Wednesday, January 6, 2016

1: Mathematics and Faith

I’ve long wrestled with the question of how my study and teaching of mathematics relates to my identity as a follower of Jesus. During my time as an undergraduate at Stanford, in particular, as I advanced in the mathematics major and started becoming seriously involved with ministry on campus, this question started dominating my thoughts. I wondered about the relationship between a discipline that sought to uncover propositional truth through proof and my commitment to the Person named Truth.

This troubled me. I wanted to take the teaching of Jesus seriously that we are to love God with the whole mind, but I was rather unclear about how doing mathematics accomplished this.

Useless by xkcd
Is mathematics part of God’s creation? Is studying algebra comparable to studying biology? Or is mathematics, especially modern mathematics, a creation of humanity? Is it just one big puzzle, like a never ending game of Sudoku? And, if so, how could I justify spending so much time doing it? Doesn’t the great want of the world demand we do more than just solve puzzles? Might much of mathematics be a distraction from meeting humanity's needs?

Over the next few posts, I offer a sketch of how I’m coming to understand the nature and value of doing and teaching mathematics from a Christian perspective. I originally shared much of this sketch during a talk I gave at Andrews University last fall during a faculty luncheon sponsored by the Center for College Faith. I must admit, though, that many of these are still live questions that I fully expect to continue wrestling with throughout my life. Over the last several years, however, I’ve gone from thinking of faith and mathematics as irrelevant to each other to finding beautiful connections between the two. Hence, my faith in Christ has become a great motivation to take mathematics seriously and my appreciation for mathematics has grown my love for God.

2: Mathematics and Reality

In Mathematics and Faith, we encountered some questions about the nature of mathematics and how doing math relates to loving God.

I cannot understate the influence a course on the philosophy of mathematics had on my thinking my last semester at Stanford. It was influential not because it answered my questions--philosophy seldom does--but because it introduced me to a new set of questions that gave me an opening to understand the relevance of faith to mathematics. Although people have been doing mathematics across diverse cultures for millennia, I learned we are without a historical consensus to what mathematics is or why it works. Many have taken the view of Hardy that “mathematical reality lies outside of 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.“ Certainly, we often speak of mathematical objects as it they are real objects that exist somewhere, but this view introduces the very hard questions of where does this mathematical reality exist and why is it that logical proof is our means of observing it.

A number of Christians have been attracted to this view, describing mathematical objects as the thoughts of God or objects existing in the “mind of God”. This can be seen as a theistic repackaging of Plato’s view that mathematical objects exist among the eternal, unchanging “forms” that he describes which could be accessed by properly trained reasoning. But Christians might do well to give some thoughtful consideration before altogether embracing this view, for the New Testament rejects reason as sufficient means to discern God’s thoughts.

Others, especially more recently, have argued that mathematics is less like the natural sciences and more like the arts; that is, mathematical theorems aren’t discovered, they’re invented. This allows us to understand Weierstrass’s statement that “the true mathematician is a poet.” While capturing the sense of creativity present in producing modern mathematics, the idea that mathematics is entirely a human construct faces some serious challenges which are perhaps best communicated by Eugene Wigner in his remarkable paper, The Unreasonable Effectiveness of Mathematics. First Wigner observes,
The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess.
In creative endeavors, especially those undertaken by a large number of people, it is important to have a clear map of where one is going. We produce blueprints and models before building skyscrapers. But it appears the case in mathematics has been a reckless addition of rooms and floors which, rather than produce incoherence, has consistently resulted in what the community describes as surprisingly elegant and beautiful.

Wigner went on to make the case that not only is mathematics surprisingly internally coherent, but it is unreasonably good at explaining the natural world. He noted that although mathematics may have been quite grounded in physical problems centuries ago when there was little distinction between a mathematician and a physicist, beginning in the 19th century mathematics became an abstract discipline and notably detached from the world the mathematicians inhibited. Non-euclidean geometries were developed, despite their counter-intuitive feel, and abstract algebraic structures were studied that seemed to have lost any attachment to phenomena in this world. Yet, Wigner observed, these became precisely the mathematical foundations that were necessary to fuel the physical discoveries of the 20th century. Wigner concluded, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

A number of individuals have offered fascinating responses to Wigner, but even these responses tend to recognize that they come short of fully satisfying the questions Wigner raised. The mystery is threefold: why are we capable of doing mathematics, why is there a deep structure to the universe, and why does our mathematics explain this structure so well? Here, as we'll see in the next post, the Christian worldview is particularly well-suited to provide a compelling framework to make sense of these miracles.

3: Mathematics and Creation

I left off in Mathematics and Reality by suggesting that Christianity is able to offer rich insights to the mystery of our ability to do mathematics and mathematics' ability to describe the natural world so effectively.

To see this, we begin with the doctrine of creation. As J. C. Polkinghorne explains, “If the world is the creation of the rational God, and if we are creatures made in the divine image, then it is entirely understandable that there is an order in the universe that is deeply accessible to our minds.” This explains why so many pioneers in modern science and mathematics, such as Newton, understood their faith to be the great motivation for the work they were doing. As C.S. Lewis observed, they expected law in nature because they believed in a Law Giver. Hence, a number of historians of science trace the historical foundation for modern science to faith in a Creator.

To make progress then on questions regarding the nature and value of mathematics, it makes sense to review the creation account. Genesis opens with God creating the world, giving it structure, and moving it from a state of confusion to one of order. After humanity--both male and female--are made in the image of God, they are then given a mandate to participate in creation: to extend creation (“be fruitful and multiply”) and rule over the creation (“have dominion”). God creates animals, Adam names them; God creates a garden, the first couple are to cultivate it. Thus we find a picture of human and divine cooperation in caring for and extending creation.

In our brief survey of the philosophy of mathematics above, we noted that there is good reason to think of mathematics both as discovery and as creation, although either account faces challenges on its own. Genesis’ image of humanity being created to care for a garden serves as a rich model to resolve this tension. A garden is both discovered and created. The plants present existed in nature already, but the way in which they are brought together, arranged, and cultivated reflects the human gardeners.

A similar story can be told of mathematics: we begin with ideas that appear very naturally within God’s creation, then, as image-bearers, we interact with creation by rationally extending these ideas. Having been made in the image of the One who made the cosmos, we quite expect some form of correspondence between the mathematical notions we develop and the structure we discover in the natural world, but we also expect our mathematics to reflect the people and societies that developed them. Mathematics then resists being classified as either a natural science or a creative art--it is both.

Some have sought to make a distinction between pure and applied mathematics, and while such a distinction of terms proves useful at times, it is rather difficult to draw that dividing line. A mathematician may pursue a line of study simply to satisfy her mathematical curiosity, but often the notions she develops are later realized to be precisely the tools needed to describe some natural phenomena. The truth that a mathematician seeks simply for its beauty or elegance is seen to fill and describe creation, bearing witness to creation’s good Creator.

One may ask--indeed many have--must mathematics find an application to the natural world to be considered valuable? Here again Genesis’ garden is helpful. Bearing fruit was not the only purpose of its trees--they’re also recorded as having been beautiful (Genesis 2:9). Beauty was valued in God’s original creation independent of any utilitarian purpose. Humanity was designed to do more than just survive; our gardens grow both fruit to be eaten and flowers to be appreciated. Similarly, the value of mathematics can be found in both its usefulness and its beauty.

4: Mathematics and Eternity

I hope Mathematics and Creation begun to paint the picture of how mathematics draws us back to our beginnings, reminding us of our role as image bearers of a good Creator. This is consistent with the proverb of Solomon, “It is the glory of God to conceal a matter; to search it out, the glory of kings.” The contemporary rendering of the proverb found in The Message replaces the word kings with scientists. I'm inclined to translate it as mathematicians. There's a deep sense in which we were created for mathematical and scientific exploration.

Perhaps the bolder claim of the doctrine of creation is that we were created to participate in the glorifying activity for eternity. Genesis pictures death as an enemy precisely because it cuts short what was meant to be never ending. Solomon teaches that eternity has been placed on the human heart (Ecclesiastes 3:11). In studying, learning, searching out those things concealed by God, and being exposed to fields of never ending discovery and development, the desire for eternity is awakened in the human heart. Mathematics, in particular, seems to endear one to the hope of eternity not only because of its study of the infinite, but also because of the way in which a few observations can quickly develop into a new field that would require eternity to exhaust.

Image from The Story of Mathematics
The New Testament closes with the vision of Christ restoring creation in a new heavens and new earth. Here it is recorded that the kings of the earth will bring their glory into the city where Christ dwells with His people (Revelation 21:23-26). I find it significant that one of the few other references to the glory of kings in Scripture is found in Solomon’s proverb recorded above. It seems we have here a vision of an eternity of God-glorifying study, discovery, and creative pursuits of humanity. Ultimately, heaven isn't about sitting on clouds or aimlessly walking streets of gold. As Ellen White reminds us, “We may be ever searching, ever inquiring, ever learning, and yet there is an infinity beyond.” This never-ending learning she offers as a vision of the world-to-come:
Heaven is a school; its field of study, the universe; its teacher, the Infinite One. A branch of this school was established in Eden; and, the plan of redemption accomplished, education will again be taken up in the Eden school.

5: Mathematics and Education

The sketch from the previous posts reveals that through the lens of faith mathematics is seen as far more than just manipulating symbols. Rather, it is an image-bearer stepping into her role of being a co-creator with God. Thus, teaching mathematics becomes about more than just helping students learn how to perform algorithms or manipulate symbols. I’ve come to think of blackboards as windows into eternity, hoping my lectures will awaken students to the sense of endless discovery and co-creation they were created for. Teaching mathematics is about helping students recognize their true identities, incredible worth, and privileged place they occupy within the universe. Mathematics testifies to the reality that we’re not here haphazardly, but were designed to discover and extend creation while appreciating its beauty. And mathematics teaches us to long for the eternity where we’ll be able to continue our education in the presence of the One who formed us in love. “In the highest sense the work of education and the work of redemption are one.”

While living in anticipation of eternity, mathematics also teaches valuable lessons of character. Over the last few years, I’ve been intentional to introduce my student to Carol Dweck’s research on growth mindset. Dweck uses this term to refer to the different ways students view their intelligence and hence respond to failure. On the one hand, students who believe intelligence is fixed--you’re either born as a “math person” or not--tend to interpret their poor performance on a math activity as evidence that they are not capable of mastering the material. On the other hand, students who are taught to understand that intelligence is something that one can develop--like a muscle--tend to interpret poor performance as indicators of what further steps they need to take in order to master the material. Intentionally helping students develop a growth mindset develop traits of character that transfer to every sphere of their lives and, through the lens of faith, are understood as having an eternal value.
Finally, in anticipating the restoration of God’s good creation, one is reminded that there is something seriously broken in this present world. Mathematics, like art, can give individuals glimpses of the world to come, but it seems that it must go a step further and actually address the suffering of this present world. Enter the great applicability of mathematics. I’ve been developing a growing burden to inspire and challenge my students to employ their training to address the great want of world in whatever life course they pursue, be it as educators, engineers, medical professionals, lawyers, or even mathematicians.