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Publisher:

Princeton University Press

Publication Date:

2005

Number of Pages:

342

Format:

Paperback

Series:

Princeton Science Library

Price:

19.95

ISBN:

0-691-12127-3

Category:

General

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by , on ]

Teri J. Murphy

03/14/2006

The subtitle captures the heart of the book, which explores infinity from a scientific perspective and from a philosophical perspective. This book is fantastic, but it is not for beginners. Despite a sentence in the preface claiming that the book "has been written with the average person in mind" (p. xix), it requires a certain level of mathematical maturity, enough to be able to follow the use of notation, precise language, and mathematical logic. I suspect that some experience with philosophy would also be useful.

The book consists of five main chapters — Infinity, All the Numbers, The Unnameable, Robots and Souls, and The One and the Many — and two excursions —The Transfinite Cardinals, and Gödel's Incompleteness Theorems. Each chapter ends with a "Puzzles and Paradoxes" section (answers are included at the end of the book). I would consider using this book in a capstone course meant to survey mathematical ideas.

Chapter 1, Infinity, introduces ideas and questions about infinity. It discusses the history of ideas about infinity, which I believe would be fun for calculus students. In fact, this whole chapter would be good for pre-service secondary teachers and other mathematics students.

Likewise, Chapter 2, All the Numbers, would be useful for mathematics majors and physics majors. As I read this second chapter, I kept thinking, "No wonder students have problems understanding these ideas." We try to get students to understand space as a collection of points without discussing any differences between this idea and their intuitions about space. That is, the typical teacher does not build a bridge between mathematical space and physical space. Rucker points out, "so great is the average person's fear of infinity that to this day calculus all over the world is being taught as a study of limit processes instead of what it really is: infinitesimal analysis" (p. 87).

Chapter 3, The Unnameable, introduces The Berry Paradox and Richard's Paradox. This chapter draws on high school geometry and constructions. The latter part of the chapter draws on infinite series. This chapter also introduces the question, "What is Truth?" Chapter 4, Robots and Souls, introduces Gödel's Incompleteness Theorems and Artificial Intelligence. Rucker sets up the first four chapters to make his points in Chapter 5, The One and the Many. In this chapter, he states, "I think it highly significant that the deepest problems of metaphysics can be given explicit set theoretic formulations" (p. 205). I think of this as a thesis sentence for the book. He went on to say, "Gödel once expressed the view that present-day philosophy is in a state comparable to that of physics before Newton. Perhaps the ultimate role of set theory will be to do for philosophy what calculus did for physics." A nice table on p. 206 summarizes Rucker's points.

I read this book in a casual way; I think, however, that the book really needs to be studied. Clearly, scholars have spent much time and energy forming and re-forming these ideas. A casual reading can give a glimpse of the beauty and power offered in them, but a deeper understanding would require a more intensive reading. I read a previous edition (1982) of this when I was in high school, more than 20 years ago. Back then, as now, the title caught my attention. As a high school student, I didn't understand many pieces of the book. As an associate professor of mathematics, I understand more, but still not all. Perhaps after another 20 years, I will understand even more. And So On.

Teri J. Murphy is Associate Professor of Mathematics at the University of Oklahoma.

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