GOD & MATH: Thinking Christianly About Math Education
BIBLICAL INTEGRATION IN MATHEMATICS: WHY AND HOW? James Sellers Assistant Professor of Mathematics Cedarville College One of the biggest issues in Christian higher education today is the issue of Biblical integration or integration of Scripture and faith. As a faculty member of Cedarville College, I am faced with the challenge to integrate in the classroom everyday. I must admit that Biblical integration in mathematics is not necessarily as easy as it is in other academic areas. In his book The Pattern of God's Truth, Frank Gaebelein refers to mathematics as "the hardest subject to integrate."1 However, I firmly believe that it is possible to integrate in mathematics even if the opportunities are not obvious. My goal in this paper is laid out in the title. I will answer the following two questions pertaining to integration:1. Why integrate Scripture and faith in mathematics courses?2. How can I integrate in mathematics? One answer to the first question is that integration is tied closely to the first objective of my teaching institution, Cedarville College: "To undergird the student in the fundamentals of the Christian faith, and to stimulate him to evaluate knowledge in the light of Scriptural truth." This objective of the college, as well as the others, guides me in the education process; it drives me to integration. But this is not the only reason to integrate. I believe that there is a need for integration driven by the influence of nonbiblical philosophies and worldviews prevalent in our society today. What are some of these nonbiblical mindsets that are infiltrating our lives? I will mention a few. Note that this list is not meant to be exhaustive. My goal is simply to highlight some of the worldviews that exist today. In the late nineteenth century the theory of naturalism emerged, whose most basic tenet is that nature is in control, and that nature determines who or what survives. Man is reduced to a mere animal, and human responsibility is reduced to virtually nothing since man is a victim of nature. This mindset opened the door for theories such as the theory of evolution proposed by Charles Darwin. Naturalism clearly denies the existence and power of God, as well as the ethical and moral codes found in His Word, since man is at the mercy of nature. The theory of pragmatism then appeared during the twentieth century as an attempt to "redeem" man from natural determinism. This, too, clashes with Scriptural beliefs in many ways. First, under pragmatism, man creates truth. Put another way, truth is a product of man's action. Man, not nature, determines, leading to a concept of free will. However, we are also led to a view of relativism or relative truth. We hear statements like "if it works, then it is truth to him." Clearly, this contradicts the concept of absolute truth or biblical truth. The concept of morality becomes relative, dependent upon each individual. But how can any of this apply to mathematics? After all, isn't mathematics "above" all of this or separate from it? Isn't it possible for mathematics to be viewed by theists and nontheists alike without disagreements? The answer, which is surprising to many, is no. The worldview of the mathematician still influences his work and the view of his work. Let me give an example. How does a mathematician view his latest result? When he proves a theorem, is he creating this result or discovering it? There have been many letters published in mathematics journals in the last few years concerning this. Most humanists seem to believe that they are creating new identities which did not exist before they "created" them. Most theists, on the other hand, view their work as discoveries of already existent, but previously unseen, mathematical truths. Most theistic mathematicians hold this view of "discovery" because of their view of God as Creator. The humanist has no God and becomes creator himself. Note now how the worldview affects one's attitude toward mathematical work, as well as the giving of credit where credit is due. A second example along these lines is important. In his recent book entitled Chance and Chaos,2 noted mathematician David Ruelle compiles several short essays dealing with a range of topics. One of the common themes throughout the book involves the issues of chance and chaos and how they should be viewed mathematically. Ruelle shares his thoughts on areas such as classical determinism, historical evolutions, quantum theory, intelligence, and even a mathematical view of the true meaning of sex. One of the elements of the book that I found most fascinating was Ruelle's constant mentioning of the theory of evolution. Indeed, the book is mathematical in nature, but deals with the subject of evolution quite often. It is easy to see how this topic fits in with the title of the book. It is clear from statements such as the following that Ruelle's worldview permeates the text and strongly guides his writing: "The structure of living organisms has changed a lot through evolution, by the process of mutation and selection, but the genetic code is so basic that it has remained essentially the same from bacterium to humans. Presumably, in the first hesitant steps of life, there was an evolution of the genetic code. When at a certain point an efficient system was evolved, it killed off the competition and survived alone." "With the advent of sex, then, the evolution of life can proceed much faster. Mutations are still occurring, of course, but a more intelligent innovative process is now also at work--the reshuffling of genetic messages. And after the reshuffling, selection operates, of course, to keep the fit and the lucky." "Possession of higher functions was of course beneficial, and encouraged by natural evolution."My goal here is not to single out David Ruelle, but to cite a recent example of mathematical discussion that is strongly influenced by the author's worldview. As we strive to equip young men and women of Christ to go into the world, we need to help them understand the differences between theistic and nontheistic worldviews. As one of the objectives of my college states, they need to "evaluate knowledge in the light of Scriptural truth" at this stage of their lives. We need to integrate so that our students can be witnesses for Christ in such a way that others will find them intellectually or logically valid. With this somewhat general answer to the first question raised at the beginning of this paper, I wish to move to the second. How can I integrate in mathematics? To answer this "how" question I want to discuss three threads which can be developed in mathematics curricula with integration in mind. The first thread comes in the area of logic. As a faculty member of Cedarville College, I strive "to enable the student to develop sound critical and analytic reasoning," which is the fourth objective of the college. A logic class is the perfect setting for this. In our mathematical logic course, students are shown the basic constructs of both propositional and predicate calculus as well as the basic methods of proving mathematical theorems. While discussing some of the basic logical forms, passages of Scripture can be used to provide examples. Students can then begin to see the logical arguments used by the biblical authors in proving points. Several biblical authors employ numerous techniques of logical proof in their arguments and students can understand these constructions through a logic course. Examples of logical constructions in the Word abound. For example, instances of universal and existential quantification of predicates appear frequently. These are statements like "for all have sinned . . ." (Romans 3:23) and "there is no one who does good" (Psalm 14:1). Constructions involving conditionals, or "if / then" statements, are also numerous. Paul's discourse concerning Christ's resurrection in 1 Corinthians 15:12-19 contains at least six conditional statements, such as "if Christ has not been raised, your faith is worthless" (1 Corinthians 15:17). John includes a conditional and its inverse, which are two nonequivalent statement forms, when he says, "He who has the Son has the life; he who does not have the Son of God does not have the life." (1 John 5:12). By combining these two statements, John develops a biconditional, or "if and only if" statement: one has eternal life if and only if one has the Son. This, of course, is one of the cornerstones of the Christian faith. Also, let me mention a proof strategy used by Christ himself which is frequently used in mathematics -- that is, proof by contradiction. In this type of proof, the negation of the desired result is assumed, proven to be absurd, and then the desired result follows. In Mark 3:22-26, the scribes believe Christ is casting out demons by Beelzebul. To prove them wrong, Christ (implicitly) assumes this, then goes on to argue that such action is absurd. As Christ says, "And if Satan has risen up against himself and is divided, he cannot stand, but he is finished!" (Mark 3:26). This then implies that Christ was not casting out demons through Satan's power, but some other source, implicitly God's power. This is a classic example of proof by contradiction. Note that I am not advocating that we look at Scripture only from a logical framework. There are many biblical truths that cannot be explained "logically" but must be accepted by faith. (Take,for example, the concept of the Triune God, three in one.) However, as I mentioned in my remarks above, I believe we need to be equipping our students with an ability to produce valid logical arguments as they go out into a highly intelligent, sophisticated world. A. W. Tozer once wrote: "There is, unfortunately, a feeling in some quarters today that there is something innately wrong about learning, and that to be spiritual one must also be stupid. This tacit philosophy has given us in the last half century a new cult within the confines of orthodoxy; I call it the Cult of Ignorance. It equates learning with unbelief and spirituality with ignorance, and, according to it, never the twain shall meet."3If a believer's logic is poor or invalid when witnessing, then the unbeliever is less likely to be receptive. We must be able to serve as apologists when needed. Frank Gaebelein writes, "Our task is not only to outlive and outserve those who do not stand for God's truth; it is also by God's grace to outthink them."4 The second thread concerns the practical application of mathematics to the physical world. One of the largest uses of mathematics is that of modeling the physical world around us, the world of our Creator. We can model the orbits of the planets, the flow of blood through an artery, the trajectory of a ball thrown in the air, and countless other phenomena. But as we do so, the Christian mathematician has an excellent opportunity to simultaneously honor the Creator and exhibit the limitations of the creature. First, modeling of many phenomena, like atmospheric conditions and weather patterns, ignores many variables. Because of the complexity of the creation, we are forced to simplify our models to a great degree in order to get a handle on them. In this we see how little man understands compared to the knowledge of God. We can merely approximate the true phenomena; in most cases we cannot achieve exact models. Most mathematical methods are indeed approximative. Does this make the mathematics useless? Certainly not. We can approximate desired values with as much accuracy as is deemed necessary and then minimize the error. (This is done, for example, when using 3.14 as an approximation of the number p, whose decimal representation does not terminate.) However, we must admit that there is often some error in the process. George Polya once wrote:"Although almost invariably in science we must beginwith what is only an approximation to the truth, weneed not rest content with it. A crudeapproximation can be made to lead to a less crudeapproximation; a good approximation to a better one.That the notion of successive approximation is a keyto more exact knowledge makes it a worthwhilestudy."5Let me also point out that much of mathematics also appears quite "exact." Clearly, for example, 2 + 3 = 5. No approximation is necessary here! But now the question should be asked: Is this absolute truth? The answer is no, for this fact is based on a set of axioms, or postulates -- the Peano Postulates. All mathematics is based on a set of axioms, whether it be Peano's for arithmetic or Euclid's for Euclidean geometry. However, if the axioms are changed, then the "mathematical truths" based on the axioms can also change. For example, in a nondecimal system, 2 + 3 may not be 5. In a base 4 system, the value of 2 + 3 is 11 (to be read "one one", not "eleven"), since, in decimal numbers, we have 1*4^1 + 1*4^0 = 1 + 4 = 5 = 2 + 3.So 2 + 3 = 5 is certainly not absolute. Our students need to see this distinction between absolute truth and truth based on axioms. Although mathematics is one of the purest of sciences, it is based on an axiom system or belief system and, therefore, does not generate absolute truths. The third thread that I will mention involves areas such as calculus and differential equations. Typically, the material in these courses does not lend itself to integration. The integration normally discussed in a calculus class is not the kind of integration that I am concerned with in this paper. However, there are some ways that integrative discussions with students can be achieved. For example, topics such as radioactive decay and carbon dating arise within the calculus course. In most textbooks, examples and homework problems involving the age of the earth or the universe appear. These can be used as springboards for discussions involving origins. Assumptions made in such problems can be discussed, and the issues of worldview and its effect on one's perception of a problem can be analyzed. I have found these interactions quite lively, especially as students with differing scientific worldviews and backgrounds interact. Similar mathematical problems have arisen when the historicity of events in the Old Testament are challenged, usually by skeptics. A classic example of this is the account of the Red Sea crossing in Exodus 14. While striving to invalidate Scripture, some have argued that it was impossible for the whole nation of Israel to have crossed the Red Sea in one day (or one night, as some interpret the passage). Another such critique of Scripture involves the growth rate of the nation of Israel while in Egypt. It seems absurd to some that Israel could have grown so large in such a short amount of time. These sorts of "rate" questions can be studied using differential equations. Using reasonable assumptions, the problems can be studied and the plausibility of such historical events in the Bible can be documented. However, I must state one strong caveat. Plausibility does not imply proof. We must not replace God's Word and its truthfulness with plausibility and probability. The fact that some event is plausible, or even highly probable, does not imply that a scientific proof has been achieved. This is especially true in the area of origins. I believe students need to realize this and should discuss this in the classroom. Ultimately, we should go back to the Bible and rely on it in any matter. As I close, let me comment on some general aspects of a Christian professor's life which I believe should be apparent, especially if one wishes to integrate. I believe that integration can take place outside of the classroom, away from the chalkboard. The courtesy with which I treat students, the justice shown in the grading process, the compassion and "listening ear" evidenced in the office, the conversations held at the gymnasium during a basketball game, and the prayers offered for students, family, and friends all should appear in the life and work of an integrating faculty member. Moreover, not all integration needs to be canned or planned. Questions that surface and discussion that occurs in the class should be guided through a pathway lined with integrative truths. I end this paper with a comment from a former student which sums up this insight: "You never had to force in the occasional bit of 'Biblical integration,' because your faith permeated everything you did." Notes 1Gaebelein, Frank, The Pattern of God's Truth: Problems of Integration inChristian Education, Moody Press, Chicago, 1968.2 Ruelle, David, Chance and Chaos, Princeton University Press, Princeton,N.J., 1991.3Gaebelein, Frank.4Gaebelein, Frank.5Polya, George, Mathematical Methods in Science, The MathematicalAssociation of America, Washington, 1977. ................
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