Portions of this blog were part of a longer essay with the same title that appeared in Perspectives on Science and Christian Faith (click here for the full article). Permission has been obtained for the duplications.
Does faith matter in mathematics? Not according to the Swiss theologian Emil Brunner. In 1937 he suggested a way to view the relationship between various disciplines and the Christian faith. Calling it the Law of Closeness of Relation, he commented,
The nearer anything lies to the center of existence where we are concerned with the whole, that is, with man’s relation relation to God and the being of the person, the greater is the disturbance of rational knowledge by sin; the further anything lies from the center, the less the disturbance is felt, and the less difference there is between knowing as a believer or as an unbeliever. This disturbance reaches its maximum in theology and its minimum in the exact sciences and zero in the sphere of the formal. Hence it is meaningless to speak of a “Christian mathematics.”1
Thus, Brunner holds a nuanced version of the doctrine of noetic depravity: sin affects the reasoning ability of humans, but does so in varying degrees depending on how “close” the object of reasoning is to their relationship with God. Mathematics, being a purely formal discipline, is beyond the reach of any adverse noetic effects. Christians and non-Christians will therefore come to the same mathematical conclusions, so that, for Brunner, the phrase Christian mathematics is an oxymoron.
On one level Brunner is correct. If one agrees to play the game of mathematics, then one implicitly agrees to follow the rules of the game. Different people following these rules will— Christian or not—agree with the conclusions obtained in the same way that different people will agree that, at a particular stage in a game of chess, white can force checkmate in two moves. In this sense mathematical practice is “world-viewishly” neutral. Moreover, the paradigm for mathematical practice has remained relatively unchanged since Euclid published his masterpiece, The Elements, in 300 B.C. That paradigm is to derive results in the context of an axiomatic system.2
It would be a mistake, however, to apply Brunner’s dictum to all areas of mathematical inquiry. One can be committed to the mathematical game, but also participate in analyzing it (and even criticizing it) from a meta level. In doing so, faith perspectives will surely influence the conclusions one comes to on important questions about mathematics.
But is the investigation of such questions really a legitimate part of the mathematical enterprise? At least three reasons can be given for an affirmative answer: (1) such questions are actually taken up at every annual meeting of the American Mathematical Society and Mathematical Association of America; (2) historically, such questions have always been investigated by the mathematical community. Indeed, David Hilbert, one of the greatest mathematicians of the twentieth century, chose two topics for discussion in conjunction with the oral defense of his doctoral degree. The first related to electromagnetic resistance. The second was to defend an intriguing proposition: “That the objections to Kant’s a priori nature of arithmetical judgments are unfounded.” Hilbert is credited as being a founder of the school of formalism, which insists that axiomatic procedures in mathematics be followed to the letter. It is thus interesting that even those who held a strict view of mathematical practice and meaning saw the investigation of important meta questions relating to mathematics as a legitimate undertaking by mathematicians. (3) Finally, for the past 44 years the Association of Christians in the Mathematical Sciences (ACMS) has devoted a great deal of energy in investigating these meta level questions, and two book projects on the subject have grown out of this organization.3 Future blogs will likely contain contributions from members of the ACMS.
Is there a helpful classification for meta level questions that Christian mathematicians might pursue as they attempt to explore the interaction between their discipline and faith? Arthur Holmes suggests four categories of faith-integration in his well-known book The Idea of a Christian College: the attitudinal, ethical, foundational, and world view. Future posts will take up issues intersecting with these categories. For this initial blog, I’d like to suggest (and define) a fifth category for consideration: the pranalogical. The term comes from examining two places in scripture where Jesus commends people for their faith.
The first one, found in Matthew 15, is the story of the Syrophoenician woman. Her daughter is demon possessed. She begs Jesus for help. In an unusual response Jesus says, “It is not good to take the children’s bread and throw it to the dogs.” The woman replies, “Yes, Lord; but even the dogs feed on the crumbs that fall from their masters’ table.” Jesus then says, “O woman, your faith is great; it shall be done for you as you wish.”
The second instance is recorded in Matthew 8 and also in Luke 7. It is the story of a Roman soldier whose servant is desperately ill. In Matthew’s version he comes to Jesus and says, “Lord, I am not worthy for You to come under my roof, but just say the word, and my servant will be healed. For I also am a man under authority, with soldiers under me; and I say to this one, ‘Go!’ and he goes, and to another, ‘Come!’ and he comes, and to my slave, ‘Do this!’ and he does it.” Jesus then says to those around him, “Truly I say to you, I have not found such great faith with anyone in Israel.” Then he heals the servant.
In addition to the praise given by Jesus in these accounts there is something else they share in common. The faith of both petitioners came, in part, from their ability to glean a practical spiritual truth by drawing an analogy from what they had learned by experience. The woman did so from behavior she observed among dogs. The soldier likewise understood what authority is by virtue of his occupation, and applied that knowledge to a trust in the authority that Jesus would have to heal.4
This analysis leads to the suggestion mentioned earlier of a pranalogical category for faith-learning, because it involves a practical application of an analogy gleaned from one’s discipline or life experience. We could also speak of a pranalogy, a word obtained by combining practical and analogy.
There are several potential pranalogical applications of mathematics that can relate to and even enhance one’s Christian faith. I’ll mention just one. It can be elucidated from two mathematical results.
The first is that, in mathematics, there are actually different sizes of infinity. Rational numbers (like 2.5) can be written as a fraction. Irrational numbers (like pi) cannot. There are infinitely many of both types, yet every student of mathematics eventually sees a proof that, in a very real sense, there are more irrational numbers than rational numbers.
The second, known as the Banach-Tarski paradox, is the result that it is possible to decompose a sphere into only five sections. Then, without distorting any of the sections in any way they can be reassembled into two completely contiguous spheres of identical size to the first.
If pressed to explain these conundrums a mathematician might say something like, “Well, that’s just how things work when dealing with mysterious concepts like infinity.”
Indeed, and if things can get so convoluted in a logically precise, carefully defined system such as mathematics, it should be no surprise when paradoxical ideas arise in the Christian faith. The study of mathematics can thus help us cope with these paradoxes, and help us to live more comfortably with mysteries that arise when thinking about God.
Developing useful analogies from one’s field of study can be fruitful, but there lurks an obvious danger. In part, it is a danger that accompanies all analogies, but it is especially prominent in mathematics: it is easy to draw analogies that are careless and trite.
Thus, one must keep in mind the limits of any model, and in dealing with mysteries ultimately return to Paul’s statement in First Corinthians, chapter 13: “For now we see in a glass darkly, but then face to face; now I know in part, but then I will know fully just as I also have been fully known.”
- Emil Brunner, Revelation and Reason (Philadelphia, PA: The Westminster Press, 1946), 383.
- A modern axiomatic system has five components: undefined terms (the basic syntactical strings); definitions (composed of undefined terms); axioms (the unquestioned assumptions from which results will be derived); propositions or theorems (the results so obtained); and rules of reasoning (the methods by which axioms and previously proved theorems will be combined to produce new results).
- Mathematics in a Postmodern Age: A Christian Perspective (Grand Rapids, MI: Eerdmans, 2001) and Mathematics Through the Eyes of Faith (San Francisco: HarperOne, 2011). Both are edited volumes by James Bradley and Russell Howell.
- This concept, taken from Luke’s account of the Roman soldier, was drawn out in a chapel talk given by Robert Brabenec of Wheaton College on March 25, 2009.