Chaos: A New Mathematical Paradigm

by Vernon Howe

Introduction

Mathematics is traditionally done in obscurity and mathematicians go about their business with the realization that hardly any non-specialist is willing to invest the time to grasp the significance of the important ideas and theorems of mathematics. Recently, however, this situation changed when the new mathematical theory referred to as chaos burst on the popular scene in an unprecedented fashion.Articles on chaos routinely appeared in the popular press. James Gleick’s book, Chaos, became a New York Times best seller. Non-specialists have jumped on the bandwagon with almost a religious fervor.
The most passionate advocates of the new science go so far as to say thattwentieth-century science will be remembered for just three things: relativity, quantum mechanics, and chaos. . . . Of the three, the revolution in chaos applies to the universe we see and touch, to objects at human scale. Everyday experience and real pictures of the world have become legitimate targets for inquiry. (Gleick 5-6)
Because of all the hype and exaggerated claims that have accompanied this phenomenon, it is important to have some idea of what the mathematical theory of chaos is. The main of this essay is to briefly explain what chaos is and to outline how this theory can inform and modify our mathematical worldview. I hope this information will help take some of the mystery out of what chaos theory is and will aid readers in applying the theory to their own disciplines.

Two Mathematical Cautions

All disciplines use jargon and technical words; for example, biologists use Latin names and chemists use symbolic names such as NaCl. In contrast, mathematicians use everyday words but give them very different meanings. Mathematicians have a completely different meaning for the word chaos and the reader should guard against thinking in terms of the way it is commonly used. For example, mathematically, the word chaos is not synonymous with disorder, clutter, pandemonium, or confusion. The mathematical meaning of the world chaos is outlined in a later section entitled “What is Chaos?” In this essay the term chaos will always be used in its technical, mathematical sense.
It is important to keep in mind that the mathematical world is a very abstract and precise concept. Many times it is not at all clear how a mathematical idea or theorem informs our real world and the tendency is overreach in the application of mathematics when there is no real connection. An example of this “metamorphic” mode of thought is to say that since mathematicians believe in the fourth dimension (which they do), therefore God must live in the fourth dimension. The point is not whether God lives in the fourth dimension but whether the abstract mathematical concept has anything directly to say about where God lives. However, if a new paradigm is discovered in one area it can direct one’s mind to the discovery of a paradigm in another area that uses similar modes of thought. The mathematical theory of chaos seems to be an important new paradigm of this type.

 

The Old Paradigm

In the rise of classical science and mathematics, several “axioms” that still affect our thinking today became part of the prevailing paradigm. Five of these are: 1) determinism-complete current knowledge yields complete predictability; 2) linearity– the standard linear mathematical models suffice; 3) reductionism– the belief that a complex system can be analyzed in terms of its constituent parts, 4) complexity–complex problems must have complex solutions; 5) randomness–seemingly random phenomena do not have natural patterns.
In many ways classical science can be viewed in terms of a search for certainty. De Jong describes this view as follows:

The Cartesian-Newtonian paradigm contends that the physical world is made up of basic entities with distinct properties distinguishing one element from another. Isolating and reducing the physical world to is most basic entities, its separate parts, provides us with completely knowable, predictable, and therefore controllable physical universe. . . .The Cartesian-Newtonian paradigm contends that the physical universe is governed by immutable laws and therefore is determined and predictable, like an enormous machine. In principle, knowledge of the world could be complete in all its details. (De Jong 100-101).

Certainly during the twentieth century this search for certainty has been under siege (e.g., by relativity and quantum mechanics in physics and Godel’s Theorem in mathematics). Chaos can be viewed as the next “nail in the coffin” in the search for certainty.

It is ironic that the Cartesian-Newtonian paradigm was motivated by “the Judeo-Christian conviction that God is a rational being and thus created a rationally knowable world to be one of the inspirations for the emergence of modern science” (Beck 154). Alfred North Whitehead said it this way:
The greatest contribution of medievalism to the formation of the scientific movement…[was] the inexpugnable belief that every detailed occurrence can be correlated with its antecedents in a perfectly definite manner. . . . How has this conviction been so vividly implanted in the European mind?. . . It must come from the medieval insistence on the rationality of God. (Whitehead 12).