Thursday, September 8, 2011

Mathematics through the Eyes of Faith: a review

James Bradley and Russell Howell have put together this exploration of the relationship between mathematics and Christian faith. The book is intended primarily for students, and could even be used as curriculum supplement for a Christian educational setting. Each chapter is completed with exercises for the student, some of which are legitimate mathematical exercises that would engage even quite advanced undergraduate mathematics students. Even without the exercises, it is a good read for anyone who wants an introduction to the philosophy of mathematics from a Christian perspective.

The main questions addressed in this book are philosophical in nature; concepts from mathematics are used primarily as instruments to stimulate thinking about the "big questions." Chapters 1 and 2 give an introduction to these big questions and an historical background in order to set the stage for the next eight chapters, which are entitled, Infinity, Dimension, Chance, Proof and Truth, Beauty, Effectiveness, Epistemology, and Ontology, respectively. These titles all allude to the "big questions" introduced in Chapter 1. The final chapter is "An Apology" for mathematics, encouraging students who might take an interest to pursue mathematics as part of a greater search for truth, and as a particular way of serving God in the world.

The authors have given a very balanced and sophisticated treatment of each of the subjects they have introduced. They have refrained from picking a side on any issue for which there appears to be room for more than one consistently Christian view (which is virtually every issue). Consider, for instance, the issue of "Chance" in the universe. It is common for believers to contrast chance with God's sovereignty; however, this can hardly be the whole story, for reasons both scientific and theological. Howell and Bradley have laid out in Chapter 5 a case for theistic determinism and a case for theistic nondeterminism. At the heart of the debate is the notion of ontological uncertainty, the state of being actually governed by chance, so that no additional knowledge could possibly remove uncertainty. Theists naturally divide on this issue as much as non-theists do; the determinist may argue that God's sovereignty and omniscience excludes ontological uncertainty, whereas the nondeterminist may argue that God has created this universe with a freedom of its own.

Perhaps that is the great puzzle of this book: is there any particular way of seeing mathematics "through the eyes of faith"? It is not so much in the answers as in the questions that Howell and Bradley demonstrate the relationship between mathematics and faith. There is no one Christian position on the ontology of numbers, or on the nature of proof; rather, there are distinctively Christian questions that we may ask. For instance, if mathematical certainty implies genuine, sure knowledge about reality, does that mean we can "know the mind of God" through mathematics? Or is mathematics merely a creaturely activity, which, just like all human thought, is in an important way eternally distinct from God's thought?

There are a few instances in the book where I find the authors drifting into speculation which may be either idle, or, perhaps, theologically disastrous. As is so typical of Christian mathematicians, the authors have indulged themselves in a little comparison between God and the three dimensional creatures in Flatland. In Chapter 4, toward the end of the chapter, we read:
And what might we expect to see if God entered the world? Think back to one of your fingers entering flatland. First, the square sees a dot: your finger has just touched down. Next, a small, tight arc which grows over time as you push your finger farther in. Is it possible that God once entered our world in much the same way, first "visible" as a very small cluster of cells, growing and changing over time as he pushed his way in until, some thirty years later, we saw in Jesus Christ the most complete picture of God that we would ever see in our world? And is it possible, even, that what we think of as unrelated individuals sitting around us in church are in fact different parts of one body whose connections can be seen in another dimension?
Personally, I find such questions mostly silly, with only a small hint of profundity. That small hint quickly loses its appeal for me once I contemplate the many abuses one may construct out of this line of reasoning: perhaps the Trinity is really just three "intersections" of an infinite dimensional God with a three (or four, or eleven, or whatever) dimensional universe. Theologians would likely protest such a description, and I have little doubt that secular mathematicians will find such analogies quite bizarre. In any case, I have never found any encouragement, or really anything useful, in the metaphorical use of higher dimensions to illustrate the relationship between God and the world.

However, such instances as the one just mentioned are rare. Overall, the discussion is sophisticated, and the authors are cautious in their conclusions. I found the last few chapters especially satisfying in this regard, particularly the chapter on the "unreasonable effectiveness of mathematics." Mathematicians have long mused on the fact that beauty seems so akin to truth, on the fact that highly abstract theories could have such robust real-world applications, and on the fact that mathematics appears to be such a universal human phenomenon. It is tempting for many Christians to try using this as a "gotcha" argument for theism. Howell and Bradley present a theistic explanation as a potentially satisfying one, but, they insist, "We want to stop short of suggesting that a naturalistic worldview cannot explain the success of mathematics." They admit the many ways in which mathematics is actually not as effective as our idealized picture suggests; for every abstract theorem that turns out to be a physical law, there are many other results which have nothing to do with reality, or may even be misleading. They also admit that "there are several evolutionary explanations for forms of human cognition, and," they add, "such explanations need not be seen as contradicting Christian beliefs." But, after all, there is still at least something to the notion that
Human aesthetic values, and their subsequent use in successful physical theories, dovetail nicely with a Judeo-Christian view that humans are created in the image of God. Whatever being in God's image exactly entails, it seems to include a rational and aesthetic capacity reflective of God's that enables humans to understand and admire his creation. In short, the implications of a Christian worldview offer an attractive explanation for the effectiveness of mathematics.
Once again, faith does not provide the "right answers," but rather helps shapes the questions. As Christian students seek to understand what being made in the image of God might mean in light of modern science, and what modern science might mean in light of faith, it is most helpful to be guided by this even-handed approach.

There is one more point on which I would like to comment. One question I have been asked in Christian circles is, "How does your faith inform how you do your work as a mathematician?" I have tried to take the question as seriously as possible, but the best answer I have been able to come up with is horribly unsatisfying: "It doesn't." From my experience, however, I do not think I am alone in this feeling among mathematicians. There is simply nothing in our field which demands that we do anything differently from any other mathematician. A sociologist or an anthropologist would surely be influenced by Christians idea about human nature; an economist or a political scientist would surely be influenced by Christian morals; a biologist or an ecologist could be greatly informed by a Christian view of creation and life. But for a mathematician, questions about first principles rarely influence work, at least so far as I am aware at this very early stage in my career.

So as I read this book, I was wondering if Bradley and Howell would have anything to say about this. Sure enough, they don't; the question never really comes up. Essentially, this is not a book about mathematics, but simply about the philosophy of mathematics. From a working mathematician's perspective, the two are highly distinct. I do not find this unsatisfying in the least. Rather, I find it very healthy that Christian mathematicians should raise deep philosophical questions, since we operate within a very interesting yet poorly understood intersection of classical and modern thought. Indeed, I think something is wrong with mathematicians who use their minds only to do mathematics, and I doubt there are really any such mathematicians in existence.

I highly recommend this book to any Christian students interested in studying mathematics or who have a general affinity for deep philosophical questions. I also think it could be a great resource for institutions like the Center for Christian Study, who are seeking ways to integrate faith and education.

You can purchase another book by Bradley and Howell from the Splintered Light Bookstore.

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