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Proof, Logic, and Conjecture: The Mathematician's Toolbox

Robert S. Wolf
W. H. Freeman
Publication Date: 
[Reviewed by
Kenneth A. Ross
, on

Before I looked at this book, I had the following two views based on my limited experience. One was that I questioned the value, from the point of view of mathematics, of a course in mathematical logic for undergraduate mathematics majors. While I agreed that logic is an important part of a complete education, I did not feel that a course in abstract logic helped many undergraduates understand and do proofs. I much preferred to teach a little logic as needed in the context of some mathematical area such as analysis, algebra, number theory, or discrete mathematics. In my long teaching career, "Logic and Set Theory" was the only course that I actively avoided having to teach a second time; I was very frustrated trying to teach Logic without mathematical context.

My second view, based on my limited experiences 20 to 30 years ago, was that none of the books written by logicians were suitable for mathematics undergraduates. The authors seemed content to build a complete logical edifice without tying it to the everyday needs of budding mathematicians.

These preconceived notions were pleasantly destroyed when I first looked at this book. Though I haven't tested it in the classroom, it is clear to me that it could be the basis of a wonderful course in which mathematics undergraduates would learn to understand and create proofs in the context of mathematics. The author has a delightful style and illustrates concepts via entertaining and pithy examples. The author obviously has very broad interests, and reading the book will add to the readers' general knowledge as well as mathematical knowledge. Also, he candidly recognizes that mathematicians want and need to know enough logic to know right from wrong, but do not necessarily want to study logic for its own sake. Indeed, in several places he points out what mathematicians know about logic and what most of them are unaware of, but he never snidely suggests that their behavior is improper. However, he does stress that students need to be aware of these practices. For example, in Chapter 4 (especially pages 75-77) he contrasts formal proofs with what mathematicians actually do. Later (pages 126-130) he provides elaborate hints for finding proofs.

This text is designed for a transition course between the applications-oriented courses such as calculus and the more theoretical upper division mathematics courses. The book is a user's guide to logic that includes fundamental concepts of mathematics (set theory, relations, functions) and introduces the basic concepts of abstract algebra, elementary analysis and number theory. It ends with a development of the complex numbers. Now I quote extensively from the Preface because the author describes the book so well.

Chapter 1 familiarizes the reader with the three main processes of mathematical activity: discovery, conjecture, and proof. While the main goal of the course is to learn to read and write proofs, this book views the understanding of the role of discovery and conjecture in mathematics as an important secondary goal and illustrates these processes with examples and exercises throughout. Chapters 2 and 3 cover the basics of mathematical logic. These chapters emphasize the vital role that logic plays in proofs, and they include numerous "proof previews" that demonstrate the use of particular logical principles in proofs. These chapters also stress the need to pay attention to mathematical language and grammar. Many of the examples and exercises in these chapters involve analyzing the logical structure of complex English statements (with mathematical or nonmathematical content) and translating them into symbolic language (and vice-versa). Chapter 4, the last chapter of Unit 1, is a thorough discussion of proofs in mathematics.

The remainder of the book is not directly about proofs. Rather, it covers the most basic subject matter of higher mathematics while providing practice at reading and writing proofs. Unit 2 covers the essentials of sets, relations, and functions, including many important special topics such as equivalence relations, sequences and inductive definitions, cardinality, and elementary combinatorics. Unit 3 discusses the standard number systems of mathematics -- the integers, the rationals, the reals and the complex numbers. This unit also includes introductions to abstract algebra (primarily in terms of rings and fields rather than groups) and real analysis. The material and the treatment in this unit are intentionally more sophisticated than the earlier parts of the book.

I would single out user-friendliness and flexibility as the main features that distinguish this book from the other available bridge course books. User-friendliness could also be called readability. One hears continually that reading is a lost art, that students (as well as the general population) don't read any more. I believe people will read books they find enjoyable to read. Every effort has been made to make this book engaging, witty, and thought-provoking. The tone is conversational without being imprecise. New concepts are explained thoroughly from scratch, and complex ideas are often explained in more than one way, with plenty of helpful remarks and pointers. There are abundant examples and exercises, not only mathematical ones but also ones from the real world that show the roles logic and deductive reasoning play in everyday life.

Criticisms of the book? I have hardly any. I don't like the format of the exercises which are labeled (1), (2), (3), etc. and are not boldface. This makes them hard to spot in the text. That's all.

There are two other things I like about the book. First, for some basic concepts such as ordered pair, binary relation and function, the author gives an "intuitive" and a "set-theoretic" definition. Second, the thoughtful "Note to the Student" advises students to approach their study of higher mathematics with a positive and active attitude, i.e., to take control of their study of higher mathematics.

A textbook that's good enough to make you want to teach a course that you used to avoid is a rare thing. Wolf's book does it right.

Kenneth A. Ross ( has been teaching at the University of Oregon since 1965. He was President of the MAA during 1995-1996. Before that he served as AMS Associate Secretary, MAA Secretary, and MAA Associate Secretary. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor. His most recent work has been on Markov chains and random walks on finite groups and other algebraic systems. He is the author of the book Elementary analysis: the theory of calculus (1980, now in 11th printing) and co-author of Discrete mathematics (with Charles Wright, 1999, fourth edition).

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