Bruce Schuman
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Synthetic Dimensionality
Synthetic Dimensionality | Forum on Conceptual Structure
Design for a Transcendental Bridge
Linkage of the universal and the particular

Introduction to the Theory of Concepts
General principles of conceptual structure

The Dimensionality of Concept Structure
Similarities and differences in category formation

The Universal Hierarchy of Abstraction
Framework for a mathematical epistemology

Synthetic Dimsionality
The recursive algebra of semantic space

Bibliography
Foundations of this model

Resume
Background and history

THE DIMENSIONALITY OF CONCEPT STRUCTURE:
Similarities and Differences in Category Formation


CONTENTS

  1. The Process of Category Formation
  2. Concept Formation in the Life of Helen Keller
  3. The Dimensional Model of Concept Formation
  4. 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  110

                                                           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 ordinal descriptors.


                 |-------------------|--------------|--------------|
       Ordinal 2:|   uncomfortable   | comfortable  | uncomfortable|
                 |----|----|----|----|----|----|----|----|----|----|
       Ordinal 1:|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.