BOX 2-1 Mathematical Structures
At various points, this chapter refers to “mathematical structures.” A mathematical structure is a mental construct that satisfies a collection of explicit formal rules on which mathematical reasoning can be carried out. An example is a “group,” which consists of a set and a procedure by which any two elements of the set can be combined (“multiplied”) to give another element of the set, the product of the two elements. The rules a group must satisfy are few: the existence of an identity element, of inverses for each element of the set, and the associative property for the combining action. Basic arithmetic conforms with this definition: For example, addition of integers can be described in these terms. But the concept of groups is also a fundamental tool for characterizing symmetries, such as in and theoretical physics. This level of abstraction is helpful in two important ways: (1) it enables precise examinations of mathematical sets and operations by stripping away unessential details and (2) it opens the door to logical extensions from the familiar. As an example of the latter benefit, note that the definition of a group allows the combination of two elements to depend on the order in which they are “multiplied,” which is contrary to the rules of arithmetic. With the explicit recognition that that property is an assumption, not a necessary consequence, mathematicians were able to define and explore “noncommutative groups” for which the order of “multiplication” is significant. It turns out that there are many natural situations that are naturally represented by a noncommutative group.
It is possible, of course, to define a mathematical structure that is uninteresting and that has no relevance to the real world. What is remarkable is how many interesting mathematical structures there are, how diverse are their characteristics, and how many of them turn out to be important in understanding the real world, often in unanticipated ways. Indeed, one of the reasons for the limitless possibilities of the mathematical sciences is the vast realm of possibilities for mathematical structures. Complex numbers, a mathematical structure build around the square root of –1, turn out to be rooted in the real world as part of the essential equations describing electromagnetism and quantum theory. Riemannian metrics, the mathematical structure developed to describe objects whose geometry varies from point to point, turns out to be the basis for Einstein’s description of gravitation. “Graphs” (these are not the graphs used to plot functions in high school) consisting of “nodes” joined by “edges,” turn out to be a fundamental tool used by social scientists to understand social networks.
A striking feature of mathematical structures is their hierarchical nature—it is possible to use existing mathematical structures as a foundation on which to build new mathematical structures. For example, although the meaning of “probability” has long vexed philosophers, it has been possible to create a mathematical structure called a “probability space” that provides a foundation on which realistic structures can be built. On top of the structure of a probability space, mathematical scientists have built the concept of a random variable, which encapsulates in a rigorous way the notion of a quantity that takes its values according to a