A couple of semesters back, I taught a “special topics” course in the history of mathematics to an audience consisting primarily of junior/senior mathematics majors. It seemed appropriate at the time to at least briefly address, at the very beginning of the course, some questions of the following sort: Just what is “mathematics”? What properties characterize it? And, as the semester progressed to more modern developments, I couldn’t resist detouring slightly to touch briefly on philosophical questions, like: What is the nature of mathematical objects like sets and numbers; do they exist, and if so, in what form? Based on the class discussion, it was apparent to me that all these were questions that most, if not all, of the students had never really given much thought to before.

Perhaps this should not have been surprising. These kinds of philosophical questions are not typically raised in mathematics courses, and Iowa State University (like, I suspect, a majority of others) does not offer a separate course in the philosophy of mathematics. Most of the students in the class had their hands full just learning how to prove things about continuous functions, vector spaces, and groups.

Because the philosophy of mathematics is not a commonly taught course, there aren’t a great many undergraduate-level textbooks available on the subject. Before getting this one to review, I had only three on my shelves: Colyvan’s *An Introduction to the Philosophy of Mathematics*, Shapiro’s *Thinking About Mathematics,* and Bostock’s *Philosophy of Mathematics: An Introduction*. The book under review is, I think, a notch or two more sophisticated than these three, and also has some features distinguishing it from them.

Linnebo’s book is introductory in the sense that it assumes no prior knowledge of the philosophy of mathematics, but it is written at a fairly sophisticated level and does assume some other background. For one thing, familiarity with philosophical reasoning and writing would be a definite advantage in reading this book; students who have not previously encountered words like “ontological” or “epistemological” will find the text rough going. In addition, the author assumes some prior background in logic; on page 33, for example, he describes “an arithmetical theory that is formulated in second-order language with three (for now) nonlogical symbols….”.

In terms of topic coverage, this text offers some alternatives to the three books listed above. Starting in the 19th century and continuing into the 20th, developments occurred in mathematics (such as the discovery of paradoxes in set theory and the development of non-Euclidean geometry) that led philosophers to reconsider the foundations of the subject. As a result, during the first half of the 20th century, the philosophy of mathematics was largely concerned with three foundational schools of thought that Colyvan calls the “big isms”: formalism, logicism, and intuitionism.

However, these topics eventually dried up as fruitful subjects of philosophical inquiry and other, more current, issues came to the fore. Linnebo’s book discusses the “big isms” in some detail but also, in the last half or so of the text, discusses some of these more contemporary issues as well. In this respect the book differs somewhat from Colyvan’s, which pays relatively short shrift to the former and focuses on the latter. The books by Shapiro and Bostock discuss the “big three” schools of thought in considerable detail, but do not discuss contemporary issues in quite the depth that Linnebo does. On the other hand, these two books spend more time discussing the earlier history of philosophy than does Linnebo.

Here is a more detailed look at the contents of Linnebo’s text: The first five chapters discuss the “big isms” referred to above. All of these approaches assume that mathematical knowledge is *a priori*, rather than (as the empirical viewpoint would have it) obtained through sense experience. Chapter 6 therefore looks at the work of John Stuart Mill and W. V. Quine, who defended an empiricist viewpoint. Chapter 7 discusses nominalism, the philosophical view that mathematical objects do not exist.

These seven chapters, which make up a bit more than half the book, cover, according to the author, “topics that tend to be included in any good course in the philosophy of mathematics.” There are still five chapters left to go, and the author uses these to discuss more specialized and advanced topics that are the subject of current philosophical debate.

Two of these chapters (10 and 12), for example, take a fairly deep look at the mathematical and philosophical foundations of set theory. In the first of these chapters, the author discusses the axioms of Zermelo-Frankel set theory, explains how sets can be created by a process of (transfinite) iteration, and addresses some of the issues that this raises. In chapter 12, the author begins with a discussion of the Continuum Hypothesis and its independence from the axioms of Zermelo-Frankel set theory and discusses the question of whether new axioms can be added to settle the truth or falsity of this hypothesis one way or the other. The mathematics invoked here is fairly substantial and a reader who doesn’t already know something about set theory will likely find these chapters difficult. (It is in chapter 12 that I found the one, relatively minor, factual error I noticed in the book — on page 170, the author credits the theorem that not all angles can be trisected by compass and straightedge to Galois, rather than Wantzel.)

The three remaining chapters look at structuralism (chapter 11), mathematical intuition (chapter 8) and mathematical abstraction (chapter 9).

In a relatively slim book like this one, it is inevitable that there will be topics omitted. While the work of Frege is discussed frequently throughout the book, the ideas of earlier philosophers (e.g., Plato, Artistotle, Kant, Descartes) are minimally discussed. (Contrast, for example, the books by Shapiro and Bostock mentioned above.) One contemporary issue that I would have enjoyed seeing a discussion of is the role of computers in mathematical proof (or other topics concerned with proof); several articles on this subject appear in the book *Proof and Other Dilemmas*, edited by Gold and Simons.

I must confess that I have always found reading the philosophy of mathematics to be somewhat difficult. Philosophy texts, even at the undergraduate/early graduate level, can be as dense, abstract and jargon-filled as mathematics books, and, to my mind at least, do not seem as intuitive as many mathematical ideas (perhaps because I’ve spent a great deal less time studying philosophy than I have studying mathematics). In this connection, it is interesting to note that at least one article (Belaguer’s “A Guide for the Perplexed: What Mathematicians Need to Know to Understand Philosophers of Mathematics,” reprinted in *Best Writing on Mathematics 2015*) has been written to help mathematicians read philosophy.

Linnebo’s book, therefore, is not an especially easy read. It is concisely written, covering a lot of material in a relatively short amount of space. It is, I think, written at a more sophisticated level, and lacks the light readable style of, Colyvan’s book. However, there are worse things that can be said of a book than the fact that the reader needs to work hard. This is a thought-provoking book, and is a useful addition to the textbook literature on this subject.

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