This book is aimed at the popular “Topics in Mathematics” course that is a requirement for the B.A. degree in many universities. This course has no prerequisites above and beyond a minimal amount of high school algebra and is meant to better acquaint the students with mathematics through an exposure to its applications and an exposition of some of its culture.
All the textbooks written for this course have adopted the survey approach and touch many topics lightly. We, however, feel that the course’s goals can be best accomplished by focusing on a few important topics and explaining both their significance and evolution in some depth. This calls for locating good mathematical ideas that are presentable at the level of the class, convincing the students of their value, and allowing enough time to permit the concepts to take root in the students’ minds through drill and development.
In our opinion, it is a serious pedagogical mistake to teach easy applications just because they show how mathematics can be used in an applied context. The superficiality of the mathematics will not escape the students and they will lose respect for the subject matter. With the respect will also go their willingness to learn. For this reason we have concentrated on a few topics whose centrality to mathematics is well known: probability, statistics, game theory, linear programming, and symmetry. The topic of social choice was added to these because its paradoxes and surprises make it a favorite with students.
While these topics have been expounded in many texts, our development of most of them is unique:
Probability and statistics: The exposition is focused on the normal curve. The evolution of this curve is set forth in detail together with explanations of the scientific needs that motivated the invention of some of the standard statistical procedures. This amounts to a comprehensible explanation of how and why the normal curve came to occupy the eminent position it does in our society – a tale in which Columbus, Galileo, the French Revolution, Gauss, and the Average Man all play their parts.
Voting systems: This chapter explores voting mechanisms that can be used to decide among a discrete set of alternatives. Although individual voters may be transitive and logical in their behavior, the aggregation of their opinions through a voting procedure can often produce problematic outcomes. The chapter includes a wide range of voting procedures, including the long-standing ones like plurality, majority, and rank-order voting, as well as some very recent proposals intended to address the weaknesses of the older methods. Several criteria for evaluation of voting procedures are considered.
Game theory: Enough game theory is taught so that the students gain an appreciation of the Nash equilibrium, a concept for which the economists voted to award the 1994 Nobel Prize to the mathematician John Nash. The popularity of the movie A Beautiful Mind makes this subject even more topical. Non-trivial applications to economics, population biology, anthropology, political science and other social situations are discussed.
Linear programming: The exposition is conventional. We included this material only because of its popularity amongst the instructors of this course and because of its importance as a tool of applied mathematics.
Symmetry: Contrary to common practice in books for this audience, we have added three-dimensional symmetry to the standard two-dimensional material. This makes possible visual/calculational problems that have just the right level of difficulty. It also provides a context for recounting to them the mysterious tale of Monstrous Moonshine – the story of the classification of the finite simple groups, MONSTER, and the mysterious connections with non-Euclidean geometry.
Map colorings: The study of map colorings passed on two surprisingly difficult problems. While both were solved a century later, both could be considered genuinely difficult problems since their proofs were long and subsequently no shorter proofs have emerged. The Four Color surely holds the distinction of producing more false proofs than any other. However, we have chosen to expound on the resolution of the Ringel-Youngs Theorem for no other reason than the exotic nature of its proof.
The evolution of the normal distribution: It is arguable that the normal distribution, otherwise known as the bell-shaped curve, has as many users as do Newton’s Three Laws. An examination of its history reveals much about the way scientists go about their business.