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 compl |