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Version: 1.0
(August 26, 2006)

DEVELOPING A CHRISTIAN PHILOSOPHY OF MATHEMATICS: INTRODUCTION

We are challenged to see Christ and His Word as Sovereign over every academic subject.  Nothing is neutral, but many insist that academics may be done autonomously, apart and separate from God and His Word.  However, the beginning of all knowledge is the fear of the Lord.  True truth can be known only when a subject is studied in relation to Him.  

To demonstrate this, I have chosen Mathematics.  It will be studied in light of the Bible and Christian Theology.  It should be read in the light of my paper on Epistemology. (also on this site)  The two subjects are inseparable.  The following are lecture notes from a Class I offered for college students at Barnabas House, in Amherst, Massachusetts in the Winter of 1979. Since that time, of course, Constructivism and Post Modernism has come on the scene and there have been new approaches to Theories of Knowledge, Logic and Math.  There is also an important relationship of these subjects to Open Theism.  I know these lecture notes need revision and updating, but they are still worth while.  I am reworking these Notes and will soon publish an updated version..

Unit I: Arithmetic

1. Properties of Numbers
2. Properties of Arithmetical Operations, eg- distribution, commutation,  association, aggregation (unity and plurality), natural /rational vs.  irrational
3. Reasoning by Recurrence, Complete Induction, principle of          Permanence  

Unit II: Infinity

1. Special Topics: Time, Continuity, Succession

Unit III: Foundations of Modern Mathematics

1. Formalism
2. Logicism
3. Intuitionism
4. Constructivism

Unit IV: Causality and Philosophy of Science

Bibliography: Beside the Scripture, the main sources for this paper, written in 1979, were

Non-Christian-

Math in Western Culture, Morris Kline
Number, Tobias Dantzig
Math for the Modern Mind, Walter Fuchs
New World of Math, George Boehm
Mathematics, David Bergammi
Mathematics for Parents, Carl Doerfer
The Common Sense of the exact Sciences, Wm Clifford

Christian-

Christian Perspectives on the Foundations of Math (a series of Papers presented over several years at an annual Conference sponsored by Wheaton College)
Foundations of Christian Scholarship (chap 8) by Vern Poythress


Unit I: NUMBERS (Basics)

A.  Definitions-

    

B.  Four Fundamental Operations with Natural (Whole) Numbers    

1. Addition
2. Multiplication
3. Subtraction
4. Division These all require certain properties-

C. Three Basic Properties of Operations

1. Distribution- multiplication is really repeated addition (for positive integers)
2. Commutation- two whole numbers may be interchanged
3.  Association

D.  More Arithmetical Philosophical Problems

1. Universals and Particulars (One and the Many or Unity and Plurality)
2. Continuity and Permanence (of natural sequences)
3.  Infinity: a never ending series

Unit II: Biblical and Theological Material Relevant to the Philosophical Issues raised so far (above)-

1.  Infinity

The basic assumption about infinity in math is that there can be an unending succession.  Consider a series of dots, making a line, that has no end.  This is a way of defining Time

 A.  Time in the Bible: Key Words




 B.  Doctrine of Time  - there will never be an end to all time, i.e.- a  timeless Eternity


C.  The Biblical Meaning of Eternity


D.  The Eternality of God [not the timelessness of God]- Deut 33:27a, Isa   57:15a, Rom 16:26 cf 1:20, Ps 29:10, 45:6 etc


The Mathematical Concept of Infinity, of a never ending series, is rightly rooted in the Nature of God.  It is Biblical and True.  Aion follows Aion without end.  God is all God is at every point on the line, forever.
   
This Concept does not come from intuition or induction- it comes directly from God.

Ecclesiastes 3:11 [God] has put eternity [Olam= a sense of the everlasting cf Chronos, a sense of indefinite or perpetual Time, ] into man's heart.  (This also means Humans know their life is not over at the grave.)

2.  Continuum: a well-ordered and perfect aggregate  (the set of all points on a line segment)

"well-ordered" = all numbers permanently arranged in greater and lesser sequence
"perfect "  = contains its own limiting values from negative infinity to positive infinity
"aggregate" = sum or collection (a group or set)

Numbers that can be represented on a line- [real numbers are believed (intuited) to be in one to one correspondence with the points of the number line, in contrast to imaginary numbers)

Whole (natural or counting) numbers: (positive integers 1 2 3  or negative: -1 -2 -3  or zero)

Rational Numbers: fractions

Irrational Numbers: not fractions, all real Numbers that are not Rational

Some Philosophers (such as Danzig) say that the concept of Continuum, as well as Infinity, rest on Intuition.  Furthermore, the mind connects Continuum with Causality, ie- it rejects the idea of chance and spontaneity and insists on necessity (if 3 than 4).  The Continuum requires belief in necessary connection, not just constant conjunction.  There must be a chain of cause and effect.  

Biblical  basis for Continuum, as well as Infinity, is the  Doctrine of Time and Causality (above)

3.  Universals and Particulars (One and the Many or Unity and Plurality)

    A.  There are 3 traditional and non-Biblical solutions to this problem.  



    B.  Questions re Universals

           a.  Do Universals exist?
           b.  What accounts for the concept of (the belief in) Universals?
           c.  Can we truly know Universals
           d.  Is this topic simply a matter of semantics?
           e.  in Math, are numbers Universals or Particulars?

    C.  Biblical Solution to the Problem of the One and the Many

Particulars are all related in a cause and effect chain and by God's Design or Plan for them.  The relationship of the One and Many is explained by the Biblical Doctrine of Causality and Providence.  The source of Unity (and meaning) is not in Universals, but in Christ: All things, that are other than Deity, hold together in Him -Col 1:17  These things are in no way divine, or part of, Deity.

The tri-unity of God is a model of particulars in unity; but, these particulars (Father, Son and Spirit) are equals in nature and not in any kind of cause and effect chain.  In the economy of God, even the roles of each Person are of equal importance and work together.  

4.  The Problem of Correspondence and Truth

We earlier defined numbers as things that correspond to other things.  What is correspondence?  This is very important as correspondence is one of two definitions of truth.  The other is "coherence". 

Correspondence is assigning every object of one collection and object of another, eg- rational numbers to points on a line [the basis of Analytic Geometry].  One thing is the arbitrary, fixed, representation of the other.  (Representation does not mean one looks like the other, but it symbolizes it.)  The problem is-  does what a symbol is said to represent, actually exist independent of its symbol?  Is mathematical truth really true?

Earlier, we defined Arithmetic as the measurement of quantitative and spatial (geometric) aspects of natural phenomena.  Arithmetic is an attempt to explain reality.  Problems developed when arithmetic seemed to fail to explain certain aspects of reality, eg- irrational numbers, transcendentals and infinite numbers.  How real are imaginary and complex numbers (numbers whos square is negative or which are a combination of real and imaginary numbers)  Do such thing really exist apart from their assigned symbols?  What is the relationship of Analytic, algebraic, descriptive, differential, projective and non-Euclidean geometries to each other.  Which one truly represents reality?

Modern Math does not attempt or claim to represent or explain reality.  It is its own reality!  Traditionally, the process of constructing math theory had four operations: [Nature/Reality => Intuition => Organization => Deduction => Application [to Nature and the real world].

In 1965, C. Allendoerfer, Professor of Mathematics at the University of Washington said, (mathematical models) "are abstract systems of undefined words, axioms, etc (that) need have nothing whatsoever to do with nature.  They are no longer models of anything, but are merely structures built by mathematicians because they are thought worth investigating.  Since these abstract mathematical systems did not spring from nature, there is no obligation for their advocates to apply related theorems back to nature and consequently we have a mathematics which exists purely of and for itself.  A large part of contemporary math is of this kind."

Modern mathematicians are not interested in Truth or non-Truth at all.  They are interested only in validity, internal consistency and non-contradiction.

All of this leads the Christian to ask What is reality?  Can math explain it?  Must math correspond to it?  Must math be concerned with Truth?  Can a Christian engage in a field which claims complete autonomy from reality and truth?  How does a Christian, who believes in the reality of Creation and its Creator, who claims sovereignty over all that is, and who, as a Christian, believes the purpose of all that is, including the designs and products of out human minds, is to glorify God, respond to this-

"Math has a reality of its own.  It is a reality that is defined in the human maind and need not be confirmed by the interpretation of a physical experiment.  The physicist is fettered to whatever his instrument tell him is real; the mathematician, however, is free to explore a much richer world, mostly of his own making."  - G. Boehm

"How then shall mathematical concepts be judged?  They shall not be judged.  Mathematics is the supreme judge. From its decisions there is no appeal"  -Dantzig

"Most researchers in math today apparently would agree with C.G. Jacoby who, when asked why he did math, replied: to glorify the human intellect."  - Boehm

The arrogant hubris of these quotes, from more than some 30 years ago, simply takes my breath away and offends me greatly as a Christian.  The attitude has spread to many non-math fields (including Theology)  and is even more prevalent today.  Consider those statements in light of-

1 Corinthians 10:31...whatever you do, do all to the glory of God.

2 Corinthians 10:5 ... take every  thought captive to obey Christ

Colossians 1:15 [Jesus Christ]  is the image of the invisible God, 16  by him all things were created, in heaven and on earth, visible and invisible, --all things were created through him and for him. 17 And he is before all things, and in him all things hold together. 18  that in everything he might be preeminent.

Nothing, including Mathematics can be autonomous.  Whatever Reality and Truth are, math must take its place within them and under the authority of Christ.  Mathematics must be done for Him.

UNIT III: Foundations of Modern Mathematics

A.  Formalism (David Hilbert, et al)


Major Weakness:


B.  Logicism (B Russell and A.N. Whitehead)


Major Weakness: though stressing Logic, this School uses many unproven assumptions in its arguments (eg- the properties of Arithmetic)   It gives no justification for what it does.

C.  Intuitionism (H. Bower)


Major Weakness:


D.  General Summary of MathematicsToday: Constructivism  (1979)


It should be readily seen that much of the common Philosophy of Mathematics is in opposition to Scripture and Reformed Christian Epistemology. Here is a challenge to the Christian Mathematician who wants to pursue this field Pro Rege.

Copyright © 2006 (CalvinFox.com)