[SCU] Chaos Theory - The Unknowalbe and Counterintuitive

This book grew out of a course on science and religion that I have taught at Santa Clara University for a number of years. This course (whose title is: “Chaos Theory Metamathematics and the Limits of Knowledge: A Scientific Perspective on Religion") is rather unique, since it is both a technical elective for engineering students and satisfies a Core Curriculum requirement in religious studies. As such, it requires two very different textbooks — one that focuses on questions related to science, aesthetics and theology (Truth, Beauty and the Limits of Knowledge), and this book, which represents its “technical companion”.

One of the most distinguishing features of this “companion” volume is the breadth of the topics that are covered. Indeed, one seldom (if ever) sees chaos theory, metamathematics, quantum mechanics and the theory of relativity united under a single title. My decision to adopt such an unorthodox approach was motivated in large part by the recognition that all four disciplines challenge the traditional Newtonian paradigm, and suggest that we must adopt a more sophisticated view of physical reality. This new outlook entails (among other things) a complete shift in the way we think, and brings into question some of our most deeply ingrained beliefs about nature. Giving up on our “common sense” world view is, of course, a difficult thing to do, and most of us have no appetite for such profound conceptual changes. In that respect, classical physics has a decided advantage, since it is capable of reconciling theoretical results with our everyday experience. This is why we still teach it in schools, and why this approach continues to be popular with the general public. The fact remains, however, that the Newtonian methodology is limited to a fairly narrow range of “well behaved” phenomena, and that it fails to describe many subtle aspects of nature. Those who have some scientific training are well aware of this deficiency, but even they usually encounter it only within a particular area of specialization. In view of that, I decided to write a book that combines insights from several different disciplines, and outlines what each of them has to say about the counterintuitive nature of physical reality, and the limitations of human knowledge. This is the common theme that binds the many diverse topics that will be discussed in the following sections.

The book is divided into four chapters, each of which focuses on a different area of modern science and mathematics. The chapters are self-contained, and can be read without reference to any other parts of the book. Although such a structure provides readers with considerable flexibility (and even allows them to "pick and choose" according to their interests), I would advise against such a selective approach. I strongly believe that reading the entire text is a worthwhile investment, since only a broad interdisciplinary perspective can provide us with a proper understanding of the intricate laws that govern the world we live in. I would therefore encourage the reader to be curious, and explore as much “uncharted territory” as possible.

This brings me to my second reason for writing this book, which has to do with the relationship between science, philosophy and theology. Although most of the topics are highly technical, the overall text is designed to serve as a sort of “catalyst” for a more general interdisciplinary discussion. Indeed, those who manage to fully grasp some of the strange and elegant results of modern science often become more open to questions regarding the nature of reality. This is usually the point when scientists can begin to appreciate philosophical and theological insights, and can engage in a meaningful conversation with these disciplines. It is my hope that those who carefully read the following chapter will become amenable to such a possibility, and might even begin to see certain subtle similarities between the scientific and theological world views. If they do, they will join a very distinguished group of individuals, which includes Newton Leibnitz, Einstein, Heisenberg, Gödel, Weyl and many other great names in the history of science and mathematics. The material covered in this book is presented in a style that is typical for upper division textbooks in science and engineering. All important ideas, and results are illustrated by examples, which should hopefully help the reader gain a better understanding of certain fundamental concepts in system theory, metamathematics and modern physics. Since chaos theory happens to be the Primary technical topic in my course, I chose to devote more attention to this field than to others. As a result, my discussion of metamathematics, quantum mechanics and relativity is more selective, and is presented with less detail.

Chapter 1, which is concerned with chaos theory, opens with an overview of linear systems. This is an essential prerequisite for understanding the dynamics of nonlinear systems, which are considered in the following section. Our discussion of this topic will include methods for computing the system equilibria, an analysis of stability properties, and a description of different kinds of attractors. Some characteristic features of chaos (such as hypersensitivity to initial conditions and the fractal dimension of strange attractors) are introduced in Section 1.3, which is followed by a discussion of various types of bifurcations. Section 1.5 examines several different mechanisms that can lead to chaotic dynamics, and the chapter concludes with two case studies - the forced pendulum, and Chua's nonlinear circuit.

Chapter 2 introduces the field of metamathematics, and provides a brief description of formal systems. This material constitutes the necessary background for understanding Gödel's Incompleteness Theorem, whose proof is outlined in Section 2.3. The final section of this chapter describes some important consequences of Gödel's result, and examines several examples of incompleteness in modern mathematics.

Chapter 3 is devoted to quantum mechanics (in its non-relativistic form). The basic mathematical formulism that is used in this discipline is introduced in Section 3.1. We then proceed to discuss Heisenberg’s Uncertainty Principle and the phenomenon of quantum entanglement, both of which impose fundamental limits on what we can know about the microscopic world. Section 3.4 also provides a brief description of Bell’s inequality, and the role that it played in resolving the famous EPR paradox.The final section of this chapter introduces some basic concepts from group theory, which is one of the most important mathematical tools used in quantum mechanics and particle physics.

Chapter 4 consists of two separate parts, the first of which deals with special relativity. This section provides an introduction to relativistic mechanics and describes some of its most important results (such as Lorentz contraction, time dilation and Einstein’s famous equation $$E=mc^2$$). In section 4.2 We turn our attention to general relativity, whose main ideas are described in the framework of differential geometry. The discussion in this section focuses on the computation of geodesics, Christoffel symbols and the Schwarzschild metric.

My work on this book was supported by a grant from the Center for Science, Technology and Society at SCU. I am grateful to the Center for its generosity and its support for this project. I am also indebted to my colleagues Ruth Davis (Computer Engineering), Radovan Krtolica (Electrical Engineering) and Betty Young (Physics) who read the who read the manuscript and provided many constructive suggestions