THE DIMENSIONALITY OF CONCEPT STRUCTURE:
Similarities and Differences in Category Formation
CONTENTS
- The Process of Category Formation
- Concept Formation in the Life of Helen Keller
- The Dimensional Model of Concept Formation
- Abstraction as the Renaming of Boundary Values
The 1987 Bantam Book publication Science, Order, and Creativity, by
David Bohm and F. David Peat, is a fertile source of ideas in the
epistemology of science. Originating in years of dialogue between these
two scientists, this book stimulates creative thinking in the science of
mind and spirit, and its open and intuitive style clarifies complex ideas
often obscured in technical writing.
As preface to this brief discussion of the dimensional approach to
conceptual modeling, I provide a basic exposition of the
fundamentals of concept formation, taken from Bohm and Peat. Their extended quotes in the
first two sections of this article are worth careful study.
The expansion of these ideas in the second two sections involves the
sort of analysis of concept structure discussed in artificial
intelligence and mathematical semantics. The most difficult challenges
today in these subjects do not involve complex or arcane technical
details, but rather fundamental underlying philosophic principles.
Mathematical theories are no better than the conceptual insights on
which they are based, and the simple overview provided by Bohm and Peat
provides a clear starting point for analysis.
1. The Process of Category Formation
From Science, Order, and Creativity, by David Bohm and F. David Peat,
Bantam Books, 1987, p.112:
Our first notions of order depend on our ability to perceive
similarities and differences. Indeed, there is much evidence which shows
that our vision, as well as other senses, works by selecting
similarities and differences. This suggests that perception begins
through the gathering of differences as the primary data of vision,
which are then used to build up similarities. The order of vision
proceeds through the perception of differences and the creation of
similarities of these differences.
In thought a similar process takes place, beginning first with the
formation of categories. This categorization involves two actions:
selection and collection. According to the common Latin root of
these two words, select means "to gather apart" and collect means "to
gather together". Hence categories are formed as certain things are
selected, through the mental perception of their differences from some
general background.
The second phase of categorization is that some of the things that have
been selected (by virtue of their difference from the background) are
collected together by regarding their differences as unimportant while,
of course, still regarding their common difference from the background
as important. Thus several birds of different size and posture may be
abstracted together from the general background of a tree without giving
particular attention to the individual differences between them. The
birds, however, clearly fall into a different category from any
squirrels which are found in the same tree. Categorization therefore
involves the combined action of selection and collection.
In the process of observing a flock of birds in a tree, the category of
birds is formed by putting things together that are simultaneously
distinguished from those that do not belong to this category -- for
example, from squirrels. In this way, sets of categories are formed,
and these, in turn, influence the ways in which things are selected and
collected. Selection and collection therefore become the two,
inseparable sides of the one process of categorization.
The determination of similarities and differences can go on
indefinitely. As some differences assume greater importance and others
are ignored, as some similarities are singled out and others neglected,
the set of categories changes. Indeed, the process of categorization is
a dynamical activity that is capable of changing in a host of ways as
new orders of similarity and difference are selected.
Most categories are so familiar to us that they are used almost
unconsciously. However, from time to time, as the result of some
important change in the way we see the world, or as our experience is
extended, new categories come into being. Categories are formed which
never existed before and new sets of similarities and differences are
considered as relevant in entirely new ways.
The creation of new categories relies on a perception that takes place
as much in the mind as through the senses. To understand the creative
nature of this process, and indeed to develop a theme which will be used
throughout this book, the idea of intelligence will be introduced.
The word intelligence is often used in a general and fairly loose way
today, but something of its original force can be found in the Latin
root intelligere, which carries the sense of "to gather in between".
It recalls the colloquialism "to read between the lines." This notion of
intelligence, which acts as the key creative factor in the formation
of new categories, can be contrasted with the intellect. The past
participle of intelligere is in fact intellect, which could then be
thought of as "what has been gathered." Intellect, therefore, is
relatively fixed, for it based primarily on an already existing set of
categories. While the intelligence is a dynamical and creative act of
perception through the mind, the intellect is something more limited and
static.
Categories therefore emerge through the free play of the mind in which
new forms are perceived through the creative act of intelligence and
then are gradually fixed into systems of categories. But this system of
categories always remains fluid and open to further change, provided
that the mind itself is open to the creative action of intelligence.
2. Concept Formation in the Life of Helen Keller
The life story of Helen Keller provides an ideal case study through
which to examine the process of concept formation, particularly because
the history of her cognitive development was so closely observed and
documented by her teacher, and also because an explanation of her
experiences can be drawn in very bold and simple strokes. From Peat and
Bohm, pp36-7:
Here the case of Helen Keller, who was taught by Anne Sullivan, is
particularly illuminating. When Sullivan came to teach this child, who
had been blind and deaf from an early age and was therefore unable to
speak, she realized that she would have to give Helen unrestricted love
and total attention.
The key step was to teach Helen to form a communicable concept. This
she could never have learned before, because she had not been able to
communicate with other people to any significant extent. Sullivan,
therefore, caused Helen, as if in a game, to come into contact with
water in a wide variety of different forms and contexts, each time
scratching the word water on the palm of her hand. For a long time,
Helen did not grasp what this was all about. But suddenly, she realized
that all these different experiences referred to one substance in many
aspects, which was symbolized by the word water on the palm of her
hand... Thus the different experiences were implied in some sense as
being equal, by the common experience of the word water being scratched
on her hand. It is worthwhile bringing out in more detail just what was
involved in this extraordinary act of creative perception. Up to that
moment, Helen Keller had perhaps been able to form concepts of some
kind, but she could not symbolize them in a way that was communicable
and subject to linguistic organization. The constant scratching of the
word water on the palm, in connection with the many apparently radically
different experiences, was suddenly perceived as meaning that, in some
fundamental sense, these experiences were essentially the same.
To return, for a moment, to the idea of a metaphor, A could represent
her experience of water standing still in a pail, while B would
represent her experience of water flowing out of a pump. As Helen
herself said, she initially saw no relationship between these
experiences. At this stage, her perception may be put as A not= B. Yet
the same word "water" was scratched on her hand in both cases. This
puzzled her very much, for it meant in some way Anne Sullivan wanted to
communicate that an equivalence existed between two very different
experiences, in other words that A = B.
Eventually, Helen suddenly perceived (of course, entirely nonverbally,
since she had as yet no linguistic terms to express her perception) that
A and B were basically similar, in being different forms of the same
substance, which was represented symbolically by the word "water"
scratched on her palm. At this point, there must have been in Helen a
state of vibrant tension, and indeed of intense creative perceptive
energy, which was in essence similar to that arising in a poet who is
suddenly aware of a new metaphor. However, in the case of Helen Keller,
the metaphor did not stop here, but went on to undergo a further rapid
unfoldment and development. Thus, as she herself said later, she
suddenly realized that everything has a name. This too must have been a
nonverbal flash of insight because she did not yet have a name for the
concept of a name.
This perception very probably had its origin in a yet higher order of
metaphor, suggested by the fact that Anne Sullivan had been playing a
similar "game" with her for many weeks, in which many different "words"
had been scratched on her hand, each associated to a number of different
but similar experiences. All these experiences were in this way seen to
be fundamentally related, in that they were examples of a single yet
broader concept, i.e., that of naming things. To Helen, this was an
astonishing discovery, for she had in this way perceived the whole
general relationship of symbol to concept, starting with water and going
on almost immediately to an indefinite variety of things that could be
extended without limit.
In summary, Bohm and Peat, p115:
When Helen Keller experienced her flash of insight, she saw the
essential similarity between all the different experiences of water.
Anne Sullivan had played a key part in this by helping Helen to select
these experiences from the general background and flux of experience, by
including them in a kind of game. Helen's moment of insight was the
perception of her first category. But this went much further than a
simple gathering of basically similar instances, for it had a name that
was communicable and which could therefore be used to symbolize the
category in thought and elevate it into a concept.
3. The Dimensional Model of Concept Formation
As characterized here by Peat and Bohm, the process of conceptualization
involves a dual discovery: the realization that there is an abstract
common factor which can link, or collect, or gather some set of distinct
factors into an abstract unity -- and the discovery that this abstract
categorical unity can itself be assigned a uniquely identifying label or
name. In the theory of synthetic dimensionality, all the distinctly
articulated symbolic elements of this process are defined and graphically
represented as dimensions: the letters are dimensions of the written word
"water", the dissimilar sensory experiences with at least one common
factor are dimensions of the general experience of water, and the abstract
general concept "water" itself -- defined in various dimensions
characterized by these experiences, and abstractly encoded or labeled or
named by those letters -- is the common factor, link, or dimension, in
which all these levels of abstraction were united as one abstract idea.
It is possible to diagram the process of symbolic abstraction,
indicating how higher levels of abstraction can be inductively assembled
from lower level dimensions. This diagram shows the process whereby the
temporally-separated and once thought to be unique experiences of water
were grouped together by Helen Keller, through an abstract feature they
held in common, showing these here as "values" along the single dimension
"water":
Experiences at different times:
| |
| |<------A water in a pail
Time | |
Invariant | W |
Common | A |<------B water running from a tap
Factor----->| T |
| E |
| R |<------C water in form of rain
| |
| |
| |<------D water on cloth to wash face
| |
Fig. 1
It is possible to further differentiate this diagram, by showing
distinct features of each of these experiences as possible values along
their own particular dimension.
| |
| |
| |<------A water in a pail (cool, deep, slow
| | moving, near metal, quiet, wet)
| W |
| A |<------B water running from a tap (hot, steamy
| T | thin, fast moving, near metal, wet)
| E |
| R |<------C water in form of rain (cool, in
| | space, drops, loud, fast moving, wet)
| |
| |<------D water on cloth to wash face (warm,
| | near cloth, quiet, damp)
| |
Fig. 2
Thus, Helen Keller's original experience might be expressed as A not= B
not= C not= D, because in many of the dimensions or indices by which these
experienced could be described, their values were quite different. One was
hot, another cold. One was quiet, another loud. In some, the water was
moving, in others, not moving. Yet there were some discernible common
factors in each of these different experiences, particularly wetness, and
perhaps others associated with sound and smell, and these allowed her to
define an equation wherein she could differentiate A and B and C and D,
and say that in a certain dimension, W (Water), these values were equal.
As we differentiate the distinct features used to describe "sensory"
experience, what we observe is that the dimensions of these descriptions
are necessarily the quantifiable linear/numeric dimensions of the physical
sciences and the "empirical plane". We see here a kind of hierarchy of
levels of abstraction, composed of types of dimensions, with the distinct
features of individual and independent experiences quantified in terms of
linear measures, such as heat, distance, rate, or viscosity.
Thus, we see the abstract general concept "water" defined at least one
level of abstraction above the sensory dimensions which index or describe
its particular qualities.
And while these values (cool, deep, slow moving...) do not suggest
themselves any common dimension for a linear ordering, each of them is
defined in a fundamental dimension of linear order, the same "quantifiable
dimensions" that characterize scientific description. Thus, any experience
characterized as "cool", or "deep", or "wet", could be associated with
some distinct ordering or measurement, addressing the questions "how
cool?", "how deep?", and "how wet?"
Conceptual Analysis:
Particular Experience A in five dimensions
| w |
| a |
| t |<-----A:\temperature\cool
| e | sensory
| r |<-----A:\depth\deep dimensions
| | and their
| i |<-----A:\speed\slow moving values of
| n | particular
| |<-----A:\touch\near metal experience
| p | A
| a |<-----A:\viscosity\wet
| i |
| l |
Fig. 3
This diagram suggests that particular experience A, "water in a pail",
can be characterized by values in at least five dimensions. And the
general concept "water", as we saw above, was a function of a series of
temporally independent experiences. There is thus a hierarchy of
independent dimensions which at a first level combined to characterize the
individual experiences A, B, C, D, which were then linked together at a
second level through the common abstract concept.
Common Factor | Independent Experience | Dimension | Value
| |
| |----------------------------------------
| | | |<-1 temperature: cool
| | |P|<-2 depth: deep
| |<-A water in pail |A|<-3 speed: not moving
| | |I|<-4 distance: near metal
| | |L|<-5 viscosity: wet
| |----------------------------------------
| | | |<-1 temperature: hot
| | |T|<-2 width: thin
| |<-B water from tap |A|<-3 speed: moving
| W | |P|<-4 touch: near metal
| A | | |<-5 viscosity: wet
| T |----------------------------------------
| E | |R|<-1 temperature: cool
| R | |A|<-2 location: in space
| |<-C water as rain |I|<-3 motion: moving
| | |N|<-4 shape: drops
| | | |<-5 viscosity: wet
| |----------------------------------------
| | |C|<-1 temperature: warm
| | |L|<-2 position: near cloth
| |<-D water on cloth |O|<-3 touch: damp
| | |T|<-4 Y:a
| | |H|<-5 Z:b
| |----------------------------------------
| | Fig. 4
Thus we have a process of "ascending hierarchical induction", where,
from the multiplicity of independent experience, the single factor common
to all these experiences was identified by its various recurring aspects.
But the conceptualization of "water" at a "higher level of abstraction"
does not somehow constrain the concept to remain divorced from experience.
The inductive bottom-up hierarchy can be immediately turned around into a
deductive top-down hierarchy, and in every category or dimension of
experience, the common factor of the experience is "water". Once
identified, there is nothing "abstract" about it: you can touch it, feel
it, wash your face with it, swim in it, go out in the rain into it --
while simply recognizing that this "it" is "the same thing every time".
And out in the rain, or when swimming, the factor of "water" may be only
one of many: temperature, friends, sounds, sun, wind, location, all
combining to form a complex experience in which water is but one factor.
But it is clear that abstract concepts such as water can be understood
as "inductively defined time invariant common factors", which have been
abstractly synthesized through a series of "temporally independent"
physical experiences. And each of these physical experiences can be
described in terms of time invariant independent dimensions of variation,
specifically characterized by fluctuations in value in these dimensions.
Immediate physical conditions can be completely described in terms of
physical dimensionality, and the abstract description by which we
characterize these experiences are assembled inductively from these
dimensional characterizations. In these terms, we are thus able to
propose the argument that "abstract qualities are hierarchical dimensional
assemblies of sensory quantities".
4. Abstraction as the Renaming of Boundary Values
In the above description, we have characterized the values for
quantitative dimensions in terms of ordinal values, such as "warm", or
"damp".
But these expressions can be defined as "boundary value cuts" on a
quantified interval scale of measurement. And this is true in general of
the verbal description of physical experience, which in day-to-day life is
seldom based on the need for exactly calibrated description. It usually
isn't necessary to distinguish between 59 degrees Fahrenheit and 60
degrees, and most of us, living in temperate climates, would be
comfortable describing a sixty degree day as "cool" or "mild". Such a
description is, in fact, a calibration on an abstract scale of values
which is "close enough" for most purposes.
Ordinal: very cold cold cool mild warm hot very hot
|----|----|----|----|----|----|----|----|----|
Interval: 10 20 30 40 50 60 70 80 90 100
Fig. 5
Clearly, these ordinal values are flexible approximations defined by
boundary values on the interval scale. We assign to the word "mild" any
meaning we choose, by selecting upper and lower boundary values. One
objective in developing the principles of synthetic dimensionality is to
allow us to formally define, and ultimately to quantify, the numeric
dimensionality of abstract qualities. This is not to say that some
abstract quality will always be defined the same every time. Quite the
opposite: the ad hoc spontaneity of synthetic dimensionality allows us to
define a unique circumstance-specific definition cascade for the values of
a qualitative dimension.
In every act of communication involving abstract ordinal values, there
is an implicit and potential decomposition of these values through a
descending hierarchical series of free-choice boundary value cuts on
lower-level dimensions.
Abstract descriptive concepts are defined in terms of lower-level
concepts, and at bottom level, ordinal values are mapped onto interval or
ratio scales of value by assigning upper and lower boundary values as
cut-off points for the meaning of the ordinal value. These first-level
ordinal values are then combined into a second level of abstraction in a
similar way, with values in a higher level of abstraction defined by
boundary value choices in the first set of qualitative descriptors.
|-------------------|--------------|--------------|
Nominal: | uncomfortable | comfortable | uncomfortable|
|----|----|----|----|----|----|----|----|----|----|
Ordinal: |very cold| cold |cool|mild|warm|hot |very hot |
|----|----|----|----|----|----|----|----|----|----|
Interval: |10 |20 |30 |40 |50 |60 |70 |80 |90 |100 |
|----|----|----|----|----|----|----|----|----|----|
Fig. 6
Gradations and shades of meanings are defined in terms of specific
values in specific sensory and quantifiable dimensions of experience.
"Wet? How wet? Cool? How cool? Red? How red?" A discriminating
wine-taster has learned to "educate his palate", and can articulate a
complex range of variation in a number of linked dimensions. By asking
enough of these kinds of questions, and defining sufficiently fine ways to
measure variation, we become discriminating and articulate connoisseurs of
sensory experience.
|