Source: https://frame-poythress.org/a-biblical-view-of-mathematics/
Timestamp: 2019-04-21 00:33:29+00:00

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The gift of logical reason was given by God to man in order that he might order the revelation of God for himself. It was not given him that he might by means of it legislate as to what is possible and what is actual. When man makes a “system” for himself of the content of revelation given him in Scripture, this system is subject to, not independent of, Scripture. Thus the idea of system employed by the Christian is quite different from the idea of system as employed in modern philosophy.
It is therefore pointless for Christians to tell non-Christians that Christianity is “in accord with the law of contradiction” unless they explain what they mean by this. For the non-Christian will take this statement to mean something entirely different from what the Christian ought to mean by it. The non-Christian does not believe in creation. Therefore, for him the law of contradiction is, like all other laws, something that does not find its ultimate source in the creative activity of God. Accordingly the non-Christian will seek to do by means of the law of contradiction what the Christian has had done for him by God. For the Christian, God legislates as to what is possible and what is impossible for man. For the non-Christian, man determines this for himself. Either positively or negatively the non-Christian will determine the field of possibility and therewith the stream of history by means of the law of contradiction.
In their world-views, Christian and non-Christian differ at fundamental points. Granted. But surely that doesn’t affect mathematics. Here, finally, is a neutral area, where Christian and non-Christian can agree. Both know that 2 + 2 = 4. How could religious differences ever affect it?
Let’s look more closely at the “neutrality postulate.” This postulate says that the knowledge and structure of a science—for example, mathematics—is not influenced by religious belief.1a Or at least science ought not to be influenced by religious belief. To put it more baldly, true scientific knowledge remains the same whether or not God exists. We intend to criticize this postulate both in terms of its fit with the actual phenomena of mathematics, and in terms of its internal self-inconsistency.
The neutrality postulate holds special attractiveness as applied to mathematics, because of the apparent widespread agreement about mathematical truths. “Everybody knows that 2 + 2 = 4.” If religious beliefs really have an influence, why is there such widespread agreement, cutting across religious lines? We intend to answer this question on several levels: (1) by showing that the agreement in mathematics is not so widespread, nor so uncorrelated with religious beliefs, as the textbooks would have you believe (§§2-7); (2) by showing that non-Christian philosophy of mathematics is involved in deep-set cleavages and antinomies, in its understanding of even so simple a truth as 2 + 2 = 4 (§§11-18); (3) by showing that only on a thoroughgoing Biblical basis can one genuinely understand and affirm the real agreement about mathematical truths (§25).
So, first of all, what differences have arisen in mathematics in connection with religious belief? Differences have arisen over arithmetical truth, over standards for proof, over number-theoretic truth, over geometric truth, over truths of analysis, over mathematical existence-not to mention the long-standing epistemological disputes over the source of mathematical truth. Let’s consider these areas one at a time.
What does this imply? Even the simplest arithmetical truths can be sustained only in a worldview which acknowledges an ultimate metaphysical plurality in the world—whether Trinitarian, polytheistic, or chance-produced plurality. At the same time, the simplest arithmetical truths also presuppose ultimate metaphysical unity for the world&mdahs;at least sufficient unity to guard the continued existence of “sames.” Two apples remain apples while I am counting them; the symbol ‘2’ is in some sense the same symbol at different times, standing for the same number.
So, at the very beginning of arithmetic, we are already plunged into the metaphysical problem of unity and plurality, of the one and the many. As Van Til and Rushdoony have pointed out, this problem finds its solution only in the doctrine of the ontological Trinity.5 For the moment, we shall not dwell on the thorny metaphysical arguments, but note only that without some real unity and plurality, ‘2 + 2 = 4’ falls into limbo. The “agreement” over mathematical truth is achieved partly by the process, described elegantly by Thomas Kuhn and Michael Polanyi, of excluding from the scientific community people of differing convictions.6 Radical monists,for example, are not invited to contribute to mathematical symposia.
A: Somewhere in the decimal expansion of pi there occurs a sequence of seven consecutive 7’s.
B: There are infinitely many primes p such that p + 2 is prime.
In 1975, no man knows whether either A or B is true. Nor is there any known procedure by which, in a finite amount of time, we could be assured of, obtaining a definite yes-or-no answer. For the intuitionists, this means that A and B should not be considered as either true or false.10 It makes no sense to talk about truth or falsehood so long as we have no way of checking. On the other hand, the Christian, on the basis of I John 3:20 (“God is greater than our hearts, and he knows everything”), Psalm 147:5, and other passages, is likely to feel that at least God knows perfectly definitely whether A or B is true. Our own limitations set no limits to His knowledge (§24) (cf. Isa. 55:8-9; Ps. 139:6, 12, 17-18).
that we know a priori the truths of Euclidean geometry. However, with the advent of the non-Euclidean geometries of Bolyai-Lobatchewsky and Riemann,11 and then the general relativity of Einstein, the “truth” of Euclidean geometry was put in question.12 Even the question of what it means for a geometry to be “true to” the world is now in dispute.13 And the deep-going philosophical differences between operationalism and realism, positivism and Platonism heavily influence one’s conclusions.
One might object that purely axiomatic geometry (as opposed to applied geometry) is at least free from these difficulties. Everyone can agree when a geometrical theorem is proved. But once again the intuitionists object to reductio ad absurdum proofs. Not only that, but one finds that a rigorous adherence to the demands of axiomatic geometry in the modern set-theoretic style requires the use of uncountable sets of points and uncountable sets of congruence transformations, thereby introducing the philosophical problematics of infinity (see §7 below).
-veloped by Errett Bishop.15 Finally, if we are committed to a more conventionist view of mathematics (see §14), or, like Leibniz, to the reality of infinitesimals, we are likely to be favorably disposed toward the nonstandard analysis of Robinson.16 Thus three different worldviews, if they do not absolutely determine, yet decisively influence one’s attitude toward alternative constructions of analysis.
produces a Dedekind cut dividing the reals into two parts, positive and non-positive; in non-standard analysis, the same definition produces a cut of reals into two parts, positive non-infinitesimal on the one hand and non-positive plus positive infinitesimal on the other; in constructive analysis, the same definition produces still a third result, namely a division of reals into absolutely positive, absolutely non-positive, and a third group of “don’t know.” Again, in constructive analysis every constructible function on [0,l] is uniformly continuous,17 while in classical analysis the cardinal number cc == 2c of discontinuous functions on [0, 1] is greater than (!) the cardinal number c of continuous functions. It can hardly be denied that, in this area at least, philosophico-religious differences have had their impact!
Does 2 exist? Does 1/4 exist? Does √2 exist? Does -2 exist? Does √-1 exist? Does dx exist? Does the transfinite number aleph-two exist? Does a measurable cardinal exist?
0pinions about mathematical existence are related to religious differences. Consider several examples: the Pythagoreans, for philosophico-religious reasons, did not want to acknowledge the existence of irrationals like √2.18 Leibniz’s philosophical convictions about infinity favorably disposed him toward using infinitesimals like dx.19 From a self-consciously Christian viewpoint, D. H. Th. Vollenhoven and Herman Dooyeweerd rejected the existence of uncountable transfinite numbers, ostensibly because of their antinomic character.20 (The author does not agree with their decision at this point, but the fact is that their mathematical convictions were religiously motivated.) Such examples show clearly that a question of mathematical existence may not be a religiously neutral affair. More generally, mathematics in the past has not been a religiously neutral science. In short, the neutrality postulate is not borne out by the history of mathematics.
The neutrality postulate makes the implicit metaphysical claims that (a) mathematical reality is not the result of God’s creating activity in any (p. 166) essential way (for if it were a result of God’s work, we could not imagine it “remaining the same” even though God didn’t exist); (b) God’s nature and the nature of number are not significantly involved in one another; they are not so related that one could infer properties of one from a study of the other. Otherwise, differing opinions about God might, for all we know, lead to differing opinions about the nature of number.
Claim (a) is already a denial of creation in its Biblical sense, as we shall see (§19); claim (b) involves a denial of the Trinity (see §19). At the moment, however, we are interested, not so much in the fact that these claims contradict orthodox Christian doctrine, as in the fact that they have a far-reaching, astonishingly dogmatic metaphysical character. The claim that metaphysics is irrelevant to mathematics turns out to be itself a metaphysical claim about mathematics. The neutrality postulate turns out to be highly “non-neutral.” To say the least, this is a paradoxical situation.
The neutrality postulate is involved in a similar paradox regarding its implicit epistemological claims. This postulate denies, in effect, that God can reveal any truths about mathematics. Suppose He could. Then conceivably He might reveal information not already established by other means. Then those people who believed what He revealed would stand in a different position in mathematics from those who did not. Such differences arising from religious belief would violate the neutrality postulate.
Now the reader may argue that all this is purely speculative, since God has not in fact chosen to record mathematical theorems in Scripture. But note the following. (1) Whether God has given us mathematical information can be determined only by an actual examination of Scripture, not (as the neutrality postulate presumably claims) in an a priori fashion. (2) Though the Bible does not contain mathematical theorems in the modern sense, it does contain teachings that instruct us, in certain cases, about what kind of mathematics is legitimate (cf. the examples in §§5-9). (3) God’s general (pre-redemptive) revealing activity is involved in every kind of mathematical knowledge (see §23). (4) In the light of (l)- (3), the neutrality postulate definitely is concerning itself with religious issues.
know its supposed truth. This very explaining constitutes a doctrine of revelation. Usually people talk about revelation from some metaphysical ultimate other than God (mind, matter, sense experience, Reason); nevertheless, men require revelation. In short, the neutrality postulate is entangled in the paradoxical net of being able to deny the relevance of (theistic) revelation only on the basis of an underlying doctrine of (secular) revelation. The neutrality postulate is epistemologically non-neutral.
Third, the neutrality postulate is involved in paradox regarding its implicit ethical claims. It makes a statement about what “ought” to be: “Mathematics “ought not” to be influenced by religious belief. Let us name this statement ‘C.’ C contradicts Christian ethics, as we shall see (§26). But, once again, let us focus on the internal paradox involved in this ethical claim. C: mathematics ought not to be influenced by religious belief. In particular, presumably it ought not to be influenced by the ethical judgments correlated with religious belief. Therefore, mathematics ought not to be influenced by the ethical judgment C that “mathematics ought not to be influenced by religious belief.” We are confronted with a self-destructive ethical claim.
The ethical claim C can rescue itself from oblivion only when it is supported by another claim, D.
D: Claim C is not a religious (albeit it is an ethical) belief.
But most people would agree that general claims like D concerning the relation of the religious to the ethical are religious claims. They are closely related to the question whether right and wrong are defined (say) by God’s commands or by conscience. Let us therefore agree that D is a religious belief.
E: mathematics ought to be influenced by C.
F: for any G, if G is a religious belief, mathematics ought not to be influenced by G.
H: if C is a religious belief, mathematics ought not to be influenced by C.
D: C is not a religious belief.
religious belief D. Hence presumably C itself is religious, a contradiction of D.
So far we have concentrated on the difficulties specific to the neutrality postulate. The neutrality postulate, we have seen, is beset by conflict with phenomena in the history of mathematics (§§2-7) and by internal self-inconsistency (§§8-10).
Other difficulties, however, the neutrality postulate shares with all non-Christian, non-theistic world-views (whether or not these views claim for themselves neutrality). To these difficulties common to all non-Christian versions of mathematics we now turn.
Of course, in a sense each particular world-view-materialism, idealism, positivism, Marxism-yes, each individual thinker, has difficulties all his own. A thoroughgoing criticism, from a Christian point of view, would thus have to deal with each thinker separately. In this paper we cannot hope to do more than sketch an outline of the way criticism should proceed. A unified procedure of criticism is to some extent possible because all non-Christian systems share similar problems, growing out of their common refusal to honor the true God.
For convenience, we divide the problems into three major areas: metaphysical, epistemological, and ethical. We expect non-Christian philosophy of mathematics (a) to have metaphysical problems because it has abandoned the true Source of being; (b) to have epistemological problems because it has abandoned the true Source of knowledge; (c) to have ethical problems because it has abandoned the true Source of value. We take up these topics in the order (b), (c), (a).
Whichever answer a person on the anti-theistic side chooses, he is bound to land himself in difficulties. Suppose that one emphasizes the a priori character of mathematical knowledge. 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? 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 priorist view is also open to more practical objections. If mathematics is indeed a priori, why do paradoxes arise? Such paradoxes come in the form of actual contradictions (Burali-forti’s paradox, Russell’s paradox) and in the form of various counter-intuitive results (the continuous space-filling curves of Peano, everywhere-continuous nowhere- differentiable functions, the downward Löwenheim-Skolem theorem, etc.). The paradoxes seem less threatening today, partly because mathematicians adopt a more conventionist attitude toward them (§14), partly because we have disposed of them by modifying our axioms (to avoid contradictions) or modifying our intuitions (to square with the theories). Nevertheless, the paradoxes illustrate that, historically considered, 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 emphasized the inductive character of mathematical knowledge. One comes to believe that 2 + 2 = 4 from repeated experience (a posteriori) of two objects plus two objects making four objects. All right, but no one has repeated experience of 2,123,955 objects plus 644,101 objects making 2,768,056 objects. So why does he believe that 2,123,955 + 644,10l = 2,768,056? “Ah,” so it is said, “he has generalized from his experience with small numbers.” Unfortunately, in the word “generalize” is concealed either an infinite regress or the specter of the a priori. 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 he conclude (generalize?) that 13 + 2 = 15 rather than 13 + 2 = 14 even 13? In terms of a consistently a posteriori viewpoint, the answer can only be, because of previous experience with other generalizations. In other words, he has generalized from previous generalizations. Why has he generalized in this particular way from those other generalizations? Because he has generalized from previous experience of generalizing from previous generalizations, and so on. Apparently, one can escape this regress only by saying at some point, “Because that’s the way the human mind operates.” And then one is confronted with an a prioriknowledge, or at least a priori heuristics.
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 imagination.
A third attempted solution to the epistemological problem deserves mention, if only because of its widespread 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 the words “two” and “four” in just that way (Wittgenstein).23 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).24 Or, “2 + 2 = 4 because it follows from our (conventionally determined) postulates” (formalists).
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 that of language. The same a priori/a posteriori problems reappear when we ask why(mathematical) language functions adequately.
At this point, we should mention that, in our opinion, certain proof-theoretic results have intensified the a priori/a posteriori problem for an anti-theistic philosophy of mathematics. We are referring particularly to Gödel’s proof of the incompleteness of Whitehead-Russell’s axioms for first-order number theory.26 This proof has been clasped to the bosom of so many philosophers of mathematics that we hesitate to read into it still one more interpretation. Nevertheless, it appears to us that the proof ought to shake confidence in any narrowly a priori or conventionalist philosophy of mathematics. For one thing, by showing that no Turing machine can be built to generate all number-theoretic truths and no falsehoods, it has raised question marks over the ability of the human mind itself to know all number-theoretic truth. And if we can’t know it all, it is certainly not a priori for us. Second, by showing that any axiom list will be incomplete, it raises question marks over any conventionalist claim that truth is defined by our choice of axioms. Assuming that number theory is consistent, the machinery of the proof produces a true statement which does not follow from the axioms; hence, this truth is not (narrowly speaking) conventionally defined by the choice of axioms.
On the other hand, Gödel’s proof gives little comfort to the a posteriori camp. For the crucial true-but-unprovable statement S exhibited in his proof cannot be “experienced” as true or a posteriori “seen” as true in any normal sense. The a posteriori camp presumably says that we learn by direct experience (two apples and two apples making four apples)and then later by proofs (the proof procedure itself being a generalization from experience of simple proofs). Yet Gödel’s S can be neither directly experienced (it’s much too complicated) nor proved. S is the first statement of its kind ever produced (no generalization from previous experience is possible), yet on inspecting the “intuitive” meaning of S, one becomes convinced that S must be true.
Because of the above difficulties, anti-theistic philosophy of mathematics is condemned to oscillate, much as we have done in our argument, between the poles of a priori knowledge and a posteriori knowledge. Why? It will not acknowledge the true God, wise Creator of both the human mind with its mathematical intuition and the external world with its mathematical properties. In sections 22-23 we shall see how the Biblical view furnishes us with a real solution to the problem of “knowing” that 2 + 2 = 4 and knowing that S is true.
Closely related to the epistemological questions are questions about the metaphysical “status” of ‘2 + 2 = 4’ in relation to other truths. What does it mean that 2 + 2 = 4? If we wanted to test whether a child understood ‘2 + 2 = 4,’ we would be satisfied only when he not only demonstrated ability to manipulate the symbols properly on paper, but also knew when to use ‘2 + 2 = 4’ in word-problems. Such a check is necessary because a child might memorize the visual shapes and manipulations of ‘2’ and ‘4’ without ever understanding what he was doing. Indeed, we might say that to know that 2 + 2 = 4 is knowing how to use those symbols in everyday life. One cannot know ‘2 + 2 = 4’ without knowing many other things in relation to that one truth. Thus we are inevitably concerned with a great plurality of experiences and truths and the relations among them.
true (2 + 3 = 5, 1 + 2 = 3, 1 + 3 = 4, etc.) and false (2 + 2 = 5,2 + 2 = 3, etc.); (b) only as a unit with a certain variational range, implying that it can be repeated in varying forms without losing its identity (two plus two makes four, two plus two equals four, two and two are four, II + II = IV, (1 + 1) + 2 = 4, etc.); (c) only as an item distributed in larger units of linguistic behavior and general human behavior (proofs using ‘2 + 2 = 4,’ word problems referring to it, ‘2 + 2 = 4’used in calculating grocery prices, income tax, and missile trajectories).
The problem here, for any anti-theistic view, is to guarantee any ultimate unity and stability to the enormous “sea” of truths and experiences in which ‘2 + 2 = 4’ is embedded. How, without knowing everything, can we really be said to know anything? ‘2 + 2 = 4’ is distributed in a larger context which, if we are to understand it, must be distributed in a still larger context, ad infinitum. Moreover, how do we know that the next thing we discover, on the borders of our knowledge, will not upset and radically overturn what we have hitherto called “knowledge”? Such a contingency seems not only to be abstractly possible (due to the necessity of defining knowledge partly in terms of its distribution in larger contexts), but actually to have occurred in the past in more than one science. Physics has radically revised its “knowledge” during the Newtonian revolution, the Einsteinian revolution, and now the quantum revolution. Even mathematics has had to revise itself at times; think of the discovery of irrationals by the Pythagoreans, of contradictions arising from reasoning with conditionally convergent infinite series, of contradictions like Russell’s paradox arising from reasoning with the naive idea of set. True, each of these three mathematical “problems” is now considered resolved, but none was resolved without a revision of the standards, yes, and the very concept of correct mathematical reasoning.
On the other hand, if the anti-theist wants to begin with an ultimate unity, rather than an ultimate plurality, of truth, he has no way to explain how plurality arises from this single truth. We have here the problem of unity and plurality which we confronted already in §2.
Yet such postulated gods can never rise above the idols of Isaiah’s description: “Tell us what is to come hereafter, that we may know that you are gods; do good, or do harm, that we may be dismayed and terrified. Behold you are nothing, and your work is nought; an abomination is he who chooses you” (41:23-24; cf. 44:6-11, etc.).
On the other hand, we can make an effort to reduce the sciences to an ultimate unity by deriving some from others. The philosophers of mathematics in the past have tried in turn to reduce mathematics (a) to linguistics (“mathematics is the science of formal languages”—the formalists),(b) to psychology (“mathematics is the study of mental mathematical constructions”—the intuitionists), (c) to logic (“mathematics is a branch of logic”—the logicists), (d) to physics (“mathematics is generalized from sense experience”—the empiricists), (e) to sociology (“mathematics is a group of socially useful statements”—the pragmatists). The form of the supposed reduction of mathematics thus gives us a rough and ready catalog of the major schools of philosophy of mathematics.
As we might expect, such attempted reductions never really succeed. At some point, they do not do justice to the distinctive character of mathematical truth, as over against physical, linguistic, psychological truth. A detailed discussion of reductionisms is beyond the scope of this work, and we must refer the reader to the extensive foundational work by Dooyeweerd and Vollenhoven, plus particular investigation of mathematics by Strauss.30 0ur suspicions should be aroused, by the very diversity of attempted reductions (to linguistics, to logic, to psychology, etc.), to question whether any of these can really be the true story. They each refute the others, by showing up a side of the picture that the others have not sufficiently acknowledged.
Finally, we should mention in passing that anti-theistic mathematics has no satisfactory ethical foundations, any more than it has metaphysical or epistemological foundations. No piece of mathematics can be written, no mathematician can ever operate, without some implicit or explicit motive, standard, and goal for the work. A mathematician may be motivated by selfishness, by fear, by altruism, or by the Lord; he may be working for money, for sheer enjoyment, or for the glory of God. But no one ever does mathematics without having this kind of factor in the background. Moreover, his motives, standards, and goals will inevitably affect what kind of problem he chooses to focus on, what relative weight he gives to pure versus applied mathematics, what standards he sets for himself in his teaching and writing, how he divides his time between teaching and research, and so on. The person who regards such factors as “extraneous” to the real business of mathematics has already lost sight of the consistent Biblical focus on the work of man as the work of man who stands before his Creator: “rendering service with a good will as to the Lord and not to men, knowing that whatever good any one does, he will receive the same again from the Lord, whether he is a slave or free” (Eph. 6:7-8). For further discussion of the Biblical view, see §26.
So far our discussion has developed in a predominantly negative direction, because we have occupied ourselves with a criticism of “neutralist”(§§1-10) and anti-theistic (§§11-18) views of mathematics. However, it is hardly possible to appreciate the true poverty of such views without some reflection on what a truly Biblical view of mathematics would look like. To this task we now turn.
In accordance with our earlier critique of anti-theistic metaphysical(§§8, 16, 17), epistemological (§§9, 12-15), and ethical (§§10, 18) foundations, we propose to discuss the Biblical viewpoint also in terms of metaphysics (§§19-21), epistemology (§§22-25), and ethics (§26). Naturally, Biblical foundations in these three areas overlap and complement one another; we take up the topics one by one in order to throw into bolder relief the radical contrasts between theistic and anti-theistic views.
In the following discussion, we are not making an attempt to use mathematics (or other sciences) as some kind of “proof” or support for the Bible. Rather, conversely, we maintain that only on the basis of obediently hearing the Word of God can we find a proper foundation for mathematics! It is God who sustains mathematics, not vice versa.
What is the metaphysical status of numbers and statements about numbers? What is the status of geometry? What is the significance of mathematics in this world? For the Christian committed to Scripture, the most important fact about mathematics must be its relation to the Lord. Since He is Creator and Sovereign over all, everything must find its meaning, yes, its very existence in Him: “in him we live and move and have our being” (Acts 17:28). “Thine, 0 Lord, is the greatness, and the power, and the glory, and the victory, and the majesty; for all that is in the heavens and in the earth is thine; thine is the kingdom, 0 Lord, and thou art exalted as head above all” (I Chron. 29:11).
identify part of the creation with God or part of God with creation, we are guilty of serious idolatry.
Therefore, the most basic question to ask about mathematical structure and laws is this: are they aspects of creation or of God? Are they, as it were, created things or God, or are they perhaps in some third category? This question is still ambiguous, because its answer depends on what we mean by “mathematics.” “Mathematics” may refer to (a) the historically growing science manifested in textbooks, articles, conferences, lectures,etc.; (b) the thoughts of the mathematicians; or (c) mathematical “structure” for the world, somehow existing independently of our thoughts (two apples and two apples making four apples; two distinct points determining a unique line between them; etc.). Mathematics (a) clearly consists in[created things and activities of created men; mathematics (b) consists in human thoughts which, as such, never have divine status (Isa. 55:8-9; Ps. 147:5). We shall have more to say about (a) and (b) in §§22, 24.
For the moment, let us concentrate on mathematics (c). Since mathemiatics (c) concerns properties of created things, we might be tempted at first glance to say, “mathematics (c) is created.” However, the Bible, while speaking over and over of God’s having created things (minerals, plants,animals, men, angels), apparently never speaks of God’s having created structures” or “laws.” A little reflection shows us that this is no accident. The Bible never represents the world as being governed by laws as such, independent of the Creator, but rather by the decrees of the King, by God Himself speaking (cf. Gen. 8:22-9:7; Jer. 33:25; Ps. 33:6-11, 18-22; 147:15-20). Because His decrees are in accordance with who He is (Ps. 19:7-9), we expect them to be wise and orderly (Ps. 104:24; Prov.8:22-31; Rom. 11:33-36).
So it is with mathematics (c). God Himself has a numerical nature. He is three in one. It is interesting that Jesus uses the plural pronoun ”we” (John 17:21; cf. John 14:23) and plural “are” (esmen, John 10:30)in speaking of the Father and the Son. Mathematics (c) is eternal because the Father, Son, and Holy Spirit (3!) are eternal (John 1:1; 17:5; Heb. 9:14). And God’s eternal numerical nature is manifested in creation much as His love, wisdom, and justice are manifested.
(Matt.11:25-30; cp. Sir. 24:25 ; 51:23-26).33 Because from the beginning the Son is God’s personal wisdom. God has no need to consult anyone else (Isa. 40:13-14). Thus we are justified in saying that the plurality of this world (the works of God) finds its basis in the plurality of the fellowship of the Father and the Son. And Psalms 104:24 also points to an origin for unity in this world when it speaks of the wisdom of the one Lord. Because there is one Lord, there is an inner consistency in everything that He does. Wisdom expresses itself in orderly rule, in justice, in proportioned love and hate (Prov. 8:13-17).
In saying “1 + 1 = 2” we are thus stating a truth about the Trinity: a truth about the Wisdom of God, and then, secondarily, a truth about the world that He governs. (Note, however, that since the Trinity and the Wisdom of God are incomprehensible, God’s own “mathematics, as it were, is not accessible to us in all its fullness. We cannot assume that our mathematics (b) is necessarily all true or exactly equivalent to God’s “mathematics.”) How far this is from a “neutralist” view of mathematics, which supposes that mathematics has nothing to do with God!
From this point of view, anti-theistic philosophies of mathematics can be classified as mathematical versions of old heresies. A strict a priori view of mathematics (§12), emphasizing that mathematical truth (b) mustbe what it is, is guilty of leveling the distinction between the Divine and human mind. What is a priori for God—namely mathematics (c)—is not a priori for us. We cannot infallibly reason to mathematical results, because God’s mind is different from ours. The leveling process ends by saying that God must conform to our a priori mathematics, resulting in Tri-theism.
Next, a strict a posteriori view of mathematics (§13), emphasizing the contingent character of mathematical truths (§16), is guilty of ignoring the fixedness of God’s numerical nature. This view ends by saying that God is non-mathematical, resulting in mystical monism.
We should also observe that the Biblical view solves, in a clear-cut fashion, the problem of the meaning of ‘2 + 2 = 4’ in relation to other truths (§16). ‘2 + 2 = 4’ finds its ultimate meaning and integration in the unchangeable fullness of the Divine Trinitarian Fellowship. Because God is unchangeable, the truth of ‘2 + 2 = 4’ is not altered by the next human discovery.
What are the characteristic features of mathematical truth, as opposed to other kinds of truth? How is mathematics related to other sciences? In answering such questions we must naturally move further away from the direct testimony of the Bible. Yet the Bible does seem to present us with grounds for a preliminary division of the sciences in Genesis 1:28-30. There God divides His creation into four major groups: mineral, plant, animal, and man; and He instructs Adam about some of the characteristic features and functions of each group. In studying such characteristic features, Adam would make beginnings in the sciences of physics, of biology, of zoology, and of anthropology. See Diagram 1.
In the course of time, we expect that further major divisions would be marked out within these sciences. For a detailed analysis of the major divisions, we refer the reader to the work of the Amsterdam philosophy and the author.35 For our present purposes, let us confine ourselves to a division of physics. Physical things share with plants, animals, and man not only so-called physical features (force relations, rigidity or non-rigidity, energy,etc.), but also kinematic features (velocity, mobility), spatial features (extension, area or volume, shape), quantitative features (how many?), and aggregative features (being potential members of various aggregates (or collections). Thus we obtain Diagram 2. The last four sciences of Diagram 2 jointly comprise mathematics.
from the same basic Source as the unity and plurality within mathematics: from the nature of God and His plan. In particular, the kinematic, spatial, and aggregative properties of this world can be traced back to the nature of God, in a manner similar to what we have already done for the quantitative aspect. As God has a numerical nature (the Trinity), so He has a kinematic, a spatial, and an aggregative nature. Of course, in using language about God’s nature we must exercise caution. God as Creator is ultimately incomprehensible to the creature. No man can understand everything about God. Thus we expect that God’s aggregative, quantitative, spatial, and kinematic nature to be, in some way,incomprehensible to us. Created analogies inevitably break down, because they are only finite images of the Infinite One. In the case of God’s numerical nature, this is obvious. God is Three Persons, yet at the same time One God. Jesus can say, “I and the Father are one” (John 10:30). No created thing is three and at the same time one in the same sublime way. We shall >see also that God’s aggregative, spatial, and kinematic nature are not strictly analogous to anything in this created world. That, perhaps, has been one reason why people have (wrongly) tended to deny that God’s nature had anything to do with space or kinematics.
works themselves” (John 14:10-11). Somehow we find that all the members of the Trinity participate, in their own ways, in even those works which we associate most distinctly with one particular member of the Trinity. In a certain sense, the members of the Trinity are not distinguished, because there is only one Lord (Deut. 6:4-5).
Second, God has a spatial nature. This expresses itself, first, in the teachings about God’s filling heaven and earth: “Can a man hide himself in secret places so that I cannot see him: says the Lord. Do I not fillheaven and earth? says the Lord” (Jer. 23:24). “In him we live and move and have our being” (Acts 17:28). See also I Kings 8:23,27; Isaiah 66:1-2; Acts 7:46-50; and passages dealing with God’s dwelling with His people, Deuteronomy 4:7,39; Isaiah 57:15; 66:2; I Corinthians 6:19; Romans 8:9-11. Note the strong stress on the fact that space offers no resistance or problem to God’s rule, but rather that God is Lord of space,doing as He pleases within it.
stability of God’s own “habitation” forms the foundation for His being the believer’s habitation: “Lord, thou has been our dwelling place in all generations. Before the mountains were brought forth, or ever thou had formed the earth and the world, even from everlasting to everlasting art God” (Ps. 90:1-2).
Similarly, some of the passages speaking of the relations of the Trinity speak in distinctively spatial terms. There are expressions of indwelling (John 14:10-11; Col. 1:19; 2:9); expressions of face-to-face relationship: “In the beginning was the Word, and the Word was with God, and the Word was God” (John 1:1). “No one has ever seen God; the only Son, who is in the bosom of the Father, he has made him known”(John 1:18); and expressions of “proceeding”: “But when the Counselor comes, whom I shall send to you from the Father, even the Spirit of truth, who proceeds from the Father, he will bear witness to me” (John 15:26). Once again, we meet with incomprehensible intra-Trinitarian relations, since, if the Father and the Son fill all and are in all (Eph. 1:23; 4:6; Jer. 23:24), the Spirit can hardly proceed from the Father in any easily comprehensible sense.
But, instead of requiring God to correspond to our ideas derived from this created world, we should rather see that, conversely, the “space” of our mathematics (b) is derived from the impress, on finite things, of God’s governing rule. The spatial and physical extent of this world is used in Scripture as one pointer to the original, uncreated immensity of God (Isa. 40:12,26; Ps. 104:25).
He judges (juridical activity: Ps. 75:2-8).
world (John 3:35; Prov. 8:30-31; Col. 1:13-16). Similarly, to use another image from Scripture, the Father has been speaking from all eternity, in Trinitarian fellowship (John 1:1-2). God’s own eternal activity and “movement” of speaking and loving is what causes the movements, and hence the kinematic character of this world: “He sends forth his command to the earth; his word runs swiftly. He gives snow like wool; he scatters hoarfrost like ashes. He casts forth his ice like morsels; who can stand before his cold? He sends forth his word, and melts them; he makes his wind blow, and the waters flow. He >declares his word to Jacob, his statutes and ordinances to Israel. He has not dealt thus with any other nation; they do not know his ordinances. Praise the Lord!” (Ps. 147:3,5-20). Note the connection of His word both with the motions of physical things, and with His love for Israel His son (a love which comes to fulfillment in His-love for the only Son).
0nce again, as in the case of other aspects of God’s nature, God’s “kinematic” nature is incomprehensible. At the same time that God is so active, He is also unchanging (Mal. 3:6; Ps. 102:27). His word is fixed(Ps.119:89); it never passes away (Luke 21:33). Thus God Himself forms the foundation, not only for the change in the world (He decrees it out of His own activity), but also for the stability of the world (He and His decrees do not change).
What is the relation among the four major subdivisions of mathematics (Diagram 2)? Between mathematics and the other sciences? Is there, indeed, any constant relation at all (§17)? Such questions must ultimately be answered in the same terms as our earlier questions about unity and plurality (§§16,19). The sciences find their unity in the personal Wisdom of God (Ps. 104:24). “He is before all things, and in him all things hold together” (Col. 1:17). This is why mathematics applies to physics. This is why the fundamental laws of physics have such simple form. We trust that mathematics will continue to find application to physics, not because of blind faith (§17), but out of the conviction that the laws of physics and mathematics are simply two diverse ways in which Christ comprehensively rules the universe.
theory). The area under curves can be calculated, either by direct geometrical approximation, or by algebraic calculation of a definite integral, whose definition involves a fundamentally kinematic limit process. The reasonings in different parts of mathematics (agorology = elementary set theory, arithmetic, geometry, kinematics [including differential and integral calculus]) agree with one another because of the unity of God. Shifts back and forth between the four major divisions of mathematics constantly exploit the fact that these sciences have their origin in the one Wisdom of God.
Because God contains both unity and plurality in Himself, there is no need for us, in the Christian framework, to resort to the futile attempts of reductionism that we discussed in §17. As a matter of fact, the reductionisms of §17 can be seen as a kind of mathematical version of an old heresy: gnosticism. Why should we say that? Well, in exploring mathematics one is exploring the nature of God’s rule over the universe, i.e., one is exploring the nature of God Himself. A reductionism thus ultimately amounts to an attempt to derive some aspects1 of God’s nature from other aspects2, an attempt to say that the latter aspects2 of God’s nature are more fundamental. The aspects2, are then somehow what is “really there,” as opposed to the only apparent existence of aspects1. I classify this as a gnostic-type heresy, because gnosticism develops a theory of emanations whereby certain inferior deities derive their being from emanations of the ultimate Deity. This gnostic derivation of being is not so dissimilar to the present-day derivation of aspects.
How do we come to know and discuss mathematics (b), that is, the thoughts and knowledge of human mathematicians? Here, for the first time, we must focus on the Christian view of man. How does man fit into the picture of mathematics? We can have no other starting point than the “definition” of man provided by Scripture: man is the image of God (Gen.1:26-30; cf. Gen. 2:7; I Cor. 11:7). As such, his talk is to imitate receptively, on a finite level, the works (naming, Gen. 2:19; 1:4; governing, Gen. 1:28; Ps. 22:28; improving, Gen. 2:15; 1:31), and rest (Gen. 2:2; Ex. 20:11) of God. Man’s mind is created with the potential, then, of understanding God (though not exhaustively). He has the capability of,understanding the aggregative, quantitative, spatial, and kinematic aspects of 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.
problem of a priori/a posteriori (§§12-15). The a priori capability of man’s created nature really corresponds to the a posteriori of what is “out there,” 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, his intuitive expectations are not always fulfilled (cf. examples in §12), because man is the image of God the infinite One. Since God is incomprehensible, His mathematics sometimes baffles us, and it is to be expected that it should. Gödel’s proof (§15) perhaps articulates one specific instance of a principial limitation on man’s knowledge in comparison to God’s.
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:l0b-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.
Note that, in the Christian framework, the a priori of man’s nature and the a posteriori of God’s universe and His revelation complement rather than compete with one another.
This is perhaps a good point to explore further the relation between human and divine “mathematics.” When God reveals Himself, He reveals Himself truly, but not exhaustively. That is one limitation. In the case mathematics, our knowledge is also limited by the fact that we see the effects of God’s decrees and rule on a finite world, without having direct access (except in the case of statements of Scripture) to those decrees and that rule itself.
Take the case of geometry. Though God has a spatial nature (§20),it would be blasphemous to say that He has the properties of Euclidean (or non-Euclidean) space. Our own mathematical systems (Euclidean or non-Euclidean) are somehow not identical with His “system.” We must say, I think, that Euclidean and non-Euclidean geometries are both exhibitions (revelations) of how God might rule the world; for they both discoveries or constructions of the human mind in the image of God.
Presumably God might have created a universe with either a Euclidean or non-Euclidean or some other geometry. Thus the variety of geometries, far from offering an obstacle to the Christian viewpoint, is simply an illustration of the freedom of God.
Is Euclidean geometry (as a system of statements) the Creator or the creature? In some sense both. The above argument shows that in some sense geometry is relative to this world (and thus created). Now, take the statement G that between any two distinct points there is exactly one straight line. (Or, equally, that the sum of angles in a triangle is 180°.) Suppose that G is actually true in our world. Then it is right to say that G expresses one of the laws of God’s creation. G is one of those things that God has ordained to be true for this world (Lam. 3:37). After all, we cannot imagine that G could be true for any other reason than because God ordained it. He is Lord! How could we know G in any other way than by reflecting (§22) God’s own original knowledge that G? Furthermore, God’s decrees, His speech, His Word (Isa. 46:9-12) say who God is (John 1:1). G says who God is. Yet G is not in every sense identical with God.
reference to points, lines, and degrees) and of characteristics of the Infinite (it is unchanging). Sometimes (as indeed is the case with the Incarnation) the created and Uncreated cannot be easily distinguished.
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(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 (cf. §26).
fashion; the cultural mandate of Genesis 1:28-30. Thus, in spite of themselves, they acknowledge God in some fashion by “imitating” Him. It is the situation described in Romans 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 §§19-21, in knowing mathematical truth they know something of God (though not exhaustively, and at places mistakenly). Nevertheless, their “knowledge” is not more beneficial to them than the knowledge of the demons (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 system. It is because Christianity is true, because God is who He is, because man is the image of God, the non-Christian knows anything.37 The supposed “common ground” of shared mathematical truth proves the very opposite of what the neutralist supposes it to prove.
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, 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).
mathematics38 (cp. the non-Christian view in §18).
find his standard in the command of God, the program which God has given man to fulfill: “Be fruitful and multiply, and fill the earth and subdue it; and have dominion . . .” (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 which it displays, for the human mind which God has enabled to understand mathematics (Ps. 145; 148). And he is 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 mathe matician’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 “neutralist” 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 The anti-Biblical presuppositions of innocent-looking agnostic statements have long been a target for Van Til’s penetrating criticism. See, for example, Cornelius Van Til, A Survey of Christian Epistemology , Volume II of the series In Defense of Biblical Christianity (Philadelphia: den Dulk Foundation, 1969), pp. 212-213.
1a Over against the neutrality postulate, we are by no means advocating a “relativity of truth” which would say that what is really true depends on the observer. On the contrary, the truth is (by definition) what God knows, and hence completely fixed from the beginning. That much we presuppose. However, in Parts I and II, and especially in Part I, we want to focus on what people believe and know about mathematical truth. Which truths they know, and what they do with them, depend on their religious convictions.
2 William K. C. Guthrie, A History of Greek Philosophy, Vol. II (Cambridge: at the University Press, 1967), p. 30.
3 “In thought should it be heeded,/Here is no plurality anywhere;/By death he is bound fast to death/Who here contemplates plurality.” Paul Deussen, The Philosophy of the Upanishads, trans, by A. S. Geden (Edinburgh: T. & T. Clark, 1906), p. 232, quoted from Brih (adâranyaka) 4.4.19. “…there is no plurality and no change. Nature which presents the appearance of plurality and change is a mere illusion (mâyâ)” (ibid., p. 237).
4 “There is however in the whole universe, alike in heaven and on earth,nothing besides the atman:-‘There is no second outside of him, no other distinct from him.’ ” ibid., p. 157, quote from Brih (adâranyaka) 4.3.23-30. Cf. also Sravepalli Radhakrishnan, Indian Philosophy , Vol. 2 (London: George Allen & Unwin Ltd.,1927), p. 535.
5 Cornelius Van Til, The Defense of the Faith , revised and abridged (Philadelphia: Presbyterian and Reformed Publishing Co., 1963), pp. 25-26; A Survey of Christian Epistemology , p. 96; and Rousas J. Rushdoony, The One and the Many: Studies in the Philosophy of Order and Ultimacy (Nutley, N. J.: Craig Press, 1971).
6 Thomas S. Kuhn, The Structure of Scientific Revolutions , 2nd ed. (Chicago: University of Chicago Press, 1970), p. 168; and Michael Polanyi, Personal Knowledge: Towards a Post-Critical Philosophy (London: Routledge & Kegan Paul, 1958),pp. 160-167.
7 Arend Heyting, “Disputation,” in Paul Benacerraf and Hilary Putnam, eds., Philosophy of Mathematics: Selected Readings (Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1964), p. 61.
8 “The . . . point of view that there are no non-experienced truths and that logic is not an absolutely reliable instrument to discover truths, has found acceptance with regard to mathematics much later than with regard to practical life and to science” (italics mine). Luitzen E. J. Brouwer, “Consciousness, Philosophy, and Mathematics,” in Philosophy of Mathematics , p. 78. Note the correlation that Brouwer makes between “life” and “science” on the one hand (expressing a religious world-view) and mathematics on the other. Elsewhere he acknowledges his philosophical debt to Kant, “Intuitionism and Formalism,” in ibid., p. 69.
9 Arend Heyting, “Disputation,” in Philosophy of Mathematics , p. 61.
10 Cf. Luitzen E. J. Brouwer, “Intuitionism and Formalism,” in ibid., p. 77, and Arend Heyting, “Disputation,” in ibid., p. 56, for intuitionistic discussion of questions similar to A and B.
11 For an evaluation of non-Euclidean geometry from a Christian viewpoint, cf. the discussion in Dirk H. Th. Vollenhoven, De Wijsbegeerte der Wiskunde van Theistisch Standpunt (Amsterdam: Wed. G. Van Soest, 1918), pp. 140-147.
12 Cf., for example, the discussion in John J. C. Smart, ed., Problems of Space and Time (New York: The Macmillan Company, 1964).
13 “Are we, following Poincaré, to attribute these findings [of physical experiment] to the influence of an external force postulated for the purpose? Or are we to take our findings at face value, and accept the geometry to which we are led as a natural geometry for physical science? The answer to this methodological question will depend largely on the universality of the geometry thus found-whether the geometry found in one situation or field of physical discourse may consistently be extended to others-and in the end partly on the predilection of the individual or of his colleagues or of his times.” Howard P. Robertson, “Geometry as a Branch of Physics,” in Paul A. Schilpp, ed., Albert Einstein: Philosopher-Scientist (Evanston,Ill.: The Library of Living Philosophers, Inc., 1949), p. 325.
14 Compare the discussion in §3.
15 Errett Bishop, Foundations of Constructive Analysis (New York: McGraw-Hill, 1967).
16 Abraham Robinson, Nonstandard Analysis (Amsterdam: North-Holland Publishing Company,1966).
17 Luitzen E. J. Brouwer, “Consciousness, Philosophy, and Mathematics,” in Philosophy of Mathematics , p. 79.
18 A History of Greek Philosophy , Vol. I, p. 265.
19 “Toute 1’entreprise de Leibniz consiste à créer une logique de 1’innni, dont toutes ses doctrines, mathématiques, physiques, métaphysiques, théologiques et morales, ne sont que des aspects divers.”-“The whole undertaking of Leibniz consists in creating a logic of infinity, of which all his doctrines, mathematical, physical,metaphysical, theological, and moral, are only diverse aspects” (italics mine). Emile Bréhier, Histoire de la Philosophie , Tome II (Paris: Presses Universitaires de France,1968), p. 210.
20 Vollenhoven, De Wijsbegeerte der Wiskunde , p. 188, and Herman Dooyeweerd, A New Critique of Theoretical Thought , Vol. II (Philadelphia: Presbyterian and Reformed Publishing Co., 1969), pp. 45, 87, 340. Dooyeweerd’s Christian philosophy considers antinomies to be a sure mark of speculative thought which does not recognize the limits of the creature (ibid., p. 38).
21 Here we follow Van Til’s practice (e.g., A Survey of Christian Epistemology,pp. 210-223) of describing non-Christian world-views, whether “neutral,” pantheistic, deistic, or atheistic, by their true colors. The only true theism is the one which worships and serves the true God, the Father of Jesus Christ (I John 2:22-24; II John 7-9), acknowledging Him as Lord of all (I Cor. 8:6; Eph. 1:21; Acts 10:36). All else is idolatry (Rom. 1:22-25). We will further explore the importance of a radically Biblical approach in §§19-26.
22 “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). Alfred Jules Ayer, “The A Priori,” in Philosophy of Mathematics , p. 295. This same article contains some discussion of Mill’s and Russell’s views of mathematical knowledge.
23 Ludwig Wittgenstein, Bemerkungen über die Grundlagen der Mathematik , herausgegeben und bearbeitet von G. H. von Wright, R. Rhees, G. E. M. Anscombe (Oxford: Basil Blackwell, 1967), pp. 4,6.
24 “The A Priori,” in Philosophy of Mathematics , p. 300.
25 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,” in Philosophy of Mathematics , p. 317.
26 Kurt Gödel, “Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I,” Monatshefte für Mathematik und Physik 38 (1931): 173-198; “On formally undecidable propositions of Principia mathematica and related systems”; English translation by B. Meltzer (Edinburgh and London: Oliver and Boyd, 1962). Cf. also Carnap’s generalization that every formal arithmetic is defective. Rudolf Carnap, The Logical Syntax of Language (New York: Harcourt,Brace and Company, 1937), §60d.
27 Kenneth L. Pike, Language in Relation to a Unified Theory of the Structure of Human Behavior , 2nd revised edition (The Hague-Paris: Mouton & Co., 1967).
28 Cornelius Van Til, A Survey of Christian Epistemology, pp. 200-201, 94-99, and The Defense of the Faith , 13-14, 41-46.
29 Albert Einstein, p. 285, quoted from Science, Philosophy and Religion: A Symposium (New York: Harper & Row, 1941). Similarly, E. P. Wigner calls this formulability of physical laws in mathematical terms an “article of faith” and a “miracle . . . which we neither understand nor deserve.” “The Unreasonable effectiveness of Mathematics in the Natural Sciences,” Communications on Pure and Applied Mathematics , 13 (I960): 10,14.
30 Dooyeweerd, A New Critique , Vol. II, pp. 47, 91, 103, 106, 385, etc.; Vollenhoven, De Wijsbegeerte der Wiskunde , pp. 20-138, 200-402; Vollenhoven, “Problemen en Richtingen in de Wijsbegeerte der Wiskunde,” Philosophia Reformata 1 (1936): 162-187; D. F. M. Strauss, “Number-Concept and Number-Idea,” Philosophia Reformata 35 (1970): 156-177 and 36 (1971): 13-42.
31 Cornelius Van Til, Christian Theistic Ethics , Vol. III of the series In Defense of Biblical Christianity (Philadelphia: den Dulk Christian Foundation, 1971).
32 Cornelius Van Til, An Introduction to Systematic Theology (Philadelphia: Westminster Theological Seminary, classroom syllabus, 1966), pp. 11-12; Dirk H.Th. Vollenhoven, Het Calvinisme en de Reformatie van de Wijsbegeerte (Amsterdam: H. J. Paris, 1933), pp. 50-51.
33 We do not believe that Jesus, or the great body of his Palestinian contemporaries, in any way thought of Ecclesiasticus, or other so-called apocryphal books,as the Word of God. Nevertheless, these passages are interesting because they show what kind of thinking was in the air on the question of wisdom, and they show the context in which Jews would have understood Jesus’ call to “come to him” and to “take his yoke.” Even if it could be shown that Jesus is specifically alluding to Ecclesiasticus (rather than merely to the Jewish wisdom tradition of which Sir. is one example), it would prove no more than Paul’s allusion to Greek poets in Acts 17:28.
34 The philosophy of the law-idea, or Amsterdam philosophy, while claiming to take a radically Christian stance, also falls victim to an old heresy-Sabellianism. According to Dooyeweerd, the numerical aspect, subject to mathematical laws, occurs only as one of the diverse aspects of the cosmic order of time. A New Critique , Vol. I,pp. 3, 24, 29, fn. 31-32. And cosmic time does not include God, the eternal One. Ibid., fn. p. 31. Hence numerical properties cannot be ascribed to God Himself. 1 + 1 + 1 = 3, as a theoretical statement, cannot speak about God. “The modal concepts of laws and of subject and object are essentially limited to a special aspect. Unlike the cosmonomic Idea, these modal concepts do not in themselves point beyond the diversity of meaning toward the transcendent origin and totality.” Ibid., p. 97. If these strictures are taken at face value, they lead to a decidedly Sabellian (modalistic) view of the Trinity. Cf. the author’s paper, “Sabellianism in the Philosophy of the Law-Idea,” Philosophia Reformata (to appear).
35 J. M. Spier, An Introduction to Christian Philosophy , 2nd ed. (Nutley, N. J.: The Craig Press, 1966), pp. 30-130. Spier popularizes the taxonomy of A New Critique ,Vol. II. Cf. also Vern Poythress, “An Approach to Evangelical Philosophy of Science,” Th.M. thesis, Westminster Theological Seminary, 1974.
36 We are not here speaking of heaven in the traditional sense, since heaven is a created place (Acts 4:24; Neh. 9:6). Since God made everything outside of Himself (Col. 1:16), His eternal habitation can be nothing else than God himself. Because, in Trinitarian fellowship, He is Himself a perfect “habitation,” He has no need of creating in order to make for Himself a habitation. Indeed, we might wonder whether the quotation marks should be placed around our “habitations,”rather than His: His Habitation is the Original.
37 The Defense of the Faith , pp. 154, 159.
38 For an extended discussion of motive, standard, and goal, cf. Christian Theistic Ethics.

References: §7
 §14
 §19
 §23
 §2
 §26
 §17
 §17
 §12
 §26
 §18
 §3
 §60