Graham Oppy is a philosopher of religion from “down under,” presently Associate Dean of Research at Monash University. This book is a scaled down version of a much more ambitious project of his (with the working title God and Infinity) to examine various views of infinity in the context of philosophy of religion, in order to then focus on “the role that the concept of infinity plays in traditional monotheistic arguments for the existence of God” (p. xi). Both God and Infinity being rather large topics, this volume has the more modest but still ambitious goal enunciated in its title, of exploring various philosophical perspectives on infinity. This is a book written by a philosopher and I think intended primarily for philosophers, but only those who can speak mathematics as a second language. Being something of a connoisseur of “the infinite” myself, and all the controversies and weirdness it spawns, I was intrigued by the author’s treatment, although I admit I read it with a dictionary of philosophy close at hand.
After a brief summary and taxonomy of puzzles and paradoxes, the journey through some of the thorny problem cases that “friends of the infinite” must confront begins with a whirlwind tour of some “Mathematical Preliminaries” from Set Theory to Non-Standard Analysis. I found the philosophical language tedious to plow through at times, and I cannot imagine that the mathematical underpinnings are easy going for anyone who is not a logician (either mathematician or philosopher). However, I did encounter some new and provocative examples beyond the standard fare of Zeno’s paradox and Hilbert’s Hotel, and the meticulous framing of each conundrum and possible resolutions was fascinating and worth the effort.
The conclusion was unsatisfying, although perhaps not surprising. For every argument made by the “enemies” of the infinite, there are good rejoinders for its supporters, and back and forth we go. It seems that there are “costs to every stance that one might take on questions about the infinite” (p. 294). So even though tangling with the infinite is fraught with difficulty, since our mathematics, science, and ordinary thought is shot through with it, we must learn to live with the problems, paradoxes and puzzles it introduces. But maybe this is not such a bad thing — it keeps us humble, and I do love a good mystery!
Bonnie Shulman is Associate Professor of Mathematics at Bates College in Lewiston, ME. Her current interests in the history and philosophy of mathematics include a fascination with the infinite and the relationship between continuous and discrete models.