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Food for Thought: The Theological Life of Pi
For Pi Day this year, chew on this: Does God know all the digits in pi? Or does he know the answer to the myriad of unresolved mathematical problems out there today?
The great fifth-century African bishop Augustine of Hippo surely would have answered with a resounding yes, though other Christian thinkers might disagree. He also would likely have taken interest in recent mathematical discoveries made possible by today’s computers, discoveries that resurrect some of these unresolved problems.
In 2016, Swiss mathematician and evangelical Christian Peter Trüb’s computer spent 105 days calculating pi out to approximately 22.4 trillion digits—the most digits ever calculated. And this past January, Jon Pace, an electrical engineer and deacon at the Germantown Church of Christ in Tennessee, learned that his computer discovered the largest known prime number and the 50th known Mersenne prime, through an effort begun in 1996 that networks thousands of computers nationwide.
Primes are numbers greater than 1 that have no proper factors—the only numbers that can divide evenly into them are 1 and themselves (e.g., 2, 3, 5, 7, and 11). A Mersenne prime is named in honor of Marin Mersenne (1588–1648), a French mathematician, theologian, philosopher, and music theorist. These primes can be expressed as 2ⁿ – 1. In the expression for Pace’s Mersenne prime, n = 77,232,917. That’s one less than the number 2 multiplied by itself over 77 million times. Just to write it down would take up the space of about 10 novels the size of Anna Karenina. Trüb’s representation of pi is 1,000 times longer than that.
Why would anyone care to engage in projects like these? Trüb said one reason is simply out of intellectual curiosity. As a boy, he was fascinated with the number pi, and he cites his belief in God as the basis for his work as a scientist.
And primes, by virtue of their mysterious nature, trigger curiosity. In 300 B.C., Euclid proved that there is an unending supply of primes, yet there is no known formula for generating them. Paradoxically, there is a formula for approximating how many primes there are up to a certain point. It estimates that there are 72,382 primes less than one million, although there are actually 78,498 such primes.
Trüb’s second reason is that mathematical constructions—seemingly having no practical application when first proposed—wind up eventually having great utility. That there is an unending supply of primes is critical for implementing many computer encryption systems.
Philosophers and theologians have long been interested in numbers. In fact, Augustine expressed interest in perfect numbers, which have a curious connection with Mersenne primes. A perfect number has the property that it equals the sum of its proper divisors. For example, the number 6 is perfect because its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. Euclid proved that if you take a Mersenne prime of the form (2ⁿ - 1) and multiply it by 2ⁿ⁻¹, you get a perfect number. For example, with n = 2, Euclid’s procedure produces the Mersenne prime of 3 (i.e., 2² - 1), which when multiplied by 2 (i.e., 2¹), gives 6, a perfect number.
Augustine commented on perfect numbers in The City of God :
These works [of creation] are recorded to have been completed in six days (the same day being six times repeated), because six is a perfect number, not because God required a protracted time, as if He could not at once create all things, which then should mark the course of time by the movements proper to them, but because the perfection of the works was signified by the number six.
At the time Augustine wrote these words, there were very few known (certainly less than seven) perfect numbers. Neither was it known how many perfect numbers there are. It may surprise some that, even today, no human knows whether there is an unending supply of perfect numbers (or Mersenne primes). Nor is it known whether there are any odd perfect numbers. Or what about the infinite decimal expansion of pi? Does God know, and can God “see” all the digits of pi?
Augustine’s affirmative answer to these questions, as well as his previous remark on perfect numbers, are reflective of his philosophy of mathematics, which is most thoroughly laid out in his work On the Free Choice of the Will.
Briefly, Augustine “Christianized” the work of Plato, who thought that mathematical objects (such as numbers) and propositions (such as 7 + 5 = 12) are eternal and necessary. They are real entities, existing independently of human knowledge, and located in a hard-to-define world of changeless forms. Augustine put these entities in the mind of God and even considered them to be part of the divine essence. According to him, as created in the image of God, humans can have access to these objects through mathematical reasoning. But even if humans are never able to discern answers to the question of how many Mersenne primes there are, or whether there are any odd perfect numbers, God certainly knows. After all, numbers are part of the divine essence, so God can fully comprehend all of them and anything about them—even though there are infinitely many—in one completed act.
Many thinkers—Christian or not—have held to some form of mathematical Platonism. The British mathematician and well-known atheist G. H. Hardy commented, in his landmark 1940 book, A Mathematician's Apology, “I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations,’ are simply our notes of our observations.”
Augustine’s ontology remains a popular view among Christians today, despite some technical problems accompanying it. For example, in Mathematics in a Postmodern Age, Texas A&M philosopher Christopher Menzel points out that viewing mathematical objects as eternal seems to conflict with the doctrine that God created all things (Col. 1:16). And their being necessary seems to infringe on God’s freedom to do what he wills. Menzel gets around the first conundrum by viewing creating as including sustaining (so that God “created” these truths by virtue of his eternally sustaining them). He gets around the second by pointing out that Christians have no trouble viewing God as being constrained in other ways that cause him to act in accordance with his nature. Just as it is impossible for God to lie (Heb. 6:18), in the same way, God's (logical) nature drives him out of necessity to “think” truths such as 5 + 7 = 12: He cannot think otherwise.
A discussion here of all the technical issues Menzel deals with would take us too far afield. For now, it is important to note that many Christians adopt an alternative view of mathematical objects, one that emerged from a student of Plato: Aristotle. Rather than viewing numbers as eternally existing entities, Aristotle saw them as abstractions that humans create in order to describe commonly shared experiences. The truths of arithmetic are logical consequences of the axioms humans have developed to help explain reality as they see it.
Of course, there are problems with this position as well, such as accounting for the awkward task of ascribing a common meaning shared by humans when they make these abstractions. Rather than dive into those kinds of issues, it should be noted that with such a view, an interesting position arises.
The 20th-century German mathematician Kurt Gödel showed that all axiom systems capable of describing arithmetic are limited. Specifically, he proved that any such axiomatic system will contain true propositions that cannot be proved. To this date, no human knows if there are infinitely many Mersenne primes, whether there are infinitely many perfect numbers, or whether there are any odd perfect numbers. If these questions are among those “Gödel-type” propositions, then not even God can answer them.
Doesn’t that position limit God’s omniscience? No, not if one views numbers as a human construction whose properties depend on the axioms they have laid out. To illustrate, consider a well-known fact about the game of tic-tac-toe: If both players use an optimal strategy, neither will win. Thus, it is impossible for God—playing in accordance with the rules of tic-tac-toe—to win a game with a human who knows how to play. In the same way, it is impossible for God to use an axiom set (the “rules”) to prove something that cannot be proven from those very axioms.
So, now that your pie plate is licked clean and you’re staring at an empty dish wondering how the bishop of Hippo could say that God knows something about mathematics that is unprovable, another position of his comes into focus. Augustine believed that numbers, being part of the divine essence, are shrouded in mystery. Followers of Augustine and those who hold Aristotle’s alternative view may well agree on this point: It is the mysteries of mathematics that lead us to ponder not only our own limitations, but also the greatness of the mind of God.
Russell Howell is a professor of mathematics at Westmont College in Santa Barbara, California. He holds degrees in mathematics and computer science. He is co-editor and contributor to Mathematics in a Postmodern Age: A Christian Perspective (Eerdmans, 2001), and Mathematics Through the Eyes of Faith (HarperOne, 2011). He enjoys tennis, playing the piano, ocean kayaking, and hiking with his wife (Kay) and yellow lab (Dickens).