General Systems Theory: A Mathematical Approach

Citation

Garro1, I. (2001), "General Systems Theory: A Mathematical Approach", Kybernetes, Vol. 30 No. 2, pp. 216-224. https://doi.org/10.1108/k.2001.30.2.216.1

In this monograph, Dr Lin presented a mathematical foundation for the general systems theory as proposed by L. von Bertalanffy (1968). The contents of this book was organized in the so‐called “top‐down” approach, launched in 1960 by M.D. Mesarovic with concepts introduced first with as few mathematical structures as possible. To show the richness of systems properties, additional mathematical conditions were added.

To motivate the establishment of his mathematical foundation for the general systems theory, Dr Lin started the book with a brief history, and various situations where the concept of general systems and related mathematical results were needed. These practical situations include: scientific history, sociology, leadership, philosophy, development of the laws of conservation, non‐standard analysis, relativity theory, Schwartz distribution theory, optimization theory, unreasonable effectiveness of mathematics, what is mathematics, structure of molecules, epistemology, foundation of mathematical modeling, differential equations.

This book contains 12 chapters. Chapter one briefly describes the history of the concept of systems and the modern systems movement, and lessons learned in the recent past in the research of systems studies. Chapters two and three are preparations on both naive and axiomatic set theory, which are needed for readers to fully understand the contents of the book. However, Dr Lin had taken special care in writing each application in such a way that mathematical symbols and abstractions are used at different levels so that readers in various fields would find it possible to read. Chapter four deals with the concept of centralizability, which was a concept first proposed by Hall and Fagen (1956). After several basic mathematical results are established, the chapter focuses on applications and implications of the theoretical results. The problems dealt with in this chapter include why there always exist leaders in any group of people, public issues of contention, importance level of problems versus media, and the growth of two modern schools of mathematics.

In Chapter five, an attempt is given to lay a systemic foundation for laws of conservation. Chapter six focuses on how a multi‐level analytic mathematics can be introduced to study structures of general systems when leveled structures are the main topic. As applications, the problem of rest mass of a photon is given a different look; and Dirac’s d function is rewritten as a function from the field GNS of generalized numbers to GNS. In Chapter seven, Bellman’s principle of optimality is seen from the angle of general systems so that several sufficient and necessary conditions are given, under which the principle holds true. Several generalizations of the principle are also given.

Chapter eight addresses the problem “Why is mathematics so ‘unreasonably effective’ when applied to the analysis of natural systems?”, as proposed by Wigner (1960). Here, Dr Lin compared mathematics with arts, music and poems, looked at the structure of mathematics, and constructed a systemic point of view of mathematics. As applications, Dr Lin developed a systemic model for the state of materials, resolutions for some age‐old philosophical problems, and introduced a challenge, named vase puzzle, facing mathematical modeling. Chapters nine and ten are devoted to the mathematical development of concepts of general systems and applications back to mathematics. One of the main results is that even though many mathematical results can be written in the form of systems, many branches of mathematics can be unified under the name of systems. This end coincides with the goal of universal algebra, as proposed by some of the first class scholars at the turn of the last century, including Whitehead, Tarski, C.C. Chang, etc. Here, many modern topics of mathematics, such as chaos, attractors, feedback transformation, etc., are studied in the light of single relation systems. At the end, as an application, an n‐dimensional control (differential equation) system is decoupled into n one‐dimensional control systems.

Chapter eleven focuses on the development of the calculus of generalized numbers in the hope that this number system can be more practically useful. The final chapter lists an array of open problems related to the present of the monograph. As Dr Lin pointed out: studies on each of these open problems will lead to establishment of a new field of scientific research, this book is organized as an “open system”, where practical situations motivate the establishment of the mathematical foundation of the general systems theory. At the same time, the mathematical rigor of the established foundation leads to more interesting and meaningful problems to everyone to think about in the years to come.

Note

1. 1.

   Communicate through Dr Ibrahim Garro c/o Jeffrey Forrest, Department of Mathematics, Slippery Rock University, Slippery Rock, PA 16057, USA.