Bruce Schuman
PO Box 23346
Santa Barbara, CA 93121
(805) 966-9515
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 UNIVERSAL HIERARCHY OF ABSTRACTION
Framework for a Mathematical Epistemology

A Project Overview and Introduction
From the ORIGIN conference on The WELL, July - August, 1991

Bruce Schuman
PO Box 23346, Santa Barbara CA 93121
bruce@originresearch.com

Contents

  1. Preface
  2. Richard Feynman on the Hierarchy of Ideas
  3. A Systematic Interpretation of Feynman's Intuition
  4. Levels of Abstraction/Levels of Ideas
  5. The Bridge Across Consciousness
  6. Epistemological Concepts Defined as Polar Opposites Across Levels of Abstraction
  7. Logical Dynamics Defined on the Universal Hierarchy
  8. The Dimensional Structure of Mind
  9. A Critique of the Top-Down Hierarchical Perspective
  10. Concluding Comments from the WELL/ORIGIN Conference
  11. References
1. Preface

From Paul Davies, God and the New Physics, Simon & Schuster, Touchstone Books, 1983, p61:

The distinction being made here is sometimes referred to as "holism versus reductionism". The main thrust of western scientific thinking over the last three centuries has been reductionist. Indeed the use of the word "analysis" in the broadest context nicely illustrates the scientist's almost unquestioning habit of taking a problem apart to solve it. But of course some problems (such as jigsaw puzzles) are only solved by putting them together, -- they are synthetic or "holistic" in nature. The picture on a jigsaw puzzle, like the speckled newspaper image of a face, can only be perceived at a higher level of structure than the individual pieces.
In this brief quote, physicist and popular science writer Paul Davies summarizes the ideas presented in this discussion of the Universal Hierarchy of Abstraction (or UHA).

The argument presented here is essentially a simple one: all conceptual structure and logical processes can be seen as organized through a single hierarchical framework that defines the relationships between parts and wholes. This hierarchical framework is implicit in all cognition, and its characteristics account for many of the observed properties of cognitive and logical systems.

The description of this hierarchical structure can become quite complex, and in other articles discussing the dimensional structure of class and category formation, I consider the technical problems involved in representing abstractions through a hierarchical decomposition cascade of what I call "synthetic dimensions".

This present article outlines the major features of what seems to me to be the general structure of ideas and categories, providing the "big picture" from a top-down and intuitive perspective, and offering a general description of this dimensional theory of mind (Section 8). I argue that all ideas and categories can be interpreted as positioned within the unitary framework of this single all inclusive universal hierarchy, which is defined across a series of levels of abstraction (levels of analysis), ranging from the microcosm to the macrocosm, or from the particular to the universal. Most epistemological ideas can be directly defined in terms of this underlying structure, and in my list of "polar opposites" defined on the hierarchy (Section 6), I provide a glossary of systematic definitions which seem intuitively appealing, and which are consistent with modern psychology, particularly right brain/left brain research.

I begin with an introductory overview from Nobel physicist Richard Feynman, who sketches out the entire concept in rudimentary and intuitive terms. I then follow his initial discussion with a systematic interpretation of his comments, and a semi-formal analytic description of the characteristics of the Universal Hierarchy and its role as the general framework for cognitive/conceptual organization.


2. Richard Feynman on the Hierarchy of Ideas

From Richard P. Feynman, The Character of Physical Law, quoted in God and the New Physics, p224:

We have a way of discussing the world . . . at various hierarchies, or levels. Now I do not mean to be very precise, dividing the world into definite levels, but I will indicate, by describing a set of ideas, what I mean by hierarchies of ideas.

For example, at one end we have the fundamental laws of physics. Then we invent other terms for concepts which are approximate, which have, we believe, their ultimate explanation in terms of the fundamental laws. For instance, "heat". Heat is supposed to be jiggling, and the word for a hot thing is just the word for a mass of atoms which are jiggling. But for a while, if we are talking about heat, we sometimes forget about the atoms jiggling -- just as when we talk about the glacier we do not always think of the hexagonal ice and the snowflakes which originally fell. Another example of the same thing is a salt crystal. Looked at fundamentally it is a lot of protons, neutrons, and electrons; but we have this concept "salt crystal", which carries a whole pattern already of fundamental interactions. An idea like pressure is the same.

Now if we go higher up from this, in another level we have properties of substances -- like "refractive index", how light is bent when it goes through something; or "surface tension", the fact that water tends to pull itself together, both of which are described by numbers. I remind you that we have got to go through several laws down to find out that it is the pull of the atoms, and so on. But we still say "surface tension", and do not always worry, when discussing surface tension, about the inner workings.

On, up in the hierarchy. With the water we have waves, and we have a thing like a storm, the word "storm" which represents an enormous mass of phenomena, or a "sun spot", or "star", which is an accumulation of things. And it is not worth while always to think of it way back. In fact we cannot, because the higher up we go the more steps we have in between, each one of which is a little weak. We have not thought them all through yet.

As we go up in this hierarchy of complexity, we get to things like muscle twitch, or nerve impulse, which is an enormously complicated thing in the physical world, involving an organization of matter in a very elaborate complexity. Then come things like "frog".

And then we go on, and we come to words and concepts like "man" and "history", or "political expediency", and so forth, a series of concepts which we use to understand things at an ever higher level.

And going on, we come to things like evil, and beauty, and hope...

Which end is nearer to God; if I may use a religious metaphor. Beauty and hope, or the fundamental laws? I think that the right way, of course, is to say that what we have to look at is the whole structural interconnection of the thing [emphasis added]; and that all the sciences, and not just the sciences but all the efforts of intellectual kinds, are an endeavor to see the connections of the hierarchies, to connect beauty to history, to connect history to man's psychology, man's psychology to the working of the brain, the brain to the neural impulse, the neural impulse to the chemistry, and so forth, up and down, both ways. And today we cannot, and it is no use making believe that we can, draw carefully a line all the way from one end of this thing to the other, because we have only just begun to see that this is a relative hierarchy.

And I do not think that either end is closer to God.

3. A Systematic Interpretation of Feynman's Intuition

After years of study, and based on the evidence I have gathered, I have become persuaded that the Universal Hierarchy of Abstraction, whether viewed as an as-yet imperfectly perceived ontological (Platonic) a priori, -- ie, an existing structure and property of being awaiting discovery, -- or merely as a very interesting and useful general engineering heuristic, is the constant background context and logical framework of all human discussion, analysis, logic and thinking. I believe that this underlying idea is essentially quite simple, and that a clear recognition of this general structure can provide a powerful and comfortable insight into the epistemology of both science and intuition.

If this general and rather simple idea were properly expounded and widely recognized, I believe the resulting insight could clear away an enormous amount of conflicting and overlapping theoretical terminology. There are thousands of philosophical theories which tend to discuss aspects and properties of this abstract general structure, each from a different point of view, and in terms of a different system of categories, -- and each of which don't quite fully perceive this underlying general principle. The Universal Hierarchy of Abstraction is basically a simple idea, but I believe it provides powerful and reliable approach to addressing most problems in cognitive psychology, linguistics, and epistemology.

Says Feynman:

We have a way of discussing the world. . .at various hierarchies, or levels. Now I do not mean to be very precise, dividing the world into definite levels, but I will indicate, by describing a set of ideas, what I mean by hierarchies of ideas.
Feynman, and many others, intuitively recognize this idea, and some people apparently assume that it is so obvious it barely needs discussing. Everywhere one looks in science, the concept of "levels of abstraction" (or analysis) is implicit and taken for granted. But nowhere that I have discovered are these levels of abstraction defined in a systematic or formal way, defining clearly the extent to which this framework is the implicit and underlying structure for all logical processes.

Any concept or comment we make is posed at some level of abstraction, for some reason which makes one particular level appropriate.

When Feynman says he doesn't mean to be precise, "dividing the world into definite levels", he is not saying that the hierarchy of ideas is not distributed across a series of "levels", but that these levels are not fixed and immutable, and are flexible, adaptive, and conformable to our purposes.

"For example, at one end we have the fundamental laws of physics."

Please note: Feynman has mentioned here, very quickly, that there are "ends" to this hierarchy of ideas, and "at one end" are the laws and ideas of physics. This is exactly what I am describing when I outline the structure of ideas in terms of the following hierarchy, through which I classify branches of knowledge in terms of the "implicit dimensionality" (ie, the number of implicit distinctions) of their concept types.


|<--Absolute Unity-Wholeness-Oneness/Absolute Abstraction/God (?)
|<--Theology/Metaphysics (ungrounded very complex composite variables)
|<--Relatively "molar" (holistic) sciences (complex composite variables)
|<--Relatively "molecular" (empirical) sciences (simple composite variables)
|<--Physics (simplest possible "atomic" variables)
|<--Actuality/Absolute physical ground of being
This hierarchy of ideas begins with the laws of physics, at the atomic level, as our most basic and fundamental ideas are mapped onto the smallest possible (one dimensional) units of conceptualization and experience, and then "ascends the hierarchy" as the conceptual units or elements become increasingly "larger", more inclusive, and more abstract, incorporating into an integrated composite block a wider and wider range of implicit dimensionality. Feynman illustrates this point by saying:
Then we invent other terms for concepts which are approximate, which have, we believe, their ultimate explanation in terms of the fundamental laws. For instance, "heat". Heat is supposed to be jiggling, and the word for a hot thing is just the word for a mass of atoms which are jiggling. But for a while, if we are talking about heat, we sometimes forget about the atoms jiggling -- just as when we talk about the glacier we do not always think of the hexagonal ice and the snowflakes which originally fell. Another example of the same thing is a salt crystal. Looked at fundamentally it is a lot of protons, neutrons, and electrons; but we have this concept "salt crystal", which carries a whole pattern already of fundamental interactions. An idea like pressure is the same.
What he is saying is that it is often times convenient to talk about "large block" conceptual structures, -- such as an ice crystal, or a far bigger block variable, such as a glacier. To describe a single object as a "glacier" involves a lot of nested implicit definition, and we don't mention all of the internal atomic structure of the glacier. The same is true for the example of the salt crystal: it is a "gestalt", a "holistic pattern", which we can discuss as a unit, even though we recognize that it possess detailed internal structure which we are choosing not to mention, perhaps for reasons of convenience and economy.

Feynman repeats this same point in terms of concepts such as "surface tension":

Now if we go higher up from this, in another level we have properties of substances -- like "refractive index", how light is bent when it goes through something; or "surface tension", the fact that water tends to pull itself together, both of which are described by numbers. I remind you that we have got to go through several laws down to find out that it is the pull of the atoms, and so on. But we still say "surface tension", and do not always worry, when discussing surface tension, about the inner workings.
Feynman uses the concept "higher", indicating that we are ascending the epistemological hierarchy. This clearly shows that there is a single linear directed quality to this hierarchy; "lower" and "higher" are well-defined linear directions, and there is nothing vague or uncertain about this. We go "up the hierarchy" to higher levels of abstraction, and "down the hierarchy" to lower levels. Pure and simple.

When Feynman says "we don't always worry about the inner workings", he is saying something very fundamental about the nature of meaning and conceptual structure. Higher-level concepts or abstractions have "implicitly nested meaning", which we don't necessary specify (ie, we use the concept "ice" without talking about atomic structure), largely for reasons of convenience and economy, even if our definitions begin to become "presumptive", in that they *imply* the implicit undefined meaning, without making it explicit.

Now Feynman begins to discuss truly complex holistic gestalts:

On, up in the hierarchy. With the water we have waves, and we have a thing like a storm, the word "storm" which represents an enormous mass of phenomena, or a "sun spot", or "star", which is an accumulation of things. And it is not worth while always to think of it way back.
He makes two important points here:

It is not "worth while to think of it way back", by which he means that we don't necessarily need to trace the "fine structure" of a phenomenon such as a star or a sunspot or a storm, but can discuss it as a gestalt, a single concept containing many implicit and non-specified distinctions. This "worth whileness" is a function of mental economy. We need a direct one-word shorthand description of a general phenomena, even if all the details are not explicitly defined. We are content, in this case, to leave these details "implicitly nested", or implicitly contained within the larger concept.

And he says "the higher we up we go the more steps we have in between, each one of which is a little weak. We have not thought them all through yet". This is probably one of the most important observations in all of epistemology, and has to do with the weak chain of definition that tends to link high-level abstractions with their empirical grounding in physics. It is the weakness of this conceptual bridgework which results in the fragmenting of the body of human knowledge into distinct and separate categories which cannot communicate with one another.

Our inability to recognize that all ideas are fundamentally distributed across one single hierarchical continuum, as a function of their block variable size (ie, number of implicit dimensions/distinctions), is probably the single most important reason for the non-scientific weakness of philosophy, and the traditional animosity between "science" and "religion". A major objective of this ORIGIN conference, and these theories, is to define this intervening conceptual bridgework with exact algebraic precision, showing how the linkage between these levels of knowledge can be accurately created.

As we go up in this hierarchy of complexity, we get to things like muscle twitch, or nerve impulse, which is an enormously complicated thing in the physical world, involving an organization of matter in a very elaborate complexity. Then come things like "frog".

And then we go on, and we come to words and concepts like "man" and "history", or "political expediency", and so forth, a series of concepts which we use to understand things at an ever higher level.

And going on, we come to things like evil, and beauty, and hope...

Here, Feynman has clearly illustrated the "position" in the hierarchy of abstraction of high-level philosophical ideas like "political expediency", and then shows that even higher, we find ideas such as "beauty and evil and hope". Recognizing that such ideals assume a very high position in this hierarchy of abstraction is, in my opinion, essential to the task of defining these abstract ideas with exact and meaningful precision. What we need to do, in order to ground abstract philosophic ideas accurately and soundly, is to trace their definition-chain up through this hierarchy, just as Feynman is suggesting we must, when he says "the higher up we go the more steps we have in between, each one of which is a little weak. We have not thought them all through yet."

Feynman concludes by asking:

Which end is nearer to God; if I may use a religious metaphor. Beauty and hope, or the fundamental laws? I think that the right way, of course, is to say that what we have to look at is the whole structural interconnection of the thing; and that all the science, and not just the sciences but all the efforts of intellectual kinds, are an endeavor to see the connections of the hierarchies, to connect beauty to history, to connect history to man's psychology, man's psychology to the working of the brain, the brain to the neural impulse, the neural impulse to the chemistry, and so forth, up and down, both ways.
Here, he outlines the fundamentally interdisciplinary nature of scientific/philosophic inquiry; an objective of this ORIGIN conference is to show how these interconnections are defined across the hierarchy, and to the extent possible, build an actual online network
(The Bridge Across Consciousness) capable of linking these traditionally disparate conceptual domains.

In his final sentences, Feynman suggests that

And today we cannot, and it is no use making believe that we can, draw carefully a line all the way from one end of this thing to the other, because we have only just begun to see that this is a relative hierarchy. And I do not think that either end is closer to God.
On this point, regarding the possibility of "drawing a line", I want to open up another line of argument, regarding "ad hoc top-down decomposition" and the algebraic structure of hierarchy, because I believe that the principles of synthetic dimensionality provide ways to overcome the top-down rigidity of "non-relativistic" Aristotelian category systems, and thus, do indeed provide us a way to "carefully draw a line from one end to the other" of this hierarchy of ideas. It is this "careful line" which I call "The Bridge Across Consciousness".


4. Levels of Abstraction/Levels of Ideas

In diagrammatic representations of the UHA, its distribution of levels can be displayed across the horizontal axis:


           Levels:  | | | | | | | | | | | | | | | | | | |
                    |<------Main Index of Levels------->|
           Concrete |                                   |   Abstract
        Actuality-->|                                   |<--Theory
This same diagram can be rotated 90 degrees, to show the main index vertically, thus representing the hierarchy of ideas shown in the previous section:

|<--Absolute Unity-Wholeness-Oneness/Absolute Abstraction/God (?)
|<--Theology/Metaphysics (ungrounded very complex composite variables)
|<--Relatively "molar" (holistic) sciences (complex composite variables)
|<--Relatively "molecular" (empirical) sciences (simple composite variables)
|<--Physics (simplest possible variables)
|<--Actuality/Absolute physical ground of being
Here, each of these domains or disciplines operates in its own range, and establishes its meaning by making interpretive connections or mappings across levels.

Rotating again:


Bottom                            Top
<----------------------------------->
| | | | | | | | | | | | | | | | | | |
|<- Concrete Actuality (lowest level)
|<--->|<- Physics
      |<--->|<- Small block variable science (chemistry, biology)
            |<------->|<- large block variable science (psychology, anthrology)
                      |<----------->|<- intuitive philosophy (very large block variable)
                                    |<- absolute unity/undefined (highest level)
Thus, in Feynman's terms, we are talking about defining the linkage or "steps between these levels", from the "high" levels at the left of this chart, to the "low" levels on the right. And our fundamental proposition regarding the epistemology of philosophic intuition is that its conceptual structures (beauty, evil, hope, political expediency) tend to be weakly or incompletely mapped to their grounding in the conceptual elements and sciences on the right side of the chart. This leaves their meaning "floating" on the left side of the chart, depending largely on social convention or the traditions of some "school of thought" (or on the dictates of some "privileged tribunal").

The ungrounded qualities of philosophic or metaphysical ideas can be graphically represented by a break in our "careful line" across levels:


        \ \
Bottom  / /                        Top
<------/ / -------------------------->
| | | |\ \ | | | | | | | | | | | | | |
        \ \
        / / <--fragmenting schism in definition chain linking
        \ \    philosophical abstraction to the empirical ground,
               creating gap between holistic intuition and science;
               can sometimes be very wide
There are many simultaneous interpretations and meanings for this general organizational chart, but two of the most important are these:
  • This chart outlines the general structure of the organization of ideas as institutionalized by science and academia (in university departmental organization, library organization, etc.)
  • It is also, and at the same time, a general description of the organizational structure of ideas in the human mind, as displayed through cognitive activity.


5. The Bridge Across Consciousness

By understanding all ideas and conceptual structure as organized within a single multi-level/hierarchical framework, we provide the essential key for establishing linkage across these levels. This link across levels of abstraction, connecting universal principles and holistic intuition at one end of the hierarchy with empirical particulars at the other, I call "The Bridge Across Consciousness". By interpreting all knowledge and cognition as organized within a single framework, we create a unifying approach the the structure of knowledge, that provides us with systematic criteria for assigning consistent definitions to all aspects of our logic and epistemology.

But as Feynman has hinted, "drawing as careful line" drawn across all levels of abstraction is a subtle challenge, and after years of working with this general intuition, I have evolved what seems to me to be the key to understanding the hierarchical organization of conceptual structure: the principles of what I call "synthetic dimensionality". This idea is is essentially a theory of measurement and concept formation based on the concept of dimension, and the similarity between the concepts of "dimension" and "ordered class".

The theory of "synthetic dimensionality" is a theory of conceptual structure, based on the idea that any ordered class, -- or element of an ordered class, -- can be interpreted as a dimension. This theory appears to provide a single universal and "recursive" (ie, defined in terms of "self-similar" units) descriptive language to model all cognitive structure. In terms of this theory, any cognitive unit (word, idea) can be understood in terms of dimensionality. An intuitive summary and overview of this idea is provided in Section 8 of this discussion, "The Dimensional Theory of Mind".

Without fully explaining this idea here, it is possible to describe the overall large-scale structure of "cognitive space" in terms of the hierarchical organization of synthetic dimensions. The following diagram is a representation of the Universal Hierarchy of Abstraction, describing the number of implicitly nested simultaneous dimensions of any particular concept, as based on the following general propositions:

  • Mind is (or can be defined as) an assembly of concepts
  • Concepts are hierarchically nested by level of abstraction
  • All concepts can be ordered on the Universal Hierarchy
  • All concepts can be "assembled from dimensions"
This diagram of the "main index" (or "main dimension") of the Universal Hierarchy of Abstraction describes the distribution of analytic "levels of abstraction" that are organized across it. The diagram shows the mapping from an "actual object" on the empirical plane, to its abstract symbolic representation X (ie, we call the actual object "X"), and then considers the higher level (increasingly general) abstract categories into which we may place X.

                         "THE BRIDGE ACROSS CONSCIOUSNESS"

 "Left Brain - Lowest level"                             "Right Brain - Highest level"

   PARTICULARS<------------------------------------------------------>UNIVERSALS
                                         |  
     zero <--few dimensions              |               many dimensions--> infinite
                                         | 
  <--Lowest level of abstraction         |  Highest level of abstraction-->
     Levels: | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
             |                                                       |
             |<---------------- Main Index of Levels --------------->|
             |                                                       |
             |-->"bottom up"                            "top down"<--|
Concrete     |                                                       |   Abstract
Actuality -->|                                                       |<--Theory,
  Actual     |-->induction                               deduction<--|   Holism,
  Object X   |-->synthesis                                analysis<--|   Metaphysics
             |<--differences                          similarities-->|
                               content<--|-->form
Many fundamental epistemological concepts can be readily defined in terms of this diagram. The "directed" nature of the hierarchy implies that one can "look two ways" through the logical organization, "up" the hierarchy from the lower levels of concrete specificity and empirical fact, towards increasingly the abstract and universal conceptual/ categorical structures of theory, or "down" the hierarchy, in the reverse order.

The concepts "top down" and "bottom up" are defined in terms of the directed quality of the hierarchy.

The "levels" in the hierarchy are directly analogous to computer pathname addresses, such as the address "WELL (conference system)/ORIGIN (particular conference in The WELL)/s49 (particular topic in ORIGIN)/#3 (particular response number in that topic)", and the "cuts" between the levels are directly analogous to the function performed by the / symbol in this address.

"Form" and "content" are polar opposite characteristics of an identical data structure element at some cut between levels. Gazing at the element "down" the hierarchy, we see it as "content". Gazing "up" the hierarchy at the element, we see it as "form".

The "directed" properties of inductive and deductive logic flow are defined on the hierarchy, as inductive processes move "up" the hierarchy from concrete empirical specifics to abstract generalities, and deductive processes move "down" the hierarchy in the opposite direction.

"Analysis" is a differentiating logic flow that moves down the hierarchy from wholes to parts, while "synthesis" is an integrative logic that moves up the hierarchy from parts to wholes.

"Similarity" and "difference" can be defined as directed properties, moving up the hierarchy as "similarities" between "two separate elements" define a single common increasingly abstract category, and down the hierarchy as this unity is differentiated.


6. Epistemological Concepts Defined as Polar Opposites Across Levels of Abstraction

Many common epistemological concepts can be defined as characterizations of data structures, distributed across a hierarchy of abstraction. This below list is one current version, and is similar to other lists found in various contemporary psychology books which describe "right brain/left brain" differences.

One can think of this list as characterizing the two sides of the "Janus-faced" / operator, which distinguishes levels of abstraction (as in computer pathnames, for example).

Each of these pairs of definitions describes slightly different aspects or properties of conceptual data structure, and merits a separate discussion at some appropriate later opportunity.


             EPISTEMOLOGICAL CONCEPTS DEFINED AS POLAR OPPOSITES
                  UP AND DOWN THE HIERARCHY OF ABSTRACTION:

                       <--Lower Level / Higher Level-->
                       <-- left brain - right brain -->
                1. Continuous----------------------Discrete
                2. Analog--------------------------Digital
                3. Numeric-------------------------Alphabetic
                4. Empirical-----------------------Abstract
                5. Particular----------------------General
                6. Specific------------------------Universal
                7. Factual-------------------------Theoretical
                8. Concrete------------------------Symbolic
                9. Local---------------------------Global
               10. Actual--------------------------Conceptual
               11. Quantitative--------------------Qualitative
               12. Reductionist--------------------Holistic
               13. Analytic------------------------Synthetic
               14. Decomposition-------------------Composition
               15. Deduction-----------------------Induction
               16. Part----------------------------Whole
               17. Subset--------------------------Superset
               18. Serial--------------------------Parallel
               19. Sequential----------------------Simultaneous
               20. One at a time-------------------All at the same
                     in single file                  time, at once
               21. Clock time/immediate------------Timeless/eternal
                     now                             now
               22. Content-------------------------Form
               23. Diversity-----------------------Unity
               24. "The Many"----------------------"The One"
               25. "The Relative"------------------"The Absolute"
               26. "Science"-----------------------"Religion"					   


7. Logical Dynamics Defined on the Universal Hierarchy

We can define most if not all logical processes as "directed" across levels of the UHA. This diagram shows the directed property of several basic kinds of logic:

     
            Levels: | | | | | | | | | | | | | | | | | | |
                    |<------Main Index of Levels------->|
                    |-->"bottom up"        "top down"<--|
        Concrete    |                                   |   Abstract
        Actuality-->|                                   |<--Theory
                    |-->induction           deduction<--|   
                    |-->synthesis            analysis<--|
                    |<--differences      similarities-->|
Across this structure we can define the following logical processes:
  • deduction/induction: deduction "descends" the hierarchy from general principles to particulars, while induction "ascends" from particulars to generals. The scientific method generally involves induction, and its conclusions are only "probable", because its certainties are particulars which do not absolutely establish the certainty of a general proposition.
  • analysis/synthesis: reductionist analysis "takes wholes apart", differentiating higher-level units into lower-level "parts", while synthesis integrally combines parts into higher-level wholes.
  • "leap of faith": an incompletely mapped or undefined bridging across many levels from lower to higher, based on a holistic intuition.
  • interpretation: the meaning or intent of some general high-level abstraction is defined in terms of some particular context. An example is the interpretation of the general laws of the U.S. Constitution by the Supreme Court in terms of specific laws.
  • interpolation: the reverse of interpretation, whereby some general or higher-level category is inductively inferred or hypothesized from lower-level instances.
This list can be expanded to include other kinds of logical processes.


8. The Dimensional Structure of Mind

This theory can be summarized as follows:

  • A dimension is a distinction that creates a range of values, from lower to higher.
  • A concept can be understood as a nested set of implicit distinctions, which we make explicit when we precisely define the concept.
  • Thus, all concepts are "made out of dimensions", in the sense that their meaning is "assembled from dimensions", across an implicit cascade with empirical measurements at the lowest level.
  • The Bridge Across Consciousness and the UHA are organized across a single "main dimension" or range of values, and all other dimensions and distinctions branch from this central form.
  • Thus, concepts at different levels of abstraction can be organized in terms of "how many implicit dimensions they contain".
  • Empirical concepts contain the fewest dimensions, and highly abstract general concepts (perhaps represented by a single word) may implicitly contain a great many dimensions.
  • Dimensions "contained within" an abstract concept are implicitly nested, not explicitly defined, and must therefore be understood through a consistent interpretive scheme which "assigns meaning" to these higher-level abstractions. If the mapping or "meaning cascade" from the abstract concept to its empirical grounding is ambiguous or undefined, the abstract concept has no exact or specific meaning.
  • The meaning of an abstract concept is thus the particular hierarchical cascade of linked and nested dimensions though which it maps to its ground in the "empirical plane" of direct experience.
  • Abstract concepts necessarily involve interpretive ambiguity because their meaning involves implicitly nested dimensionality, which must be made explicit and exact before an abstract concept has an unambiguous meaning. The "empirical plane" is the foundation for stable hierarchical concept assembly, because empirical concepts are one-dimensional, and involve an isomorphic or directly proportional analogic mapping from symbolic element to physical element, whereas abstract concepts involve (perhaps undefined) intervening levels or layers of abstraction.
The Universal Hierarchy of Abstraction is the "conceptual mainframe" for all cognition. It is the macro-concept, which contains all other concepts, positioning their organization within a single framework. It defines the general "operating system" for cognition, providing the basic structures and relationships through and in terms of which all conceptual processing occurs. It is the implicit and unconscious background and framework for all of our model-building and analysis, and for all our perception. All of academic and scientific disciplines are organized within it, and our language is organized in terms of its structure. Logical processes move across its levels, from higher to lower and back again. Analog/empirical conceptual processes are embedded within it, and it provides the context which gives them their meaning.


9. A Critique of the Top-down Hierarchical Perspective

The cognitive/AI theorist Douglas Hofstadter, author of Goedel, Escher, Bach, and Metamagical Themas, among other books, has energetically explored and expounded hierarchical ideas. In Metamagical Themas, he discusses his own original notion that consciousness and conceptual structure is strictly hierarchical, and his eventual conversion away from this idea.

The algebraic structure of the LISP programming language is intimately related to the general principles of Synthetic Dimensionality, generally because its basic data structure is a "list" (a linear/sequentially-ordered set of elements), any element of which may be another list. That definition is isomorphic to (ie, exactly the same thing as) the definition of a synthetic dimension. In Ch. 19, "LISP: Recursion and Generality", he says (p452):

A programmer's instinct says that you can cumulatively build up a system, encapsulating all the complexity of one layer into a few functions, then building the next layer up by exploiting the efficient and compact functions defined in the preceding layer. This hierarchical mode of buildup would seem to allow you to make arbitrarily complex actions be represented at the top level by very simple function calls. In other words, the functions at the top of the pyramid are like "cognitive" events, and as you move down the hierarchy of lower-level functions, you get increasingly many ever-dumber subroutines, until you bottom out in a myriad calls of trivial "sub-cognitive ones". All this sounds very biological, -- even tantalizingly close to being an entire resolution of the mind-brain problem. In fact, for a clear spelling-out of just that position, see Daniel Dennett's book Brainstorms, or perhaps worse, see parts of my own Goedel, Escher, Bach!

Yes, although I don't like to admit it, I too have been seduced by this recursive vision of mechanical mentality, resembling nothing so much as an army, with its millions of unconscious robot privates carrying out the desires of its top-level cognitive general, as conveyed to them by large numbers of obedient and semi-bright intermediaries...

This, in the opinion of my current self, is a crazy vision, and my reasons for thinking so are presented in Chapter 26, "Waking up from the Boolean Dream".

In Ch. 26, he continues this discussion (p653), which incidentally is reprinted from the book Interdisciplinary Messages:
In a normal program, you can account for every single operation at the bit level by looking "upward" towards the top-level program. You can trace a high-level function call downward: It calls subroutines that call other subroutines that call this particular machine-language routine that uses these words and in which this particular bit lies. So there is a high-level, global reason why this particular bit is being manipulated.

By contrast, in an ant colony, a particular ant's foray is not the carrying out of some global purpose. [How do we know that??] It has no interpretation in terms of the overall colony's goals; only when many such actions are considered at once does their statistical quality then emerge as purposeful or interpretable. Ant actions are not the "translation into machine language" of some "colony-level program". No one ant is essential; even large numbers of ants are dispensable. All that matters is the statistics; thanks to it, the information moves around at a level far above that of the ants. Ditto for neural firings in brains. Not ditto for most current AI architecture.

AI researchers started out thinking that they could reproduce all of cognition through a 100 percent top-down approach: functions calling subfunctions calling subsubfunctions and so on, until it all bottomed out in some primitives. [note: one problem with this method, as per Roger Schank, is that the "primitives" are not very primitive...!] Thus intelligence was thought to be hierarchically decomposable, with high-level cognition at the top driving low-level cognition at the bottom. There were some successes and some difficulties, -- difficulties particularly in the realm of perception. Then along came such things as production systems and pattern directed inference. Here, some bottom-up processing was allowed to occur within an essentially top-down context. Gradually, the trend has been shifting. But there is still a large element of top-down quality in AI.

It is my belief that until AI has been stood on its head and is 100% bottom-up, it won't achieve the same level or type of intelligence that human have. To be sure, when that kind of architecture exists, there will still be high-level global, cognitive events, -- but they will be epiphenomenal, like those in a brain. They will not in themselves be computational. Rather, they will be constituted out of, and driven by, many smaller computational events, rather than the reverse. In other words, subcognition at the bottom will drive cognition at the top. And, perhaps most importantly, the activities that take place at that cognitive top level will neither have been written nor anticipated by any programmer. This is the essence of what I call "statistically emergent mentality".

Hofstadter goes on to mention that connectionist and PDP research falls under the heading of the bottom-up statistical method.

But with deserved respect for Hofstadter and the PDP connectionist schools of thought, I continue to be persuaded that consciousness, as we experience it, and as we direct our own minds and bodies through it, is a top-down intentional process, and that the problems with this method that Hofstadter (and others) mention can be overcome through two basic principles inherent in the methods of synthetic dimensionality: 1) the method of ad hoc top-down decomposition, and 2) the reduction of all "primitives" to only one: synthetic dimension. When we show that *all* conceptual structure can be built or assembled from this one single element, and when we show the absolutely flexibility of this method through ad hoc assembly, we overcome all the objections I know about to the very advantageous and intuitively comfortable top-down methods of classical AI.


10. Concluding Comments from the WELL/ORIGIN Conference

I picked up my copy of Quarterman's The Matrix last night, and bought a few other books, including Wilber's Eye to Eye, (because of his hierarchical discussion of mandala, as mentioned in ori 89), -- and also a pioneering book in ecology/biology, entitled Hierarchy: Perspectives for Ecological Complexity (University of Chicago Press, 1982), by T. F. H. Allen and Thomas B. Starr.

Hierarchy becomes one more jewel in my slowly growing collection of materials that vigorously advocate hierarchical models of being, mind, or reality, and reinforce and further illustrate the many aspects of this underlying universal theme.

As their conclusion makes clear, they struggled with this book, saying things that sound to me so very familiar. Their concluding paragraph:

If this book has been hard going in the reading, we promise it was no easier in the writing. But think back, how much easier could we have made it? We contend, not much. Intuition led us to some strange places, and we have done our best to explain how we got there. We realize that our treatment is probably cumbersome and it is certainly primitive. Perhaps we read like men possessed; we feel better after the exorcism. We feel this book was written because the times imposed it upon the authors. There is a paradigm shift occurring in biology, and this book is one of the manifestations of that change. We recognize that changes in scientific worldviews are important political events within the scientific community, and so offer ourselves as commentators as the struggle ensues. (Allen and Starr, p260)
In this ORIGIN conference, I turn loose the methodology of algebraic hierarchical analysis on not only biology or ecology, but the entire spectrum of academic scientific and spiritual disciplines, that spectrum which is itself a hierarchy (of levels of abstraction). This book, with its roots in Arthur Koestler's The Ghost in the Machine, and The Act of Creation, -- and, of course, in von Bertalanffy's General System Theory, -- offers a rich banquet of insights into universal formal structure.

And their ontology of this idea is identical to the position I took at the beginning of this discussion:

We do not mean to imply that reality, independent of our cognizance, is in its nature hierarchical; in fact, we are not sure what that could mean let alone what it does mean. What we are trying to say is that somewhere between the world behind our observations and human understanding, hierarchies enter into the scheme of things. Simon (1962) makes the point that

If there are important systems in the world that are complex without being hierarchic, they may to a considerable extent escape our observation and our understanding. Analysis of their behavior would involve such detailed knowledge and calculation of the interactions of their elementary parts that it would be beyond our capacities of memory or computation.

Simon could be interpreted as suggesting that hierarchical structure is a consequence of human observations. Even if this is seen as limiting the significance of hierarchical conceptions (we take an opposite view), hierarchical approaches are at least of heuristic value.

The classical notion of hierarchy involves discrete levels. Although they may be both conceptually and pedagogically helpful, the implicit discontinuities between even levels that appear singularly real and discrete are to some extent arbitrary. Discrete levels need to be recognized as convenience, not truth. Even so, some arbitrary levels of organization are of more general application than others.

Philosophers are careful to identify whether an argument pertains to existence in terms of objective external reality, -- that is to say, whether an argument is an ontological discussion, -- or is concerned with experience and is restricted to that which is knowable, -- an epistemological discourse. We do likewise. Throughout this book we do not address questions of ontological reality for given levels but prefer to take an epistemological stance in a utilitarian philosophy. (Allen and Starr, p6)

I could fill up 10 pages tonight with quotes from this book. We'll get to "The Janus-faced Holon" (ch2) and "Functional and Structural Boundaries" (ch6), -- and some other topics, -- when we have time...


References

Allen, T. F. H., and Thomas B. Starr (1982) Hierarchy, Perspectives for Ecological Complexity, University of Chicago Press

Booch, Grady (1991) Object Oriented Design with Applications, Benjamin/Cummings, Redwood City, CA

Bohm David, and F. David Peat (1987) Science, Order, and Creativity, Bantam Books, New York

Davies, Paul (1983) God and the New Physics, Touchstone Books, Simon & Schuster, New York

Gardner, Howard (1986) The Mind's New Science, A History of the Cognitive Revolution, Basic Books, New York

Hofstadter, Douglas R. (1980) Goedel, Escher, Bach: An Eternal Golden Braid, Random House, New York

_____ (1985) Metamagical Themas: Questing for the Essence of Mind and Pattern, Basic Books, New York

Minsky, Marvin (1986) The Society of Mind, Simon and Schuster, New York

Posner, Michael I. (1989) Foundations of Cognitive Science, MIT Press, Bradford Books, Cambridge, Mass

Rand, Ayn (1990) Introduction to Objectivist Epistemology, Second Edition, Meridian, Penguin Books, New York

Rosch, Eleanor (1988) "Principles of Categorization", in Readings in Cognitive Science, edited by Collins and Smith, Morgan Kaufmann, San Mateo, CA

Schuman, Bruce (1993) Synthetic Dimensionality: A Universal Recursive Algebra for Describing Semantic Space

Smith, Edward E., and Douglas L. Medin (1981) Categories and Concepts, Harvard University Press, Cambridge, Mass

Sowa, John F. (1984) Conceptual Structures: Information Processing in Mind and Machine , Addison-Wesley, Reading, Mass.