To Infinity and Beyond

To Infinity and Beyond:  A Cultural History of the Infinite, Eli Maor. 1991 (reprint of Birkhauser edition.1987) 284 pp., 162 illustrations, paper $16.95 ISBN 0-691-02511-8, Princeton University Press, Princeton, NJ, 08540  http://pup.princeton.edu.

David Hilbert (1862-1943), the great German mathematician, once remarked:  “The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite…”  Indeed, shedding light on the infinite has been a daunting task for mathematicians and philosophers down through the ages.  In his book, To Infinity and Beyond, Eli Maor does a very admirable job of clarifying the conceptual underpinnings of this enigmatic topic. 

The book is divided into four parts:  Mathematical Infinity, Geometric Infinity, Aesthetic Infinity, and Cosmological Infinity.  The first part, Mathematical Infinity, provides the reader with a brief, but accurate overview of the historical evolution of the concept of infinity supported by many well-known examples from the history of mathematics, including Zeno’s paradoxes, the magnificent work of Georg Cantor, and examples of convergent infinite series.  The second part, Geometric Infinity, explores a variety of interesting topics in geometry including circle inversion, geometric paradoxes involving infinity, and a discussion of some unusual curves.  The third part, Esthetic Infinity, is a fascinating collection of chapters including a substantial discussion of the role of infinity as related to the art of M.C. Escher.  There is a rich examination of many cultural aspects of infinity encompassing areas such as the ornamental designs of the Moslems and the theological and philosophical perceptions of infinity dating back to the twelfth century kabbalists, a mystic movement of Jewish devotees whose belief in the transcendence of God led them to the Ein Sof (the infinite).  The fourth and final part, Cosmological Infinity, is for the most part a brief history of astronomy from the ancient Greeks to the present and an account of how astronomers and cosmologists have struggled with the notion of a universe that at once is viewed as infinite, and at other times as finite and bounded. Of particular interest in this part is the saga of Giordano Bruno (1548-1600) who was neither scientist nor philosopher, but made his mark by his profession of an infinite universe, populated by an infinity of stars, each surrounded by planets on which intelligent beings thrive and prosper.  These beliefs met with the wrath of the Roman Inquisition and he was burned at the stake in 1600.

One of the most attractive features of Maor’s book is its versatility.  Mathematicians and non-mathematicians alike will find many aspects of this book to be informative and interesting.  Actually, even many high school students will find enjoyment throughout the book. Maor is to be commended for his clarity, accuracy, and enthusiasm.  This is an excellent book to add to your library

Anthony Piccolino, Associate Professor of Mathematics, Montclair State University, Montclair, NJ