Most philosophical writing about mathematics falls into one of two categories: philosophers who have never done any mathematical research, writing for other philosophers; or mathematicians who have read little of the literature in philosophy of mathematics, writing expositorily. The former group tends to ignore how mathematics is actually done (although this has improved some in recent years); the latter writes about questions that interest mathematicians, but they write at a philosophically superficial level.
The original impetus for work in foundations of mathematics came from philosophical questions, but much of the work in the last fifty years in logic and set theory has been primarily mathematical, not philosophical. Some people working in the foundations of mathematics have the background to write thoughtfully about both mathematics and its philosophy, however, and a few do write occasional philosophical articles. Recently two collections of essays have been published by such people. In 1998, Solomon Feferman published In the Light of Logic, a collection of his essays over a twenty-year period. Now William Tait, Professor Emeritus at the University of Chicago has published The Provenance of Pure Reason: Essays in the Philosophy of Mathematics and Its History, a collection of his philosophical essays written over a similar time interval.
The range of content is very wide, from a discussion of the philosophical significance of his work on finitism, to discussion of traditional philosophy of mathematics questions such as the existence of mathematical objects and how we acquire mathematical knowledge ("Truth and Proof: The Platonism of Mathematics", "Beyond the Axioms: The Question of Objectivity in Mathematics") to philosophy of logic and set theory ("The Law of Excluded Middle and the Axiom of Choice", "Constructing Cardinals from Below") to two articles discussing Plato’s philosophy of mathematics, to discussion of the contributions to foundations and philosophy of mathematics of Gödel, Wittgenstein, Frege, Cantor and Dedekind.
Since Tait’s philosophical views have evolved during this period, he didn’t simply reprint these essays. Rather, he has, in one way or another, updated them. He begins with a substantial introduction that would stand on its own as a contribution to the subject. In it he eloquently discusses the importance of the axiomatic conception of mathematics, both for mathematics itself and for applications of mathematics. He also has a discussion of the question of the existence of mathematical objects that is a major contribution to that topic.
The first two essays, which discuss philosophical implications of his work on finitism, are reprinted without change, but are followed by an appendix that updates his views in light of recent work. Others, such as "On the Concept of Number," contain both footnotes from the original article and footnotes updating the original (the latter placed in square brackets to distinguish them from those in the original article). In one case, his views have changed sufficiently that he wrote a completely new chapter, "Constructing Cardinals from Below", rather than reprinting his earlier articles on the topic.
The articles are all very carefully and thoughtfully written. Most articles will be of interest to both mathematicians interested in philosophy and philosophers of mathematics. I believe that chapters 3 ("Truth and Proof: The Platonism of Mathematics") and 4 ("Beyond the Axioms: The Question of Objectivity in Mathematics") — and the introduction — have the widest general interest, but all the chapters have significant contributions to understanding mathematics and its relationship to the rest of our understanding.
Given that the chapters came from separate sources, I was surprised to run into some difficulty when I started (after reading the introduction) by reading Chapter 3. While the earlier part of the chapter was quite readable, when I got to section 13 I found a lot of notation that I didn’t recognize. The index wasn’t helpful. Later, when I went back to chapters 1 and 2, I found that the notation was defined there. The notation is also used early in chapter 5. Thus, the reader is warned that the chapters are not as independent as one would expect. Even if your interest in finitism is minimal, you need to either be familiar with it or read the first chapter before reading some of the later chapters.
Because the chapters are written for a scholarly audience, this is not a book for light dabbling. However, the care with which it was written rewards those willing to put in the effort to read its chapters carefully. Due to the range of topics discussed in the book, and the clarity of the exposition, anyone interested in the philosophy of mathematics will find at least a few of the chapters of this book (and certainly the introduction) well worth reading.
Bonnie Gold (firstname.lastname@example.org) is Professor of Mathematics at Monmouth University.