How Faith Influences Mathematics

Why Math Works

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by John D. Mays

Back in 1999 when I began teaching in a classical Christian school, one of the first books I heard about was James Nickel’s little jewel, Mathematics: Is God Silent? Must reading for every Christian math and science teacher, the book introduced me to a serious problem faced by unbelieving scientists and mathematicians. Stated succinctly, the problem is this: Mathematics, as a formal system, is an abstraction that resides in human minds. Outside our minds is the world out there, the objectively real world of planets, forests, diamonds, tomatoes and llamas. The world out there possesses such a deeply structured order that it can be modeled mathematically. So how is it that an abstract system of thought that resides in our minds can be used so successfully to model the behaviors of complex physical systems that reside outside of our minds?

For over a decade now this problem, and the answer to it provided by Christian theology, has been the subject of my lesson on the first day of school in my Advanced Precalculus class. But before jumping to resolving the problem we need to examine this mystery – which is actually three-fold – more closely.

In his book Nickel quotes several prominent scientists and mathematicians on this issue. In 1960, Eugene Wigner, winner of the 1963 Nobel Prize for Physics, wrote an essay entitled, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Wigner wrote:

The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and…there is no rational explanation for it…It is not at all natural that ‘laws of nature’ exist, much less that man is able to discern them…It is difficult to avoid the impression that a miracle confronts us here…The miracle of appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.

Next Nickel quotes Albert Einstein on this subject. Einstein commented:

You find it surprising that I think of the comprehensibility of the world…as a miracle or an eternal mystery. But surely, a priori, one should expect the world to be chaotic, not to be grasped by thought in any way. One might (indeed one should) expect that the world evidence itself as lawful only so far as we grasp it in an orderly fashion. This would be a sort of order like the alphabetical order of words of a language. On the other hand, the kind of order created, for example, by Newton’s gravitational theory is of a very different character. Even if the axioms of the theory are posited by man, the success of such a procedure supposes in the objective world a high degree of order which we are in no way entitled to expect a priori.

One more key figure Nickel quotes is mathematician and author Morris Kline:

Finally, a study of mathematics and its contributions to the sciences exposes a deep question. Mathematics is man-made. The concepts, the broad ideas, the logical standards and methods of reasoning, and the ideals which have been steadfastly pursued for over two thousand years were fashioned by human beings. Yet with this product of his fallible mind man has surveyed spaces too vast for his imagination to encompass; he has predicted and shown how to control radio waves which none of our senses can perceive; and he has discovered particles too small to be seen with the most powerful microscope. Cold symbols and formulas completely at the disposition of man have enabled him to secure a portentous grip on the universe. Some explanation of this marvelous power is called for.

The first aspect of the problem these scientists are getting at is the fascinating fact that the natural world possesses a deep structure or order. And not just any order, mathematical order. It is sometimes difficult for people who have not considered this before to get why this is so bizarre. Simply put, the order we see in the cosmos is not what one would expect from a universe that started with a random colossal explosion blowing matter and energy everywhere.

Many commentators have written about this and professed bafflement over it. All of the above quotes from Nickel’s book and many, many more are included in Morris Kline’s important work, Mathematics: The Loss of Certainty, which explores this issue at length. In his book The Mind of God, Paul Davies, an avowed agnostic, prolific popular writer and physics professor, takes this issue as his starting point. Davies finds the order in the universe to be incontrovertible evidence that there is more “out there” than the mere physical world. There is some kind of transcendent reality that has imbued the Creation with its mathematical properties.

The second aspect to the problem or mystery we are exploring is that human beings just happen to have serious powers of mathematical thought. Now, although everyone is happy about this, I rarely find anyone who is shocked by it. Christians hold that we are made in the image of God, which explains our unique abilities such as the use of language, the production of art, the expression of love, self-awareness, and, of course, our ability to think in mathematical terms. Non-Christians don’t accept the doctrine of the imago Dei, but seem to think that our abilities can all be explained by the theory of natural selection.

But hold on here one minute. Doesn’t it seem strange that our colossal powers of mathematical imagination would have evolved by means of a mechanism that presumably helped us survive in a pre-industrial, pre-civilized environment? Our abilities seem to go orders of magnitude beyond what evolution would have granted us for survival.

I know all about the God-of-the-gaps argument, and I’m not going to fall for it here. It may be that some day the theory of common descent by natural selection will be able to explain how we became so smart. That’s fine, and I’m not threatened by it. All I’m saying is that for now Darwinism still has a lot of explaining to do. And getting back to the concerns in this essay, I for one do not take Man’s amazing intellectual powers for granted. They are wonderful.

The third aspect to our problem is the most provocative of all. Mathematics is a system of symbols and logic that exists inside of our heads, in our minds. But the physical world, with all of its order and structure, is an objective reality that is not inside our heads. So how is it that mathematical structures and equations that we dream up in our heads can correspond so closely to the law-like behavior of the independent physical world? There is simply no reason for there to be any correspondence at all. It’s no good saying, Well we all evolved together, so that’s why our thoughts match the behavior of reality. That doesn’t explain anything. Humans are a species confined since Creation to this planet. Why should we be able to determine the orbital rules for planets, the chemical composition of the sun, and the speed of light? I am not the only one amazed by this correspondence. All those Nobel Prize winners are amazed by it too, and they are a lot smarter than I am. This is a conundrum that cannot be dismissed. John Polkinghorne said it well in his Science and Creation: The Search for Understanding:

“We are so familiar with the fact that we can understand the world that most of the time we take it for granted. It is what makes science possible. Yet it could have been otherwise. The universe might have been a disorderly chaos rather than an orderly cosmos. Or it might have had a rationality which was inaccessible to us…There is a congruence between our minds and the universe, between the rationality experienced within and the rationality observed without. This extends not only to the mathematical articulation of fundamental theory but also to all those tacit acts of judgement, exercised with intuitive skill, which are equally indispensable to the scientific endeavor.” (Quoted in Alister McGrath, The Science of God.)

Which brings us to the striking explanatory power of Christian theology for addressing this mystery. As long as we ponder only two entities, nature and human beings, there is no resolution to the puzzle. But when we bring in a third entity, The Creator, the God who made all things, the mystery is readily explained. As the figure here indicates, God, the Creation, and Man form a triangle of interaction, each interacting in key ways with the other. God gives (present tense verb intentional) the Creation the beautiful, orderly character that lends itself so readily to mathematical description. And we should not fail to note here that the Creation responds, as Psalm 19 proclaims: “The heavens declare the glory of God.” (I have long thought that when the Pharisees told Jesus to silence his disciples at the entry to Jerusalem, and Jesus replied that if they were silent the very stones would cry out, he wasn’t speaking hyperbolically. Those stones might have cried out. They were perfectly capable of doing so had they been authorized to. But I digress.)

why math worksSimilarly, God made Man in His own image so that we have the curiosity and imagination to explore and describe the world He made. We respond by exercising the stewardship over nature God charged us with, as well as by fulfilling the cultural mandate to develop human society to the uttermost, which includes art, literature, history, music, law, mathematics, science, and every other worthy endeavor.

Finally, there is the pair of interactions that gave rise to the initial question of why math works: Nature with its properties and human beings with our mathematical imaginations. There is a perfect match here. The universe does not possess an order that is inaccessible to us, as Polkinghorne suggests it might have had. It has the kind of order that we can discover, comprehend and describe. What can we call this but a magnificent gift that defies description?

We should desire that our students would all know about this great correspondence God has set in place, and that considering it would help them grow in their faith and in their ability to defend it. Every student should be acquainted with the Christian account of why math works. I recommend that every Math Department review their curriculum and augment it where necessary to assure that their students know this story.

John D. Mays is the founder of Novare Science and Math in Austin, Texas. He also serves as Director of the Laser Optics Lab at Regents School of Austin. John entered the field of education in 1985 teaching Math in the public school system. Since then he has also taught Science and Math professionally in Episcopal schools and classical-model Christian high schools. He taught Math and 20th Century American Literature part-time at St. Edwards University for 10 years. He taught full-time at Regents School of Austin from 1999-2012, serving as Math-Science Department Chair for eight years. He continues to teach on a part time basis at Regents serving as the Director of the Laser Optics Lab. He is the author of many science textbooks that I invite you to explore further on the Novare website.

What If Learning

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Last week I attended the Kuyers Institute Conference on “Virtues, Vices, and Teaching.” The focus of the conference was on pedagogical practices that instill virtue in students. The big question addressed in the sessions was “how do we as educators build the character of our students (focusing more on who students become rather than what students learn) while still being faithful to the content of our discipline?” It was a great conference and I’m still processing many of the ideas. There was an entire session of papers that focused specifically on this issue in math education. Hopefully I will be able to cajole the authors of those papers to share their ideas here. For now, I would like to share one resource that was presented at the conference: the website www.whatiflearning.com.

From the website, here is a summary of this new resource:

This site is for teachers who want their classrooms to be places with a Christian ethos or atmosphere, whatever the subject or age group they teach. It explores what teaching and learning might look like when rooted in Christian faith, hope, and love. It does this by offering over 100 concrete examples of creative classroom work and an approach that enables teachers to develop their own examples. “What if Learning” is a “distinctively Christian” approach developed by an international partnership of teachers from Australia, the UK, and the USA. It is based on the premise that a Christian understanding of life makes a difference in what happens in classrooms. Its aim is to equip teachers like you to develop their distinctively Christian teaching and learning strategies for their own classrooms.

I still need to explore this resource in more detail, but from everything that I have seen thus far, I am quite impressed. On the website’s Examples Page there are 10 concrete examples of what it looks like to teach math from a distinctly Christian perspective. Here are the titles of those 10 lessons with link to their content:

A shortcut link to these lessons will be included under the resource tab as well as the link side bar on the right.

Creation Care as a Focus for a General Mathematics Course

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By Dr. John Roe, Penn State University

The following is from a talk given by Dr. John Roe at the ACMS Conference this past June. It is with his gracious permission that I am sharing it here. I invite you to check out Dr. Roe’s blog “Points of Inflection.”

1. INTRODUCTION

Christian higher education institutions typically expect their faculty members to demonstrate “the integration of faith and learning” in their classrooms. (While the specific language of “integration” is most characteristic of the Reformed tradition, a requirement to exhibit some kind of wholeness between academic subject-matter and faith commitments is both natural and widespread.) Some subjects seem to lend themselves to such “integration” more easily than others—philosophy or history or even economics, for instance, seem to lie closer to traditional faith concerns than mathematics. Indeed, it may seem that mathematics is one of the toughest nuts for the would-be “faith and learning integrator” to crack!

In his article on mathematics in [9], Harold Heie expresses his strategy for initiating students into the integrative project as follows:

The goal of initiation requires that I help students see that the worldview project is worthwhile. That is much easier said than done. How does a teacher do that? A strategy that I have found to work, some of the time, is to pose an integrative question. By “integrative question” I mean a question that cannot be addressed adequately without formulating coherent relationships between academic disciplinary knowledge and biblical/theological knowledge.

Various kinds of integrative questions have been proposed int the mathematical realm. For instance, one could ask questions like:

  • What is the nature of mathematical objects. Are they created or discovered? In what ways is it appropriate to say that mathematics provides the “laws” of God’s creation? (ontological and epistemological).
  • Can mathematical insights provide analogies to help us understand theological truths? Is the deductive method of the mathematician useful for defending the Christian faith? (apologetic). This is Heie’s own proposal.
  • What does it mean to practise mathematics ethically? What are the values informing the practice of mathematics? In what ways can they support, and in what ways can they conflict with, other values?(axiological)

It’s the contention of this note that the complex of questions surrounding environmental sustainability or creation care [2] provide a timely and appropriate integrative opportunity for mathematicians. This is because sustainability questions are at the same time quantitative questions (to say “sustainable” is to invite the questions: how long?, how much?, how do you measure?) and theological questions (to say “sustainable” is to invite the question what is worth sustaining? and how does it relate to the One who, according to Scripture (Hebrews 1:3; Colossians 1:17), “sustains everything by the word of his power”.) The diagram below (various versions of which can be found online by searching for “triple bottom line” or related phrases) suggests how sustainability can serve as an integrative or cross-cutting theme, even from a quite secular perspective.

sustainabilityA persistent line of argument in contemporary discourse (ranging from [14] in the 1960s to [1] this year) suggests that Christian faith has impeded a proper concern for the integrity of the created order. Addressing this claim will combine technical analysis with ontological, ethical and apologetic motifs to yield a powerful integrative theme.

2. A COURSE PROPOSAL

I teach at a large, secular school. My calling does not allow me to “integrate faith and learning” in a way that presumes or encourages student commitment to a particular faith tradition. Nevertheless, the preceding considerations have impelled me to develop a course proposal which will use “sustainability” as an integrative theme to press beyond the confines of a conventional mathematics course. I also feel that it’s important to offer this material to non science majors. Sustainability should be a showcase to students across the disciplines about the importance and value of a mathematical understanding. We’ll see how successful this is!

There is nothing surprising in wanting to develop a new course — it is what math professors do all the time. But usually, when I and my colleagues dream of new courses, we are thinking of small classes of eager graduate students to whom we can explain our latest research ideas. Here, I’m after something a bit different.

The goal will be through a General Education Mathematics course, to enable students to develop the quantitative and qualitative skills needed to reason effectively about environmental and economic sustainability. Let me unpack that a bit:

  • General Education Mathematics At most universities (including PSU), every student, whatever their major, has to take one or two ”quantitative” courses — this is called the ”general education” requirement. I want to reach out to students who are not planning to be mathematicians or scientists, students for whom this may be the last math course they ever take.
  • quantitative and qualitative skills I want students to be able to work with numbers (”quantitative”) – to be able to get a feeling for scale and size, whether we’re talking about gigatonnes of carbon dioxide, kilowatts of domestic power, or picograms of radioisotopes. But I also want them to get an intuition for the behavior of systems (qualitative), so that the ideas of growth, feedback, oscillation, overshoot and so on become part of their conceptual vocabulary.
  • to reason effectively A transition to a more sustainable society won’t come about without robust public debate — I want to help students engage effectively in this debate. I hope to do this partly by using an online platform for student presentations. Engaging with this process (which includes commenting on other people’s presentations as well as devising your own) will count seriously in the grading scheme.
  • environmental and economic sustainability I’d like students to get the mathematical idea that there are lots of scales on which one can ask the sustainability question—both time scales (how many years is ”sustainable”) and spatial scales. We’ll think about global-scale questions (carbon dioxide emissions being an obvious example) but we’ll try to look at as many examples as possible on a local scale (a single building, the Penn State campus, local agriculture) so that we can engage more directly.

We’ll address these goals through four general themes: Measuring (quantification and estimation), Changing (dynamical systems), Risking (probabilities), and Networking (graph theory ideas). It seems to me that these four
themes are the central ones in a mathematical understanding of ecology and of sustainability more broadly.

I am hoping to develop my own materials for this course. The following texts have, however, been an inspiration in various ways and anyone interested in developing a course of this kind might find it worth consulting one or more of them: [4], [5], [6], [7], [8], [13].

3. AN EXAMPLE

One does not have to develop a whole new course in order to use sustainability themes in mathematics education. Here is a simple example showing how a sustainability unit can be integrated within a calculus class. Many calculus examples are hackneyed (how often do you really prop a ladder against a wall with its foot in the bed of a truck driving at constant velocity?) The design of wind turbines provides the focus for a nice optimization problem in which the extrema of a cubic polynomial arise naturally. For more detail on this, see [3], Section 5.2.

Two key parameters govern the amount of energy available to a wind turbine: the area S swept out by the turbine blades, and the speed v1 of the incoming wind. A third relevant quantity is a constant, the air density ρ. In one second, a cylinder of air of cross-section S and length v1 approaches the turbine. The mass of this air is then ρSv1 and the kinetic energy it contains is half the mass times the square of the speed, that is

energyThis is the power (energy per unit time) theoretically available in the cylinder of wind incident on the turbine.

But the turbine can’t possibly extract all that power! To understand why not, imagine what would happen if we extracted all the energy from some cylinder of the incoming wind. That cylinder of air would stop moving. Then, there would be nowhere for the next lot of incoming wind to go! So, while the air must slow down some as it passes the turbine and loses energy, it can’t slow down completely. How much of the theoretical power can we extract from the incoming wind?

An idealized wind turbine may be modeled as follows. Wind arrives from one side (say the left) the left at speed v1, passes through the turbine of area S at speed v(avg), and leaves at the right at speed v2. As the air slows down, it spreads out, because the total volume V of air (per unit time) passing each “slice” must be the same. That is, if A1 is the cross-sectional area of the cylinder of incoming air, and A2 that of the corresponding cylinder of outgoing air, we must have

volumeThe power available to the turbine is the difference between incoming and outgoing kinetic energies, that is,

powerIt can be shown (using conservation of momentum as well as energy; the details are in [3]) that v(avg) = 1/2 (v1 + v2). Thus the available power is

available powerwhere  λ = v2/v1.

In this equation, S and v1 are fixed, but the ratio λ between [0, 1] can be changed by varying the turbine design. What choice of  will make the energy output as great as possible?

It is a simple calculus exercise to show that the maximum value occurs when λ= 1/3 ; at this point

maximum valueand the maximum power available to the turbine is 16/27  or roughly 59% of the naive estimate

energyThis is called the Betz limit on wind turbine efficiency.

Note. The MAA ran a workshop this spring aimed at developing a portfolio of freely accessible units on mathematics and sustainability; the above is based on one of my contributions to this workshop. The whole portfolio can be found at [10], see the URL http://serc.carleton.edu/sisl.

REFERENCES

[1] David C. Barker and David H. Bearce. End-Times Theology, the Shadow of the Future, and Public Resistance to Addressing Global Climate Change. Political Research Quarterly, 66(2):267–279, June 2013.

[2] Richard Bauckham. The Bible and Ecology: Rediscovering the Community of Creation. Baylor University Press, August 2010.

[3] Egbert Boeker and Rienk van Grondelle. Environmental Physics: Sustainable Energy and Climate Change. Wiley, 3 edition, September 2011.

[4] John Harte. Consider a Spherical Cow. University Science Books, June 1988.

[5] John Harte. Consider a Cylindrical Cow: More Adventures in Environmental Problem Solving. University Science Books, February 2001.

[6] Lee R. Kump, James F. Kasting, and Robert G. Crane. Earth System, The. Prentice Hall College Div, 1 edition, March 1999.

[7] Greg Langkamp and Joseph Hull. Quantitative Reasoning & the Environment. Pearson, 1 edition, July 2006.

[8] Robert L. McConnell and Daniel C. Abel. Environmental Issues: An Introduction to Sustainability. Benjamin Cummings, 3 edition, April 2007.

[9] Arlin Migliazzo. Teaching as an act of faith : theory and practice in church-related higher education. Fordham University Press, 1st ed. edition, 2002.

[10] Debra Rowe. Sustainability Improves Student Learning.

[11] Francis A Schaeffer, Udo Middelmann, Lynn White, and Richard L Means. Pollution and the death of man. Crossway Books, Wheaton, Ill., 1992.

[12] Paul Tillich. The Shaking of the Foundations:. Wipf & Stock Pub, reprint edition, May 2012.

[13] Martin E Walter. Mathematics for the environment. Chapman & Hall/CRC Press, London; Boca Raton, FL, 2011.

[14] Lynn White. The Historical Roots of Our Ecologic Crisis. Science, 155(3767):1203–1207, March 1967. PMID: 17847526.

Related Post: Stewards of the Created Order