SYNTHETIC DIMENSIONALITY
The Recursive Algebra of Semantic Space
Bruce Schuman
PO Box 23346, Santa Barbara, CA 93121
March, 1991
"Nobody can say what a variable is."
-- Hermann Weyl
CONTENTS
- Introduction and project history
- A classification of dimensions
- The isomorphism of "dimension" and "class"
- Categories defined by boundary values in n
dimensions
- Graphic models of synthetic dimensions
- Ad hoc top-down decomposition
- Synthetic dimension as universal primitive
Synthetic dimensionality is an algebraic model of class and category
structure based on the concept of dimension. In this model, actual
objects are quantitatively described in terms of values in n dimensions.
A class of objects is defined in these dimensions, where the similarities
or common properties of the objects are defined in dimensional values, as
are their distinguishing differences.
If the objects of a class can be (sequentially) ordered by their values
in their defining dimensions, this class is itself algebraically
isomorphic to a dimension, and the objects (elements, members) of the
class can be thought of as values of the dimension. A "synthetic"
dimension is thus defined as an ordered class of abstract objects
(elements, members) themselves described in more than one simultaneous
dimension/value.
The recursive properties of this approach to class specification (ie,
dimensions are defined in terms of dimensions) appears to lead to a
surprising and elegantly simple characterization of semantic space. This
paper introduces the basic ideas of this method, and argues that a
synthetic dimension is a "universal primitive" from which all conceptual
structure can be assembled.
1. INTRODUCTION AND PROJECT HISTORY
In 1987, equipped with a small personal computer and a hypertext-type
outline processor, I began organizing and collating a glossary of
epistemological concepts and definitions, attempting to build a general
model of cognitive or semantic space. I was guided by my basic instinct
that conceptual structure is generally organized across a "hierarchy of
abstraction", and in previous years I had explored the issues involved in
the algebraic representation of this structure. In these terms, for
example, there is a nested hierarchical relationship between the concepts
of "furniture", "chair" and "rocking chair". Clearly, a chair is a type of
furniture, and a rocking chair is a type of chair. These three classes of
object are thus related to one another across a descending series of
levels of abstraction, which might have at its bottom level a particular
actual rocking chair. My guiding instinct was that the entire conceptual
structure of cognition is more or less organized in similar terms.
But my attempts to analytically map the whole of cognitive space in
terms of this hierarchical structure initially met with little success. I
had drawn hundreds of tree-structured diagrams that characterized this
relationship, and I was especially interested in "mandala-like" diagrams
that modeled these relationships over a series of redifferentiated
concentric circles. These diagrams modeled (defined) "absolute unity"
and "the highest level of abstraction" at the center of these concentric
circles, and the infinite multiplicity and diversity of particular things
and objects as distributed at the periphery. I found this diagram highly
intuitive, and consistent with many schools of psychology. But try as I
might, I could not find a useful or reliably accurate way to map or
describe what was apparently a highly "multi-dimensional" set of
relationships through this diagram. Multi-faceted abstract objects simply
could not be classified or interpreted "the same way every time".
Any two-dimensional tree-structured diagram drawn on paper suffers from
the rigidity of what has been called an "Aristotelian type hierarchy". A
clear example of this limitation can be found in a university library,
where each book has to occupy only one exact shelf position, addressed
through the Dewey Decimal System. In today's scientific environment, where
the most interesting subjects often involve the intersection of several
traditional academic disciplines or departments, a particular book might
be categorized in any number of alternative ways, but can only occupy one
shelf position. Today, librarians can overcome this limitation by
searching for the book electronically through any number of relevant
descriptive keywords.
Though I failed in my initial attempts to diagram the hierarchical
relationship between abstract and empirical concepts, I was persuaded that
this fundamental principle, highly intuitive and natural, is not wrong.
As I saw it, the best and most desirable answer to this difficult problem
does not lie in so- called "semantic networks", which tend to be arbitrary
heuristics, but in terms of a flexible and adaptive general hierarchy,
which is algebraically defined so as to satisfy the many simultaneous
constraints governing the linkage of concepts across levels of
abstraction.
This is an enormously subtle and challenging algebraic problem, which
has yet to be overcome by researchers working in this field. As I see it,
the problem is akin to a mysterious and fabulously complex lock for which
we as yet have no key -- and I believe that the only hope of opening this
lock, and meeting these simultaneous constraints, is a relentless
trial-and-error approach that considers hundreds of alternatives and is
guided by deep intuition. Since we are searching for a general
interpretive model or set of rules, which will be applicable to all
instances throughout semantic space, the opening of this lock must be
pursued "from the top down", and no key will work consistently until every
detail is perfect. Having pursued this trial and error approach for
several years, it is currently my sense that the recursive methods of
synthetic dimensionality do provide a means to overcome this problem. By
defining the elements of semantic space in terms of abstract dimensions, I
believe it is now possible to elegantly integrate the otherwise
overwhelming complexity of conceptual structure and terminological
diversity into a single set of general principles.
The Epistemological Dictionary
As I continued to work with my outline processor, I built a vocabulary
of approximately 300 fundamental epistemological concepts, which I
attempted to systematically define in consistent algebraic terms,
free-associating for hours at a stretch as I worked with these ideas. I
was deeply involved with the concept of "information", and the digital
bit-structure of symbolic representation. For me, a concept was an
"information structure" (as per Peter Wegner's 1968 classic
Programming Languages, Information Structures, and Machine
Organization), and I wanted to map or diagram the meaning of that
structure across its levels of abstraction.
Going over and over this material, from many alternative points of
view, and factoring in the most fertile scientific literature I could
find, one single idea began to appear over and over again: the concept of
dimension. I intuitively felt that all abstract categories or concepts
could be linearly factored into their constituent dimensionality, and that
the meaning of an abstract concept could be defined in terms of a
(adaptive and ad hoc) cascade of dimensions usually grounded in the
"empirical dimensionality" of quantitative measurement. Clearly, when we
wish to describe an actual object exactly, we describe it in terms of
quantitative measurement. But how do we descriptively characterize
abstractions in terms of dimensions?
We might, for example, distinguish between various kinds of rocking
chairs in terms of quantitative dimensions, such as height, length,
weight, cost, age, color, etc. And we might pose more abstract dimensions
or attributes in terms of which we might describe more subtle or
qualitative ranges of variation. These might include such variables as
"beauty", "character", or "workmanship".
In 1987, I first conceptualized an intuitive proposition which was to
guide much of my work that followed. Put simply: "All concepts are made
out of dimensions -- and dimensions themselves are made out of
dimensions." It is this recursive hypothesis that has led to most of what
I have discovered about conceptual dimensionality, and which forms the
ontological foundation of my experimental chain of definitions.
A Rosetta Stone for Cognitive Science?
The purpose of this present paper is simply to provide an initial
overview of this work, to generally describe the methods and principles of
this way of looking at conceptual structure, and to provoke some
discussion of the controversial and subtle points raised by these
propositions. The algebraic consequences of these ideas are complex and
detailed, and this paper is only intended as a general survey of the major
themes.
As it stands today, Synthetic Dimensionality is not a working computer
program or language, and has not been defined in terms of any practical
application, such as a database management system, or natural language
interpreter. Somewhat like Marvin Minsky's Society of Mind, it
is merely a body of philosophical and algebraic ideas. But it seems to me
that the discovery (or hypothesis) that all conceptual structure can be
perfectly modeled in terms of a single linear algebraic primitive element
almost certainly has powerful and far-reaching consequences. If I am
correct, and it is true that all conceptual structure can be defined in
these extremely simple and elegantly recursive terms, I would suppose that
the possible applications for this method would be limitless and extremely
potent.
Is Synthetic Dimensionality a "Rosetta Stone" for cognitive science?
Until this method has enjoyed substantial expansion and development in
terms of practical applications and testing, this question cannot truly be
answered in the affirmative. But the algebraic simplicity of this method
is tantalizing and intriguing, and unless there are hidden errors in this
untested method, it seems likely that the representation of conceptual
abstractions in terms of linear dimensions will eventually become a
powerful and effective tool in our high-speed information processing
environment.
2. A CLASSIFICATION OF DIMENSIONS
A "dimension" is usually thought of as describing the continuous
variation of some numeric variable, and the values of the dimension are
the numeric values taken by the variable as it changes, defined as
multiples of the "units" of the dimension. "Dimension" and "variable" are
almost identical concepts, and a classification of types of variables can
be understood as a classification of types of dimensions. Variables (or
dimensions) can be classified by the size and properties of their range of
values, and by the type of scale which describes them.
The representation of "qualitative" values in terms of dimensions is
not a new idea, and "ordinal" variables are common in social and
behavioral science. There is an ascending hierarchical relationship
between types of variables, which depends on the precision (and the
simultaneous dimensionality) with which they are defined. The below
classification of dimensions is taken from Cluster Analysis for
Applications, by Michael R. Anderberg, Academic Press, New York,
1973, p.26:
A systematic and comprehensive classification of variables [or
dimensions] provides a convenient structure for identifying essential
differences among [types of] data elements. This section presents a
cross-classification of variables based on two familiar descriptive
schemes.
Classification According to Size of the Range Set
From a mathematician's point of view, it is natural to distinguish
among variables on the basis of the number of variables in the range
set, that is, the number of distinct values the variable may assume.
To explore this approach adequately it is useful to have the following
concepts about counting the number of elements in a set.
1. A set is finite if its elements may be put one-to-one
correspondence with a sub-set of the positive integers, the latter
containing a largest number. Less formally, a set is finite if its
elements can be counted and some definite integer is given as the
number of elements in the set.
2. A set is countably infinite if its elements may be put into
one-to-one correspondence with the set of positive integers. The
latter set is infinite since there is no largest integer (the claim
that some large number, say K, is the largest integer is refuted by
exhibiting K + 1, a larger integer). Less formally, the set may be
counted, at least in principle, but it would take infinitely long to do
so. However, between any two given elements of the set there is a
finite number of other elements in the set.
3. A set is uncountably infinite if its elements cannot be put
into
one-to-one correspondence with the positive integers. Fundamentally,
between any two real numbers there are infinitely many real numbers as
opposed to the finiteness characterizing an interval in a countably
infinite set. Given a starting point in the set, it is not meaningful
to speak of "a next number". If the set consists of all real numbers
between 1.5 and 2.0, what is "the next number" after 1.5? It is not
1.5000001, since 1.50000009 lies between the two numbers. Indeed, any
candidate for "next number" fails since infinitely many numbers may be
found between 1.5 and the candidate. Hence, the set is uncountable.
With these concepts, a familiar classification scheme for variables is
as follows:
1. A continuous variable has an uncountably infinite range
set.
Typically such a variable may assume any value in an interval (say 1.5
to 2.0) or a collection of such intervals.
2. A discrete variable has a finite, or at most a countably
infinite
range set.
3. A binary or dichotomous variable is a discrete
variable which
may take on only two values.
Classification According to Scale of
Measurement
In the social and behavioral sciences one frequently encounters a
classification of variables based on their scale of measurement. It
will be convenient to illustrate this scheme with a variable X and two
objects, say A and B, whose scores on X are X(A) and X(B) respectively.
1. A nominal scale merely distinguishes between classes. That
is,
with respect to A and B one can only say X(A) = X(B) or X(A) not= X(B).
2. An ordinal scale induces an ordering of the objects. In
addition
to distinguishing between X(A) = X(B) and X(A) not= X(B), the case of
inequality is further refined to distinguish between X(A) > X(B) and
X(A) < X(B).
3. An interval scale assigns a meaningful measure of the
difference
between two objects. One may say not only that X(A) > X(B), but also
that A is X(A)-X(B) units different than B.
4. A ratio scale is an interval scale with a meaningful zero
point.
If X(A) > X(B), then one may say that A is X(A)/X(B) superior to B.
These scale definitions are ordered hierarchically from nominal up to
ratio. Each scale embodies all the properties of the scales below it in
the ordering. [These scales as listed here are in ascending order]
Therefore, by giving up information one may reduce a scale to any lower
order scale.
Frequently variables on nominal and ordinal scales are referred to as
categorical variables or qualitative variables, often with ambiguity as
to whether any order relation exists. For contrast, variables on
interval or ratio scales are then referred to as quantitative
variables.
It is worth making a special note of Anderberg's comment: "by giving up
information one may reduce a scale to any lower order scale." The study
of synthetic dimensionality can be understood as a study of the
relationship between classes of variables, and involves the precise
algebraic analysis of the manner by which information is included or
"given up" in the definition of a variable.
3. THE ISOMORPHISM OF "DIMENSION" AND "CLASS"
When my pursuit of algebraic epistemology began to converge onto the
concept of dimension, I began collecting definitions of dimension from as
many sources as possible. I read books on dimensional analysis, as well
as in related fields such as cluster analysis and multi-dimensional
scaling. And, of course, I also did what I could to work my way through
the best literature on fractals, including the original work by Benoit
Mandelbrot.
I built lists of alternative definitions of dimension, and for a time I
was working with twenty or thirty different ways to define the concept.
From mathematics, I brought in the concepts of "set", "class", "cut" (as
in "Dedekind cut"), and "variable", as well as "genus and differentia" and
"similarity and difference" from the theory of classification, and the
concepts of "information", "information structure" and "bit" from computer
science.
In my outline processor, I would free-associate my way back and forth
between these alternative definitions, looking for every possible linkage
and connection, fertilizing my search with the most seminal literature I
could find. As this linkage continued to build, and I began to find
confirmations of my basic intuitions from a great many alternative points
of view, I became increasingly convinced that my basic conviction that
"all concepts are built from dimensions" was indeed on the right track.
But as they say in the psychological theory of creativity, there are
times when one simply has to abandon one's efforts, and let the creative
work bubble in the "unconscious idea processor" of the mind. And my
periodic approach to this work took just this form: I would work hard on
the project for a week or two, then leave it alone for a couple of months,
only to pick it up again. And as I did this, over the course of a couple
of years I found my natural instincts and intuitions seeming to converge
to one single central definition of dimension, from which it seemed I
could then build all others.
As it stands now, that fundamental and underlying definition takes the
following form:
Algebra of dimensions: fundamental definition
A dimension is an ordered class of values, with the following
properties:
- It is a set of distinct elements.
- This set of elements have in common one or more similarities.
- These similarities among the elements are defined in terms of
values in dimensions; ie, elements are "similar" to the degree that
they have identical values in identical dimensions.
- These similar elements may have distinguishable differences.
- These differences are defined in terms of values in dimensions;
ie, elements are "different" to the degree that they are defined in
non-identical dimensions, or have non- identical values in identical
dimensions.
- If these similar elements do not have distinguishable differences
(ie, they are identical), the dimension is a "quantitative
dimension" of the type which describes physical measurements, and
the elements are the units of measure (such as "feet" or "pounds" or
"apples").
- These distinct elements can be ranked in serial/linear order,
according to their values in the dimensions in which their differences are
defined.
- A consequence of this ranking is that each ranked element or
object in a dimension/class can be interpreted as a value of the
dimension.
This definition has a number of properties or consequences which are worth
noting:
- It provides a systematic and precise way to define the somewhat
ambiguous concepts of similarity and difference.
- It defines a dimension as a class in such a way as is simultaneously
consistent with the ordinary intuitive definition of a dimension (such as
length as measured in inches) and the intuitive definition of a class (a
set of objects with one or more properties in common). If the elements of
the class are identical, the dimension is quantitative, and the elements
are the units of measure.
- Both the common properties (the similarities), and the
distinguishing differences of the elements of the class are defined in
terms of dimensions and values in dimensions.
- This approach allows us to use one highly compact and recursively
defined algebraic element (ie, dimension) to both abstractly classify and
fully describe any object, to any desired degree of specificity. Clearly,
when we wish to exactly describe any object, we give its exact
measurements in some set of dimensions. Both abstractly classifying and
empirically describing any object in terms of a single algebraic concept
is elegant, compact, and convenient.
- A dimension is "recursively compositional" -- which is to say that
a dimension, like a fractal, is built out of "self- similar" elements.
There is a complementary duality between a value and a dimension. Every
value is itself a dimension; every dimension is a value.
- Since a dimension is a class of values, and a value is a dimension,
"dimensions are built out of dimensions".
- "Class" and "dimension" are isomorphic concepts: an ordered class is
a synthetic dimension, and a synthetic dimension is an ordered class.
- As an ordered class, a dimension is not merely any set of objects
which we can group together by common properties. In order for a class to
be defined as a dimension, we must be able to order (or "sort") the
elements of this class into an unambiguous serial/linear sequential list.
A class of elements defined in four different dimensions can be sorted in
four different ways, according to the values of the elements in each of
these dimensions. (An example is the sorting of files on a floppy disk by
a personal computer operating system, according to their values in four
descriptive parameters: name, date, size, type.)
- A dimension is thus a (unambiguously sequential) list of
values.
- Just as any value is itself a dimension, any element or member or
value of this list (in this definition, "element", "member" and "value"
are equivalent and interchangeable concepts) may itself be either a single
undifferentiated "unit", or can be itself another list.
- It is interesting to note that these above two points make the
definition of dimension intimately related to the fundamental definitions
in the LISP ("list processing") programming language, oftentimes
considered the primary language for artificial intelligence (see Douglas
Hofstadter, Metamagical Themas, pp 396-454).
The definition of dimension is recursive (ie, "defined in terms of
itself") in some profound and subtle ways. Not only is there an
equivalence between dimension (a class of values) and an ordered class (a
class of abstract objects), but each of the values of the dimension are
themselves recursively definable as dimensions. It is this recursion
which allows me to argue that not only are "all concepts built from
dimensions -- but dimensions themselves are built from dimensions".
This can be illustrated in detail by systematically demonstrating that
all elements of this above definition can be defined in terms of
dimensions. That is, the concepts of "class", "set", "distinction",
"similarity", "difference", "category", "element", "member", "value",
"rank", and "unit" can all be defined in terms of dimensions -- as can
any other basic concept from epistemology. Additionally, I have found that
all the basic "data structures" of computer science and linear algebra can
be defined in terms of dimensions. The process begins simply by noting
the fact that a dimension can be represented as a row vector. Thus, every
row vector in a data structure is a dimension of the structure.
In a "quantitative dimension" such as "length in inches", all the
inches are the same. Each inch is exactly identical to every other inch,
-- with the one exception that each inch is labeled or identified as a
particular numeric multiple; ie, the first inch, the second inch, the
third inch, etc. This dimension is thus a scale of values, like a ruler
or yardstick, where the values are "one dimensional".
I use the phrase "synthetic dimension" to describe any dimension which
involves multi-dimensional (or linearly decomposable) values. A
"synthetic" dimension is a range of values, just like any other dimension,
but its values are not simply identical units, but are instead "similar"
units which nevertheless have some distinguishable difference. In this
sense, the concept "synthetic dimension" includes the normal intuitive
definition of dimension, but is more general, and is defined at a higher
level of abstraction.
A consequence of the above fundamental definition is that any ordered
class can be thought of as a dimension. Thus, a "set of tea cups", if the
cups can be placed in serial order according to some criteria inherent in
their description, becomes a (synthetic) dimension. In this dimension,
the unit is "tea cups", and they are ordered or sorted by their value in
some criteria of their description, such as height or weight or volume.
Synthetic dimensionality offers a way to not only define or fully
characterize and describe all objects in terms of quantitative dimensions,
but also defines a consistent way that all abstract features, properties,
characteristics, and attributes of any object can be defined as
(synthetic) dimensions.
4. CATEGORIES DEFINED BY BOUNDARY VALUES IN N
DIMENSIONS
There are any number of ways to define the meaning of "concept", and
from my point of view, the words "concept", "class", and "category" are
closely related and almost interchangeable. In the context of modern
cognitive science, there are generally three approaches taken to the
definition of concepts and categories: the "classical" or Aristotelian,
the probabilistic, and the "prototypical", associated with Wittgenstein
and Zadeh. A good overview text on cognitive science, such as Howard
Gardner's The Mind's New Science, can introduce the reader to a
full discussion of these approaches. Two other excellent sources are Smith
and Medin's Categories and Concepts, and John Sowa's
Conceptual Structures: Information Processing in Mind and
Machine.
My dimension-based approach to this subject might be described as
"modified classicism". That is, I generally approach the definition of
classes and categories in Aristotelian terms, but then adapt these terms
to a flexible and adaptive algebraic scheme which I believe incorporates
the advantages of the other approaches to categorization. This scheme is
intended to retain the clarity and simplicity of the classical approach,
while avoiding the limitations which these others methods seek to
overcome.
John Sowa characterizes these three basic approaches to conceptual
definition. From Conceptual Structures, p 16:
For most of the concepts of everyday life, meaning is determined not by
definition, but by family resemblance or a characteristic prototype.
In a study of concepts, Smith and Medin (1981) summarized three views
on definitions:
1. Classical. A concept is defined by a genus or supertype and a set
of necessary and sufficient conditions that differentiate it from other
species of the same genus. This approach was first stated by Aristotle
and is still used in formal treatments of mathematics and logic. It is
the approach that Wittgenstein presented most vigorously in his early
philosophy, but rejected in his later writings.
2. Probabilistic. A concept is defined by a collection of features and
everything that has a preponderance of those features is an instance of
that concept. This is the position taken by J. S. Mill. It is also
the basis for the modern techniques of cluster analysis.
3. Prototype. A concept is defined by an example or prototype. An
object is an instance of a concept c if it resembles the characteristic
prototype of c more closely than the prototypes of concepts other than
c. This is the position taken by Whewell and is closely related to
Wittgenstein's notion of family resemblances.
In fuzzy set theory, Zadeh (1974) tried to formalize the probabilistic
point of view. His related theory of fuzzy logic extends uncertainty
to every step of reasoning. In prototype theory, however, judgments
are made in a state of uncertainty, but once a plant is classified as a
member of the rose family, further reasoning is done with discrete
logic. Fuzzy set theory has important applications to pattern
recognition, but fuzzy logic is problematical. As an adaptation of the
classical scheme, I approach the definition of categories and concepts
as follows: a category is defined by boundary values (lower and upper)
in n (any number) of simultaneous dimensions. Common and
distinguishing properties of any member of the category are defined by
values in dimensions. If an object is characterized by values which are
within these boundary values in all of these simultaneous dimensions,
the object is "in" that class or category. If one of the values that
characterize the description of the object is not within this boundary
value range, the object is not in the category.
This method of defining categories requires some clarification, and
involves defining exactly what is meant by a dimension. Several initial
points should be considered, and each of these requires some clarifying
discussion:
1. Any object (whether abstract or concrete) can be symbolically
represented (or "modeled") by a set of values in a set of dimensions. In
a database, we might represent a category such as "employee" by several
dimensions or ranges of value, such as age, education, marital status,
seniority, skill level, or any other descriptors which we find useful.
2. The choice of boundary values in categorization involves "drawing a
line" along some possibly arbitrary range of values. At what point along a
hypothetical continuum does the value "red" become "pink"? This cut-off
point or boundary value is defined stipulatively and possibly ad hoc, and
not necessarily in terms of some apriori or necessarily system-wide
scheme.
3. Any characteristic, feature, attribute or property of an object can
be defined as a (synthetic) dimension of the object. If the characteristic
or feature is defined in "qualitative" terms, it may have to be linearly
factored into a set of quantitative dimensions. This is an important
aspect of the doctrine of synthetic dimensionality.
4. The (top-down and linear) factoring of abstract or "qualitative"
concepts (or dimensions/attributes) is ad hoc, arbitrary, free-form, and
stipulative, and is custom-tailored by a speaker to the exact requirements
of usage in an immediate context. It is this principle which enables us
to overcome a rigidity in definition which requires some word or concept
to "always mean the same thing". I argue that in the context of actual
usage, the potential meaning of any word or concept is stipulatively
defined ("dimensioned") by the speaker in a way that is fitted to the
needs of the immediate moment, and this definition involves a descending
hierarchical cascade of increasingly specific linear factors which are
initially implicit and unspoken, but can be made explicit, depending on
the need for a higher level of precision.
Thus, the dimension-based approach to defining the meaning of a concept
is entirely stipulative (ie, is defined by the user of the concept to suit
his/her purposes), and involves two major degrees of freedom: 1) the
choice of which dimensions compose the concept, and 2) the choice of
boundary values in those dimensions. This approach to definition is
fundamentally distinguished from approaches which presume that there is
some "best" or "correct" single definition, howsoever conceived. In this
context, word meanings are taken from a loosely defined social pool of
approximate commonly held meanings, and given exact context- specific
values by the speaker. Thus meaning is in part "conventional", but is
precisely shaped by stipulation in the context of usage to a specific
exact form, characterized by the specific choice of dimensions of
composition, and boundary values in those dimensions.
What is a "cup"?
These principles can be illustrated by example. Let us consider the
abstract concept of "cup", and ask whether or not some particular actual
object is or is not a cup. My dictionary defines a cup as "a small, open
container for beverages, usually bowl-shaped and with a handle."
In our nested hierarchy of abstractions, we note that, in general, a
"cup" is a specific type of "container for beverages" (ie, a species of
the genus "beverage containers", itself a species of "container"),
distinguished from other beverage containers by its specific attributes.
In our discussion here, we will show how those attributes and distinctions
can be defined by values in dimensions, thus determining what is or is not
a cup by a series of boundary value ranges which constrain any object
within the class "cup".
The general hierarchical structure of the concept cup, according to
this definition, is that it is a type of container. Specifically, it is a
"beverage" container. And it is distinguished from other beverage
containers by certain specific properties.
In the broader class (genus) "beverage container", we might include the
following possible members: buckets, pots, cans, glasses, bottles, bowls,
cups, and pans -- and we might omit objects such as plates, trays or
vases (on the grounds that their dimensional values do not lie within the
boundary value range of "beverage container". Clearly, determining
whether or not a "plate" is a "beverage container" involves drawing an
arbitrary line or cutoff point in some defining dimension, such as the
height of the sides of the plate. At what point (along a dimension) does
a concavely curved plate become a large shallow bowl? Or, at what point
along some range of variation does a liquid, such as gravy, become a
"beverage"? Is "soup" a "beverage" -- and if you put enough water in
gravy, does it become "soup"?
These distinctions and definitions are generally constrained by social
convention and expectation, but in actual usage are exactly defined by a
speaker in an arbitrary and ad hoc way. If we say "the gravy was soupy",
we are characterizing the gravy in terms of the dimensions of soup,
whatever those may be. Determining these in any detail is entirely up to
the speaker.
The dimensionality of the concept "cup", as defined by Webster's, can
be diagrammed. This scheme interprets the above definition of the generic
class "cup" in terms of values in five dimensions:
|---------------------------|------------------------------|
| Object class: "cup" | |
|---------------------------|------------------------------|
| Dim 1: "size?" | Value: "small" |
|---------------------------|------------------------------|
| Dim 2: "open or closed?" | Value: "open" |
|---------------------------|------------------------------|
| Dim 3: "shape?" | Value: "usually bowl-shaped" |
|---------------------------|------------------------------|
| Dim 4: "type of object?" | Value: "beverage container" |
|---------------------------|------------------------------|
| Dim 5: "handle?" | Value: "usually" |
|---------------------------|------------------------------|
Fig. 1
Defining dimensions and values in these terms may seem unfamiliar, but
these values (and dimensions) can be grounded in the familiar quantitative
dimensionality through a process of linear factoring. As per Anderberg's
scheme, the values "small", "open", and "usually" are ordinal or nominal,
while the values "usually bowl-shaped" and "beverage container" require
additional factoring (and this factoring, like all conceptual factoring in
this scheme, is top-down, linear, and ad hoc).
It is helpful to note that we assign values all the time in such
"non-quantitative terms". We might consider "ordinal" and "interval"
values describing temperature in terms of the following scheme:
----------|-------------------|--------------|--------------|
Nominal:| uncomfortable | comfortable | uncomfortable|
----------|----|----|----|----|----|----|----|----|----|----|
Ordinal:|very cold| cold |cool|mild|warm|hot |very hot |
----------|----|----|----|----|----|----|----|----|----|----|
Interval:|10 |20 |30 |40 |50 |60 |70 |80 |90 |100 |
----------|----|----|----|----|----|----|----|----|----|----|
Fig. 2
For the sake of convenience, and because we may be operating within a
wide boundary value range with no need to be exact (and also because we
may be factoring in other simultaneous dimensions as we do so), in some
contexts we may be quite content to define temperature as "very cold",
with no need to specify that we are referring to a boundary value range
between 10.0000 and 29.9999 degrees. Factoring in additional but perhaps
non-explicit dimensions, we might define the temperature value as
"uncomfortable" -- the various dimensions of "comfort" including not only
air temperature, but perhaps how we are feeling, the clothes we are
wearing, the humidity or wind-chill, or any other factor.
Thus, in our determination of what is or is not a "cup", we might be
defining the values in the various dimensions which describe a cup as
"small", or perhaps "shallow", or "tall", but we could assume that these
values implied some type of ordinal-to-interval conversion, as per some
appropriate implicit scheme, which, if necessary, we could make explicit.
Linear dimensional factoring of definition values:
"Cup"
|---------------------------|------------------------------|
| Synthetic value | Linear dimensional factors |
|---------------------------|------------------------------|
| Type: beverage container | Beverages? Containers? |
|---------------------------|------------------------------|
| Shape: bowl-shaped | Bowl shape? |
|---------------------------|------------------------------|
| Size: small | In what dimensions? |
|---------------------------|------------------------------|
| Top: open | (nominal two-state variable) |
|---------------------------|------------------------------|
| Handle: usually | (ordinal variable) |
|---------------------------|------------------------------|
Fig. 3
To fully define a cup as a "beverage container", we may have to fully
dimension the concepts of "beverage" and "container". We can do this in
the same way. What is a container? What is a beverage? Both of these
concepts can be linearly factored into their synthetic dimensions.
The value "bowl-shaped" has a hierarchical decomposition; "bowl-shaped"
is a type of "shape". To completely specify the dimensions of this
concept, we have to first of all decide what are the dimensions of "shape".
Then, we must decide what are the dimensions of "bowls". Clearly, this is
a process akin to that of defining "cups", and without some grounding in
stipulation, convention, or example, we are caught up in a circular
definition (ie, a cup is bowl-shaped, and a bowl is cup-shaped). But if
pressed to be exactingly specific, we can avoid this problem simply by
descending in levels of abstraction, and defining an exact set of specific
quantitative dimensions to characterize a particular object.
My dictionary defines "shape" as "the outline or characteristic surface
configuration of a thing". Defining this general definition in terms of
quantitative dimensions would not be difficult, though in some cases
perhaps rather mechanically complex. One might stipulatively create an
ordinal dimension that linearly varies by "degree of concavity", and
across this dimension distribute such values as plates (almost no
concavity), shallow cups and bowls (some concavity), and tall, thin vases
and glasses (high concavity). "Bowl-shaped" can thus be seen as an
approximate ordinal boundary value range in some posited dimension of
linear variation.
The concept "container" can be linearly factored in the same way. The
dictionary defines container as "a thing in which material is held or
carried." The broader class that includes "container" is "things", a very
broad class indeed, and the specific distinguishing characteristic
(differentia), that this object holds or carries materials, still defines
a very broad class. By narrowing the type of "material" to that of
"beverage", we have narrowed the class considerably.
Thus, to answer our question, is this specific object c a "cup", we
have to take its measurements in the dimensions of its definition. Is it
a "beverage container"? Is it "small"? Is it "open"? Is it
"bowl-shaped"? Does it have a handle?
The answer to each of these questions is a value in a class with its
own dimensional specifications. Stipulative boundary values must be
assigned to determine the exact meaning of words such as "small" -- or
the point at which an object become sufficiently concave to be regarded as
"bowl-shaped" and not "plate-shaped".
If a specific object is within the boundary value range in each of
these defining dimensions, we can define the object as a "cup". If not, we
will have to define the object as a member of some other class, such as
"plate", or "bowl", or "glass".
We can show each of these properties as a boundary value range in some
dimension, whether nominal, ordinal, or interval:
cup
Size: -|----|----|----|----| |----|----|----|-> (ordinal)
Shape: -|----|----|----|----| |----|----|----|-> (ordinal)
Type: -|----|----|----|----| |----|----|----|-> (ordinal)
Handle: ---------------------| ---------------|-> (nominal)
Open: ---------------------| ---------------|-> (nominal)
Fig. 4
Ordinal values on a dimension such as "size" ("small", "large", etc.)
can be factored in various ways, such as the following quantitative
descriptors:
cup
Height: --|----|----|----|----| |----|----|----|->
Width at diameter: -|----|----| |----|----|----|->
Volume contained: --|----|----| |----|----|----|->
Ratio of height to width: ----| |----|----|----|->
Fig. 5
In any of these descriptive dimensions, a "cup" is a class of objects
within a boundary value range. Any object that is "too wide" or "too
narrow", or "too tall" or "too short" is not a cup.
A "cup" is a highly constrained object, defined within an n-dimensional
envelope. Objects within that envelope are cups. Objects that are outside
the envelope are not cups.
5. GRAPHIC MODELS OF SYNTHETIC DIMENSIONS
The graphic representation of a synthetic dimension is inherently
two-dimensional. Unlike a conventional quantitative dimension, often
modeled on a Euclidean straight line, a synthetic dimension has "width"
or "thickness". This is because the elements being characterized by the
dimension are being described in two simultaneous dimensions -- their
difference and their similarity -- which we can algebraically
characterize in the X and Y axes.
As per the fundamental definition, a synthetic dimension is an ordered
class of distinct elements, where these elements can be interpreted as the
values of the dimension. These elements have one or more similarities or
common properties, these common properties being defined as a common
boundary value range in a common dimension. That is to say, all the
elements of the class (the values of the synthetic dimension) share a
boundary value range in at least one common dimension.
But the elements of the class (values of the dimension) are also
distinguished from one another by taking differing values (occupying
different boundary value ranges) within another common dimension.
We can thus graphically represent a synthetic dimension as a row
vector, or linear/sequential list of adjacent cellular addresses, as
follows:
|----|----|----|----|----|----|----|----|----|----|----|
| a | b | c | d | e | f | g | h | i | j | k | D
|----|----|----|----|----|----|----|----|----|----|----|
Fig. 6
where the (a,b,c...) represent the elements of the class (values of the
dimension). Let us call this dimension D.
This cellular row-vector structure is a convenient graphic model of a
synthetic dimension, and allows us to show how similarity can be defined
in the Y axis, and difference in the X axis.
Let us presume that X is some quantitative dimension, such as "height",
in which all the elements (a,b,c...) are defined. Thus, these elements
can be ordered in the X dimension, according to their values in X.
Thus, our row-vector characterization defines the boundary value range
in X for each of the elements (a,b,c...), and the "walls" or "edges" of
the address cells define these boundary values, thus:
|----|----|----|----|----|----|----|----|----|----|----|
| a | b | c | d | e | f | g | h | i | j | k |
X--|----|----|----|----|----|----|----|----|----|----|----|-->
Fig. 7
The cuts in X defined by the boundary values can be defined thus:
X----|----|----|----|----|----|----|----|----|----|----|-->
Fig. 8
These cuts are identical to the "Dedekind cut" which characterizes
points on the Real Number Line. They can be assigned numeric values,
thus:
a b c d e f g h i j k
X--|----|----|----|----|----|----|----|----|----|----|----|-->
0 1 2 3 4 5 6 7 8 9 10 11
Fig. 9
Therefore, we say that element a has a boundary value range from 0 to
1, element b a range from 1 to 2, c a range from 2 to 3, and so forth.
As we are diagramming these values here, we are showing the
distinctions among the (a,b,c...) as being defined by integer values and
evenly distributed, but there is no reason for this to be the case. To
show a distinction between the elements in the dimension X, it is enough
that none of the elements take the same value (share the same boundary
value range).
It is perhaps worth noting that there is no real difference between a
"value" and a "bounded range of values", since any value is only accurate
to within a specified degree of precision, or number of decimal places. A
value defined as 3.2563986 is thus still a bounded value range, since we
have not specified what the next decimal place might be (and we can, in
fact, understand any "next" decimal place as a "species" of the genus
defined by the previous decimal place).
Thus, in the graphic representation of synthetic dimensions, we define
the differences among elements in terms of values in the X dimension.
The similarity (or common property) which brings the elements together
into a class, on the other hand, is defined as a boundary value range in
the Y dimension.
Thus, of our elements (a,b,c...) in the dimension D, we define the
horizontal boundaries of the address cells, which they all share in
common, as a common boundary value range in Y, as in Figure 10:
Y
6-|
|
5-|
|
4-|----|----|----|----|----|----|----|----|----|----|----|
| a | b | c | d | e | f | g | h | i | j | k |
3-|----|----|----|----|----|----|----|----|----|----|----|
| 1 2 3 4 5 6 7 8 9 10 11 X
2-|
|
1-|
|
0-| Fig. 10
Continuing to characterize the elements (a,b,c...) in terms of
quantitative (numeric) values, we can say that while the differences among
the elements are defined by the differing values in the X dimension, all
of these elements share a common bounded value range, from 3 to 4, in Y.
Though we are here numerically defining the values of D, there is no
reason why we could not define these values ordinally. Both value ranges,
in both the X and Y dimensions, could be ordinal, as well as interval,
ratio, or nominal.
Objects in a class can share several common properties, which can be
represented synthetically in the Y dimension as a single value,
characterized by a single class name or label, such as "cup". The single
value "cup" can be one of a linear distribution of "beverage containers",
which have been sorted according to linear/quantitative criteria in their
definition, such as average height or weight or diameter -- or even
alphabetical order.
We can thus define the intersection of a class and a "superclass" as
the intersection of two synthetic dimensions, as follows, where D* is the
superclass "beverage containers", D is the class "cups", and d* is some
sub-class. Elements in D can be specific types of cups, such as mugs, tea
cups, coffee cups, demitasse, etc., as ordered by some linear/quantitative
criteria in their definition.
D* = genus (beverage containers)
D = species (specific cups)
d* = subspecies/sub-class
(a,b,c...) = members of the species, or types of cups
F = cups
X = linear range of variation distinguishing cups
Y = linear range of variation distinguishing
beverage containers
(1,2,3...) = either specific cups of type g, or list of n
dimensions characterizing all members of
(a,b,c...)
Y
|
D* |
|----|
| A |
|----|
| B |
|----|
| C |
|----|
| D |
|----|
| E | d* |
|----|----|----|----|----|----|----|----|----|----|----|
| F | a | b | c | d | e | f | g | h | i | j | D
|----|----|----|----|----|----|----|----|----|----|----|-->X
| G | | 1 |
|----| |----|
| H | | 2 |
|----| |----|-------------
| I | d** | 3 | ?
|----| |----|-------------
| J | | 4 |
|----| |----|
| 5 |
|----|
| 6 |
|----| Fig. 11
This enables us to show the "inheritance" of features as the orthogonal
intersection of two dimensions, rather than the more common representation
as a (two-dimensional Aristotelian) tree. The flexibility of this approach
comes in part from the fact that each element in these lists is a value on
a scale. We can generalize and argue that "similarity is perpendicular to
difference".
In response to Hermann Weyl's challenge ("Nobody can say what a
variable is."), we might reply by suggesting that the very concept of
variable is inherently at least two-dimensional, since the data
elements
(the values of the variable at different times t) all have at least one
something in common (they are all values of "the same thing".) Weyl's
challenge is at least partly based on the question, "If a variable is
constantly changing, how can we say it is always the same thing?" In
other words, how does a variable retain a consistent time-invariant
identity, while at the same time undergoing state transformations, or
taking different possible values? The answer is: a variable is a
synthetic data structure, in which, simultaneously, something changes and
something remains the same. That alone requires at least two dimensions
to describe. If we factor in the problem of concretely representing this
data structure in some symbolic medium (such as a computer or a piece of
paper), we must factor in any number of additional dimensions of
representation...
This discussion only begins to characterize the algebraic consequences
of defining synthetic dimensions. There are hundreds or thousands of
implications of these definitions, which can and probably should be worked
out in the context of a specific applications-oriented engineering and
research environment.
6. AD HOC TOP-DOWN DECOMPOSITION
Any synthetic value (ie, any concept defined at a level of abstraction
above the quantitative) can be linearly factored into its constituent
dimensionality. As we saw, the word "cup" can be defined in five
dimensions, and several of these dimensions can and must themselves be
factored, in order to ground the definition of "cup" in an exact
quantitative description which filters out all the objects in the world
that are not cups.
Not all words, of course, can be linearly factored into quantitative
dimensions. A word like "beauty" has many meanings, which can vary from
context to context and from speaker to speaker, and in most contexts
cannot be given an exact quantitative definition. In some highly
specialized situation, there might be some exceptions to this rule, but,
in general, the concept "beauty" is defined in highly "qualitative"
dimensions which cannot be realistically grounded in quantitative
measurement -- though perhaps there are some mathematical principles that might
be invoked, such as "the golden ratio" or principles of symmetry, balance,
or other qualities that can be described precisely.
This incomplete or partial dimensional cascade, which extends across
descending levels of abstraction, but which does not terminate in
quantitative measurement, is typical of a broad class of concepts, which are
generally described as "intuitive" or "metaphysical", or even "religious".
This incomplete decomposition cascade is the reason that there exists a
gulf or divide or fundamental separation between the domains of "science"
and of "religion". Generally speaking, the concepts of religion, as they
are conceived today, simply cannot be mapped to quantitative measurement.
Instead, their meaning lies suspended in a highly abstract space that must
be interpreted by faith and belief, rather than in terms of testable and
reproducible knowledge. A fuller discussion of dimensional decomposition
across the Universal Hierarchy of Abstraction can outline these principles
in detail.
The below diagram characterizes the hierarchical decomposition of a
concept considerably less abstract than "beauty", but still involving some
highly qualitative factors. From H.J. Zimmerman, Fuzzy Sets, Decision
Making, and Expert Systems, Kluwer Academic Publishers, Boston, 1987,
p22:
Conformity with social
Business and economic standards
behavior---------->|
| Economic thinking
Personality------->|
| | Motivation
| Potential--------->|
Credit- | Physical and mental
worthiness------->| potential
|
| Continuity of margin
| Liquidity--------->|
Financial | Income minus expenses
basis------------->|
| Other net property
Security---------->|
Property minus
long-term debts
Fig. 12
This schematic provides an approximate "quantification" of the abstract
concept of "credit-worthiness", as it might be used without explicit
definition in some business context.
Some points to consider:
1. This differentiation is arbitrary and stipulative. It is
consciously created in one way rather than another, perhaps by a single
individual, perhaps by an organization. It is a "rigid hierarchical
cascade of meaning assignments", but it could be defined in any
number
of other ways. It is, then, in a sense an ad hoc definition,
rather
than some ontologically-grounded absolute. The entire cascade could be
redefined at any time.
2. The last four variables at the lowest level of abstraction (on the
far right) can be numerically quantified. The four other variables
(associated with "personality") could probably also be numerically
quantified, if we were to extend the hierarchical cascade a few more
levels, defining what "we mean by" those still rather qualitative
descriptors.
3. Any such ("rigid hierarchical") cascade we might create would
also
be arbitrary, stipulative, and ad hoc. We would create it in one way
rather than another to suit some purpose -- and not because we had
discovered "the ontologically absolute descriptors of personality".
4. "Credit-worthiness" can thus be defined as at least in part a
function of boundary values in the four numerically-quantified variables;
each of those variables would have some lowest acceptable value (and,
presumably, no highest value).
5. If all of our variables were quantified, we could then define an
n-dimensional envelope of boundary values. The credit- worthiness of any
individual would be defined as "within" that bounded perimeter. If we are
willing to "exercise judgment" in defining the boundary values of
qualitative dimensions, we can define the n-dimensional envelope in both
qualitative and quantitative dimensions.
This diagram illustrates the two major dimensions of ad hoc meaning
specification: the choice of which dimensions compose the meaning of a
concept, and the choice of boundary values in those dimensions. In the
ascertainment of the credit-worthiness of any individual, we would take
the "measurements" of that individual, and load them into the bottom level
of our (here, eight dimensional) model, and do a synthetic algebraic
compilation, which would have at its top level a two-state (dichotomous or
binary) dimension, its two values being: 1) is credit-worthy, and 2) is
not credit-worthy. In this framework, depending on the assignment of
boundary values (how much of any of these factors is "enough"?), and on
the descriptive measurements of the individual, this decision would be
algebraically determined.
In the dimensional specification of what is or is not a "cup", we are
also free to chose boundary values, the choice of which determines whether
a particular object is a "cup" or a "bowl" or a "glass" or a "mug" or a
"stein" -- or, in fact, a "plate" or a "table" or a "Dalmatian"...
The structure of a categorical decomposition cascade across levels of
abstraction can be generalized, as per Figure 13.
LEVELS OF ABSTRACTION CHARACTERIZED BY TYPE OF DIMENSION
<-----Nominal--------Ordinal 2-------Ordinal 1-----Quantitative->
|------|------
|-------------->|
| |---------|---
|------------->|
| | |---|---------
| |-------------->|
| |-----------|-
Abstract Concept->|
|
| |-----|-------
| |-------------->|
| | |-|-----------
|------------->|
| |-----|-------
|-------------->|
|----------|--
Fig. 13
In this diagram, the horizontal lines represent variation in the X axis
(ie, categorical differentia), and the vertical lines represent variation
in the Y axis (categorical genus). Thus, each differentiation point takes
the following form:
Y
| |
|----|
value in Y--->| |-----------|-----------> X
|----| ^
| | value in X
Fig. 14
At each level of abstraction, an ordinal value is selected from a range
of possible values (the genus), and that ordinal value is
redifferentiated, either into lower-level ordinal values, or, at the
lowest level, into quantitative values. Figure 13 shows a "binary
differentiation" at each level (ie, the cascade splits into only two
dichotomous parts), but each level could be differentiated into several
dimensions. Each differentiation point represents a choice of value (or
boundary value range) at a particular level of abstraction. Thus, at each
point where the analysis descends a level of abstraction or precision,
there is a differentiation similar to that shown in Figure 14, such that
the value in X, shown as a "cut" in Fig. 14, becomes a bounded range in
*Y, as per Fig. 15:
Y *Y
| | | |
|----| |----|
value in Y--->| |-----------| |------> X
|----| |----|
| | | |
|----|
| |
|----|
| |---|--------------->*X
|----| ^Value in *X
^
Value in X
Fig. 15
Thus, the "quantitative" value defined by a "cut" in X shown in Fig. 14
becomes a bounded range in *Y at a lower level of abstraction. This
process occurs at each descending level of abstraction, until some "lowest
motivated level" of analysis or precision is defined. That might take an
ordinal value, or a quantitative value defined in a specific number of
decimal points. Since each new decimal point repeats the differentiation
of the boundary value range, this same differentiation (Figures 14 and 15)
takes place as each new decimal point value is defined.
The nominal (or dichotomous) determination of whether or not a
particular object IS or IS NOT a member of a particular abstract category
is determined in a bottom up manner, by taking the "measurements" of the
actual object at the lowest level of abstraction (quantitative
dimensionality). This decision is algebraically determined by the
boundary value choices in the decomposition cascade from the abstract
concept to its "meaning" in quantitative dimensions.
The assignment of meaning to an abstract concept, on the other hand,
takes place "in the opposite direction", as the abstract concept is
linearly factored across a series of descending levels into its
quantitative dimensions.
All definition and meaning assignment can be characterized in these
terms, and a comprehensive theory of conceptual structure, including a
classification of types of concepts, can be characterized in terms of an
ad hoc linear decomposition cascade across levels of dimensionality.
This method retains the dichotomous clarity of Aristotelian categories
(an object either IS or IS NOT a member of a particular class), but
overcomes the paralyzing rigidity which results from attempting to design
a single universal categorical taxonomy that is applicable to all
situations under all circumstances.
Instead, the method of ad hoc top-down decomposition permits a speaker
to emphasize just those shades or dimensions of meaning that are pertinent
to a particular context, adjusting those meanings by the exact and
context-specific assignment of boundary values.
This analytic description of categorical factoring is entirely
consistent with observation from empirical psychology, and appears, in
fact, to be "the way we actually do it in real life". This algebraically
determinate method overcomes the rigidity of the Aristotelian type
hierarchy, and the peculiarities and uncertainties of methods based on
"fuzzy logic", as well as the indeterminism of methods based on
prototypes.
It is true that any top-down decomposition is inherently rigid and
inflexible. But in this model, that inflexibility lasts but for a single
moment, under the drive of highly specific motivation, as a single act of
meaning-specification is undertaken. When that particular act of
communication is completed, that particular cascade of exact meaning
assignments is discarded.
7. SYNTHETIC DIMENSION AS UNIVERSAL PRIMITIVE
It can be argued that the fundamental intellectual act is the process
of "drawing a distinction" -- or of analytically subdividing some
previously unanalyzed "whole unit" into at least two constituent
sub-elements.
This act takes the abstract form of a "cut", analogous to the "Dedekind
cut" used to define continuity in the Real Number Line, as the previously
unitary data element or abstract object is divided into at least two
elements, and possibly more.
We have shown this cut in the form of lower and upper boundary values
in some range of variation, where "cut-off points" in a dimension are
defined to separate categories, as in Figure 9.
But in our indefinitely decomposable (ie, "potentially bottomless")
recursive scheme, we are defining the boundary value cuts between classes
in terms of dimensional values which themselves may possess a potential
"width" in an orthogonal dimension. Thus, every dimension is itself a cut
in some "higher" dimension, of which it becomes a sub-class or
sub-species.
In the quantitative characterization in Figure 9, the elements
(a,b,c...) are being numerically characterized in terms of multiples
("units") of X, but we have not specified the dimensionality of X, which
could be linear/quantitative, or multi-dimensional/qualitative.
The generic intersection of D* and D, as per Figure 16, is simply that
of a cut defined on a dimension, a boundary value range assigned in Y as
common to all the elements (a,b,c...) in X.
Y
D* |
|----|
| A |
|----|
| B |
|----|----|----|----|----|----|----|----|----|----|----|
| C | a | b | c | d | e | f | g | h | i | j | D
|----|----|----|----|----|----|----|----|----|----|----|-->X
| D |
|----|
| E |
|----| Fig. 16
This definition of dimension is recursive. That is, a dimension is
defined as an ordered class of dimensions -- as possibly constrained by
some "lowest actual level of analysis", where we might find at the bottom
of our decomposition cascade some atomic "undifferentiated unit" or
"modular part" which we subject to no further decomposition.
As shown in Fig. 13, the decomposition cascade of an abstract concept
terminates thus at some "lowest motivated level", usually as defined in
some type of quantitative dimensionality.
But the exact specification of this value involves defining a boundary
value range, which may be a function of the accuracy of the process of
measurement. A "one dimensional" quantitative dimension is analogous to
the "real number line" or "continuum", generally regarded as the
foundation of mathematics. But the process of actual measurement always
involves some degree of error, and actual measurement is always accurate
to some specific number of decimal places.
Each new level of decimal place can be understood to represent a new
value assignment in a range of possible values (ie, 0, 1, 2, 3, 4, 5, 6,
7, 8, 9). That range of values is a synthetic dimension, with the common
factor or "similarity" being that these values all represent
differentiations of the previous decimal place.
Thus, a number such as 3.25634 can be understood to define a "genus"
with 10 potential differentiations. Whether or not these differentiations
are "invoked" or called upon depends upon the motivation involved in
determining that level of analysis. Is this level of accuracy required
for some reason? Is the rest of the analytic process at the same level of
accuracy? Is the measurement process actually this accurate? At some
lowest motivated level of decomposition, the analytic process stops, and
the value assigned to the variable is perceived as "accurate enough for
this application".
Analog-to-Digital Conversion at the Lowest Level of Analysis
The discrete decimal-place structure of numbers means that the process
of measurement involves an "analog-to-digital conversion", as the
measurement of the "actual object" is taken by analog means (comparison
with a template), and converted to decimal (digital) format, naturally
incurring an undefined "round-off error". In general, this process takes
the following form, where Y = last decimal place of measurement on some
template (unit of measure):
Y** Y* Y
----|----|----|- Y**, Y*, and Y are decimal places.
| | 9 |
| |----|- "6" and "3" are sample values, and could
| | 8 | be any value in the range 0-9.
| |----|-
| | 7 | This represents the decimal series .63?
| |----|- with the location of the decimal point
| | 6 | left unspecified, and the value "4"
| |----|- taking the last (?) decimal place.
6 | 3 | 5 |
| |----|-
| | 4 | -----
| |----|- |
| | 3 | |
| |----|- |
| | 2 | |
| |----|- |
| | 1 | |
| |----|- |
| | 0 | |
----|----|----|----|---
^ ^ ^ ^
| | | Actual object being measured
| | Lowest calibration of template or "measuring stick"
| Second lowest level of calibration
Third lowest level of calibration
Fig. 17
In Figure 17, as we have drawn it here, we show the "actual height" of
the object being measured as somewhere around .6343, but since our
"measuring stick" is only accurate to the values .004 or .005, we would
probably assign the value .004 to this object. This process converts the
"continuous analog value of the actual object" to some discrete digital
value defined on our template measuring stick, at its lowest level of
calibration.
At some lowest motivated level of analysis, where we decide to
terminate the decomposition -- and in a quantitative dimension, decide on
a specific number places -- we assign value by a process akin to a
"Dedekind cut". Value assignment simply takes this form:
|----------------|-------------------------------|------> X
| V |
lowest possible highest possible
value value
Fig. 18
where the dimension X is presumed to be continuous (or "potentially
continuous"), taking the form of the Real Number Line.
But we must always recognize that the "analog-to-digital conversion"
involved in assigning this value means that the value given to V is
"actually" a bounded range, as characterized by the unknown "next decimal
place" in the series .634?
The implication of this for the study of dimensionality is that "each
new decimal place is a new dimension" of measurement, since that potential
range of values (0-9) is a synthetic dimension. Thus, the "cut" in the
Real Number Line X shown in Figure 18 is potentially a bounded range in Y,
as per Figure 14.
The Dimensional Assembly of All Conceptual Structure
Given that we can thus ground our lowest level of concept definition
and specification in the Real Number Line by a sequential process of
orthogonal synthetic dimension cuts, we can then reverse the order of
definition, and "compositionally assemble" any concept or data structure
by ascending the hierarchical order of levels of abstraction.
It so happens that both an alphabet and a vocabulary are synthetic
dimensions (ie, they are both a linear/sequential list of distinct
elements belonging to a common class that can be linearly ordered -- in
this case, by alphabetical order).
Thus, as we ascend the hierarchical structure of abstraction, we assign
labels to levels of abstraction we have created by synthetically combining
lower-level dimensions and their values. We name these categories with
nouns or proper nouns, identified by unique combinations from the elements
of the vocabulary dimension (themselves defined by unique combinations
from the alphabet dimension).
In this way, abstract ("language-based") symbolic abstraction is
"smoothly mapped" into quantitative dimensionality, and from there to the
Real Number Line. The implication of this for the study of conceptual
structure, whether algebraic or defined in language, is that any
conceptual structure can be wholly assembled, in every detail, from the
generic primitive element "synthetic dimension", as defined in the
fundamental definition.
Synthetic dimensions are the cuts or distinctions themselves, and they
also define that which is cut. It is true that an abstract decomposition
cascade takes the form of a "cut on a cut on a cut on a cut", in a
"self-similar" structure akin to that of a fractal. But it also true that
every element of this cascade, including the symbolic elements (or
"terminal characters") such as numbers and letters and words, all can
be perfectly and exactly defined as synthetic dimensions.
CONCLUSION
The concluding argument of this paper is a simple one: all abstract
conceptual structure is assembled from synthetic dimensions.
Terminological diversity, overlap and inconsistency can be eliminated from
the study of conceptual structure by the adoption of this schematic
interpretation.
Synthetic dimensionality appears to define the "generic form" of
semantic space, and it can probably be argued that every "natural
language" is an adaptation and "relabeling" of values and dimensions in
this general space, as shaped and individualized by the miscellaneous
stresses and influences of cultural evolution and psychological economy.
This paper has outlined only the major aspects of this theory, and has
not developed any detailed discussion of specific algebraic implications.
These implications, it would seem, can and should receive a full
discussion and expansion, in the context of a motivated and applications-oriented
engineering and research environment.
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