Mathematics Through the Lens of Faith: A Path For Integration

We appreciate this latest contribution to our “Through the Lens of Faith” series. Matt Lunsford argues that integration is a two way street, so perhaps it would be fitting to add “and Faith Through the Lens of Mathematics” to the article title! We would also like to acknowledge that this article first appeared in the Fall 2023 edition of the Journal of the Union Faculty Forum under the title “Mathematics & Faith: A Path For Integration“. We are grateful to Matt Lunsford and the Journal for permission to re-post this article.

As members of the Academy who are employed in Christian higher education, we have a peculiar requirement in our vocation. We are called to think Christianly about our work in general and about our discipline in particular. This path for integration of faith and discipline often can seem to be a one-way road where faculty only use the Bible or a prescribed canon taken from the Christian intellectual tradition (e.g., Augustine, Aquinas, Luther, Lewis, etc.) to understand their discipline but not the reverse. It is generally left to the faculty member to discover how to integrate this additional content within the context of the discipline. Here, I postulate the path to a robust form of integration is, in fact, a two-way street, in which Holy Scripture and the Christian intellectual tradition inform our view of the discipline and how we teach it, and our discipline illuminates our understanding of the Christian faith and our ability to communicate this faith effectively. The goal of this essay is to explore a path for integration of faith and the discipline of mathematics. To do so, I will begin by discussing three aspects of my discipline: language, nature, and contribution.

Three Aspects of Mathematics

Language

Creation is a gallery, full of intrinsic beauty. The feat of expressing this beauty with fidelity requires rigorously defined terminology. An inherent preciseness is required to ensure that we humans, as universally as possible, comprehend the terminology uniformly. Mathematics provides such a language. Defining terms with prudence is the starting point for conventional mathematics. Mathematical language is characterized by clarity, precision, universality, and, whenever possible, simplicity. Not unlike an ordinary dictionary, which defines words using other words, the mathematical vocabulary, to avoid circular reasoning, must have terms which remain undefined. Obviously, these undefined terms should constitute as small a set of words as possible. For example, a mathematician studying geometry must define “right angle” before using that term in any formal way. Assuming the terms used here either have been defined previously or have been designated as undefined terms, a mathematician defines “right angle” as an angle formed by two lines intersecting to form equal adjacent angles. Continuing in this same manner, precise mathematical definitions of more challenging concepts such as symmetry, similarity, congruence, dimension, infinity, chance, paradox, contradiction, order, and chaos can be established. In many ways, the language of mathematics is ideal for describing creation’s beauty.

Nature

The nature of mathematics is discovering mathematical truth. The mathematician follows a rigorous method for establishing mathematical truth:

definition → axiom → conjecture → proof → theorem.

As we have seen, definitions are essential to the language of mathematics. Axioms are statements that are assumed to be true without the requirement of a proof. For example, the statement “all right angles are equal” is considered an axiom of geometry by mathematicians. This claim seems self-evident from the aforementioned definition of right angle; therefore, no additional logical argument is required. Beyond a minimal set of mutually agreed upon axioms, any statement that is claimed to be true requires a mathematical proof. A mathematician often begins with a conjecture, that is, a statement that seems plausible considering the definitions, axioms, and theorems that already have been established. If an unerring logical argument can be presented (i.e., a mathematical proof), then the statement moves from the category of conjectures to the category of theorems. Typically, a proof requires not only deductive reasoning but also creativity and imagination. Mathematical truth produced in this manner exhibits permanence, in that, if we continue working within the same framework of definitions and axioms, known as an axiomatic system, then the theorems proven will remain true. Conversely, if a statement is false in the axiomatic system, then it always will be false within that system.

An example of a timeless mathematical truth is the Pythagorean theorem, which states that a square constructed upon the hypotenuse (the longest side) of a right triangle (i.e., a triangle with a right angle) is equal in area to the sum of the areas of the squares constructed upon the two legs (the remaining sides) of the right triangle. It is believed that the Greek philosopher Pythagoras discovered this theorem around 400 B.C. If he or any of his followers gave a proof of this theorem, then that proof has been lost. Fortunately, an ingenious proof of this theorem appears in Elements, written by Euclid around 300 B.C. After constructing squares on each side of the right triangle, Euclid sections the square on the hypotenuse into two smaller rectangles. He then proves that the area of each rectangle corresponds exactly to the area of one of the squares constructed on the legs.

Two millennia later, the character Scarecrow in the classic movie The Wizard of Oz, upon receiving his diploma—which apparently signified the miraculous appearance of a missing brain—says these words: “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side” [1]. By the way, an isosceles triangle is a triangle for which exactly two of its three sides have the same length. How is one to decide if this statement is true or false? The nature of mathematics dictates that one should be able to demonstrate either a proof of this statement, thus establishing it as mathematical truth, or provide a counterexample to this statement and assign it to the trash heap of falsehoods. In this case, a counterexample is readily produced. In fact, I challenge the reader to find even one isosceles triangle for which Scarecrow’s conjecture is true.

Unfortunately, mathematics has limits when it comes to establishing its own truth. Because of Gödel’s Incompleteness Theorems, there exist well-formed mathematical statements within a given axiomatic system that lie outside the purview of a proof. Hence, the truth value of these mathematical statements is unknown. Mathematicians generally believe that these statements are rather complex and therefore are not a hindrance to the routine work of producing mathematical truth. However, if it makes sense to talk about absolute truth within a given axiomatic system, then mathematicians are aware that absolute truth within a given axiomatic system is not synonymous with mathematical truth since the latter can never encompass the former. That is, if we assume that mathematical statements are either true or false, then there exist true statements within the system that cannot be proved.

Contribution

A remarkable contribution of mathematics to human society is its ability to impart understanding. How does mathematical truth discovered within an axiomatic system convey knowledge of the real world? One primary way involves the creation of mathematical models of reality. Working within well-formed models, mathematicians, engineers, scientists, financial analysts, and others can analyze and obtain truths about the real world and even predict what will happen in the future. Mathematics advances our understanding of the past, present, and future status of whatever the researcher attempts to model. Of course, these mathematical models are not exact.

The knowledge derived from mathematics has changed our everyday lives. We drive cars, ride in subways, fly in airplanes, use laptops and smartphones, receive urgent health care, yet somehow we remain unaware that the understanding mathematics brings is essential for all of these technologies. There is a recent joke of a person on social media lamenting that another day has passed without having to use the Pythagorean theorem. Actually, the Pythagorean theorem has been in continual use since its discovery. The theorem yields the mathematical formula for measuring distances in two and three dimensions. In current times, because the result is so fundamental, the actual statement of the theorem is often hidden from view. For instance, our society is enamored with flat screens, and the instructions for the pixels in those flat screens are determined in part by their relative distance from an initial locus. In fact, any technique for lighting an image or scene that calculates illumination for each pixel, such as per-pixel lighting, necessarily utilizes the relative location of each pixel on the rendered image. So, whether one is watching digital TV, or playing a 3-D video game, or just tapping, swiping, or dragging on a smartphone, the Pythagorean theorem is there, working in the background, making this technology function. 

A Path for Integration 

How can one integrate mathematics and the Christian faith? How does one think Christianly about mathematics? Is it possible that not only the skills acquired from studying mathematics (critical thinking, problem solving, struggling with a difficult problem, etc.) but also the content of mathematics is transferable to an integrated life? To address these questions, consider a passage of scripture from the New Testament. 1 Peter 3:15 commands, “but sanctify Christ as Lord in your hearts, always being ready to make a defense to everyone who asks you to give an account for the hope that is in you, but with gentleness and respect” [2]. Serious consideration of this passage reveals that words (“to give an account”) and truth (“to make a defense”) and understanding (“for the hope that is in you”) are key components of a faithful life. 

Communicating the richness of the Christian faith demands language sufficiently robust to articulate our thoughts fully. Christian thinkers such as Augustine, Pascal, and Lewis have used mathematical language to enlighten and clarify their thoughts. Similarly, we can use mathematical language to clarify, to construct analogies, and to make claims. One literary example is Edwin Abbott’s novella Flatland [3]. In this work, Abbott uses the subject matter of geometry and the concept of dimensionality to create an analogy for the gulf that exists between a lost unregenerate person and a spiritually regenerate one [4].

We are truth-seekers, and fortunately for us, we can draw upon the work of theologians over two millennia who have studied the scriptures to aid us in our defense of the faith. However, when doctrinal differences arise, as they certainly have in the past, how does one decide which path leads to the best orthodoxy? One potential tool in our arsenal is the model used by mathematicians for seeking mathematical truth. Like the nature of mathematics, the systematic theologian seeks to define terminology precisely (definitions), to state in advance all suppositions believed to be true without additional justification (axioms), and then proceeds to deduce doctrinal statements (theorems) from this structure. The 20th century British scholar C. S. Lewis exemplifies this approach in his apologetic work Mere Christianity. Applying this model, we have a valuable resource to enhance our defense of the faith. 

Mathematics enhances our understanding of God’s creation and therefore has contributed greatly to the advancement of human civilization. As believers, we acknowledge the often-positive contributions of mathematics to human society, but we also realize that “the hope that is” in each of us is not to be found in human knowledge or its applications. We place our hope neither in mathematics nor in the technologies it facilitates, for these will not provide the redemption for which the creation groans. Our hope is in “Christ as Lord” and in Him alone. Though our hope is in Christ and not in mathematics, we remain grateful for the ways mathematics facilitates the purposes of Christ’s kingdom here on earth.

Thinking Christianly about the discipline of mathematics is a two-way road. The language of mathematics allows us to use an expanded vocabulary to communicate the beauty of creation as well as a deeper understanding of our faith. The nature of mathematics provides us with a coherent model for demonstrating truth through rational means. The contribution of mathematics to human understanding gives us new ways to appreciate the design of creation and to improve the physical condition of our human society. Mathematics, through both its content and the skills its study produces, remains an integral part of a liberal arts education and facilitates a vibrant path for the integration of faith and discipline.

REFERENCES

[1] Vidor, King, et al. The Wizard of Oz. Metro-Goldwyn-Mayer (MGM), 1939.

[2] Holy Bible: New American Standard Bible. 1995, 2020. LaHabra, CA: The Lockman Foundation.

[3] Abbott, Edwin A. Flatland: A Romance of Many Dimensions. New York: Dover, 1992. 

[4] Lunsford, Matt D. Dialogue Between Dimensions: The Communication Dilemma in Flatland. Journal of the Union Faculty Forum Vol. 32 (2012), pp. 57-62

Previous Articles by the Author at ESN

Matt Lunsford has written previously for ESN, publishing a series of articles titled “A Christian Mathematician’s Apology.” Here are links to that series:

Finding My Vocation. Matt describes how he discerned his calling to study and teach mathematics.

Integrating Faith and Mathematics. Lunsford describes what it means for him to be a “Christian mathematician” and what it means for him to live the integrated life.

Redeeming the Discipline. Matt outlines his ideas of what “redeeming the discipline” looks like for him as a mathematician.

Thinking Christianly. A description of how Lunsford “thinks Christianly” about both his scholarship and pedagogy.