The philosophy of mathematics is undergoing a significant revival, resulting in a recent spate of books representing the beginnings of new subfields of study. This book is an example of this: it is the first book I know of (certainly in the last couple of centuries) in the subfield of religious perspectives on mathematics and its philosophy. In this review, I will discuss both the aims and value of this subfield, as well as the aims and value of the book itself.
Religious perspectives on the philosophy of mathematics have two foci, as well as two quite different potential audiences. One focus is on the issues in the philosophy of mathematics from the perspective of the given religious viewpoint: how might the religion contribute to the general discussion of the philosophy of mathematics. When this is done well, it is if interest to both those interested in philosophical issues related to mathematics, and to those sharing the faith of the authors of the book. When the discussion reconceptualizes some questions or concepts in ways which can make sense in the absence of that shared faith, it is of interest to the wider community. When, however, the issues are resolved by appeal to a deus ex machina particular to that faith, the discussion loses its wider appeal.
The second focus is how mathematics and its philosophy can help the faithful understand issues with which that particular religion and its believers are struggling: just as philosophers over the centuries have used mathematics as an archetypical example for assorted epistemological and ontological discussions, various religious traditions may also want to make use of mathematics. The audience for this second agenda, however, is primarily restricted to those sharing the particular religious faith, or at least, actively interested in its discussions.
This book attempts to introduce both aspects of these issues from a Christian perspective. There are people interested in such explorations from other religious perspectives as well: for example, at the first POMSIGMAA Contributed Paper Session in the Philosophy of Mathematics in January, 2003, M. Anne Dow, from the Maharishi University of Management, discussed, from the perspective of Transcendental Meditation, "A Unifying Principle Describing How Mathematical Knowledge Unfolds."
In reviewing this kind of work, there are two separate questions: of what interest is the discussion to the general mathematical or philosophical community, and of what interest is it to members of the given faith. As I am not a Christian, and am writing for a mathematical publication, I will not try to address the second question at all.
The aim of the book is to introduce the concept of examining mathematics from a Christian perspective. The book succeeds admirably in its main project: to demonstrate that, while Christians don't have a different mathematics than do atheists, Jews, or Hindus, there is a distinctive Christian perspective on mathematics. However, while the book's nearly 400 pages would seem to imply — contrary to the comment of a friend of one of the editors, quoted in the Conclusion, "That's going to be one short book!" — that there is a lot to be said about this topic, there is in fact not a large proportion of the book devoted to what Christianity has to say about mathematics. This is presumably partly due to a desire of the editors to keep the book largely self-contained. Therefore, in many chapters most of the space is devoted to giving the general background of the philosophical issues, or the relevant pieces of mathematics, or the historical background, and only the final 10% of the chapter is spent on the Christian perspective on what has been introduced.
Many of these summaries are quite nicely done. I'm not an expert in world intellectual history, but having spent twenty years at a liberal arts college where mathematics faculty take their turn teaching discussion sections of "Cultures and Traditions," I have some familiarity with the subject. The summaries of mathematics and its relation to the development of science and culture which constitutes part II of the book, "The Influence of Mathematics," seem quite balanced and well presented, for the very brief summaries they are. There is little which is specifically Christian in what is presented. The author of chapters 5 and 6 suggests that the pre-modern and modern idea that mathematics represents a form of certainty independent of God may be somewhat problematic for Christianity. So is the vision of mathematics as the structure within which one frames questions of whether an investigation is scientific and thus entitled to serious consideration. In the chapter on the mathematization of culture, the danger (both for Christians and for the world) of mathematics presenting itself as a value-free, impersonal basis for decision making is discussed. The first chapter of the book describes the modern versus postmodern world views, taking Frege as an example of the former and Paul Ernest as an example of the latter. It is basically well-written, although it starts with a list of common answers to "Why are the theorems of mathematics true?" not one which is the one which a mathematical "platonist" would give: "because they correctly describe facts about and relationships between mathematical objects." The second chapter's comparative study of the role of mathematics in ancient Greece, medieval Islam, and pre-modern China gives a good background for considering the issue of justification in mathematics, as well as to what extent is mathematics culturally determined.
Overall, the quality of the discussion in this book is very uneven. The book is a collection of articles, each written primarily by one or two of its ten different authors at nine different universities on a dozen different topics in one way or another involving the philosophy of mathematics — and a bit of the sociology, psychology, and pedagogy of mathematics — from a Christian perspective. However, the editors of the book have chosen to only indicate in the acknowledgements section who was the primary author of which chapter (and have not assigned the primary responsibility of chapter 12 — Teaching and Learning Mathematics: The Influence of Constructivism — to anyone). Had they exercised sufficient editorial oversight that the book read like a uniform whole, this might have been more understandable. As is, I found myself spending a lot of time speculating, as I was reading the book, why they chose to present the book this way. (Their explanation in the acknowledgements is not very satisfactory.) There are many places where their choice is extremely awkward. For example, chapter 10, "The Possibility of Detecting Intelligent Design," refers, especially in its first half, over and over again to the work of one William Dembski. As I found this chapter particularly irritating, I finally turned to the Acknowledgements and found that he was, in fact, the principal author of the chapter!
Unfortunately, the two chapters which are largely devoted to philosophical issues are the least satisfactory. In "God and Mathematical Objects" (Chapter 3) Christopher Menzel sets out to establish mathematical objects as objective and independent of human minds. This is a view I'm sympathetic to, but there are many challenges confronting this viewpoint, such as, if mathematical objects are not physical objects, and yet they're not in our minds, where are they; how can people have knowledge of these objects, etc. Menzel sets out to establish numbers as properties situated in the mind of God. To do so, he gives a thorough exposition of how Russell's paradox can be applied not just to sets, but to properties and propositions. To resolve this paradox, God must continually reconstructing all the levels of the set-theoretic hierarchy (and equivalent ones for properties) and, in his beneficence, share an understanding of this with man. One of the dangers of religious philosophy is the "pulling the rabbit out of the hat" nature of many of the arguments — as soon as there is a potential contradiction or complexity, it's dealt with via God's omnipotence, omniscience, etc.
Chapter 4, "The Pragmatic Nature of Mathematical Inquiry," has nothing particularly relevant (that I can find) to Christianity in it, but it is full of rash, unsupported statements, such as "from 'no contradiction has to date been derived from B' (the proscriptive support) mathematicians conclude that 'no contradiction is in fact derivable from B' (the proscriptive generalization)." (p. 108) The point of the chapter seems to be to demonstrate that there is no more certainty in mathematics than anywhere else, since we use unsupported hypotheses as much as any science.
The third part of the book, "Faith Perspectives in Mathematics," is, except for its first chapter ("Mathematics and Values," which discusses intrinsic and extrinsic arguments for the value of mathematics) about issues other than philosophy on the border of mathematics which are relevant for Christians. It seems primarily aimed at the faithful rather than at the larger community. One chapter, "Creativity and Computer Reasoning," attempts to explain why it's unlikely that we can build a computer which can think like a human being. Another, "The Possibility of Detecting Intelligent Design," attempts to shore up proofs of the existence of God by delineating what would constitute proof that the world has been designed by some intelligence rather than arising randomly. "A Psychological Perspective on Mathematical Thinking and Learning" introduces only a small part of that subject and seems more out of place than most chapters in this book. The final chapter, "Teaching and Learning Mathematics: The Influence of Constructivism," explains what constructivism means for mathematical pedagogy and where it can come into conflict with Christian values.
Overall, the book is an intriguing, though not unflawed, introduction to this subject. I look forward to more discussion of these topics by some of the authors in the future.
Bonnie Gold (firstname.lastname@example.org) is professor of mathematics at Monmouth University. Her interests include alternative pedagogies in undergraduate mathematics education and the philosophy of mathematics.