Yi Lin. General Systems Theory: A Mathematical Approach
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Contents
5. A Theoretical Foundation for the Laws of Conservation
. . . . . . . . .
111
5.1. Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
5.2. Brief History of Atoms, Elements, and Laws of Conservation . . .
112
5.3. A Mathematical Foundation for the Laws of Conservation . . . . . .
115
5.4. A Few Final Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
6. A Mathematics of Computability that Speaks the Language of
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121
6.1. Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121
6.2. Multilevel Structure of Nature . . . . . . . . . . . . . . . . . . . . . . . . . .
122
6.3. Some Hypotheses in Modern Physics . . . . . . . . . . . . . . . . . . . .
124
6.4. A Non-Archimedean Number Field . . . . . . . . . . . . . . . . . . . . . .
125
6.5. Applications
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
129
6.6. Some Final Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
134
7. Bellman’s Principle of Optimality and Its Generalizations
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135
7.1. A Brief Historical Note
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135
7.2. Hu’s Counterexample and Analysis
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136
7.3. Generalized Principle of Optimality
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138
7.4. Some Partial Answers and a Conjecture
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140
7.5. Generalization of Fundamental Equations
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143
7.6. Operation Epitome Principle
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147
7.7. Fundamental Equations of Generalized Systems
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150
7.8. Applications
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157
7.9. Conclusion
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161
8. Unreasonable Effectiveness of Mathematics: A New Tour
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163
8.1. What Is Mathematics?
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163
8.2. The Construction of Mathematics
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166
8.3. Mathematics from the Viewpoint of Systems
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171
8.4. A Description of the State of Materials
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174
8.5. Some Epistemological Problems
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176
8.6. The Vase Puzzle and Its Contradictory Mathematical
Modelings
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177
8.7. Final Comments and Further Questions
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188
9. General Systems: A Multirelation Approach
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191
9.1. The Concept of General Systems
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191
9.2. Mappings from Systems into Systems
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200
9.3. Constructions of Systems
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204
9.4. Structures of Systems
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214
9.5. Hierarchies of Systems
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224
9.6. Controllabilities
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235
Levels
Contents
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9.7. Limit Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
245
9.8. References for Further Study
. . . . . . . . . . . . . . . . . . . . . . . . . .
254
10. Systems of Single Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
255
10.1. Chaos and Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
255
10.2. Feedback Transformations
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
263
10.3. Feedback-Invariant Properties
. . . . . . . . . . . . . . . . . . . . . . . . . .
271
10.4. Decoupling of Single-Relation Systems . . . . . . . . . . . . . . . . . . .
280
10.5. Decomposability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .
291
10.6. Some References for Further Study
. . . . . . . . . . . . . . . . . . . . . .
309
11. Calculus of Generalized Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . .
311
11.l.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
311
11.2. Continuity
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
317
11.3. Differential Calculus
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
321
11.4. Integral Calculus
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
325
11.5. Schwartz Distribution and
GNS
. . . . . . . . .. . . . . . . . . . . . . . .
335
11.6. A Bit of History
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
344
11.7. Some Leads for Further Research
. . . . . . . . . . . . . . . . . . . . . . .
345
12. Some Unsolved Problems in General Systems Theory . . . . . . . . . . .
347
12.1. Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
347
12.2. The Concept of Systems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
348
12.3. Mathematical Modeling and the Origin of the Universe
. . . . . . .
351
12.4. Laws of Conservation and the Multilevel Structure of Nature
. . .
356
12.5. Set-Theoretic General Systems Theory
. . . . . . . . . . . . . . . . . . .
358
12.6. Analytic Foundation for an Applicable General Systems
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
363
12.7. A Few Final Words
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
367
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
369
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
379
CHAPTER 1
Introduction
Although people often talk about “modern science” and “modern technology,”
what really are their characteristics? The most important and most obvious char-
acteristic is that the forest of specialized and interdisciplinary disciplines can be
easily seen, yet the boundaries of disciplines become blurred. That is, the “overall”
trend of modern science and technology is to synthesize all areas of knowledge into
a few major blocks, as evidenced by the survey of an American national committee
on scientific research in 1985 (Mathematical Sciences, 1985). Another character-
istic, which is not obvious but dominates the development of modern science and
technology, is that the esthetic standard of scientific workers has been changing
constantly. Because of the rapid progress of science and technology, researchers
and administrators of scientific research have been faced with unprecedented prob-
lems: How can they equip themselves with the newest knowledge? How can they
handle new knowledge, the amount of which has been increasing in a geomet-
ric series? If traditional research methods and esthetic standards are employed
without modification, then today’s unpublished scientific results could become
outdated tomorrow. In this new environment, a new scientific theory — science
of sciences — was born (Tian and Wang, 1980). In particular, administrators of
scientific research are interested in whether science itself can be studied as a social
phenomenon.
Parallel with the science of sciences some natural questions arise. Can the
concept of wholeness or connection be used to study similar problems in different
scientific fields? Can the point of view of interconnections be employed to observe
the world in which we live? In the second decade of this century, von Bertalanffy
(1934) wrote: since the fundamental character of living things is its organization,
the customary investigation of the single parts and processes cannot provide a
complete explanation of the vital phenomena. This investigation gives us no
information about the coordination of parts and processes. Thus the chief task
of biology must be to discover the laws of biological systems (at all levels of
organization). We believe that the attempts to find a foundation for the theoretical
point at a fundamental change in the world picture. This view, considered as a
method of investigation, we call “organismic biology” and, as an attempt at an
explanation, “the system theory of the organism.” From this statement it can
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Chapter 1
be seen that parallel to the challenge of modern science and technology, a new
concept of “systems” had been proposed formally. Tested in the last seventy some
years, this concept has been widely accepted by the entire spectrum of science and
technology [for details, see (Blauberg et al., 1977)].
In this chapter we analyze the historical background for systems theory and
several important research directions. For a detailed treatment on the history of
systems theory, please refer to (von Bertalanffy, 1972; von Bertalanffy, 1968).
1.1.
Historical Background
Before discussing the historical background of systems theory, let us first talk
about what systems methodology is. Scholars in the field understand it in different
ways. However, their fundamental points are the same, loosely speaking. In
the following, let us first look at typical opinions of the two important figures in
modem science: H. Quastler and L. Zadeh.
Quastler (1965) wrote: generally speaking, systems methodology is essen-
tially the establishment of a structural foundation for four kinds of theories of
organization: cybernetics, game theory, decision theory, and information theory.
It employs the concepts of a black box and a white box (a white box is a known
object, which is formed in certain way, reflecting the efficiency of the system’s
given input and output), showing that research problems, appearing in the afore-
mentioned theories on organizations, can be represented as white boxes, and their
environments as black boxes. The objects of systems are classified into several
categories: motivators, objects needed by the system to produce, sensors, and
effectors. Sensors are the elements of the system that receive information, and ef-
fectors are the elements of the system that produce real reactions. Through a set of
rules, policies, and regulations, sensors and effectors do what they are supposed to
do. By using these objects, Quastler proved that the common structure of the four
theories of organization can be described by the following laws: (1) Interactions
are between systems and between systems and their environments. (2) A system’s
efficiency is stimulated by its internal movements and the reception of information
about its environment.
Zadeh (1962) listed some important problems in systems theory as follows:
systems characteristics, systems classification, systems identification, signal rep-
resentation, signal classification, systems analysis, systems synthesis, systems
control and programming, systems optimization, learning and adaptation, systems
liability, stability, and controllability. The characteristics of his opinion are that the
main task of systems theory is the study of general properties of systems without
considering their physical specifics. Systems methodology in Zadeh’s viewpoint
is that systems theory is an independent scientific discipline whose job is to de-
velop an abstract foundation with concepts and frames in order to study various
Introduction
3
behaviors of different kinds of systems. Therefore, systems theory is a theory of
mathematical structures of systems with the purpose of studying the foundation of
organizations and systems structures.
Even though the concept of systems has been a hot spot for discussion in almost
all areas of modern science and technology, and was first introduced formally in
the second decade of this century in biology (von Bertalanffy, 1924), as all new
concepts in science, the ideas and thinking logic of systems have a long history.
For example, Chinese traditional medicine, treating each human body as a whole,
can be traced to the time of Yellow Emperor about 2800 years ago; and Aristotle’s
statement “the whole is greater than the sum of its parts” has been a fundamental
problem in systems theory. That is, over the centuries, mankind has been studying
and exploring nature by using the thinking logic of systems. Only in modern times
have some new contents been added to the ancient systems thinking. The method-
ology of studying systems as wholes adequately agrees with the development trend
of modern science, namely to divide the object of consideration into parts as small
as possible and studying all of them, seek interactions and connections between
phenomena, and to observe and comprehend more and bigger pictures of nature.
In the history of science, although the word “system” was never emphasized,
we can still find many explanatory terms concerning the concept of systems. For
example, Nicholas of Cusa, that profound thinker of the fifteenth century, linking
Medieval mysticism with the first beginning of modern science, introduced the
notion of
coincidentia oppositorum,
the opposition or indeed fight among the parts
within a whole which nevertheless forms a unity of higher order. Leibniz’s hier-
archy of monads looks quite like that of modern systems; his mathesis universalis
presages an expanded mathematics which is not limited to quantitative or numeri-
cal expressions and is able to formulate much conceptual thought. Hegel and Marx
emphasized the dialectic structure of thought and of the universe it produces: the
deep insight that no proposition can exhaust reality but only approaches its coin-
cidence of opposites by the dialectic process of thesis, antithesis, and synthesis.
Gustav Fechner, known as the author of the psychophysical law, elaborated, in the
way of the natural philosophers of the nineteenth century, supraindividual organi-
zations of higher order than the usual objects of observation — for example, life
communities and the entire earth, thus romantically anticipating the ecosystems of
modern parlance. Here, only a few names are listed. For a more comprehensive
study, see (von Bertalanffy, 1972).
Even though Aristotelian teleology was eliminated in the development of
modern science, problems contained in it, such as “the whole is greater than the
sum of its parts,” the order and goal directedness of living things, etc., are still
among the problems of today’s systems theory research. For example, what is a
“whole”? What does “the sum of its parts” mean? All these problems have not
been studied in all classical branches of science, since these branches have been
established on Descartes’ second principle and Galileo’s method, where Descartes’
second principle says: divide each problem into parts as small as possible, and
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Chapter 1
In (von Bertalanffy, 1972), von Bertalanffy described the scientific revolution
in the sixteenth century as follows: “(The Scientific Revolution of the sixteenth
to seventeenth centuries) replaced the descriptive-metaphysical conception of the
universe epitomized in Aristotle’s doctrine by the mathematical—positivistic or
Galilean conception. That is, the vision of the world as a teleological cosmos
was replaced by the description of events in causal, mathematical laws.” Based
on this description, can we describe the change in today’s science and technology
as follows? At the same time of continuously using Descartes’ second principle
and Galileo’s method, systems methodology was introduced in order to deal with
problems of order or organization.
Galileo’s method implies simplifying a complicated process as basic portions and
processes [for details see (Kuhn, 1962)].
From this superficial discussion, it can be seen that the concept of systems we
are studying today is not simply a product of yesterday. It is a reappearance of
some ancient thought and a modern form of an ancient quest. This quest has been
recognized in the human struggle for survival with nature, and has been studied at
various points in time by using the languages of different historical moments.
Ackoff (1959) commented that during the past two decades, we witnessed the
appearance of the key concept “systems” in scientific research. However, with the
appearance of the concept, what changes have occurred in modern science? Under
the name “systems research”, many branches of modern science have shown the
trend of synthetic development; research methods and results in various disciplines
have been intertwined to influence overall research progress, so one feels the ten-
dency of synthetic development in scientific activities. This synthetic development
requires the introduction of new concepts and new thoughts in the entire spectrum
of science. In a certain sense, all of this can be considered as the center of the
concept of “systems .” One Soviet expert described the progress of modern science
as follows (Hahn, 1967, p. 185): refining specific methods of systems research
is a widespread tendency in the exploration of modern scientific knowledge, just
as science in the nineteenth century with forming natural theoretical systems and
processes of science as its characteristics.
Should we continue to use Descartes’ second principle and Galileo’s method?
Yes, for two reasons. First, they have been extremely effective in scientific research
and administration, where all problems and phenomena could be decomposed into
causal chains, which could be treated individually. That has been the foundation
for all basic theoretical research and modern laboratory activities. In addition, they
won victories for physics and led to several technological revolutions. Second,
modern science and technology are not Utopian projects as described by Popper
(1945), reknitting every corner for a new world, but based on the known knowledge
base, they are progressing in all directions with more depth, more applicability,
and a higher level of difficulty.
On the other hand, the world is not a pile of infinitely many isolated objects,
where not every problem or phenomenon can be simply described by a single
Introduction
5
causal relation. The fundamental characteristics of this world are its organizational
structure and connections of interior and exterior relations of different matters.
The study of either an isolated part or a single causality of problems can hardly
explain completely or relatively globally our surrounding world. At this junction,
the research progress of the three-body problem in mechanics is an adequate
example. So, as human race advances, studying problems with multicausality or
multirelation will become more and more significant.
In the history of scientific development, the exploration of nature has always
moved back and forth between specific matters or phenomena and generalities.
Scientific theories need foundations rooted deep inside practice while theories
are used to explain natural phenomena so that our understanding can be greatly
enhanced. In the following, we discuss the technological background for systems
theory — that is, the need which arose in the development of technology and
requirement for higher level production.
There have had been many advances in technology: energies produced by vari-
ous devices, such as steam engines, motors, computers, and automatic controllers;
self-controlled equipment from domestic temperature controllers to self-directed
missiles; and the information highway, which has resulted in increased communi-
cation of new scientific results. On the other hand, increased speed of communi-
cation furthers scientific development to a different level. Also, societal changes
have brought more pressing demands for new construction materials. From these
examples, it can be seen that the development of technology forces mankind to
consider not only a single machine or matter or phenomenon, but also “systems”
of machines and/or “systems” of matter and phenomena. The design of steam
engines, automobiles, cordless equipment, etc., can all be handled by specially
trained engineers, but when dealing with the designs of missiles, aircraft, or new
construction materials, for example, a collective effort, combining many differ-
ent aspects of knowledge, has to be in place, which includes the combination of
various techniques, machines, electronic technology, chemical reactions, people,
etc. Here the relation between people and machines becomes more obvious, and
uncountable financial, economic, social, and political problems are intertwined to
form a giant, complicated system, consisting of men, machines, and many other
components. It was the great political, technical and personnel arrangement suc-
cess of the American Apollo project, landing humans on the moon, that hinted that
history has reached a point where all aspects of science and technology have been
maturely developed so that each rational combination of information or knowledge
could result in unexpected consumable products.
A great many problems in production require locating the optimal point of
the maximum economic effect and minimum cost in an extremely complicated
network. This kind problem not only appears in industry, agriculture, military
affairs, and business, but politicians are also using similar (systems) methods to
seek answers to problems like air and water pollution, transportation blockages,
decline of inner cities, and crimes committed by teenage gangs.
6 Chapter 1
In production, there exists a tendency that bigger or more accurate products
with more profits are designed and produced. In fact, under different interpreta-
tions, all areas of learning have been faced with complexity, totality, and “systemal-
ity.” This tendency betokens a sharp change in scientific thinking. By comparing
Descartes’ second principle and Galileo’s method with systems methodology and
considering the development tendency, as described previously, appearing in the
world of learning and production, it is not hard to see that because of systems
concepts, another new scientific and technological revolution will soon arrive. To
this end, for example, each application of systems theory points out the fact that the
relevant classical theory needs to be modified somehow [see (Klir, 1970; Berlin-
ski, 1976; Lilienfeld, 1978)]. However, not all scientific workers have this kind of
optimistic opinion. Some scholars believe that it is an omen that systems theory
itself is facing a crisis [see (Wood-Harper and Fitzgerald, 1982) for details]. Only
time and history will have the right to tell who is right and who is wrong.
1.2. Several Aspects of Systems Theory
In the past half century or so, since the concept of systems was first introduced
by von Bertalanffy, systems thinking and methodology have seeped into all corners
of science and technology. Because of such a wide range of applications, scholars
in systems science have been divided into two big groups. One group follows
the direction pointed out by Quastler and Zedah, and the other contains scholars
with the desire to discover general principles while solving practical and specific
problems. Thus, under the name “systems methodology,” many problem-oriented
systems approaches have appeared. Here we look at the taxonomy of different
approaches, made available by Wood-Harper and Fitzgerald based on the field
of application of each approach. For the original work, see (Wood-Harper and
Fitzgerald, 1982).
1.2.1.
General Systems Methodology
General systems methodology is a theoretical methodology whose purpose is
to realize the ideal model of a systems theory described by von Bertalanffy (1968),
Quastler (1965) and Zadeh (1962) — that is, establish a general theory applicable
to explain phenomena in various specific systems. Based on the idea and logic of
establishing such a general systems theory, many scholars have tried to employ
known results in general systems theory to solve practical problems (Klir, 1970).
However, Berlinski (1976) and Lilienfeld (1978) thought that all these attempts
were extremely unsuccessful, since if a conclusion were obtained by applying
results from general systems theory, it would require modification of the relevant
Introduction
7
theory. However, from the previous discussion, it can be seen that it is natural
for the application of results from general systems theory to bring about such a
phenomenon, because we are using a different methodology than our forefathers
used. On the other hand, if an application of results of general systems theory
requires a complete rebuilding of the structure and rules of the relevant theory, we
surely need to study the correctness of the application of the results from general
systems theory. Also, because of this phenomenon, some systems analysts believe
that considering applications of general systems theory is not practical. At the
same time, many scholars hope to locate the reason why this phenomenon ever
appeared and to fix the problem so that the resulting general systems theory will
be more like what von Bertalanffy, Quastler, and Zadeh dreamed about, and much
easier to use to solve practical problems.
1.2.2. Human Activity Systems Approach
Checkland (1975) tried to establish a systems theory in order to solve problems
with unclear meanings and incomplete definition of structures. He believes that
this kind of system appears in environmental protection, administration of orga-
nizations, and business. He also attempted to look for a specific solution among
many candidates of solutions so his approach would be more meaningful in the
sense of application, because when practical problems are being solved often the
effort is difficult to define and causes controversy.
Checkland established his methodology based on general systems theory and
modified some aspects of systems theory so that the results would be more ap-
plicable in the real world. This methodology has been applied to many different
problems and obtained some obvious consequences. It can be classified into
general systems methodology, with its emphasis on problems without definite
structures, and complicated environments. Before solving a problem, this method
requires effort to explore and test the problem structure.
1.2.3.
The Participant Approach
The participant approach emphasizes the importance of the researcher(s) in
systems analysis (Mumford et al., 1978). Mumford believes that men must be
considered as a part of the system under consideration, either as participants of the
activities of the system or as designers of the system.
1.2.4.
The Traditional Approach
The traditional approach was introduced by the American National Computer
Center. To a certain degree, this method is accepted by many systems analysts.
8 Chapter 1
Its basis is that certain problems are solvable with computers. Each application is
considered individually; and problems are usually solved by finding and designing
the optimal subsystems. Based on research on the properties of systems of interest,
the condition of optimality is determined. According to the desired output, the
input is designed. This process is employed to complete the systems design. See
(1978–1979) for details.
1.2.5.
Data Analysis Approach
Data analysis is established on the principle that the fundamental structure
of systems is data [see (1978) for details]. The hypothesis is that if we can,
under certain conditions, classify the data, then we will be able to comprehend the
essential properties of the system under consideration. For the same set of data,
different mathematical models can be established. However, all these models must
possess some common properties. This implies that the given set of data can be
the initial point for studying properties of the underlying structure, and there is
no need to define various models for different applications. Also, if the relations
between the data have been established, then the essential structure of the system,
represented by the set of data, will appear naturally. Therefore, data analysis can
be considered as the “median” steps leading to the ultimate understanding of the
structure of organization.
1.2.6. Structured Systems Analysis Approach
The structured systems analysis approach was described individually by
de Marco (1980) and Gane and Sarsons (1979). It has been attracting the attention
of more and more businesses and organizations. This method was developed to
solve problems contained in classical research methodologies. For example, how
can the concept of subsystems be used to coordinate a large group of analysts and
to solve complicated problems concerning big machines and large systems? This
method offers new tools for analysis and literature editing. For instance, concepts
such as data flow diagrams, data dictionary, and constructed English have been
introduced. These concepts have allowed the literature of known systems to be
edited more clearly; consequently discoveries of new structures of the systems can
be expected.
From the aforementioned classification of systems approaches due to Wood-
Harper and Fitzgerald, it can be seen that the basic assumptions of various systems
methodologies can be considered as axioms or as results of general systems theory.
In a certain sense, all the systems approaches listed are applications of general
systems theory. At the same time, the first three approaches emphasize under-
standing of systems structures and theoretical analysis of systems, while the other
three approaches concentrate on problem solving with the hope that the results will