Distinctively Christian Mathematical Instruction: A Hopeful Imagination
I am a Christian mathematics professor at a Christian university. When I tell people what I do, there are two typical responses. One is said with a laugh: “Does the answer to 1+1 depend on whether or not you’re a Christian?” To which I reply, “No, the answer depends on whether or not you’re a mathematician.” 1 The other is simply, “Oh . . .” said with a facial twitch that uniquely combines a blank expression with a cringe. In both cases, the conversation quickly moves on to more interesting topics, such as the weather.
. . . science permits, through mathematics, the possibility of a God who acts in the world, who answers prayers, who heals, and who responds to evil but also gives the world freedom to rebel.
In my experience, it is rare to find someone with sufficient imagination to wonder seriously whether mathematics can be taught Christianly. It is rarer still to find someone who can articulate such a vision. I’m not always sure whether I can, myself. A graduating student once said to me, “I’m still not sure that a Christian perspective makes a difference in mathematics, but I do know that you personify a Christian mathematics teacher,”—surely a mixed compliment, as it was precisely the former that I had been attempting to teach!
It has been almost ten years since I became open to the possibility of mathematical instruction that is distinctively Christian. The journey on which I then embarked has been surprisingly filled with wonder. I am grateful to those who have walked with me—my fellow members of ACMS (Association of Christians in the Mathematical Sciences), and my students at CMU, who more often than not have been the ones to drag me along from one sight to the next. Though the path ahead remains dim, a glance over my shoulder at the beauty behind instills such a joy in my heart that I have a confidently hopeful imagination for the future.
By Christian mathematical instruction, I do not mean the kind of simplistic Christian overlay typified by word problems such as, “If the animals went into the ark two-by-two, and Noah counted six different types of animals, how many animals are on the ark?” Nor am I referring to devotional analogies, like, “Just as a triangle has three sides, yet is one triangle, so the Godhead has three persons, yet is one God.” These examples of “Christian math” may have some value, but cannot be passed off as university-level instruction.
Rather, my hopeful imagination has been inspired by a profound connection between mathematics and creation, by opportunities to work side-by-side with people of other religions on issues of shared concern, by a truly two-way dialogue between mathematics and Christian theology, and even by an interchange between mathematics and Anabaptism. I will demonstrate each of these, primarily from various classroom experiences.
Recognizing that my readers are (in general) not mathematical, I have explained some of the simple mathematics that I use. On the other hand, complicated mathematics, while kept to a minimum, is marked with an asterisk (*) but left unexplained. This permits me to demonstrate theological interactions using “real math,” without losing the reader in an attempt to simplify very complex material. If you wish to investigate further, wikipedia.org is reasonably reliable for mathematical concepts. But you will lose very little, and likely gain much in terms of your mental health, if you simply read asterisked concepts (e.g. *mathematical induction) as “one of those instances of real math that I don’t need to understand.”
MATHEMATICS AND CREATION
One way in which math might be thought of as Christian is a refusal to separate the sacred from the secular. If we believe with Paul that our Creator made all things visible and invisible, that in God we live and move and have our being, that Christ is all and is in all, then naturally the study of mathematics itself, is sacred. Since this is a particular strength of Reformed Christian academics, it is not surprising that they are predominant among those who integrate faith and mathematics. The best example is the book Mathematics in a Postmodern Age: A Christian Perspective, 2 edited and written by members of ACMS.
Consider, for example, the following ways of expressing pi (the ratio of a circle’s circumference to its diameter). Each is a sum of infinitely many fractions.
Even non-mathematicians can see patterns in these formulas. 3 Mathematicians, ruminating over the aesthetic beauty of each individual formula and of the connections between them, experience a profound feeling of awe, frequently enhanced by something like epiphany as they begin to understand (usually in second year university Calculus) the reasons for these relationships. The feeling is not dissimilar to hearing the early morning call of a loon across a glassy mist-covered lake, or participating chorally in a Mahler symphony, or strolling through a good art gallery, or completing a no-look give-and-go, or whatever particular sacred creative pleasure tickles your own fancy.
If you cannot even remotely empathize with those of us who experience aesthetic beauty in abstract mathematics (yes, I recognized that look on your face!), consider the more concrete connection between mathematics and physical creation. Examples abound, but here is a fun one, often introduced already in high school. I discuss it in my Discrete Mathematics course, when introducing proof by *mathematical induction.
The Fibonacci numbers, popularized by Dan Brown in The Da Vinci Code, are formed by following a simple rule: To get the next number in the sequence, always add together the previous two. If the first two numbers are both 1, then the sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34 . . . It is surprising to most people that these particular numbers are so copious in phyllotaxis, the study of patterns in plants. For example, the number of petals on a flower, the arrangement of leaves on a stem, the number of shoots on a stalk, the number of spirals in a sunflower seed head (or a cauliflower head, a pinecone, a pineapple, a cactus . . . ), are commonly Fibonacci numbers. This seems to be because *Phi, the “golden ratio” , represents an optimal growing arrangement—e.g. the best exposure of lower leaves to sunlight, or the most efficient packing of seeds—and Phi is best approximated by the ratios of consecutive *Fibonacci numbers. (Incidentally, in Brown’s novel, the relationship between Fibonacci numbers and Phi is one of many “facts” that is entertainingly fudged.)
It is a bit of a philosophical puzzle that there is such a fundamental connection between mathematics and the physical universe. Since 1960, this puzzle has been known as “the unreasonable effectiveness of mathematics,” 4 but mathematicians, scientists, and philosophers began pondering it long before that. Einstein, for example, used *non-Euclidean geometry to develop relativity theory. Ironically, non-Euclidean geometry, a pure mathematics project of the nineteenth century, was generally considered the best example of useless mathematics—until Einstein used it, of course! In his words, “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?”
Today’s thinkers continue to struggle with the question. For example, in 2007 physicist Max Tegmark proposed a solution called the Mathematical Universe Hypothesis (MUH), which states that the physical world is in reality an abstract mathematical structure. 5 However, to anyone who believes that the one true God created both the universe and the human mind/spirit in God’s own image, the puzzle of unreasonable effectiveness is neither puzzling nor unreasonable. Indeed, we expect mathematics to reveal awe-inspiring features of our Creator’s world, as mathematics is precisely that activity of the human mind most closely connected to that which we call science—the study of creation. Christian students of mathematics are thus empowered to celebrate their activity as vocation, and their vocation as gift—the gift of the ability to comprehend, at least in part, the work of God’s hands.
MATHEMATICS AND RELIGION
Profound as it is to think of mathematics as a sacred participation in creation, I suggest that this is not necessarily distinctively Christian. Many other religions similarly emphasize the sacredness of the whole cosmos. If there is a Christian distinction, it may be in the particular meaning we give to the concept of sacred. Of course, to share this vocation with those of other religions does not in any way diminish its value. It is rather a commonality to be celebrated.
The same can be said for other connections between math and Christianity. For example, in some ways mathematics has achieved a near god-like status in popular culture. I regularly encounter those who think of it as the last bastion of absolute truth, who thus use it to refute postmodernism. Math is also assumed to be society’s most effective problem-solving tool, both in terms of problem-analysis and problem-correction. Though it may seem self-defeating, many mathematicians are skeptical of this uncritical embrace of the power of mathematics. In fact, it troubles especially mathematicians with religious beliefs, motivating us to work together to communicate to our students the limitations of mathematics, including: that much of our modern science is based on approximated solutions to *mathematical models, which typically are themselves approximations of approximations; that *Gödel’s Incompleteness Theorem proves mathematical foundations to be much shakier than most people think; that our overconfidence in Western (Greek) mathematics leads to an undervaluing of the field of *ethnomathematics, thereby contributing to the cultural colonization and oppression of those whose mathematics has developed along different lines; and that *statistical models exacerbate the marginalization of the poor, both by homogenizing a population and by ignoring all nonquantitative roots of poverty.
Or consider the ethical choices of applied mathematicians, which I discuss particularly in Calculus and Differential Equations. For example, my own specialties of *mathematical modeling, *computational algorithms, *differential equations, and *optimization can most easily (and with greatest financial reward!) be applied to militarization (e.g. military tactics, weaponry design) or to materialism (e.g. financial market prediction, shareholder profit for insurance companies). I have chosen instead to apply them to energy conservation, through the design of more efficient fan blades; and to healthcare, through research into effective treatments for epidemics like HIV/AIDS, malaria, pandemic influenza, and the West Nile virus. The fundamental reasons for my choice are faith-based. In that sense at least, I share common ground with mathematicians of many religions—even secular humanism.
MATHEMATICS AND CHRISTIANITY
To share common ground with other religions does not, of course, make the activity any less Christian. But several years ago it struck me that most apologetics for “Christian mathematics” went no further than this, and I wondered whether that was necessarily the case. So my students and I began to exercise our imaginations in, experimenting with connections between mathematics and Christian theology itself. In our opinion, the experiment has been fruitful. Here are some examples of ways in which we feel that mathematics intersects very particularly with Christianity. In a brief article like this there isn’t space to expand upon these, but you can at least get the flavor of the kinds of conversations we have.
The first is another example of mathematics interacting with creation, but notice that in this case the interaction is distinctively Christian. I recently became aware that a branch of mathematics called *Chaos Theory has been used by scientists and theologians to make claims about Christian theology. Knowing very little about Chaos Theory, and wanting to learn enough of the mathematics to understand the theological arguments, I did what any self-respecting academic would do: I taught a course on it. My students and I discovered a wide variety of theological arguments based in part on Chaos Theory, regarding divine action, open theism, prayer, healing, miracles, theodicy, and more. As we worked through both the mathematical and the theological literature, we became convinced that the arguments all came down to one fundamental metaphysical claim: that the mathematics of Chaos Theory demonstrates an indeterministic universe, with which God can interact in a very real way, without violating any laws of physics.
If this claim is true, if in John Polkinghorne’s words creation is “subtle and supple,” 6 then science permits, through mathematics, the possibility of a God who acts in the world, who answers prayers, who heals, and who responds to evil but also gives the world freedom to rebel. Whether or not Chaos Theory indeed implies metaphysical indeterminism is a question that has been hotly debated in the Science and Theology circles in the last ten years. 7 Also during that time, theologians like Clark Pinnock, 8 Gregory Boyd, 9 and John Sanders 10 have assumed the metaphysical claim in their theological defense of open theism. Unfortunately, they seem not to have understood the mathematical subtleties of this part of their argument. Our classroom interaction thus had a cutting-edge relevance, which enhanced the pure pleasure experienced from simply having the conversation itself.
Second, in my Discrete Mathematics course, when introducing *propositional and predicate logic, we attempt to perform what might be called mathematical exegesis. Consider, for example, the Pauline statement, “For if Abraham was justified by works, he has something to boast about, but not before God” (Rom. 4:2). It is often not clear, for contemporary readers, to what the phrase “but not before God” refers. Analyzing the statement using mathematical logic reveals that the form of the argument is an incomplete *Modus Tollens (If p, then q. Not q. Therefore, not p.), also known as an enthymematic hypothetical destructive syllogism. The argument is missing its conclusion, a rhetorical device—Paul expects the readers to complete it themselves. For the statement to work logically, the phrase “but not before God” must refer to boasting. So the complete, implied argument is: “If Abraham was justified by works, then he has something to boast about. No one can boast before God. Therefore, Abraham was not justified by works.”
Or consider the fruit and vine analogy, in the middle of the Johannine farewell discourse, which is built primarily of *conditional propositions. Some are explicitly “if p then q” propositions (e.g. John 15:7). Others use an alternative but equivalent conditional form, such as “not q unless p” (e.g. John 15:4). Still others implicitly contain conditional logic (e.g. John 15:2a, “He removes every branch in me that bears no fruit,” is implicitly equivalent to the conditional statement, “If a branch in me bears no fruit, then He removes it”). After reducing each proposition in the argument to its logical form, we discover that it can be restated as the *biconditional proposition: “A branch remains in the vine, if and only if it bears fruit.” Thus, fruit-bearing is the quintessential and exclusive characteristic of being connected with the divine.
I claim that this is a justifiably helpful tool for analyzing such texts, since the logic of mathematics is in fact the logic of the ancient Greeks. So a thorough understanding of mathematical logic can help us interpret ancient texts in which the implied author and audience are strongly influenced by Greek thought and culture—such as the Fourth Gospel and the Letter to the Romans. Clearly, this is distinctively Christian mathematical instruction, interacting as it does with the sacred text of Christianity itself.
Finally, interpreters such as John Howard Yoder 11 and Walter Wink 12 allow us to view mathematics as one of the Pauline “principalities and powers.” Several authors have claimed that the humanities, and Western culture more generally, have been unwitting victims of mathematization (e.g. our widespread reliance on statistical reasoning). We discuss this trend, but also consider ways in which theology itself has been susceptible. For example, systematic theology (and much pop theology) is rooted in the mathematical ideals of consistency, abstraction, and universality; and is developed through reasoning that looks suspiciously similar to the *axiomatic-deductive method of mathematics. To demonstrate this, I give first-year students an example of what many like to call an “apparent contradiction” in scripture, and ask them what they think. Invariably they attempt some kind of harmonization. I then announce that they have been successfully “mathematized,” and point out some of the ways in which this focus on consistency distracts us from the particularities of each text.
I could expound at length (and do so in the classroom) about how participants in some of the classic Christian debates are influenced unknowingly by mathematics. For example, young-earth and day-age creationists are typically unaware of how mathematical are their reading lenses, as are those who interpret the admonition of 1 Timothy 2:12 as supracultural absolute truth.
Notice that in treating mathematics as a Pauline power, we have a truly two-way conversation between mathematics and theology. Hermeneutics informs our perception of mathematics, opening our eyes to its influence as a cultural force. Accepting this, mathematics then informs our theology, opening our eyes to ways in which theology itself has been subject to that influence.
MATHEMATICS AND ANABAPTISM
To find connections between mathematics and Anabaptism is, for me, the next frontier. So in some ways this is the most hopeful section of my argument. Of course, some of the discussion above has already been influenced by my Anabaptist convictions—for example a sympathetic awareness of the poor and the oppressed, and a rejection of military applications of mathematics.
Another way I connect mathematics with Anabaptism is what I call mathematical narrative theology. One example is my telling the story of mathematician Blaise Pascal in the ACMS online journal, 13 in which I highlight aspects of Pascal’s life and thought that resonate with Anabaptists. A second example is this wonderful story about the Pythagorean community—likely apocryphal, but nonetheless truth-telling in the mythical sense. It’s a story I tell my Calculus class when using the *Intermediate Value Theorem to prove the existence of a real number x such that x2 = 2, and to my Discrete Mathematics class when introducing the concept of *proof by contradiction.
The Pythagoreans were a Greek religious cult, in about the fifth century BCE. Their religious practices included the so-called Pythagorean Maxim—namely abstinence from beans. (Some scholars attribute this to the belief that bean pods were a conduit to the underworld, though Herman Melville, in Moby Dick, highlights an added benefit: “In this world, head winds are far more prevalent than winds from astern (that is, if you never violate the Pythagorean maxim).” Little details like that add to the enjoyment of classroom storytelling—at least for me. Readers whose sense of humor precludes the scatological are forgiven this deficiency.)
The prime tenet of the Pythagorean religion was a belief in the rationality of the world, where rationality is defined in mathematical terms. That is, everything that exists can be expressed as a ratio of natural numbers. (Natural numbers are the integers 1, 2, 3, 4, 5, etc.) For example, musical intervals are rational. When a vibrating string is touched at its midpoint, the frequency (the rate at which the string vibrates) exactly doubles. We now call this the “octave.” To the Pythagoreans it was the most pleasing interval, precisely because it was based on the simplest ratio, 1:2. Interestingly, 2500 years later Western music is still based on this mathematically rational philosophy. The three basic Pythagorean intervals—the octave (1:2), fifth (2:3), and fourth (3:4)—remain fundamental to the tonality of our worship songs, whether inspired by classical or pop music.
Assume for the moment that this religious foundation is true. Then all constructible lengths must be expressible as ratios of natural numbers. Herein lies the problem. Most readers will be familiar with the Pythagorean theorem, one of many mathematical insights that comes from this ancient religious community. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the remaining two sides. Most of us learned this as a2 + b2 = c2. Clearly, one can construct a triangle in which two sides of equal length create a right angle. If those two sides have length one, then the third side must have length sqrt 2 (a = 1, b = 1, c = sqrt 2, and so a2 + b2 = c2 = 2). Following Pythagorean reasoning, then, sqrt 2 must be a rational number (i.e., a fraction). But it can be shown that sqrt 2 is not rational. Since this is obviously a contradiction (a number cannot be both rational and not rational), therefore the initial assumption is false. Therefore, the religious foundation of Pythagoreanism is false. (The argument form of this paragraph is called a proof by contradiction, or reductio ad absurdum, Latin for “reduction to the absurd.”)
Notice the irony. When the Pythagoreans discovered this (an inevitable outcome, given the very bright mathematical minds in the community), they were put in the unenviable position of having proved their religion false—using the tenets of that very religion! By the time of this discovery, though, their religious community had gained much respect throughout the Greek world, imparting influence and power to the Pythagoreans themselves. This power they were unwilling to relinquish, and so they committed among themselves to keep the discovery secret. According to legend, the secret was so tightly held that when one priest tried to escape the community to share the “good news” with the general Greek population, the unfortunate evangelist-to-be was chased down and drowned at sea.
One need not think long and hard to discover distinctively Anabaptist Christian truth in the story. It unmasks the seductive nature of religious power, and the tendency to protect that power with coercion (which may be physically violent, but may also be psychologically violent). It points out the tendency of religious communities to defend certain beliefs even long after those beliefs have been proven false. It reveals the dangers of secret dealings by authoritative leaders behind closed doors. More positively, it affirms Anabaptism’s emphasis on community, openness, honesty, and trust; and its communal hermeneutic in which all voices are heard and valued. These are ways in which Christians follow the logic of the cross, rather than the logic of the empire.
In conclusion I wish simply to repeat that the journey along the path of distinctively Christian mathematical education has been full of joyful and fruitful discovery, and to emphasize that the destination is still nowhere in sight. Anyone wishing to walk with me—to point out hidden sights along the main path, to nudge me down a promising rabbit trail, or to turn me away from a fruitless dead end—will find me to be a very willing conversation partner.
- Through no fault of their own, most people confuse mathematics with decimal arithmetic. (Usually, they have been taught to do so.) Of course, in decimal (base-10) addition, it is always true that 1+1=2. But in mathematics, “1,” “+,” and “=” are symbols that take on different meanings in different contexts. A simple example: if the context is binary addition, then 1+1=10. A more complicated example: if the context is addition modulo 2, and “=” refers to congruence classes, then 1+1=.
- Russell Howell and James Bradley, eds., Mathematics in a Postmodern Age: A Christian Perspective (Grand Rapids, MI: Eerdmans, 2001).
- If symbols like these produce feelings of panic, let me suggest a book such as Calvin C. Clawsen’s, Conquering Math Phobia: A Painless Primer (New York: John Wiley & Sons, 1991).
- Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications in Pure and Applied Mathematics 13, no. 1 (February 1960).
- Max Tegmark, “Mathematical Cosmos: Reality by Numbers,” The New Scientist, 14 September 2007, 38–41. For a more technical version, see Max Tegmark, “The Mathematical Universe,” Foundations of Physics (accepted), currently available at http://arxiv.org/pdf/0704.0646.
- John Polkinghorne, Quarks, Chaos, and Christianity: Questions to Science and Religion (New York: Crossroad Classic, 1996), 67.
- Some of the debate can be found in Chaos and Complexity: Scientific Perspectives on Divine Action, 2nd ed., eds. Robert John Russell, Nancey Murphy, and Arthur R. Peacocke (Vatican City: Vatican Observatory Publications, 2000).
- Clark Pinnock, Most Moved Mover: A Theology of God’s Openness (Grand Rapids, MI: Baker Academic, 2001).
- Gregory Boyd, God of the Possible: A Biblical Introduction to the Open View of God (Grand Rapids, MI: Baker Books, 2000).
- John Sanders, The God Who Risks: A Theology of Providence (Downers Grove, IL: InterVarsity Press, 1998).
- John Howard Yoder, The Politics of Jesus, 2nd ed. (Grand Rapids, MI: Eerdmans, 1994). See especially Chapter 8, “Christ and Power.”
- Walter Wink, The Powers that Be: Theology for a New Millennium (New York: Galilee Trade, 1999).
- Tim Rogalsky, “Blaise Pascal: Mathematician, Mystic, Disciple,” Journal of the Association of Christians in the Mathematical Sciences, 2006. Available at http://www.acmsonline.org/journal2006.htm.