Infinity: A Very Short Introduction

While there is certainly no shortage of books on infinity, many of which have been reviewed on MAA Reviews, this Infinity, as part of the Very Short Introduction Series, offers a concise and compelling invitation to think about some of the deeper issues behind basic questions about infinity. Where do we encounter infinity? What is infinity? What are some logical challenges and subtleties that arise in dealing with infinity? How have human thoughts about infinity evolved? For a particularly tantalizing question, Stewart proffers: if in practice, all the numbers ever used by humans thus far are within a finite range, do we even need infinity and why?

Written in an accessible style, yet not skirting ideas of convergence or an occasional proof by contradiction, Infinity should appeal to a broad audience of math fans. I would especially recommend the book as supplementary reading for students of calculus or introductory set theory and to anybody who has ever found themselves baffled or awed by the mysteries of infinity.

Setting the stage with ample motivation, Stewart begins with a chapter of nine puzzles, “proofs”, and paradoxes followed by a chapter to clarify a few potential misconceptions regarding infinity vs. large numbers. Throughout, readers will encounter many classics (e.g., Zeno’s paradoxes, Grandi’s creation series \(1-1+1-1+\cdots\), Gabriel’s horn, Koch’s snowflake, Hilbert’s hotel, Cantor’s diagonalization argument) with accompanying discussion carefully placing each mathematical dilemma or idea in greater context to serve the author’s stated aim to “raise the reader’s awareness of the subtle distinctions that philosophers, theologians, and mathematicians have been forced to make when contemplating infinity.”

The two most technical chapters are on calculus (including a two-page primer on the origins of nonstandard analysis) and on cardinals. Infinity is not bound to mathematics, and readers will also learn about the historical intertwining of religion, philosophy, and mathematics; foray into perspective paintings; ponder infinity in optics, relativity, and Newtonian gravity; and contemplate the size of the universe.

Perhaps a greatest success of the book is the extent to which it conveys the human element of the lengthy process of grappling with and trying to resolve issues surrounding infinity. Reading along, one feels justified and in good company about any lingering insecurities or misconceptions they themselves might have about infinity today. After all, as Stewart writes,

It took more than two thousand years to learn how to handle infinity without it exploding in our faces, and even then, it can still cause trouble.

As a bonus, with the OUP edition measuring in at roughly six and a half by four and a half inches, you might even be able to carry Infinity in your pocket.

Briana Foster-Greenwood is an Assistant Professor of Mathematics at Cal Poly Pomona.