INTEGRATION OF FAITH AND MATHEMATICS

1. INTRODUCTION

     This is a half-serious paper, exploring the vast opportunities for Christian teachers to discover illustrations of biblical truths with mathematical and scientific truths. Even when such illustrations are never perfect, they are useful in crystalizing some aspect of our faith, sometimes in a very beautiful manner.

     I have been lecturing Calculus III for 10 semesters in USC as a graduate student from 1963-68, and again at Biola during 1980, and from 1987 till now. Experiences at Calculus II, linear algebra, differential equations, modern algebra, fundamentals of math, set theory, all offered opportunities to discover ways of relating mathematical insights to Christian beliefs. Usually they are discovered at the spur of the moment. They serve several purposes:
     (a) They wake up all those students falling asleep.
     (b) They make students feel awed at how truths in various academic disciplines are related, as well as appreciate the exquisite beauty in, of all things, mathematics, just like art or music.
     (c) They may lead some listener to think more deeply concerning divine truths as taught in the Bible, which will inevitably affect his life.
     (d) They make me feel that perhaps I am playing some useful role in God's plan in the area of Christian education, and such a feeling of being in the right direction under His personal guidance is most gratifying.

2. WHAT IS TRUTH?

     I am not a philosopher, but I see plenty of opportunities to spur the listeners to start some philosophical thinking, which has great implications for our Christian faith. Here is an example:

     I was teaching foundations of mathematics. "Truth is objective rather than subjective" came to my mind when I thought about the concept of the cardinal number of a set. In Sesame Street, once Ernie and Bert were quarrelling over how many oranges there were in the basket. Bert counted 8, but Ernie counted 10. Ernie played the Devil's advocate by counting some oranges twice. Said Bert: "Hey, you counted that orange already." Said Ernie, "Why am I not supposed to count any orange twice?" Bert: "Because it is not counting unless each orange is counted once and once only." Ernie: " Even if I do so, how can you guarantee my count and yours ought to be the same?" Bert: "Each basket of oranges have a definite count, regardless of who counts it". Ernie: "Why has it to be so exact?" Bert: "Because it is the objective truth." Ernie: "Come to think of it, you said I counted this orange twice, that is only because your memory remembered I counted it already. But what if my memory told me I did not? How do you know your memory is not fooling you?"

     The above story opened a can of worms. It almost makes plausible the Eastern philosophy that "something is real only because you believe it is so". Such beliefs will cast doubt on the truthfulness of mathematics, and perhaps on Christian doctrines as well. My solution for such doubting Thomases is to tell them that mathematics is a logically consistent model and tool that has 'concurred with personal experiences of many individuals. If only Ernie had marked each orange that he counted with a marker pen, he would have arrived at the same count as Bert or any one else, which gives credence to the belief that each basket of oranges does have a unique number as its "count"'. It is still only a belief, yet a belief that is never refuted by personal experiences for centuries. Such beliefs are called 'objective truths'. So objective truths are beliefs that can stand the test of individual experiences. This definition of objective truths can apply to Christian beliefs.

3. EXAMPLES OF SOME ILLUSTRATIONS

3.1 GOD'S OMNISCIENCE

     (a) The human body is topologically equivalent to the unit sphere S: x² + y² + z² <= 1.
     Let P(x₀, y₀, z₀) be a point inside the sphere. Embed the 3- dim space in a 4-dim. space with variables x, y, z, w. Then the parametric equations:
     x = x₀ + t, y = y₀ + t, z = z₀ + t, w = t describe a line in 4-dim space that intersects S at one and only one point, namely P. In 3-dim space, it is impossible for a straight line to intersect S at only one interior point, and yet it is possible in 4-dim. If we assume God lives in a space at least 4- dimensions in nature, then He can see through every atom and particle in you and I, and He can surely record all our thoughts and schemings, on His computer cassette tapes, for future scrutiny at the Judgment Seat.

     (b) Taylor's theorem in Calculus says for most functions f(t), where t can be interpreted as "time", if you know enough about f at the present moment, such as f(0), f'(0), f''(0), f'''(0), etc., you can predict the future and describe the past with alarming accuracy. Thus

     f(t) = f(0) + f'(0) t + f''(0) t²/ 2! + f'''(0) t3/ 3! + ... Psalm 139 says ``Thine eyes have seen my unformed substance, and in Thy book they were all written.'' God took one look at your genes, and He knew instantly that you are a hopeless sinner, no matter how hard you try.

3.2 FOREKNOWLEDGE VERSUS FREEWILL

     Think of Green's theorem equating a double integral to a line integral and the Cauchy integral formula relating the value of a complex function f(z) with the integral of f along any simple curve C that goes around the point z.

     (a) Green's theorem says no matter what f(x,y) may change within a bounded 2-dim area A, you need not measure it minutely everywhere in order to calculate the double integral of f over A. All you need is to measure f on the boundary of A and that suffices. God similarly need not laboriously monitor your actions and motives from day to day, hour to hour. He has his means of taking a survey around the periphery of our existence in space-time, and He knows our accumulated works and deeds immediately.

     (b) Cauchy's integral formula is very interesting:
     f(z) = (1/2 p i) ò_C f(w)/(w-z) dw, integrated along any simple closed curve (e.g., a big circle) C in the complex plane around the point z. Suppose the x-axis represents time, and y-axis represents our physical existence, albeit one dimensional. Then the half-plane x<0 represents the past, the half-plane x>0 represents the future. Suppose the w-plane is enmeshed in a grid whose points have x and y coords that are multiples of a small distance c > 0. Suppose further that f(z) has been defined in the time past at grid points within C's interior. Now for the immediate future you have all the freedom that you want in defining f at the grid points in C. All this freedom does not change the past history, yet somehow if C is taken to be the boundary of your life in space-time, from the values of f on C we can recompute everything you have done at all the grid points. c can be as small as, say, a nanosecond.

     Does the distant past determines the present? Cauchy's theorem says no, if we only look at f(w) at the points on C in the past. However, what you do in the immediate future does have long term effects on values of f(w) on the future part of C. The mystery is that by recording f(w) on the boundary curve C, God can tell all that we are doing in our little time-space region of "personal free will".

     (c) Theorem of residues is interesting. Suppose C is a simple closed curve in the z-plane around the origin, and the half plane x < 0 represents the past, the half plane x > 0 represents the future, as before. If f(z) represents the spiritual values of your actions in your life, and the y-coordinates represents your activities, physical locations, etc. Let L = ò_C f(z) dz represent the sum total of your entire life before God. Your freedom is to define f(z) at each grid point as a finite value, yet the value of the line integral is dependent only on the line integral of f(z) around some points p1, p2, ... within C, called the poles. Such values of line integrals, called residues, around a pole p is independent of the curve, denoted by Res (p). which we can take as an infinitely small circle around the pole.

     The theorem then says L = Res (p1) + Res(p2) + ... One of the poles occur at our birth, its residue being negative, inherited from Adam. No matter how much is your subsequent effort to define f(z) at grid points, your net worth L before God will be negative, which deserves hell. When you accept Jesus Christ into your life, a super human event happens: another pole with a greater positive residue value is introduced into your life. Henceforth your worth L before God becomes positive, and can never go negative, till we see Him face to face. Hallelujah.

3.3 ETERNITY

     There is a problem: put four 2's together to form as large a number as possible. No +, -, *, / signs or parentheses, only indexing. What is the largest number? It represents a glimpse of eternal life in heaven.

     The smallest answer is 2222, a greater number is 222 squared, or 222². Then 22²² is a number with about 30 digits, 2²²² is a number with about 70 digits, something like the total number of electrons that can fill up a galaxy. = 2¹⁶ = 65536, a very small number. However, the correct answer is , which = 2^4194304, which in turn is a number with 1262612 digits, the first few being 2152... It has about the same number of characters as the NASB New Testament, i. e., printing it will take as much paper and ink as the New Testament. The thought of suffering that many years in hell if I have not received salvation brought me nightmares.

     The hyperbolic geometry of the space within the unit circle C brings us also some insight into eternity in time. We shall use Klein's model, where distance s(P₁,P₂) between two points in C is defined as s(P₁,P₂) = m(P₃,P₁) m(P₄,P₂) / m(P₄,P₁) m(P₃,P₂) where m( ) is the Euclidean distance between two points, and P₃, P₄ are points where the straight line through P₁,P₂ cuts C. s is called the Cross- ratio of the points (P₁, P₂, P₃, P₄).

     Let x-axis with s as distance represents time, and lines parallel to the y-axis represent everything happening in the world at the same time. Let s(P₁,P₂) represent the hyperbolic distance between two points whose Euclidean coordinates are (x₁,y₁) and (x₂,y₂) . Then if P₁, P₂, P₃ are 3 points, then s(P₁,P₃) = s(P₁,P₂) + s(P₂, P₃) only if they are collinear and P₂ is between P₁ and P₃. Also s( P₁,P₂) goes to infinity if either one of them moves closer and closer to C. Thus the x-axis diameter represents time from infinite past to infinite future. To us infinite future is unimaginable, but in God's eye, when He looks at our space time with a different metric, say m instead of s, then infinity of time and space is only a piece of His handywork with finite scope and size in His hand. When Christ became incarnate, He dove into our space-time universe with definite, historical, instances of birth, death, and resurrection. Yet when He talked about the "glory which I had with Thee before the creation of the universe" (John 17), He simply referred to the "time" before He dove into our space-time. He was with the Father before the foundation of our time. The Jehovah's Witnesses wasted their breath preaching that we must not worship the Son, because He was created in time. No, the Son was not created in space-time. Rather, He created our space-time. He was the architect, as well as the builder.

3.4 GOD DWELLS IN UNAPPROACHABLE LIGHT.

     Let us take the xy-plane, and define the hyperbolic metric
     s(P₁,P₂) = s( (x₁,y₁),(x₂,y₂) ) = Ö( (x₂-x₁)² - (y₂-y₁)² ). We shall use the convention of letting (xn,yn) be the coordinates of the point Pn. The distance is defined only for point pairs where | y₂-y₁| < |x₂-x₁|. This metric space has the triangle inequality in reverse, so that for three points P₁, P₂, P₃, such that x₁ < x₂ < x3 and the segments joining any two of them have slopes < 1, we will have s(P₁,P₂) + s(P₂,P₃) <= s(P₁,P₃), with equality only when they are collinear. This space is called the Lorentz space.
     If we let x represents time as before, and y represents our physical position, choosing 1 second as the unit along x-axis and distance travelled by light in 1 second as the unit along y-axis, then this is Einstein space-time model. The lines y = ± x are the 45° lines through the origin, which represent speed of light. All physical objects moves with less than the speed of light, hence |dy/ dx| <1 always. The trajectory of any physical object is a curve whose slope is always between -45° and 45°.

     Let us hypothesize that spiritual beings live in the same space-time universe as ours, with the rule that their speeds are always faster than the speed of light. Then their metric is
     s(P₁,P₂) = s( (x₁,y₁),(x₂,y₂) ) = Ö( -(x₂-x₁)² + (y₂-y₁)² ), which is defined only when the segment P₁ P₂ has slope >1 or <-1. Their universe and ours normally cannot communicate. Their trajectories are curves with slopes > 45° or < -45°, they can even go parallel to the y-axis, thus being at different places at the same instant. However, in rare moments when angelic or demonic beings appear in some physical form to humans, they cannot stay stationary in a line segment parallel to the x-axis, but must "flutter"in a zigzag curve whose slopes are >1 and <-1 alternatively, yet maintaining the same y coordinates over some time inteval. Thus they should appear somewhat transparent, ethereal. The Incarnation is really a miracle, because the Son of God departed from His universe and come to dwell, not flutter, in our world.

3.5 SUBSTITUTIONARY ATONEMENT

     The following example is most amusing:
     Integrate ò ( 1 / xÖ(x²+2x+2) ) dx.
     A second or third semester calculus student would try to match it up with standard formulas such as ò ( 1 / Ö(x²+a²)) dx or ò ( 1 / xÖ (x²+a²) ) dx , etc. all to no avail. He may try to complete the square by expressing x² +2x +2 as (x+1)² +1, then substituting u = x +1. Yet the little x in front of the square root sign give him fits. He cannot split the fraction as the sum of two fractions either. Now he realizes he is hopelessly lost, like a sinner. That little x is his "sin burden" which he could not get rid of. Here comes the good news, from an old (calculus) Book: Submit yourself humbly under a new Substitution:
     Let u = 1/x (The substitution of Christ). With a bit of calculation, the integral transforms into ò ( 1 /Ö (2 u² +2 u +1) ) du . The amazing thing is that in front of the square root sign there is no more sin burden. The sin is gone, and that breathes New Life into his prospects. Completing the square again, then it will fit some arc sinh formula, and the sin problem is solved. That is salvation.

4. CHRISTIAN PERSPECTIVE OF HISTORY

     Christian math teachers cannot shirk from the responsibility of teaching the tremendous role mathematics played in development of modern science and hence modern history. If we don't impart the Christian point of view, others will introduce their materialistic or humanistic or Darwinian views.

     One does not need to teach a course of history of maths to make our views known. One can introduce it from time to time, even in secular schools. One question I like to ask is: "Why is the western world so much stronger than cultures with a long history, such as Arabia, China, India?" Could it be God's plan that mathematicians such as Descartes, Vieta, Galileo, and Newton were born in the West, so that the Western world can lead the way to scientific discoveries, which eventually hastened the spreading of the Gospel? Western science could not have advanced so rapidly were it not for Newton's discovery of laws of mechanics and gravitation plus the invention of calculus to put these laws in precise mathematical footing. We should get students to appreciate what the world would be if not for the discovery of these math and physical laws. Of course, if Newton were not born, others would have discovered the laws of mechanics just the same, perhaps some 50 years later. Leibnitz already discovered calculus.

     Meanwhile, China's lead in mathematical research came to a standstill by 14th century, due to many reasons. (Remember they calculated p to 7 figures some 900 years ahead of the West, and recorded Pascal's triangle some 400 years ahead of Pascal.) It is good to study the social and political reasons for China's slowing down. Now we can extrapolate: If God had His plan to get the Chinese people, and even former Soviet KGB members, to admit their spiritual bankruptcy, and to accept the Gospel in unprecedented numbers today, He can and will do the same for the Muslim world.

     The ability to relate historical milestones in math with other milestones in the history of astronomy, physics, chemistry, and the sciences, in human social history, in political movements, and ultimately, in history of evangelical Christianity, should be cultivated by every Christian math and science teacher, even though not many of us can be like Francis Sheaffer.

5. CONCLUSION

     Math is like the sciences, there are so many beautiful things to discover. Our task is to help students dispel their fears of abstraction and logical proofs. Once they have gotten used to deriving things logically, day in and day out, just like a pianist gets used to practising 3 or 4 hours a day, they can have the heart to appreciate the beauty that is everywhere. Music can open up a person's soul, and math can open up a person's mind to appreciate the Creator.