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
- Preface
- Richard Feynman on the Hierarchy of Ideas
- A Systematic Interpretation of Feynman's
Intuition
- Levels of Abstraction/Levels of Ideas
- The Bridge Across Consciousness
- Epistemological Concepts Defined as Polar
Opposites Across Levels of Abstraction
- Logical Dynamics Defined on the Universal
Hierarchy
- The Dimensional Structure of Mind
- A Critique of the Top-Down Hierarchical
Perspective
- Concluding Comments from the WELL/ORIGIN
Conference
- 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.
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