This groundbreaking book lays a theistic foundation for the study of mathematics, exploring everything from simple concepts such as addition and subtraction to more complex topics such as set theory and the nature of infinity.
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About the Author
Vern S. Poythress (PhD, Harvard University; ThD, University of Stellenbosch) is professor of New Testament interpretation at Westminster Theological Seminary in Philadelphia, Pennsylvania, where he has taught for nearly four decades. In addition to earning six academic degrees, he is the author of numerous books and articles on biblical interpretation, language, and science.
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God and Mathematics
Let us begin with numbers. We can consider a particular case: 2 + 2 = 4. That is true. It was true yesterday. And it always will be true. It is true everywhere in the universe. We do not have to travel out to distant galaxies to check it. Why not? We just know. Why do we have this conviction? Is it not strange? What is it about 2 + 2 = 4 that results in this conviction about its universal truth?
All Times and All Places
2 + 2 = 4 is true at all times and at all places. We have classic terms to describe this situation: the truth is omnipresent (present at all places) and eternal (there at all times). The truth 2 + 2 = 4 has these two characteristics or attributes that are classically attributed to God. So is God in our picture, already at this point? We will see.
Technically, God's eternity is usually conceived of as being "above" or "beyond" time. But words like "above" and "beyond" are metaphorical and point to mysteries. There is, in fact, an analogous mystery with respect to 2 + 2 = 4. If 2 + 2 = 4 is universally true, is it not in some sense "beyond" the particularities of any one place or time?
Moreover, the Bible indicates that God is not only "above" time in the sense of not being subject to the limitations of finite creaturely experience of time, but is "in" time in the sense of acting in time and interacting with his creatures. Similarly, 2 + 2 = 4 is "above" time in its universality, but "in" time through its applicability to each particular situation. Two apples plus two more apples is four apples.
Divine Attributes of Arithmetical Truth
The attributes of omnipresence and eternity are only the beginning. On close examination, other divine attributes seem to belong to arithmetical truths.
Consider. If 2 + 2 = 4 holds for all times, we are presupposing that it is the same truth through all times. The truth does not change with time. It is immutable.
Next, 2 + 2 = 4 is at bottom ideational in character. We do not literally see the truth 2 + 2 = 4, but only particular instances to which it applies: two apples plus two apples. The truth that 2 + 2 = 4 is essentially immaterial and invisible, but is known through manifestations. Likewise, God is essentially immaterial and invisible, but is known through his acts in the world.
Next, we have already observed that 2 + 2 = 4 is true. Truthfulness is also an attribute of God.
The Power of Arithmetical Truth
Next, consider the attribute of power. Mathematicians make their formulations to describe properties of numbers. The properties are there before the mathematicians make their formulations. The human mathematical formulation follows the facts and is dependent on them. An arithmetical truth or regularity must hold for a whole series of cases. The mathematician cannot force the issue by inventing a new property, say that 2 + 2 = 5, and then forcing the universe to conform to his formulation. (Of course, the written symbols such as 4 and 5 that denote the numbers could have been chosen differently. And a mathematician can define a new abstract object to have properties that he chooses. But we do not "choose" the properties of natural numbers.) Natural numbers conform to arithmetical properties and laws that are already there, laws that are discovered rather than invented. The laws must already be there. 2 + 2 = 4 must actually hold. It must "have teeth." If it is truly universal, it is not violated. Two apples and two apples always make four apples. No event escapes the "hold" or dominion of arithmetical laws. The power of these laws is absolute, in fact, infinite. In classical language, the law is omnipotent ("all powerful").
2 + 2 = 4 is both transcendent and immanent. It transcends the creatures of the world by exercising power over them, conforming them to its dictates. It is immanent in that it touches and holds in its dominion even the smallest bits of this world. 2 + 2 = 4 transcends the galactic clusters and is immanently present in the behavior of the electrons surrounding a beryllium nucleus. Transcendence and immanence are characteristics of God.
The Personal Character of Law
Many agnostics and atheists by this time will be looking for a way of escape. It seems that the key concept of arithmetical truth is beginning to look suspiciously like the biblical idea of God. The most obvious escape, and the one that has rescued many from spiritual discomfort, is to deny that arithmetical truth is personal. It is just there as an impersonal something.
Throughout the ages people have tried such routes. They have constructed idols, substitutes for God. In ancient times, the idols often had the form of statues representing a god — Poseidon, the god of the sea, or Mars, the god of war. Nowadays in the Western world we are more sophisticated. Idols now take the form of mental constructions of a god or a God-substitute. Money and pleasure can become idols. So can "humanity" or "nature" when it receives a person's ultimate allegiance. "Scientific law," when it is viewed as impersonal, becomes another God-substitute. Arithmetical truth, as a particular kind of scientific law, is also viewed as impersonal. In both ancient times and today, idols conform to the imagination of the one who makes them. Idols have enough similarities to the true God to be plausible, but differ so as to allow us comfort and the satisfaction of manipulating the substitutes that we construct.
In fact, however, a close look at 2 + 2 = 4 shows that this escape route is not really plausible. Law implies a law-giver. Someone must think the law and enforce it, if it is to be effective. But if some people resist this direct move to personality, we may move more indirectly.
Scientists and mathematicians in practice believe passionately in the rationality of scientific laws and arithmetical laws. We are not dealing with something totally irrational, unaccountable, and unanalyzable, but with lawfulness that in some sense is accessible to human understanding. Rationality is a sine qua non for scientific law. But, as we know, rationality belongs to persons, not to rocks, trees, and subpersonal creatures. If the law is rational, as mathematicians assume it is, then it is also personal.
Scientists and mathematicians also assume that laws can be articulated, expressed, communicated, and understood through human language. Mathematical work includes not only rational thought but symbolic communication. Now, the original law, the law 2 + 2 = 4 that is "out there," is not known to be written or uttered in a human language. But it must be expressible in language in our secondary description. It must be translatable into not only one but many human languages. We may explain the meaning of the symbols and the significance and application of 2 + 2 = 4 through clauses, phrases, explanatory paragraphs, and contextual explanations in human language.
Arithmetical laws are clearly like human utterances in their ability to be grammatically articulated, paraphrased, translated, and illustrated. Law is utterance-like, language-like. And the complexity of utterances that we find among mathematicians, as well as among human beings in general, is not duplicated in the animal world. Language is one of the defining characteristics that separates man from animals. Language, like rationality, belongs to persons. It follows that arithmetical laws are in essence personal.
The Incomprehensibility of Law
In addition, law is both knowable and incomprehensible in the theological sense. That is, we know arithmetical truths, but in the midst of this knowledge there remain unfathomed depths and unanswered questions about the very areas where we know the most. Why does 2 + 2 = 4 hold everywhere?
The knowability of laws is closely related to their rationality and their immanence, displayed in the accessibility of effects. We experience incomprehensibility in the fact that the increase of mathematical understanding only leads to ever deeper questions: "How can this be?" and "Why this law rather than many other ways that the human mind can imagine?" The profundity and mystery in mathematical discoveries can only produce awe — yes, worship — if we have not blunted our perception with hubris (Isa. 6:9–10).
Are We Divinizing Nature?
But now we must consider an objection. By claiming that arithmetical laws have divine attributes, are we divinizing nature? That is, are we taking something out of the created world and falsely claiming that it is divine? Are not arithmetical laws a part of the created world? Should we not classify them as creature rather than Creator?
I suspect that the specificity of arithmetical laws, their obvious reference to the created world, has become the occasion for many of us to infer that these laws are a part of the created world. But such an inference is clearly invalid. The speech describing a butterfly is not itself a butterfly or a part of a butterfly. Speech referring to the created world is not necessarily an ontological part of the world to which it refers.
The Bible indicates that God rules the world through his speech. He speaks, and it is done:
By the word of the Lord the heavens were made,
and by the breath of his mouth all their host. (Ps. 33:6)
For he spoke, and it came to be;
he commanded, and it stood firm. (Ps. 33:9)
And God said, "Let there be light," and there was light. (Gen. 1:3)
God also continually sustains the world by his word: "he upholds the universe by the word of his power" (Heb. 1:3). God's word has divine wisdom, power, truth, and holiness. It has divine attributes, because it expresses God's own character. God expresses rather than undermines his own deity when he speaks words that address the created world.
We may then conclude that the same principle applies in particular to numerical truths about the world. God governs everything, including numerical truth. His word specifies what is true. The apples in a group of four apples are created things. What God says about them is divine. In other words, his word specifies that 2 + 2 = 4.
The key idea that the law for the world is divine is even older than the rise of Christianity. Even before the coming of Christ people noticed profound regularity in the government of the world and wrestled with the meaning of this regularity. Both the Greeks (especially the Stoics) and the Jews (especially Philo) developed speculations about the logos, the divine "word" or "reason" behind what is observed. In addition the Jews had the Old Testament, which reveals the role of the word of God in creation and providence. Against this background John 1:1 proclaims, "In the beginning was the Word, and the Word was with God, and the Word was God." John responds to the speculations of his time with a striking revelation: that the Word (logos) that created and sustains the universe is not only a divine person "with God," but the very One who became incarnate: "the Word became flesh" (1:14).
God said, "Let there be light" (Gen. 1:3). He referred to light as a part of the created world. But precisely in this reference, his word has divine power to bring creation into being. The effect in creation took place at a particular time. But the plan for creation, as exhibited in God's word, is eternal. Likewise, God's speech to us in the Bible refers to various parts of the created world, but the speech (in distinction to the things to which it refers) is divine in power, authority, majesty, righteousness, eternity, and truth.
The analogy with the incarnation should give us our clue. The second person of the Trinity, the eternal Word of God, became man in the incarnation but did not therefore cease to be God. Likewise, when God speaks and says what is to be the case in this world, his words do not cease to have the divine power and unchangeability that belong to him. Rather, they remain divine and in addition have the power to specify the situation with respect to creaturely affairs. God's word remains divine when it becomes law, a specific directive with respect to this created world. In particular, 2 + 2 = 4 remains a divinely ordained truth when it becomes a specific directive with respect to four apples on the kitchen table.
The Goodness of Law
Is 2 + 2 = 4 morally good? An arithmetical truth is not directly a moral precept. But indirectly it requires us to conform to it. We have an ethical constraint to believe the truth, once we have become convinced of it. We can also say that in a wider sense it is "good" for the universe and for us that 2 + 2 = 4. It never lies. We would not be able to live, nor would the universe hold together, without the consistency of arithmetical truths.
The Beauty of Law
Is 2 + 2 = 4 beautiful? I think so. But not everyone is good at seeing the beauty in mathematics. I think there is beauty in the simplicity of 2 + 2 = 4. It is in harmony with the world. It is beautiful that its truth is displayed repeatedly, in four apples, four pencils, and four chairs. It is beautiful in its harmony with other arithmetical truths, with which it can be combined.
The beauty in arithmetic shows the beauty of God himself. Though beauty has not been a favorite topic in classical expositions of the doctrine of God, the Bible shows us a God who is profoundly beautiful. He manifests himself in beauty in the design of the tabernacle, the poetry of the Psalms, and the elegance of Christ's parables, as well as the moral beauty of the life of Christ.
The beauty of God himself is reflected in what he has made. We are accustomed to seeing beauty in particular objects within creation, such as a butterfly or a lofty mountain or a flower-covered meadow. But beauty is also displayed in the simple, elegant form of some of the most basic physical laws, like Newton's law for force, F = ma, or Einstein's formula relating mass and energy, E = mc. The same goes for the simple beauties in arithmetic and the more profound beauties that mathematicians discover in advanced mathematics.
The Rectitude of 2 + 2 = 4
Another attribute of God is righteousness. God's righteousness is displayed preeminently in the moral law and in the moral rectitude of his judgments, that is, his rewards and punishments based on moral law. Does God's rectitude appear in mathematics? The traces are somewhat less obvious, but still present. People could try to disobey arithmetical laws, for example, when they are trying to balance their checkbook. If they do, they may suffer for it. There is a kind of built-in righteousness in the way in which arithmetical laws lead to consequences.
In addition, the rectitude of God is closely related to the fitness of his acts. It fits the character of who God is that we should worship him alone (Ex. 20:3). It fits the character of human beings made in the image of God that they should imitate God by keeping the Sabbath (vv. 8–11). Human actions fitly correspond to the actions of God.
In addition, punishments must be fitting. Death is the fitting or matching penalty for murder (Gen. 9:6). "As you have done, it shall be done to you; your deeds shall return on your own head" (Obadiah 15). The punishment fits the crime. There is a symmetrical match between the nature of the crime and the punishment that fits it. In the arena of arithmetical law we do not deal with crimes and punishments. But rectitude expresses itself in symmetries, in orderliness, in a "fittingness" to the character of arithmetic. This "fitness" is perhaps closely related to beauty. God's attributes are involved in one another and imply one another, so beauty and righteousness are closely related. It is the same with the area of arithmetical law. Arithmetical laws are both beautiful and "fitting," demonstrating rectitude.
Law as Trinitarian
Does 2 + 2 = 4 specifically reflect the Trinitarian character of God? Philosophers have sometimes maintained that one can infer the existence of God, but not the Trinitarian character of God, on the basis of the world around us. Romans 1:18–21 indicates that unbelievers know God, but how much do they know? I am not addressing this difficult question, but rather reflecting on what we can discern about the world once we have absorbed biblical teaching about God.
God has specified by his word that 2 + 2 = 4. Thus, in its origin the truth that 2 + 2 = 4 is a form of the word of God. Hence, it reflects the Trinitarian statement in John 1:1, which identifies the second person of the Trinity as the eternal Word. In John, God the Father is the speaker of the Word, and God the Son is the Word who is spoken. John 1 does not explicitly mention the Holy Spirit. But earlier Scriptures associate the Spirit with the "breath" of God that carries the word out.(Continues…)
Excerpted from "Redeeming Mathematics"
Copyright © 2015 Vern S. Poythress.
Excerpted by permission of Good News Publishers.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Diagrams and Illustrations 9
Introduction: Why God? 11
Part I Basic Questions
1 God and Mathematics 15
2 The One and the Many 29
3 Naturalism 35
4 The Nature of Numbers 42
Part II Our Knowledge of Mathematics
5 Human Capabilities 57
6 Necessity and Contingency 63
Part III Simple Mathematical Structures
7 Addition 73
8 The Idea of What Is Next 81
9 Deriving Arithmetic from Succession 87
10 Multiplication 92
11 Symmetries 96
12 Sets 100
Part IV Other Kinds of Numbers
13 Division and Fractions 109
14 Subtraction and Negative Numbers 116
15 Irrational Numbers 121
16 Imaginary Numbers 126
17 Infinity 129
Part V Geometry and Higher Mathematics
18 Space and Geometry 135
19 Higher Mathematics 142
Appendix A Secular Theories about the Foundations of Mathematics 151
Appendix B Christian Modifications of Philosophies of Mathematics 161
Appendix C Deriving Arithmetic 168
Appendix D Mathematical Induction 173
Appendix E Elementary Set Theory 178
General Index 191
Scripture Index 199
What People are Saying About This
“Redeeming Mathematics is a valuable addition the growing literature on the relationship between mathematics and Christian belief. Poythress’s treatment of three distinct dimensions of mathematicsas transcendent abstract truths, as part of the physical world, and as comprehensible to human beingsis a unique and helpful addition to the conversation on this relationship. The book is accessible to non-specialists, but even those who are well-versed in these matters will find much to interest and challenge them.”
James Bradley, Professor Emeritus of Mathematics, Calvin College; author, Mathematics Through the Eyes of Faith; Editor, Journal of the Association of Christians in the Mathematical Sciences