# A Mathematics Course for Political and Social Research

Will H. Moore
David A. Siegel
Edition: STU - Student edition
Pages: 456
https://www.jstor.org/stable/j.ctt32bbhk

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. List of Figures
(pp. xi-xii)
4. List of Tables
(pp. xiii-xiv)
5. Preface
(pp. xv-xx)
6. ### I Building Blocks

• Chapter One Preliminaries
(pp. 3-27)

Math is a formal language useful in clarifying and exploring connections between concepts. Like any language, it has a syntax that must be understood before its meaning can be parsed. We discuss the building blocks of this syntax in this chapter. The first is the variables that translate concepts into mathematics, and we begin here. Next we cover groupings of these variables into sets, and then operators on both variables and sets. Most data in political science are ordered, and relations, the topic of our fourth section, provide this ordering. In the fifth section we discuss the level of measurement...

• Chapter Two Algebra Review
(pp. 28-43)

Of all the chapters in this book, this is the one most safely skipped. Most of this chapter is taken up by a review of arithmetic and algebra, which should be familiar to most readers. If you feel comfortable with this material, skip it. If it is only vaguely familiar, don’t. The third section briefly discusses the utility of computational aids for performing calculations and checking work.

There are several properties of arithmetic that one uses when simplifying equations. These arise from the real numbers or integers for which the variables stand. In other words, because the variables we use...

• Chapter Three Functions, Relations, and Utility
(pp. 44-80)

Hagle (1995, p. 7) opines that “functions are valuable to social scientists because most relationships can be modelled in the form of a function.” We would add that functions are valuable for those political scientists who want to make specific theoretical claims and/or use statistics to test the implications of theories of politics. In other words, functions are valuable because they are explicit: they make very specific arguments about relationships. In addition, functions play a key role in developing statistical models.

What is a function? Functions may be defined in several ways, each developed more fully below. To get us...

• Chapter Four Limits and Continuity, Sequences and Series, and More on Sets
(pp. 81-100)

We have now covered most of the building blocks we need to introduce the more complex topics in the remainder of the book, but there remain a few more important ideas to address. Specifically, we must describe properties of functions and sets related to limit behavior, the behavior of these mathematical constructs as we approach some point in or out of their domains. Those readers with a stronger math background will likely have seen many of these ideas before, though they may have not seen them presented in this fashion, and so could benefit from skimming this chapter. We expect...

7. ### II Calculus in One Dimension

• Chapter Five Introduction to Calculus and the Derivative
(pp. 103-116)

In our experience, calculus and all things calculus-related prove the most stressful of the topics in this book for those students who have not had prior calculus coursework. We conjecture that this is due to the foreignness of the subject. While probability and linear algebra certainly have some complex concepts one must internalize, much of the routine manipulations students perform in applying these concepts use operations they are used to: addition, multiplication, etc. In contrast, calculus introduces two entirely new operators, the derivative and the integral, each with its own set of rules. Further, these operators are often taught as...

• Chapter Six The Rules of Differentiation
(pp. 117-132)

In the previous chapter we introduced the derivative as an operator that takes a function and returns another function made up of the instantaneous rate of change of the first function at each point. We also presented the definition of the derivative in terms of the limit of the discrete rate of change in a function across two points as the difference between the points went to zero. From this definition we could, with some algebra, compute derivatives of some polynomial functions. We’ll need to calculate derivatives for more complex functions in political science, however. Rather than go back to...

• Chapter Seven The Integral
(pp. 133-151)

Recall from Chapter 5 that our primary use of calculus will come in allowing us to deal with continuity usefully. The derivative provides us with the instantaneous change in a continuous function at each point. The derivative, then, permits us to graph the marginal rate of change in any variable that we can represent as a continuous function of another variable. In the next chapter we make extensive use of the derivative to find maxima and minima.

But what if we care less about change than about the net effect of change? Say we had some continuous function that represented...

• Chapter Eight Extrema in One Dimension
(pp. 152-172)

We have noted repeatedly over the past few chapters that a major motivation for computing the derivative is to find the maximum or minimum of a function. Maxima and minima are both types of extrema, and this chapter is devoted to finding them. Thus, in some ways this chapter is the payoff to this part of the book.

Finding extrema is useful in optimization theory, a topic that comes up fairly often in political science, and one to which we return in Chapter 16. Theorists often assume that an actor wants to maximize or minimize something (e.g., power, utility, time...

8. ### III Probability

• Chapter Nine An Introduction to Probability
(pp. 175-197)

Probability is central to the work of political scientists, as uncertainty is prevalent in every aspect of the social world and probability is the formal language of uncertainty. You need to care about probability because it can help you develop explicit statements about uncertainty. This will be necessary when you are trying to develop a theory to explain situations where people are acting under uncertainty. For example, individuals could be uncertain of each other’s benefits or costs for taking some action, which of a number of potential strategies they might use in any instance, or even just what outcome is...

• Chapter Ten An Introduction to (Discrete) Distributions
(pp. 198-241)

As discussed in Chapter 1, variables are indicators of concepts, and they take several values. If we look at a population (or sample), we often want to know how many people or states or other variables of interest hold each value. Put differently, we want to know the distribution of cases across the values of the variable. One can use mathematics to develop (1) an understanding of how the cases are distributed in a given population (sample) or (2) an expectation of how cases should be distributed given one’s beliefs about the process that produces the variation in that concept...

• Chapter Eleven Continuous Distributions
(pp. 242-272)

In the previous chapter we covered the concepts of random variables and their distributions, but used only discrete distributions in our discussion and examples. We did this to keep a chapter very important for the development of both empirical and theoretical political science free of calculus, for those readers who might want to skip over Part II of the book. However, there is little in the previous chapter specific to discrete distributions. Indeed, as we show below, replacing sums with integrals gets you much of the way toward representing the distributions of continuous random variables.

In this chapter we make...

9. ### IV Linear Algebra

• Chapter Twelve Fun with Vectors and Matrices
(pp. 275-303)

You first encountered mathematics as whole numbers and learned how to add, subtract, multiply and divide them. That field of inquiry is arithmetic. A loose distinction between arithmetic and algebra is that the latter replaces numbers with variables. So, for example, we can let the variable x take any value within the set of whole numbers. What you may not know is that the algebra you learned in middle school (in the United States) could actually be called scalar algebra, which is to say algebra concerning single variables. Scalar algebra in this usage is a subset of vector algebra, which...

• Chapter Thirteen Vector Spaces and Systems of Equations
(pp. 304-326)

Having read the previous chapter, one could easily have gotten the impression that linear algebra is a bunch of disconnected rules. Sure, they might be useful in some (a lot of) circumstances, but they may not seem to have a lot of coherence to them. That’s a perfectly reasonable inference given our presentation thus far, which was geared toward providing the tools needed to do most of the manipulations of vectors and matrices that you’ll need to do during your first couple of statistics classes. But, it turns out, this characterization is hardly fair. The rules of vector and matrix...

• Chapter Fourteen Eigenvalues and Markov Chains
(pp. 327-352)

Unlike the previous two chapters, this one provides neither key tools for computation nor core insights into linear algebra that are widely useful for all students. Rather, it introduces a handful of more advanced topics that are coming into increasing use in both statistics and formal theory in political science, and contains material one needs before tackling these more advanced topics. Because of this, we suggest that only readers tackling a full semester math course, as opposed to a math camp, should take on this chapter now, with the rest returning later should these topics arise.

The first section of...

10. ### V Multivariate Calculus and Optimization

• Chapter Fifteen Multivariate Calculus
(pp. 355-375)

The majority of this book has thus far dealt with functions of a single variable, and operations such as differentiation on these functions. This is sufficient for many applications; however, we do not want to limit our use of calculus in the social sciences to functions of a single variable. For example, we might believe multiple independent variables act in concert to determine the value of a dependent variable, or a political actor might have to make multiple choices at once. In these situations, we must use multiple variables to model the real world. This introduces some complexity, as we...

• Chapter Sixteen Multivariate Optimization
(pp. 376-399)

Multivariate optimization is needed in statistics when trying to find the vector of coefficients that minimize least squared error or maximize a likelihood function, and in game theory (and decision theory) when trying to make the best choice among several options. Thus, it is central to quantitative analysis in the social sciences. Because of this, it was our primary justification for delving into multivariate calculus in the previous chapter. The purpose of this chapter is to provide techniques to optimize functions of multiple variables.

We consider two types of optimization here: unconstrained and constrained. Unconstrained optimization, the topic of the...

• Chapter Seventeen Comparative Staticsand Implicit Differentiation
(pp. 400-412)

So far in this part of the book we have learned how to compute the rate of change of functions of multiple variables and use the tools we developed to do this to compute the maxima, and sometimes the minima, of functions of more than one variable. Sometimes this is all we want to do. In statistics, we may be interested in the vector of coefficients that maximize a likelihood function or minimize squared error. In decision or game theory, we may want to know what the optimal actions are for one or more individuals, given the others’ actions and...

11. Bibliography
(pp. 413-422)
12. Index
(pp. 423-430)