Review of Mathematics Through the Eyes of Faith, by James Bradley and Russell Howell

The Council for Christian Colleges & Universities has published several volumes in a series called Through the Eyes of Faith. I have read only the volume on mathematics, but the general idea behind the series seems to be to provide textbooks that professors at Christian colleges can use to teach a course that provides a Christian perspective on their field of academic study.

Writing a textbook in this series for the subject of mathematics is an extremely challenging task, for several reasons. First, it is hard even to state some of the main issues without some technical knowledge of mathematics beyond what is normally taught at the high-school level. Since the target audience cannot be assumed to have this knowledge, it must be introduced on the fly. Second, it is virtually impossible to discuss a Christian perspective on mathematics without introducing some basic concepts from the philosophy of mathematics, which can be unfamiliar and difficult to grasp even for those with some general background in philosophy. Third, as the authors themselves recognize, there is a widespread presumption, even among Christians, that the whole concept of a “Christian perspective on mathematics” is absurd. Chapter 2 begins with the following sentences:

Ask some of your friends if they think there is a relationship between mathematics and Christian belief. You’ll probably get answers like, “No, not at all; are you kidding?”

Given these challenges, I believe that the authors have succeeded admirably. Of course, the real test of any textbook is how well it works in the classroom, and I do not even have any secondhand, let alone firsthand, knowledge of that. However, as far as I know, there is no other book on the market that comes close to filling this important niche. The authors write clearly and provide some interesting exercises at the end of each chapter. Furthermore, I did not detect any errors that I would consider to be really serious; that alone is quite a feat for a book that straddles several technical disciplines and tries to keep the discussion as simple as possible.

A short review cannot do justice to a book that covers this much material, so I will compromise by skipping over what I consider to be “standard” mathematics and philosophy of mathematics, focusing only on the interplay between Christianity and mathematics that is, after all, the main point of the book. Rather than discuss the chapters of the book in order, I will group them into four categories, and discuss them in roughly decreasing order of the proportion of material that pertains specifically to the relationship between Christianity and mathematics.

Epistemology and Ontology (Chapters 2, 6, 9 and 10)

The authors discuss two contrasting views on the nature of mathematics that they loosely label “Platonic” and “Aristotelian.” Mathematical platonism is the idea that mathematical entities have a real, objective existence independent of human thought. The Christian angle on mathematical platonism, which the authors mostly attribute to Augustine (though they also note that some aspects were anticipated by the Pythagoreans), is that mathematical entities reside eternally in God’s mind, and that the reason we are able to access them is that God created us with that capacity.

In response to the question of how humans could perceive anything in the universe of ideas, Augustine proposes his theory of illumination. He wrote, “We must know this [the truth of mathematical statements that go beyond experience] by the inner light, of which bodily sense knows nothing.”

Also in this camp are Kepler, who famously spoke of “thinking God’s thoughts after him,” and Newton and Leibniz.

Isaac Newton (1642–1727) saw his great work on the laws of gravitation as explaining those ideas in God’s mind that provided the basis of God’s relationship with the physical universe. Gottfried Leibniz (1646–1716), with Newton the founder of calculus, argued that God could not abolish mathematical truths without abolishing himself.

In support of this view, the authors sketch four arguments from revealed theology and four arguments from natural theology. Without going into detail, the four arguments from revealed theology are (1) God desires to reveal himself to us, (2) God is triune and hence his nature is necessarily mathematical, (3) God is omniscient so all mathematical truths must be in his mind, and (4) God cannot deny himself and so his nature supports the law of noncontradiction in logic. The four arguments from natural theology are (1) mathematics is indispensable for understanding the world, (2) mathematics is universal across all cultures, (3) mathematical facts are necessary and not contingent, and (4) mathematical knowledge is a priori.

The authors also briefly mention two other realists with a slightly different perspective from Augustine’s: the Jewish thinker Philo of Alexandria and the twentieth-century philosopher Michael Polanyi.

Philo contended that God created a “noetic cosmos” of immaterial realities as a precursor to his creation of the sensible world. Mathematical forms exist in this noetic cosmos. Contrary to Augustine, Philo did not argue that these forms exist in the divine reason. Hence, learning mathematics does not imply knowing the mind of God with certainty; rather it involves knowing aspects of this noetic cosmos.

As for Polanyi, they mention only that “by referring to what he calls tacit knowledge, he carefully analyzes the role of intuition and establishes a language of perception for our awareness of a priori knowledge.”

What the authors loosely call the Aristotelian perspective emphasizes how mathematics arises from human cognitive activity, as humans interact with creation.

This view emphasizes that mathematics is more than pure deduction; the concepts, axioms, and definitions used in mathematics derive ultimately from the human experience of creation. There is always a connection back to the creation, broadly conceived, no matter how far mathematical ideas have been abstracted from their source. … One might refer to the position just outlined as a type of mathematical empiricism [emphasis in original].

Names that the authors associate with this point of view are D. H. Th. Vollenhoven and H. Dooyerweerd. They also draw a connection with nominalism and scholastic theology.

The key difference between this perspective and Christian Platonism is that, for Platonists, some mathematics as well as the laws of logic are part of God’s nature; this alternative perspective states that they are created and hence contingent on God’s; will. They could have been different and God is not constrained to act in accord with them, except as bound by his promises.

A key argument for this view is that what seem to us to be absolutely necessary and certain truths may not in fact be intrinsic to God’s nature. For example, as Imre Lakatos has argued, mathematical statements are context-dependent. The famous results of Kurt Gödel have also contributed to the “loss of certainty” in mathematics. Interestingly, the authors also say that “intuitionism has been particularly attractive to Christian thinkers who regard the Augustinian claim that we can know the mind of God as presumptuous; rather, they emphasize that mathematics is a human activity and see intuitionism as a more faithful expression of Christian humility than realism.”

Wigner’s “Unreasonable Effectiveness of Mathematics” (Chapter 8)

Many readers of this review will have read, or at least heard of, Eugene Wigner’s famous essay on the unreasonable effectiveness of mathematics in the natural sciences. My personal take on Wigner’s essay is that trying to come up with a “rational explanation” of the effectiveness of mathematics is misguided. The first step in seeking a rational explanation would be to confirm that there is something that needs explaining, and that in turn would require a quantitative account of how much effectiveness would be reasonable and why, as well as a demonstration that the amount of effectiveness in the real world exceeds that threshold by a statistically significant amount. Neither Wigner nor any subsequent author has provided such a quantitative account, and in its absence, how could we even assess whether a proposed explanation succeeds? What Wigner’s essay does is help us cultivate our sense of wonder and aesthetic pleasure when encountering mathematics, and as such, I think that the title would be less misleading if it were changed to something like, “The Wondrous Effectiveness of Mathematics.”

My personal view on the topic aside, the authors present an interesting argument by Mark Steiner that the success of applying mathematics to the physical world supports a theistic worldview. Steiner focuses on the success of the strategy of seeking beauty in physical theories.

If humans evolved without the purposive action of a creator, there would be no reason to think that they were in any way “privileged.” Thus, for Steiner, a naturalist would not expect human aesthetic preferences to have any significant bearing on explanations regarding how the universe works, for that would indicate that some special privilege does indeed reside within the human species.

The authors consider several arguments and counterarguments, and then conclude, “We want to stop short of suggesting that a naturalistic worldview cannot explain the success of mathematics. We do want to suggest, however, that a theistic explanation is very compelling and might well be the best one to account for the continuing success of mathematical theories that ultimately grow out of human aesthetic criteria.”

Theological Topics Not Specific to Mathematics (Chapters 5 and 7)

Chapter 7, on beauty, might be eye-opening to those unfamiliar with the notion that mathematics is beautiful, but the connection that the authors make with the Christian faith applies to beauty of all kinds, and not just with the beauty in mathematics.

Why is aesthetic contemplation itself of any worth? Why should we contemplate the beauty of mathematics, or art, of nature, of love, or even of God himself? Nicholas Wolterstorff contends that it gives us a foretaste of the joy of shalom. This word is often translated as “peace,” which makes us think of a lack of war and conflict. However, in Christian theology there is much more to the concept than just lack of strife. Shalom denotes a reality—present at creation, lost through the fall, and one day to be restored—in which there is wholeness and completeness of individuals, societies, and indeed of the whole created order. Wolterstorff writes, “Aesthetic delight is a component within and a species of that joy which belongs to the shalom God has ordained as the goal of human existence and which here already, in this broken and fallen world of ours, is to be sought and experienced.”

Chapter 5, on chance, does touch on some deep theological and philosophical questions, but in my opinion they do not have much to do with mathematics. Probability theory provides technical tools for certain kinds of calculations, but is only tangentially related to the concept of chance that is discussed here.

Mathematical Tools for Thinking About God (Chapters 3 and 4)

Chapter 3, on infinity, and Chapter 4, on dimension, have considerable mathematical content, but relatively little theological content. The main point seems to be to suggest that the mathematical study of these topics can provide some theological insights. For example, the study of higher dimensions can help us understand how God transcends the universe that we can observe with our senses. I think that this is true, although I do not think the notion of higher dimensions takes us very far theologically.

The mathematical study of infinity, on the other hand, is in my opinion very relevant to some classical arguments for the existence of God. Most notably, the so-called “Kalam cosmological argument” has been revived in recent years (by William Lane Craig and others). I believe that this argument is seriously unconvincing, and that the past century of mathematical experience with reasoning about infinity shows us why the argument is unconvincing. I hope to develop this point in more detail elsewhere (EDIT added January, 2023: I have now published an article on the kalam cosmological argument in the Mathematical Intelligencer), but here let me just say that I do think that it is good that the authors include a chapter on infinity; however, they do not discuss the cosmological argument, and I see that as a major omission. Perhaps this will be rectified in future editions.

Concluding Remarks

As I mentioned above, the authors have succeeded admirably in providing a college-level introduction to a difficult topic. Students who master the material here will be well-positioned to pursue further study of the subject if they so wish.

There is really only one thing that makes me a little uneasy about the book. In the end, the reader may still come away with the impression that the connections between mathematics and the Christian faith, while not non-existent, are arcane and not really relevant to everyday life. Even professional mathematicians are unlikely to make any adjustments to their mathematical research programs or their spiritual lives on the basis of the issues presented here. Should we conclude that mathematics has little practical relevance to the life of the ordinary Christian?

Speaking for myself, I think that the most powerful effect that my mathematical training has had on my non-mathematical life is that it has taught me that I can apply a mathematical way of thinking to almost anything. By this I mean that idea that any statement that is not axiomatic can be derived from the axioms via a public chain of reasoning that everyone can verify for himself or herself. While one may sometimes choose not to go through the time-consuming process of verifying everything, and instead take for granted the word of an authority for the sake of expediency, in principle there is never any need to do so. (This is arguably not just a mathematical way of thinking but more generally a scientific way of thinking, but nowadays, as scientific experiments get increasingly sophisticated, requiring equipment and data available only to a select few, mathematics is the discipline that remains closest to the ideal of complete reproducibility without the need to trust authority.)

Sadly, the instinct for taking the time to verify statements that one has the power to verify, instead of always trusting what some authority says, has largely been lost nowadays. One might think that evangelical Protestants, with their long tradition of personally checking the Bible to validate theological claims, would have what I am here calling a “mathematical” mindset, but my anecdotal impression is that they are increasingly abandoning it. As I write this, Oxford Dictionaries has just announced that their word of the year is “post-truth.” In times like these, I believe that mathematics is able to offer a unique perspective on truth, personal verification, and respect for authority that is sorely lacking in society in general and the church in particular. The authors briefly touch on this point in the final chapter of the book (Chapter 11, “An Apology”), but I hope that they will consider saying more in future editions.

Posted December 2016

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