Conceptual Foundations for Multidisciplinary Thinking

By Stephen Jay Kline | Go to book overview

CHAPTER 9
The Theory of Dimensions

In this chapter, we will develop a second, independent demonstration of the impossibility of both the reductionist and the synoptic programs. In Chapter 10 we will develop a third independent demonstration of this result. Why is this task done three times over? There are several reasons. The existence of three independent demonstrations makes the result more robust than one. Each demonstration speaks more strongly to particular groups of scholars. Chapter 8 may be persuasive for biologists; this chapter for scientists, engineers, and mathematicians; and Chapter 10 for computer scientists and individuals concerned with brain functions. The demonstrations in Chapters 8 and 10 are structural and verbal; the demonstration in this chapter is mathematical and meets current standards of proof. Finally, the three demonstrations taken together provide more tools for use in multidisciplinary discourse than any one alone.

What is the theory of dimensions? It is a theory that insists on a particular kind of consistency in the formation of equations modeling physical reality. The basis for the theory of dimensions is nothing more than the familiar idea that it is an error to equate apples and oranges. As we saw, when we fail to make "needful" distinctions for a given situation, even between apples and oranges, we cannot understand that situation well. In mathematical sysreps we make this idea of "needful" distinctions precise by insisting that each separate term in the equations we use to describe physical systems have the same dimensions. This idea, the need for the same dimensions in each separate term of the equations we use as sysreps, is called "the Principle of Dimensional Homogeneity." The words "separate term" indicate any group of symbols following a plus or minus sign. For example, in the equation A B/C = D + E F, each of A B/C, D, and E F may be forces, but some cannot be forces and others temperatures.

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