.
19.
Feature Selection and Weighting. Watanabe's (1985)
"Ugly Duckling Theorem"
captures the same insight. He describes how, considered
only logically, there is no basis for saying that
the "ugly duckling" -- the odd swanlet among the several ducklings in
the Hans
Christian Anderson fable -- can be said to be any less similar to any
of the
ducklings than the ducklings are to one another. The only reason it
looks as if
the ducklings are more similar to one another than to the swanlet is
that our
visual system "weights" certain features more heavily than others -- in
other
words, it is selective, it abstracts certain features as
privileged. For
if all features are given equal weight and there are, say, two
ducklings and a
swanlet, in the spatial position D1, S, D2, then although D1 and D2 do
share
the feature that they are both yellow, and S is not, it is equally true
that D1 and S share the feature that they
are
both to the left of D2 spatially, a feature they do not share with D2.
Watanabe
pointed out that if we made a list of all the (physical and logical)
features
of D1, D2, and S, and we did not preferentially weight any of the
features
relative to the others, then S would share exactly as many features
with D1 as
D1 shared with D2 (and as D2
shared with S). This is an exact analogue of Borges's and
Luria's memory
effect, for the feature list is in fact infinite (it includes either/or
features too, as well as negative ones, such as "not bigger than a
breadbox,"
not double, not triple, etc.), so unless some features are arbitrarily
selected
and given extra weight, everything is equally (and infinitely) similar
to everything
else.
But
of course our sensorimotor systems do not give equal weight to all
features;
they do not even detect all features. And among the features they do
detect,
some (such as shape and color) are more salient than others (such as
spatial
position and number of feathers). And not only are detected features
finite and
differentially weighted, but our memory for them is even more finite:
We can see,
while they are present, far more features than we can remember
afterward.
20.
Discrimination Versus Categorization. The best
illustration of this is the
difference between relative and absolute discrimination that was
pointed out by
George Miller in his famous 1956 paper on our brains'
information-processing
limits: "The Magical Number 7+/-2". If you show someone an unfamiliar,
random
shape, and immediately afterward show either the same shape again or a
slightly
different shape, they will be able to tell you whether the two
successive
shapes were the same or different. That is a relative
discrimination, based
on a simultaneous or rapid successive pairwise comparison. But if
instead one
shows only one of the two shapes, in isolation, and asks which of the
two it
is, and if the difference between them is small enough, then the viewer
will be
unable to say which one it is. How small does the difference have to
be? The "just-noticeable-difference" or JND is the smallest difference
that we can
detect in pairwise relative comparisons. But to identify a
shape in
isolation is to make an absolute discrimination (i.e., a
categorization), and Miller showed that the limits on absolute
discrimination
were far narrower than those on relative discrimination.
Let
us call relative discrimination "discrimination" and absolute
discrimination "categorization." Differences have to be far greater in
order to identify what
kind or individual something is than for telling it apart from
something else
that is simultaneously present or viewed in rapid succession. Miller
pointed
out that if the differences are all along only one sensory dimension,
such as
size, then the number of JNDs we can discriminate is very large, and
the size
of the JND is very small, and depends on the dimension in question. In
contrast, the number of regions along the dimension for which we can
categorize
the object in isolation is approximately seven. If we try to subdivide
any
dimension more finely than that, categorization errors grow.
This
limit on categorization capacity has its counterpart in memory too: If
we are
given a string of digits to remember we -- unlike Luria's S, who can
remember a
very large number of them -- can recall only about 7. If the string is
longer,
errors and interference grow.
21.
Recoding and Feature Selection. Is
there any way to increase our capacity to make categorizations? One way
is to
add more dimensions of variation; presumably this is one of the ways in
which
S's synaesthesia helped him. But even higher dimensionality has its
limits, and
never approaches the resolution power of the JND of sensory
discrimination.
Another
way of increasing memory is by recoding. Miller showed that if we have
to
remember a string of 0's and 1's, then a string of 7 items is about our
limit.
But if we first learn to recode the digits into, say, triplets in
binary code,
using their decimal names -- so that 001 is called "one", 010 is called
"two,"
011 is called "three" etc., and we overlearn that code, so that we can
read the
strings automatically in the new code, then we can remember three times
as many
of the digits. The 7-limit is still there, but it is now operating on
the
binary triplets into which we have recoded the digits: 101 is no longer
three
items: it is recoded into one "chunk," "five." We have learned to see
the
strings in terms of bigger chunks -- and it is these new chunks that
are now
subject to the 7-limit, not the single binary digits.
Recoding
by overlearning bigger chunks is a way to enhance rote memory for
sequences,
but something similar operates at the level of features of objects:
Although
the number of features our sensory systems can detect in an object is
not
infinite, it is large enough so that if we see two different objects,
sharing
one or a few features, we will not necessarily be able to detect that
they
share features, hence that they are the same kind of object. This is
again a
symptom of the "underdetermination" mentioned earlier, and is related
to the
so-called "credit assignment problem" in machine learning: How to find
the
winning feature or rule among many possibilities (Sutton 1984)?
To be
able to abstract the shared features, we need supervised categorization
training (also called "reinforcement learning"), with trial and error
and
corrective feedback based on a large enough sample to allow our brains
to solve
the credit-assignment problem and abstract the invariants underlying
the
variation. The result, if the learning is successful, is that the
inputs are
recoded, just as they are in the digit string memorization; the
features are
re-weighted. The objects that are of the same kind, because they share
invariant
features, are consequently seen as more similar to one another; and
objects of
different kinds, not sharing the invariants, are seen as more
different.
This
within-category enhancement of perceived similarity and
between-category
enhancement of perceived differences is again the categorical
perception (CP)
described earlier in the case of color. The sensory "shadows" of light
frequency, intensity and saturation were recoded and re-weighted by our
evolved
color receptors so as to selectively detect and enhance the spectral
ranges
that we consequently see as red, yellow, etc.
22.
Learned Categorical Perception and the Whorf Hypothesis. When CP is an
effect of
learning, it is a kind of a Whorfian effect. Whorf (1956) suggested
that how
objects look to us depends on how we sort and name them. He cited
colors as an
example of how language and culture shape the way things look to us,
but the
evidence suggests that the qualitative color-boundaries along the
visible
spectrum are a result of inborn feature detectors rather than of
learning to
sort and name colors in particular ways. Learned CP effects do occur,
but they
are subtler than color CP, and can only be demonstrated in the
psychophysical
laboratory (Goldstone 1994, 2001; Livingston et al. 1998).
Figure
2 below illustrates this for a task in which subjects learned texture
categorization. For an easy categorization task, there was no
difference before
and after learning, but for a hard one, learning caused within-category
compression and between-category separation. (From Pevtzow & Harnad
1997).
Figure 2. Left :
Examples of the Easy (upper) and Hard (lower) texture categories.
Right :
Ratio of discrimination accuracy after/before learning (Post/Pre) in
the Easy
and the Hard task for Learners only. Separation is indicated by a ratio
>1
and compression by a ratio <1. Error bars indicate standard error.
There is
a significant compression within and near-significant separation
between for
the Hard task but nonsignificant separtation only for the Easy Task.
(From
Pevtzow & Harnad 1997.)
Yet
learned CP works much the way inborn CP does: Some features are
selectively
enhanced, others are suppressed, thereby bringing out the commonalities
underlying categories or kinds. This works like a kind of input filter,
siphoning out the categories on the basis of their invariant features,
and
ignoring or reducing the salience of non-invariant features. The
supervised and
unsupervised learning mechanisms discussed earlier have been proposed
as the
potential mechanisms for this abstracting capacity, with sensorimotor
interactions also helping us to converge on the right affordances,
resolving
the underdetermination and solving the credit-assignment problem.
Where
does this leave the concrete/abstract distinction and the
vanishing-intersections problem, then? In what sense is a primrose
concrete and
a prime number abstract? And how is "roundness" more abstract than
"round," and "property" more abstract still? Identifying any category
is always based on
abstraction, as the example of Funes shows us. To recognize a wall as a
wall
rather than, say, a floor, requires us to abstract some of its
features, of
which verticality, as opposed to horizontality, is a critical one here
(and
sensorimotor interactions and affordances obviously help narrow the
options).
But in the harder, more underdetermined cases like chicken-sexing, what
determines which features are critical? (The gist of this
underdetermination is
there in the Maine joke: "How's your wife?" "Compared to what?")
23.
Uncertainty Reduction. Although
categorization is an absolute judgment, in that it is based on
identifying an
object in isolation, it is relative in another sense: What invariant
features
need to be selectively abstracted depend entirely on what the alternatives
are, amongst which the isolated object needs to be sorted. "Compared to
what?"
The invariance is relative to the variance. Information, as we learn
from
formal information theory, is something that reduces the uncertainty
among
alternatives. So when we learn to categorize things, we are learning to
sort
the alternatives that might be confused with one another. Sorting walls
from
floors is rather trivial, because the affordance difference is so
obvious
already, but sorting the sex of newborn chicks is harder, and it is
even
rumoured that the invariant features are ineffable in that case: They
cannot be
described in words. That's why the only way to learn them is through
the months
or years of trial and error reinforcement training guided by feedback
under the
supervision of masters.
24.
Explicit Learning. But
let us not mistake the fact that it is difficult to make them explicit
verbally
for the fact that there is anything invisible or mysterious about the
features
underlying chicken-sexing -- or any other subtle categorization.
Biederman did
a computer-analysis of newborn chick-abdomens and identified the
winning
invariants described in terms of his "geon" features (Biederman &
Shiffrar
1987). He was then able to teach the features and rules through
explicit
instruction to a sample of novices so that within a short time they
were able
to sex chicks at the brown-belt level, if not the black belt level.
This
progress should have taken them months of supervised trial-and-error
training,
according to the grandmasters.
So if
we accept that all categorization, great and small, depends on
selectively
abstracting some features and ignoring others, then all categories are
abstract. Only Funes lives in the world of the concrete, and that is
the world
of mere passive experiential flow from one infinitely unique instant to
the
next (like the sand in the wind). For to do anything systematic or
adaptive
with the input would require abstraction, whether innate or learned:
the
detection of the recurrence of a thing of the same kind.
25.
Categorization Is Abstraction. What about degrees
of abstractness? (Having, with G.B. Shaw,
identified categorization's profession Ð abstraction -- we are now
merely
haggling about the price.) When I am sorting things as instances of a
round-thing
and a non-round-thing, I am sorting things. This thing is round, that
thing is
non-round. When I am sorting things as instances of roundness and
non-roundness, I am sorting features of things. Or rather, the things I
am
sorting are features (also known as properties, when we are not just
speaking
about them in a sensorimotor sense). And features themselves are things
too:
roundness is a feature, an apple is not (although any thing, even an
apple, can
also be a part, hence a feature, of another thing).
26.
Sensorimotor
Grounding:
Direct and Derivative. In principle, all
this sorting
and naming could be applied directly to sensorimotor inputs; but much
of the
sorting and naming of what we consider more abstract things, such as
numbers,
is applied to symbols rather than to direct sensorimotor interactions
with
objects. I name or describe an object, and then I categorize it: "A
number is
an invariant numerosity" (ignoring the variation in the kinds or
individuals
involved). This simple proposition already illustrates the adaptive
value of
language: Language allows as to acquire new categories indirectly,
through "hearsay," without having
to go through the time-consuming and risky process of direct
trial-and-error
learning. Someone who already knows can just tell me the
features of
an X that will allow me to recognize it as an X.
(This
is rather like what Biederman did for his experimental subjects, in
telling
them what features to use to sex chickens, except that his method was
not pure
hearsay, but hybrid: It was show-and-tell, not just tell, because he
did not
merely describe the critical features verbally; he also pointed
them out
and illustrated them visually. He did not first pretrain his subjects
on
geon-naming, as Miller's subjects were pretrained on naming binary
triplets.)
27.
The Adaptive Advantage of Language: Hearsay. If Biederman had
done it all with words,
through pure hearsay, he would have demonstrated the full and unique
category-conveying power of language: In sensorimotor learning, the
abstraction
usually occurs implicitly. The neural net in the learner's brain does
all the
hard work, and the learner is merely the beneficiary of the outcome.
The
evidence for this is that people who are perfectly capable of sorting
and
naming things correctly usually cannot tell you how they do it.
They may
try to tell you what features and rules they are using, but as often as
not
their explanation is incomplete, or even just plain wrong. This is what
makes
cognitive science a science; for if we could all make it explicit,
merely by
introspecting, how it is that we are able to do all that we can do,
then our
introspection would have done all of cognitive science's work for it
(see
Appendix 1). In practice we usually cannot make our implicit knowledge
explicit,
just as the master chicken-sexers could not. Yet what explicit
knowledge we do
have, we can convey to one another much more efficiently by hearsay
than if we
had to learn it all the hard way, through trial-and-error experience.
This is
what gave language the powerful adaptive advantage that it had for our
species
Cangelosi & Harnad 2001; see Figure 3).
Figure 3. An artificial-life simulation of
mushroom foragers. Mushroom-categories could be learned in two
different ways,
by sensorimotor "toil" (trial-and-error learning with feedback from the
consequences of errors) or linguistic "theft" (learning from
overhearing the
category described; hearsay). Within a very few generations the
linguistics "thieves" out-survive and out-reproduce the sensorimotor
toilers. (But note
that the linguistically based categories must be grounded in
sensorimotor
categories: it cannot be theft all the way down.) (From Cangelosi &
Harnad
2001.)
Where
does this leave prime numbers then, relative to primroses? Pretty much
on a
par, really. I, for one, do not happen to know what primroses are. I am
not
even sure they are roses. But I am sure I could find out, either
through direct
trial and error experience, my guesses corrected by feedback from the
masters,
and my internal neural nets busily and implicitly solving the
credit-assignment
problem for me, converging eventually on the winning invariants; or, if
the
grandmasters are willing and able to make the invariants explicit for
me in
words, I could find out what primroses are through hearsay. It can't be
hearsay
all the way down, though. I will have had to learn some things
ground-level
things the hard, sensorimotor way, if the words used by the
grandmasters are to
have any sense for me. The words would have to name categories that I
already
have.
Is it
any different with prime numbers? I know they are a kind of number. I
will have
to be told about factoring, and will probably have to try it out on
some
numbers to see what it affords, before recognizing that some kinds of
numbers
do afford factoring and others do not. The same is true for finding out
what
deductive proof affords, when they tell me more about further features
of prime
numbers. Numbers themselves I will have had to learn at first hand,
supervised
by feedback in absolutely discriminating numerosities, as provided by
yellow-belt arithemeticians -- for here too it cannot be hearsay all
the way
down. (I will also need to experience counting at first hand, and
especially
what "adding one" to something, over and over again, affords.)
28.
Absolute Discriminables and Affordances. But is there any
sense in which primroses
or their features are "realer" than prime numbers and their features?
Any more
basis for doubting whether one is really "out there" than the other?
The sense
in which either of them is out there is that they are both absolute
discriminables: Both have sensorimotor affordances that I can detect,
either
implicitly, through concrete trial-and-error experience, guided by
corrective
feedback (not necessarily from a live teacher, by the way: if, for
example,
primroses were edible, and all other flowers toxic, or prime
numerosities were
fungible, and all others worthless, feedback from the consequences of
the
sensorimotor interactions would be supervision enough); or explicitly,
through
verbal descriptions (as long as the words used are already grounded,
directly
or recursively, in concrete trial-and-error experience; Harnad
1990).The
affordances are not imposed by me; they are "external" constraints,
properties
of the outside world, if you like, governing its sensorimotor
interactions with
me. And what I do know of the outside world is only through what it
affords (to
my senses, and to any sensory prostheses I can use to augment them).
That 2+2
is 4 rather than 5 is hence as much of a sensorimotor constraint as
that
projections of nearer objects move faster along my retina than those of
farther
ones.
29.
Cognitive Science is Not Ontology. Mere cognitive
scientists (sensorimotor
roboticists, really) should not presume to do ontology at all, or
should at
least restrict their ontic claims to their own variables and terms of
art -- in
this case, sensorimotor systems and their inputs and outputs. By this
token,
whatever it is that "subtends" absolute discriminations -- whatever
distal
objects, events or states are the sources of the proximal projections
on our
sensory surfaces that afford us our capacity to see, recognize,
manipulate,
name and describe them -- are all on an ontological par; and subtler
discriminations
are unaffordable.
Where
does this leave goodness, truth and beauty, and their sensorimotor
invariants?
Like prime numbers, these categories are acquired largely by hearsay.
The
ethicists, jurists and theologians (not to mention our parents) tell us
explicitly
what kinds of acts and people are good and what kind are not, and why
(but the
words in their explicit descriptions must themselves be grounded,
either
directly, or recursively, in sensorimotor invariants: again, categories
cannot
be hearsay all the way down.). We can also taste what's good and what's
not
good directly with our senses, of course, in sampling some of their
consequences. We perhaps rely more on our own sensory tastes in the
case of
beauty, rather than on hearsay from aestheticians or critics, though we
are no
doubt influenced by them and by their theories too. The categories
"true" and "false" we sample amply through direct sensory experience,
but there too, how
we cognize them is influenced by hearsay; and of course the formal
theory of
truth looks more and more like the theory of prime numbers, with both
constrained by the affordances of formal consistency.
30.
Cognition Is Categorization. But, at bottom, all
of our categories consist in ways we
behave differently toward different kinds of things, whether it be the
things
we do or don't eat, mate with, or flee from, or the things that we
describe,
through our language, as prime numbers, affordances, absolute
discriminables,
or truths. And isn't that all that cognition is for -- and about?
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Appendix
1.
There
is nothing
wrong with the "classical theory" of categorization.
Eleanor
Rosch has suggested that because we cannot state
the invariant basis on which we categorize, that invariance must not
exist
(Rosch & Lloyd 1978), and hence there is something wrong with the
so-called "classical" theory of categorization, according to which we
categorize on the
basis of the invariant features that are necessary and sufficient to
afford
categorization.
Not
only do I not think there's anything the least bit
wrong with that "classical
theory," but I am pretty confident that there is no non-magic
alternative to it. Rosch's alternative was to vacillate rather vaguely
between
the idea that we categorize on the basis or "prototypes" or on the
basis of "family resemblances". Let's consider each of these candidate
mechanisms in
turn:
To categorize
on the basis of prototypes would be to identify a bird as a bird
because it
looks more like the template for a typical bird than the template for a
typical
fish. This would be fine if all, many, or most of the things we
categorize
indeed had templates, and our internal categorization mechanism could
sort
their sensory shadows by seeing which template they are closest to; in
other
words, it would be fine if such a mechanism could actually generate our
categorization capacity.
Unfortunately
it cannot. Template-matching is not very successful among the many
candidate
machine-learning models, and one of the reasons it is unsuccessful is
that it
is simply not the case that everything is a member of every
category, to different degrees. It is not true ontologically that a
bird is a
fish (or a table) to a certain degree; nor is it true functionally that
sensory
shadows of birds can be sorted on the basis of their degree of
similarity to
prototypical birds, fish or tables. So prototype/template theory is a
non-starter as a mechanism for our categorization capacity. It might
explain
our typicality judgments Ð is this a more typical bird than that --
but being
able to make a typicality judgment presupposes being able to
categorize;
it does not explain it: Before I can say how typical a bird this is, I
first
need to identify it as a bird!
So if
not prototypes, what about family-resemblances, then? What are family
resemblances? They are merely a cluster of either/or features: This is
an X, if
it has feature A or B or not C. Either/or features (disjunctive
invariants) are
perfectly classical (so forget about thinking of family-resemblances as
alternatives to classical theories of categorization). The problem is
that
saying that some features are either/or features leaves us no closer to
answering "how" than we were before we were informed of this. Yes, some
of the
affordances of sensory shadows will be either/or features, but what we
need to
know is what mechanism will be able to find them!
The
last Roschian legacy to category theory is the "basic object" level,
vs. the
superordinate or subordinate level. Here too it is difficult to see
what, if
anything, we have learned from Roschian theory.
If
you point to an object, say, a table, and ask me what it is, chances
are that I
will say it's a table, rather than a Biedermeyer, or furniture, or
"Charlie".
So what? As mentioned earlier, there are many ways to categorize the
same
objects, depending on context. A context is simply a set of
alternatives among
which the object's name is meant to resolve the uncertainty (in
perfectly
classical information-theoretic
terms). So when you point to a table and ask me what it is, I
pick "table" as the uncertainty-resolver in the default context (I may
imagine that
the room contains one chair, one computer, one waste-basket and one
table. If I
imagine that it contains four tables, I might have to identify this one
as the
Biedermeyer; and if there are four Biedermeyers, I may have to hope you
know
I've dubbed this one "Charlie."
So
much for subordinate levels. The same path can be taken for
superordinate
levels. It all devolves on the old Maine joke, which comes close to
revealing a
profound truth about categories: "How's Your wife?" Reply: "Compared to
what?"
If we were just discussing the relative amount you should invest in
furniture
in your new apartment, as opposed to accessories, and you forgot what
was in
the adjacent room and asked what was in there (when there was just a
table) I
might reply furniture. If we were discussing ontology, I might say
"vegetable"
(as opposed to animal or mineral). Etc.
So
citing the "basic object level" does not help explain our
categorization
capacity; that's just what one arbitrarily assumes the default context
of
interconfusable alternatives to be, given no further information. The
only
sense in which "concrete" objects, directly accessible to our senses,
are
somehow more basic, insofar as categorization is concerned, than more
"abstract" objects, such as goodness, truth or beauty is that
sensorimotor
categories must be grounded in sensory experience and the content of
much of
that experience is fairly similar and predictable for most members of
our
species.
Appendix Two. Associationism
begs the question of
categorization.
The
problem of association is the problem of rote pairing: an object with
an
object, a name with a name, a name with an object. Categorization is
the
problem of recognizing and sorting objects as kinds based on
finding the
invariants underlying sensorimotor interactions with their shadows.
Associationism had suggested that this was just a matter of learning to
associate tokens (instances, shadows) of an object-type with tokens of
its
type-name -- as indeed it is, if only we can first figure out which
object-tokens are tokens (shadows) of the same object-type! Which is in
turn
the problem of categorization. Associationism simply bypassed the real
problem,
and reduced learning to the trivial process of rote association,
governed by
how often two tokens co-occurred (plus an unexplicated influence of how
"similar" they were to one another).
Some
associative factors are used by contemporary unsupervised learning
models,
where internal co-occurrence frequencies and similarities are used to
cluster
inputs into kinds by following and enhancing the natural landscape of
their
similarities and dissimilarities. But this is internal association
among
representational elements and patterns (e.g., units in a neural
network), not
external association between input tokens. And its scope is limited, as
we have
seen, for most of the shadows of most of the members of most of the
categories
in our dictionary could not be sorted into their respective categories
by
unsupervised association alone, because of underdetermination. Nor is
supervised learning merely rote association with the added associative
cue of
the category-name (as provided by the supervisory feedback). The hard
work in
these learning models is done by the algorithm that solves the
credit-assignment problem by finding the winning invariance in the
haystack --
and no model can do this at human categorization-capacity scale just
yet.
Critics of associationism, however, drew the incorrect conclusion that
because
(1) we don't know what the invariance is in most cases and (2)
association is
ill-equipped to find it in any case, it follows that there either is
no
invariance, or our brains must already know it innately in some
mysterious way.