BIBLICAL INTEGRATION IN MATHEMATICS: WHY AND HOW?
http://www.iclnet.org/pub/facdialogue/24/sellers24
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.
following two questions pertaining to integration:
I will answer the
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."3
If 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 begin
with what is only an approximation to the truth, we
need not rest content with it. A crude
approximation can be made to lead to a less crude
approximation; a good approximation to a better one.
That the notion of successive approximation is a key
to more exact knowledge makes it a worthwhile
study."5
Let 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 in
Christian 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 Mathematical
Association of America, Washington, 1977.