by Vern S. Poythress
[Published in The Journal of Christian Reconstruction 1/1 (1974) 128 - 140. Used with permission.]
What does the Bible say about mathematics? The superficial answer, given by too many Christians, is that the Bible and mathematics are unrelated. Mathematics and religion are two separate spheres; neither has need of or connection with the other. However, such an answer has not really reckoned with the biblical teaching on creation, fall, and redemption.
According to Genesis 1, God created everything, and ordained man to a special place in creation. Now God sustains everything that he has created: “In him we live and move and have our being” (Acts 17:28). Does this mean that he has also created mathematics? We would be inclined to answer “yes,” except that mathematics is not a “thing” like a man, an animal, or a plant. It is therefore more correct to say that God says what shall be true for his creation (Eph. 1:11; Lamen. 3:37-38), and mathematical truth is part of this truth. Moreover, God teaches men whatever they know of this truth (Ps. 94:10-11).
But the teaching of Genesis carefully distinguishes man the creature from God the Creator. This implies that God is the originator of mathematical truth, while man is only a recipient. God knows all mathematical truth and never errs, while man knows in part and does err.
Next, we learn from Colossians 1:16 together with Genesis that creation is a Trinitarian activity. The Father, the Son, and the Holy Spirit each has his own role with respect to it. Creation is by the Father (I Cor.8:6), through and in the Son (Col. 1:16; John 1:3), with the movement of the Holy Spirit within it (Gen. 1:2; Ps. 104:30). What does it mean that creation is in the Son, in Christ? We get at least a partial answer to this in Colossians 1:17: “He is before all things, and in him all things hold together.” He is before all things, both chronologically and in pre-eminence and honor. This is the significance of “first-born” (vs. 15). He infuses life into creation as he does into the church (vs. 18).
Second, “in him all things hold together.” The word sunestēken, “hold together,” can be used transitively to mean “set or band together, establish.” Intransitively it means “to cohere.” The universe is banded together
or coheres in Christ. Hebrews 1:3 says also, “He reflects the glory of God and bears the very stamp of his nature, upholding the universe by his word of power.” This idea can be applied also to mathematical truth concerning creation. The relations of mathematics both within itself and to other truths, we must say, are due to the sustenance of Christ. Mathematics “coheres” in Christ.
Finally, creation has a purpose: it is “for him” (Col. 1:16). In Eph.1:10, Paul announces that the plan of God is to unite all things in Christ. Mathematical truth also, therefore, will serve his glory.
If these things are so, it cannot be surprising that all creation, and hence mathematical truth concerning creation, reveals God. Theologians call it general revelation. Romans 1:20 says, “His invisible attributes, his eternal power and deity, have been visible since the creation of the world, being discerned in the things he has made.” But the reaction to this true revelation on man’s part is universally negative (Rom. 1:18). The revelation only serves to condemn men for their foolishness (vs. 21) rather than lead them to God. This inverted response is, of course, due to the fall. The creation has been put out of joint (Rom. 8:20), by the abdication and rebellion of its proper master, man.
The Reformational expression “total depravity” of man is used to say that no area of man’s nature and no area of his involvement with the world is left untouched or undefiled by man’s rebellion (Rom. 3). For an exposition of this, one could not do better than turn to the second book of Calvin’s Institutes. “The whole man, from the crown of the head to the sole of the foot, is so deluged, as it were, that no part remains exempt from sin, and therefore, everything which proceeds from him is imputed as sin.”1 Therefore, we now expect mathematical truth also to be misunderstood and perverted by fallen man.
How does the perversion take place? We may start with Romans 1:18-21. Mathematical truth clearly exhibits God’s eternal power and deity, in that it testifies to its origin in God. Men receive the gift of mathematical knowledge from God, but refuse to acknowledge the giver (Rom. 1:21). They are blind to the revelation of the glory of God (II Cor. 4:3; Eph. 4:17). General revelation is shut out by man’s sinful blindness.
This is the reason for special revelation, the revelation of Scripture. Accordingly, Calvin likens the Scripture to a kind of spectacles through which proper vision of the creation can be restored.2 God’s covenant with his people is the means of enlightening them through Scripture. Because of this enlightenment, the psalmist can see God in the manifold works of nature (Ps. 19; 104; 147; 148, etc.).
Psalm 8 is a good illustration of the restoration of man. “Yet thou hast made him [man] a little less than God, and dost crown him with glory and honor. Thou hast given him dominion. . . .” The spectacles here have restored the proper picture of man and creation, the picture of Genesis 1. God has given man dominion. However, in Hebrews 2 we find Psalm 8 applied to Christ. Why is this? Hebrews 2 itself points out that man has forfeited proper relation to creation at the fall; hence Psalm 8 is no longer strictly true of man in general but only of Christ. The picture of Psalm 8 has not only to be seen, but to be retrieved or restored. Christ by his perfect obedience and death retrieved what man could not. Only in Christ is a man’s relation restored, and only in Christ does he regain the dominion of Psalm 8 and Genesis 1.
It is with such attitudes that we should approach the concrete subject matter of mathematics and the concrete problems raised by philosophers of mathematics. In summary, we have these theses:
– 1. Mathematical truth is part of what God says in creating and sustaining the world (Ps. 33:6).
– 2. Mathematics “holds together” in Christ.
– 3. We therefore expect mathematics to reveal his glory.
– 4. However, after the fall we “see” the revelation only with the eyes of faith, enlightened by the Spirit speaking in Scripture.
Questions: (1) What is to be the relation between Scripture and experiment in obtaining knowledge about the world? (2) Why does Scripture say much about an area like ethics or anthropology, but leave us, we feel, more “on our own” in mathematics? Can we speak of some areas being less centrally related to the needs of salvation?
2. 2+2 = 4
In this section we deal with the philosophical context and presuppositions of mathematics.
Needless to say, those philosophers who are not Christians would take issue with the interpretation of mathematics in section 1. But are the disagreements important? Can’t a person do mathematics and come out with the right conclusion without any philosophy? Isn’t the substance of mathematics philosophically and religiously neutral?
Yes, non-Christians can do mathematics, but only because God enables them to do so. Only because the Christian God exists and sustains them and teaches them are they able to do mathematics, and to act as if God didn’t exist.
We can see this by considering the statement “2+2 == 4.” Everyone agrees to this, right? Isn’t this real and true knowledge, independent of a person’s religious presuppositions? Not quite. A radical monist might dispute the truth of 2+2 == 4. Moreover, we have just made statements
that depend weightily on a world view, on presuppositions about this world. 2+2 == 4 is true. What is truth? What is the “everyone” that “agrees” to this truth? (What is man?)
Furthermore, in order for someone to be a mathematician or even to write down or to think “2+2 == 4,” he must have a confidence in his memory and in the reality of time. And there are more questions. What, after all, does “2+2 == 4″ mean? A man with another language or another set of symbols would not understand it.
Well, you may say, put two apples in an empty bag, add two more, and then count that there are four. At the very least, this involves someone remembering that he has already added the first two apples, and knowing how to count. It involves believing that time is not, e.g., circular. (Otherwise, adding the last two apples might be the very same event as adding the first two apples.) It also involves assumptions about the identifiability, stability, and constancy of apples in time. And so on. Furthermore, if we are involved in any kind of mathematical or even physical reasoning, we have to cope with an explanation or interpretation of logic.
Now, all this may seem quite obvious. However, the point is that even the most elementary mathematical content involves an intrinsic commitment to a certain vaguely defined philosophy or world view. Every mathematician must have it to get off the ground. This vague world view assumes, among other things, that the world “makes sense” or that it “hangs together.” Reality is not a complete flux, and man’s mind is not a complete flux. What happens is not completely random and without sense. Man’s mind, logic, and external reality cohere.
Note that we have run across the word coherence before. We have seen that the Scriptures attribute the coherence of the world to the creative and sustaining activity of God, who has created man with a capacity for understanding the world, and who governs the world in an orderly way (for example, Ps. 104:20-23 speaks of the orderliness of day and night). Specifically, all things cohere in Christ (Col. 1:17). That is why it is possible to count, to remember, to communicate by language, to know truth. God bears witness to himself (Rom. 1:18) even in 2+2 = 4.
We must say that mathematicians receive this witness, since they act on the basis of belief in coherence. They act as if 2+2 == 4 has real meaning, meaning that it can receive only as a truth related in manifold ways to the creation of God and to the other truths that he ordains. But this truth is suppressed, because it is denied that God is the Author of 2+2 = 4. Today, one of the methods of suppression is secularized philosophy of mathematics, at which we will look next.
Questions: (1) The above analysis of 2+2 = 4 makes it plain that 2+2 = 4 is not something that we know a priori (prior to all experience)
in the strictest sense. Is there some reason why philosophers have always wanted to assign mathematical and logical truths to the category of a priori truths? Is there indeed something special about them? (2) Why cannot a truth like 2+2 == 4 be used as “common ground” with an unbeliever in an apologetic argument?
3. The Relation of Mathematics to Sciences
Now we would like to trace how true and false philosophies work out in views of the relation of mathematics to sciences. How is mathematics related on the one hand to logic, and on the other hand to physics and to experiment? In modern times this question has been further elaborated by the various schools of mathematical philosophy, so that we can ask, “How is mathematics related on the one hand to physics and on the other hand to (1) logic in the sense of a standard for our reasoning, (2) language, both in thought and written, and (3) mental mathematical constructions, and the larger area of imagination?”
The Christian says that the bond, or coherence, of these areas is the creative and sustaining activity of God in Christ. God has created the world a plurality, as we can see from Genesis 1. Nevertheless, it is a plurality that is structured, related. And hence truths of various kinds are related.
Thus we can use physical pictures in mathematics (e.g., geometry), we can apply mathematics to the world of motion of objects (physics), we can apply logic to mathematics (in proofs), and so on. We are assured that things await to be discovered about the relations of these fields because of Christ the Order of creation. For the same reason we are not surprised to find close relations between different areas of mathematics: for example, set theory, group theory, complex variable theory, and topology.
This is pluralism. Each of these areas has its proper meaning and significance; none is reduced to another. But it is pluralism with a unity due to the plan of God. Non-Christian philosophies of mathematics, as one might expect, do not have the same resources. Hence, generally speaking, they adopt either an ultimate plurality or an ultimate unity, without ever getting the two together. They cannot, because they refuse to acknowledge the true God and his act of creation.
First, let us take a brief look at non-Christian pluralism. This type of philosophy acknowledges the distinctiveness of mathematics in comparison with physics or logic. Kant is a good representative. For him, there is such a thing as mathematical intuition. It is not identified with pure logic (which he calls “analytic a priori“), or with the faculty of concept formation (“understanding”), or with language, or with physics.
How are these things related? What constitutes the unity? Non-Christian pluralism is problematic, because if plurality is foundational,
there remains no way to introduce relations afterward. In effect, there is no bond between (say) mathematics and physics. One would expect, under this view, that neither mathematics nor logic would have any application to physics. Kant’s answer is that the mathematical forms of space and time are universal forms for all experience. Consequently, mathematics must hold for that experience. But this amounts to little better than saying that man’s mind imposes mathematics on a world that, in itself, may very well be nonmathematical. Hence the mathematical and the non-mathematical are not really brought together.
Now let us look at a non-Christian philosophy of mathematics that emphasizes unity. Unity can be obtained (or so it is supposed) by claiming that mathematics is “merely” logic, or formal language, or physics. Mathematics is seen as unified with, rather than distinct from, some other area.
As an example of this unifying direction, or monism, let us take logicism, which includes mathematics under logic. The chief representatives of this school are Bertrand Russell, Carl Hempel, and Rudolf Carnap (though Carnap shows tendencies toward “formalism,” which includes mathematics under formal languages). Hempel says, “Mathematics is a branch of logic. It can be derived from logic in the following sense:
a. All the concepts of mathematics, i.e., of arithmetic, algebra, and analysis, can be defined in terms of four concepts of pure logic.
b. All the theorems of mathematics can be deduced from those definitions by means of the principles of logic.3
Thus for Hempel all is unified under the heading “logic.”
What Hempel does not say here is that this “reduction” of mathematics to logic has been achieved only by expanding the term “logic” to include axioms and concepts that were formerly not called “logical.”4 Hence one may still ask why the “old” logic (e.g., prepositional logic) and mathematics (like 2+2 = 4) harmonize. As long as one attempts to explain everything in terms of one principle, any remaining diversity is a thorn in the side.
Thus non-Christian philosophies do not really succeed in explaining the unity and diversity in the relation of mathematical truth to other truth. By contrast, the Christian’s ultimate reference-point is the God of truth who is both one God and Lord (unity of truth) and three persons (implying, among other things, diversity of truth).
Questions: (1) How does one criticize the other “monistic” phi-
losophies of mathematics-formalism (reducing mathematics to formal languages) and empiricism (reducing mathematics to physics or to “experience”)? (2) What kinds of effect have these non-Christian philosophies had on the more practical, day-to-day work of mathematicians? Can we detect some influence on mathematical textbooks? (3) What useful insights can these philosophies give to the Christian who knows the truth?
4. Views of Knowledge of Mathematics
Non-Christian philosophers have tried to give an account of how we know that 2+2 = 4. Do we know it simply by pure reflection (a priori, independent of experience)? or by observing the world (a posteriori, derived from experience)? Do we gain the knowledge by remembering what we once knew but have forgotten (Plato)? by logical argument (Russell?)? Or do we gain it by repeated experience of (say) two apples and two apples (John S. Mill)? Or some combination of these? Or is “2+2 = 4″ not real “knowledge” at all, but simply a linguistic convention about how we use “2″ and “4″ (A. J. Ayer)?5
Non-Christians have tried all of these answers. But, as we shall see, without the biblical doctrine of creation they cannot adequately account for mathematical knowledge.
Let us first look at the “a priori” answer. This answer says that we know 2+2 = 4 independent of experience of the world around us. In a sense, there are as many versions of this answer as there are philosophers, but I will oversimplify and consider them as all one answer.
According to this view, then, “2+2 = 4″ is some kind of universal, eternal truth. But why, in that case, should two apples plus two apples usually, in experience, make four apples? The apriorist separates 2+2 = 4 radically from all experience and all contingency. But his problem is then to explain what it can mean for 2+2 = 4 to apply to a world of apples, baseballs, houses, and the like. If the truth 2+2 = 4 is made too diversefrom the world of “contingency,” then it can never be unified with the world. Once again, the problem of unity and diversity that we considered in section 3 appears.
Let us continue to press the questions. We grant that 2+2 = 4. But why should an admittedly contingent world offer us repeated instances of this truth, many more instances than we could expect by chance? If
the external world is purely a chance matter, if anything can happen in the broadest possible sense, if the sun may not rise tomorrow, if, as a matter of fact, there may be no sun, or only a sputnik, when tomorrow comes, if there may be no tomorrow, etc., can there be any assured statement at all about apples? Why, for instance, don’t apples disappear and appear randomly while we are counting them?
If, on the other hand, the external world has some degree of regularity mixed in with its chance elements, why expect that regularity to coincide, in even the remotest way, with the a priori mathematical expectations of human minds? Such questions can be multiplied without limit. Once one has made the Cartesian separation of mind and matter, of a priori and a posteriori, one can never get them back together again.
A strict a priori view is also open to more practical objections. If mathematics is known a priori, why have paradoxes sometimes arisen within mathematics? In the past actual contradictions have sometimes arisen from starting assumptions that appeared to be a priori true. The contradictions obtainable from conditionally convergent sums and integrals, and Russell’s paradox concerning the set of all sets that do not contain themselves, are cases of this kind. Mathematical theory has also produced various counter-intuitive results, like the space-filling curves of Peano and everywhere-continuous nowhere-differentiable functions.
The paradoxes seem less threatening today, partly because mathematicians adopt a more conventionalist attitude toward them. Whatever axioms may be “convenient” are adopted, and the results are simply conventions of language. (See below on this.) Partly the paradoxes have been disposed of by modifying the axioms (to avoid contradictions) or modifying one’s intuitions (to square with the theories). Nevertheless, the history of the paradoxes illustrates that supposedly a priori mathematical convictions are not always reliable.
It is understandable that these difficulties on the a priori side have led people to cast about for an a posteriori solution. In this case one emphasizes the supposed inductive character of mathematical knowledge. Mathematical knowledge is viewed as a generalization from experience in the world. One comes to believe that 2+2 = 4 from repeated experience of two objects plus two objects making four objects.
So far so good. But no one has repeated experience of 2,123,955 objects plus 644,101 objects making 2,768,056 objects. So why do people believe that 2,123,955+644,101=2,768,056? The consistent reply would be, “People generalize on the basis of their previous experience with small numbers.” Unfortunately, in the word “generalize” are concealed all the problems that we began with. We may ask, “Why does a person “generalize” in one way rather than another?” Why, after observing that 3+2 = 5, 4+2 = 6, . . . , 12+2 = 14, does a person conclude (gen-
eralize?) that 13+2 = 15 rather than 13+2 = 16 or even 13+2 = 13? Doubtless this seems natural to any educated person nowadays, partly because he was taught such things. But how did he come to know it at the beginning?
In terms of a consistently a posteriori viewpoint, the answer can only be that a person generalizes this way because of previous experience with other generalizations. He has had experience before with detecting patterns. In other words, the step to 13+2 = 15 is based on generalization from previous experience, previous experience of other generalizations. But why has he generalized in this particular way from those other generalizations? Because he has generalized from previous experience of generalizing from previous generalizations. Etc. Apparently, one can escape this regress only by saying at this point, “Because that’s the way the human mind operates.” And then one is confronted with an a priori knowledge, or at least a priori heuristics, that is, ways of arriving at knowledge.
The a posteriori solution is also open to more practical, prosaic objections. What about the constantly growing quantity of abstract, non-visualizable mathematical entities? To claim that transfinite numbers, topological spaces, and abstract algebras are somehow impressed on us from sense experience takes some stretch of the imagination.
A third attempted solution to the problem of mathematical knowledge deserves mention, if only because of its wide-spread popularity among mathematicians themselves. This is the view that mathematics is, in some sense, a mere convention of our language, and thus not “knowledge” at all. 2+2 = 4 because we have agreed in our language to use words “two” and “four” in just that way.6 Or, to put it another way, in saying “2+2 = 4” we are just saying “A is A” in a roundabout way (A. J. Ayer).7 Or, “2+2 = 4 because it follows from our (conventionally determined) postulates” (formalists).
All these “conventionalist” answers are really so many variations of the a priori solution, inasmuch as one can still ask the same unanswerable questions about why mathematics should prove so useful in dealing with the external world. If it is pure convention, why should this be? Or if one says that the conventions are chosen because they are useful, one moves into the a posteriori camp, where he is confronted with the same unanswerable questions about the role of generalization.8
The fact that the conventionalist answer can be used either in an a priori or an a posteriori direction points up another factor: that the conventionalist “answer” may not really be an answer at all, but simply a shifting of the question from the area of mathematics to the area of language. The same a priori-a posteriori problems reappear when we ask why mathematical language functions adequately.
Because of the above difficulties, non-Christian philosophy of mathematics is condemned to oscillate, much as we have done in our argument, between the poles of a priori explanation and a posteriori explanation of mathematical knowledge. Why? It will not acknowledge the true God, wise Creator of both the human mind with its mathematical intuition and insight and the external world with its mathematical properties. In the next section we shall see how the biblical view furnishes us with a satisfying answer to the problem of “knowing” that 2+2 = 4.
Questions: (1) In what ways does one’s explanation of how he knows mathematical truth affect what he knows? (2) Can a study of how children learn mathematics help us to obtain insight into the a priori-a posterioriproblems?
5. The Biblical View of Knowledge
To understand how men come to know 2+2 = 4, we must appreciate the biblical view of man. We have already seen in section 1 that man was appointed to a special role in the created world. More specifically, Scripture says that man is the image of God (Gen. 1:26; cf. Gen. 2:7; I Cor. 11:7). As such, his work is to imitate receptively, on a finite level, the works (naming, Gen. 2:19; 1:4; governing, Gen. l:28;Ps. 22:28; improving. Gen. 2:15; 1:31), and rest (Gen. 2:2; Ex. 20:11) of God. Man is created with the potential, then, of understanding God and his works (though not exhaustively). He has the capability of understanding the mathematical aspects of God’s truth and God’s rule, since he himself is a ruler like God. Thus he can generalize with confidence from 2+2 = 4, etc., to 2,123,955 + 644,101 = 2,768,056.
Here we have the first step in a Christian answer to the problem of mathematical knowledge. The a priori capabilities of man’s created nature, and his potential for mathematical insight, really correspond to the a posteriori of what is “out there.” This is because man is in the image of the One who ordained what is “out there.” At the same time, man’s mathematical reasoning is not always right, and his intuitive expectations are not always fulfilled, because man is the image of God the infinite One.
Next, we should ask how a man comes to know mathematical truths that he hasn’t known before. This, one might say, is part of the a posteriori
side of mathematics. The Bible answers that God reveals to men whatever they know: “He who teaches men knowledge, the Lord, knows the thoughts of man, that they are but a breath. Blessed is the man whom thou dost chasten, 0 Lord, and whom thou dost teach out of thy law” (Ps. 94:1 Ob-12). “But it is the spirit in a man, the breath of the Almighty, that makes him understand. It is not the old that are wise, nor the aged that understand what is right” (Job 32:8-9; cf. Prov. 8). The Lord’s instruction sometimes comes, of course, by way of “natural” revelation (Ps. 19; Isa. 40:26; 51:6; Prov. 30:24-28). Thus we can do justice to the real novelty that is sometimes found in a new mathematical theorem.
6. Public Science
The existence of a science of mathematics depends upon the ability of men to communicate with one another, and on the availability of a medium of communication. Both of these factors go back to creation. Men have one racial origin (Acts 17:26). They share a common nature (the image of God, Gen. 1:26-27; 5:1-3), and they have been given the gift of language as part of their equipment to fulfill the cultural mandate of Genesis1:28-30 (see Gen. 2:19-23). This furnishes us adequate grounds for believing today that others understand us, and that our language is adequate to the cultural task that God has given us.
This also helps to explain why mathematicians can have much agreement in spite of religious differences. Men cannot cease to be in the image of God, even if they rebel against him (Gen. 3:5, 22). They either imitate God in obedience or “imitate” him by trying to become their own lord. Neither can they escape the impulse to fulfill, in some fashion, the cultural mandate of Genesis 1:28-30. Thus, in spite of themselves, they acknowledge God in some fashion (cf. Rom. 1:18-22; James 2:19).
Hence non-Christians, in the image of God, can and do make significant contributions to mathematics. They can know many mathematical truths. As we have seen in section 1, in knowing mathematical truth they know God (though not exhaustively, and at places mistakenly). Nevertheless, their “knowledge” is not more beneficial to them than the knowledge that demons have (James 2:19). Hence, Christian and atheist, indeed all kinds of religious people, share mathematical truths, but for all non-Christians it is only in spite of their religion. It is because Christianity is true, because God is who he is, because man is the image of God, that the non-Christian knows anything.9 The supposed “common ground” of shared mathematical truth proves the very opposite of what non-Christians suppose it to prove.
Questions: (1) What limitations does the necessity of communication place on human mathematics? (2) How should the involvement of communication with mathematics be appreciated in the classroom?
7. Ethics of Mathematics
Finally, we give a brief sketch of how biblical ethics applies to work in mathematics. A Christian recognizes that he lives under the Lordship of God, in the light of God’s present commands and God’s coming judgment. He sees that, as in the case of Abraham and the nation of Israel, his whole life—marital, political, economic, social, spatial—ought to be structured and determined by his covenantal relation to God. All of life should be a response of service to God (I Cor. 10:31).
Thus, work in mathematics can have relevance to the Christian only insofar as it is motivated by the love of God, commanded by the law of God, and directed to the glory of God and the consummation of his kingdom. These are the motive, the standard, and the goal of work in mathematics.10
To be more specific, we must take into account the fact that men have a diversity of callings (I Cor. 7:17-24). Not all men are called to be specialists in mathematics. For the one who does so specialize, using the gifts that God has given him (Luke 19:11-26), how does Christian ethics come to bear? How should the biblical motive, standard, and goal affect him?
(a) The mathematician should be motivated by the love of God to understand the mathematical truths that God has ordained for this world (and so understand something of God himself, section 1); love of neighbor should also motivate him to apply mathematics to physics, economics, etc.
(b) The mathematician should find his standard in the command of God, the program that God has given man to fulfill (Gen. 1:28). Part of this program is that man should understand God’s works (Gen. 2:18-23).
(c) The mathematician should work for the glory of God. He should praise God for the beauty and usefulness that he finds in mathematics, for the incomprehensible nature of God that it displays, for the human mind that God has enabled to understand mathematics (Ps. 145, 148). And he should endeavor to exhibit ever more fully and clearly to others that “from him and through him and to him are all things. To him be glory for ever. Amen” (Rom. 11:36).
We intend, by the above description, to delineate not only what a mathematician’s inward attitudes should be, but also what his work, his words, and his writings should express overtly and covertly. A man’s words
normally express what he is: “For out of the abundance of the heart the mouth speaks. The good man out of his good treasure brings forth good, and the evil man out of his evil treasure brings forth evil. I tell you, on the day of judgment men will render account for every careless word they utter; for by your words you will be justified, and by your words you will be condemned” (Matt. 12:34b-37). If a man is working for the glory of God, he won’t be a “secret” believer; he will say so as he talks mathematics.
How far this is from a “neutral” stance! The man who ignores God as he does his mathematical task is not “neutral,” but rebellious and ungrateful toward the Giver of all his knowledge.
1 John Calvin, Institutes of the Christian Religion (Grand Rapids: Eerdmans,1964), II, ch. 1,9.
2 Ibid., I, ch. 6, 1.
3 Paul Benacerraf and Hilary Putnam, eds., “On the Nature of Mathematical Truth,” Philosophy of Mathematics: Selected Readings (Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1964), p. 378.
4 For example, the relation of membership in a set, and the axioms of infinity and reducibility, would not be considered as “purely logical” by many people.
5 “When we say that analytic propositions [among which Ayer includes mathematical propositions] are devoid of factual content; and consequently that they say nothing, we are not suggesting that they are senseless in the way that metaphysical utterances are senseless. For, although they give us no information about any empirical situation, they do enlighten us by illustrating the way in which we use certain symbols” (italics mine; from Alfred Jules Ayer, “The A Priori,” Philosophy of Mathematics, p. 295). This same article contains some discussion of Mill’s and Russell’s views of mathematical knowledge.
6 Cf. Ludwig Wittgenstein, Bemerkungen über die Grundlagen der Mathematik (Oxford: Basil Blackwell, 1967), pp. 4,6.
7 “The A Priori,” p. 300.
8 Cf., for example, Ernest Nagel, “The choice between alternative systems of regulative principles [in logic and mathematics] will then not be arbitrary and will have an objective basis; the choice will not, however, be grounded on the allegedly greater inherent necessity of one system of logic over another, but on the relatively greater adequacy of one of them as an instrument for achieving a certain systematization of knowledge” (“Logic Without Ontology,” Philosophy of Mathematics, p. 317).
9 Cornelius Van Til, The Defense of the Faith (3rd, rev. ed.; Philadelphia: Presbyterian and Reformed Publishing Co., 1967), pp. 154, 159.
10 For an extended discussion of motive, standard, and goal in Christian versus non-Christian ethics, cf. Cornelius Van Til, Christian Theistic Ethics, In Defense of Biblical Christianity, vol. III (Philadelphia: den Dulk Christian Foundation, 1971).