This text (an English translation of a German book that has gone through several editions) is one of the more attractive books on the philosophy of mathematics that I have encountered recently. Its principal distinguishing feature is that it is written by mathematicians, not philosophers. This distinction manifests itself in several ways in the text. The book provides a lot of mathematical content while still keeping mathematical prerequisites reasonably low. The book is also, I think, somewhat more accessible to those of us whose training is in mathematics rather than philosophy. The jargon of philosophers can sometimes make for heavy reading, but the authors’ writing style here is refreshingly clear. Additionally, mathematical problems and issues are used to motivate philosophical questions, thus making the book somewhat more down to earth than a text that just plunges into abstract philosophical ideas.

The book begins with a look at the real numbers and some of the philosophical issues presented by them. The authors begin with the standard proof of the irrationality of the square root of 2, explain the problems this caused the ancient Greeks, and then discuss the “nested interval” approach to the construction of the real numbers. They explain how these ideas raise philosophical questions concerning, for example, the meaning of number, the nature of axiomatic reasoning and the concept of infinity.

Chapter 2 is an historical overview of mathematical philosophy, strongly linked to the history of mathematics. The chapter begins with individual sections (starting with Pythagoras) on famous mathematicians and philosophers, and then (once the early 20th century has been reached) changes focus from people to ideas. There are sections on the main “isms” of mathematical philosophy (intuitionism, formalism, etc.) as well as some concluding chapters on modern (post-1960) topics in the area.

The next chapter is also in the nature of an overview, this time focusing on mathematical issues. Questions that were raised previously (such as the nature of infinity) are returned to and discussed in more detail.

Chapters 4 and 5 address foundational issues: first set theory, then logic. The mathematical discussions here are fairly extensive: we are exposed, for example, to both Zermelo-Fraenkel and Von Neumann-Bernays-Gödel set theory. Gödel’s theorems are of course discussed, as is Tarski’s work on the concept of truth. Mathematical history is again not slighted, and there are sections, for example, on the history of logic and the history of axiom systems.

Chapter 6 is an introduction to the ideas of nonstandard analysis, a topic in mathematics that very few undergraduates get to see. The authors give a brief overview of the subject and discuss some philosophical issues related to the idea of things being “infinitely small”, a nice complement to previous discussions about some of the problems inherent in things being “infinitely big”.

The seventh and final chapter of the book is a brief (roughly ten page long) summary and retrospective on what has come before. The authors recall the various questions that have been raised and conclude with a short discussion of what the philosophy of mathematics is and what it can accomplish.

In addition to the textual content, there is an extensive bibliography (more than 14 pages long, with 386 references, a great many of them not in English) and also an 18-page section of mini-biographies of some of the people discussed in the book, proceeding literally from A (Aristotle) to Z (Zermelo).

There are a number of times when the translation from the German results in poor grammar (“The significance of Georg Cantor’s (1845–1918) works… are enormous.”; “Dedekind attempts to make the foundations of mathematics precise and to arrange them were often criticized.”) or awkwardly phrased sentences (“Relevant is first of all their content.”) but this is not a serious problem. Idiosyncratic English notwithstanding, I found this book somewhat easier to follow than many other standard books on the philosophy of mathematics.

This is a fairly large book (more than 400 pages of text) and surely contains enough — indeed, more than enough — material for a one-semester course in the philosophy of mathematics. It has other uses as well, however: mathematicians or mathematics students who want to be exposed to some of the high points of the subject should find much of interest here, and anybody teaching a course in the history of mathematics might find this a good source of interesting lecture material.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.