Mathematical Modeling in the Social and Life Sciences

CHAPTER 1

Mathematical Models

The status of a science is commonly measured by the degree to which it makes use of mathematics.

—S. S. Stevens

It is still an unending source of surprise to me to see how a few scribbles on a blackboard or on a sheet of paper could change the course of human affairs.

—Stanislaw Ulam

I. Mathematical Systems and Models

A. Mathematical Systems

Science studies the real world. In their role as scientists, human beings want to discover the laws that govern observed phenomena. When we better understand phenomena, then we may make valid predictions about future behavior. In a more active capacity, such understanding can lead to intelligent efforts to control phenomena, or at least influence them.

In this book, we will examine how we can use mathematical systems as tools to help achieve some of these aims. Although you will examine some examples from the physical sciences, most of our attention will be on problems of primary interest to social and life scientists, philosophers, and humanists.

A mathematical system consists of a collection of assertions from which we derive consequences by logical argument. We commonly call the assertions the axioms or postulates of the system. They always contain one or more primitive terms that are undefined and that hence have no meaning inside the mathematical system.

A familiar mathematical system is that of plane geometry. Two of the primitive terms in this system are “point” and “line.” As examples of axioms in this ...