I recently set for myself what I thought would be the simple task of cataloguing my copies of mathematics journal articles by topic, since they are arranged in the drawers by author. Immediately the difficulty of the task becomes apparent. Usually the adjective describes the methods used, and the noun describes the subject matter being discussed. Thus algebraic geometry is geometry in which such algebraic concepts as rings and ideals are the tools; geometric algebra is algebra in which the visual aspect of such notions as orthogonality guides the intuitions. Are Hilbert spaces algebraic or geometric? Spectral theory spans both the discrete and the continuous. Do topological proofs of theorems in mathematical logic get filed under topology or under logic?

It is to some of these same questions that Kuyk addresses himself, not as a librarian but as a philosopher. His answer in a phrase is that subjects within mathematics can never be parceled out into neat packages, and that always there will be the discrete, the continuous, and the interplay between them. We must view mathematics in such a way that different aspects of it are seen to complement each other, hence the title of his book, Complementarity in Mathematics. In so doing, he reaches deep into the work of the reformed philosopher Herman Dooyeweerd, and touches on themes that have cropped up in such diverse places as linguistics, artificial intelligence and semantics.

Dooyeweerd's philosophy includes the notion that there are "spheres" of knowledge--the numerical, spatial, kinematic, physical, biological, psychical, logical (or theoretical), historical, linguistic, social, economic, aesthetic, juridical, ethical, and faith dimensions. Three of these modalities, as they are often called, are of immediate importance for the mathematician: the numerical, the spatial, and the logical. Kuyk does not discuss the latter on the same level as the first two, but not because the theoretical subsumes all of the other spheres. Such hubris Dooyeweerd calls the pretended autonomy of theoretical thought. Dooyeweerd argues that these spheres are irreducible and complementary: irreducible in that no one of them can be explained in terms of the others; complementary in that they all apply in any given situation. Kuyk suggests that the discrete/continuous dichotomy and the numerical-algebraic/geometric dichotomy are just two of the ways that mathematicians in practice show that Dooyeweerd was right. The theoretical sphere of knowledge comes in, if I understand Kuyk rightly, via the interplay of the two. Thus for Kuyk "set theory logic" and category theory stand outside this polarity. I would imagine that Kuyk considers them to be in the theoretical sphere. For a specific mathematical example, I would see compactness arguments as being among those that bridge the gap between the discrete and the continuous. There are others who are working on a Dooyeweerdian approach to the philosophy of mathematics. See for example Poythress (1974).

I am immediately struck by the fact that a secular publisher with a prestigious editorial board has published a philosophy of mathematics with clear roots in a Christian world-view. It is true that there are secular critiques of theoretical thought as well that share Dooyeweerd's view insofar as it bears on Kuyk.*

Ever since the ancient Greeks fell in love with geometry, philosophical thought about the nature of knowledge has been dominated by models derived from mathematics and theoretical physics. This fact has had two regrettable consequences. On the one hand--to stand Whitehead's epigram on its head--it has doomed the whole of subsequent philosophy to being a series of footnotes to Plato. On the other hand, it has tempted philosophers to concentrate on questions of logical form, to the neglect of questions about rational function and intellectual adaptation.

So, too, Kuyk has occupied himself with questions about the "rational function and intellectual adaptation" of mathematics. This Kuyk carries further in a more recent paper (see references).

But further when one looks about for other examples of this philosophy of complementarity, one finds again that it is the Christians that are defending the position. Kenneth L. Pike regards in one of his earlier papers the necessity of linguistics dealing with speech acts as irreducibly and complementarily "particle, wave, and field" (Pike, 1959). That is, as the discrete, the continuous, and their interaction. Long before linguists in general began to recognize the value of going beyond the syntax and beyond the sentence in their grammatical analysis, Pike refused to take such a narrow view. As early as 1959 in his Language in Relation to a Unified Theory of the Structure of Human Behavior, which at 761 pages is affectionately called the "blue doorstop" in our household, Pike maintained (1967:588):

In my view, a principle of complementarity, in which a description must eventually be repeated from the viewpoints of particle, wave, and field, must be given priority over a single approach which may be simplistic in its outline, and in its non-overlapping phases, but is less able to treat various phases of a system.

In his most recent book (1977), he is able to cite even more evidence that such a complementarist view is important at every level of grammatical analysis. Now the idea is more fashionable (cf., e.g., Keesing, 1979).

Before going on to my other two examples, there is an important but tangential point to be made in connection with this last comment on the fashionableness of ideas. Kuhn in science and Lakatos in mathematics specifically have argued the case that there is an important social component in the development of any science in which not only do the theorems change, but the definitions change at the same time through a historical process of periods of revolution and other periods of successive refinement. I take it from the fact that Kuyk cites both Kuhn and Lakatos favorably that he sees these views as congenial with his own. At least it is clear that these views do support the view of Dooyeweerd that on the one hand even the most abstract theory has a social component, and on the other hand even the most widely respected and broadly based theory has no claim to any autonomy. To tie the whole matter in with Kuyk by way of a metacomment, if Kuhn is right that paradigms are shifting, then the time is ripe for a paradigm of mathematics such as that of Kuyk which highlights shifting paradigms.

For my second example of the way in which a complementarist viewpoint has been marshalled in support of a way of looking at a discipline, I cite Donald MacKay. MacKay is probably best known to the Christian community of this country as the author of The Clockwork Image: A Christian Perspective on Science. He argues that to explain how the mind works in a mechanistic way is not to explain it away. Similarly, one is correct to say while pointing to a blackboard, "those are chalk marks," and one is equally correct to say "that is the law of Pythagoras." In Dooyeweerd's terms, the logical is not reducible to the physical, neither is the physical reducible to the logical. They stand as complementary explanations of the same event.

I have noted a similar phenomenon in the theory of semantics. In observing a child's development, one finds that the same event is viewed at one time as a process and at another time as a thing. The two concepts of thing and process are fundamental and irreducible to each other. The distinction persists in language as the noun/verb distinction. So for example words like chair and meet can be viewed as both noun and verb:

He chaired the meeting.
He ran in the meeting.
He was in the running.
He met the chair before the business meeting began.

Workers in artificial intelligence discuss this distinction in terms of procedural and declarative knowledge, or more simply the distinction between a computer program and its data. But ever since von Neumann, it has been recognized that a program can be data or the data can be a program. One can treat a concept as a noun and reference it as a single concept without analysis, or one can treat the same concept as a process, in which case one analyzes it into a sequence of steps. That these two views are complementary is developed further in my dissertation (see references). Perhaps all of modern philosophy is doomed to be footnotes to Aristotle as well as Plato!

In this short essay I have attempted to show that Kuyk's view of mathematics is rooted in a theological framework, that it parallels what other Christians have written about in their disciplines without consciously drawing on that same theological framework, and that the time is ripe for such a viewpoint. As a concluding unscientific postscript, I personally find Kuyk's viewpoint congenial because I tend to be a synthesizer. "Stop! Stop! You're all right!" I'm always anxious to claim. Whether this stems from a philosophical commitment, the desire to be a peacemaker, or just the teacher in me is beyond the scope of this discussion.

REFERENCES

Gene B. Chase. An information-processing model for mathematics education. Ph.D. dissertation. Cornell University, 1979.

_____. Skolem's paradox and the predestination/free-will discussion. A Christian Perspective on the Foundations of Mathematics, proceedings of the conference held at Wheaton College, April 28-30, 1977. 75-81. 1977.

Herman Dooyeweerd. A New Critique of Theoretical Thought. Philadelphia. 1953.

Roger M. Keesing. Linguistic knowledge and cultural knowledge: some doubts and speculations. American Anthropologist, 81, 1, 14-36, 1979.

Thomas S. Kuhn. The Structure of Scientific Revolutions. Chicago: University Press, 1962.

Willem Kuyk. Complementarity in Mathematics. Boston: Reidel, 1977.

_____. Dynamic variegations of mathematical development (A complementarist approach). To appear, 1979 (?).

_____. The irreducibility of the number concept. Ms.

Imre Lakatos. Proofs and Refutations. Cambridge: University Press, 1976.

Donald M. MacKay. Choice in a mechanistic universe: a reply to some critics. British Journal for the Philosophy of Science, 22, 275-285, 1971.

_____. The Clockwork Image: A Christian Perspective on Science. Downers Grove: Inter-Varsity Press, 1974.

_____. Freedom of Action in a Mechanistic Universe. Cambridge: University Press, 967.

_____. The logical indeterminateness of human choices. British Journal for the Philosophy of Science, 24, 405-408, 1973.

Kenneth L. Pike. Language as Particle, Wave, and Field. The Texas Quarterly, 2, 2, 37-54. 1959.

_____. Language in Relation to a Unified Theory of the Structure of Human Behavior. The Hague: Mouton, 1967. (Preliminary edition, 1959).

_____ and Evelyn G. Pike. Grammatical Analysis. Summer Institute of Linguistics Publications 53. Dallas: Summer Institute of Linguistics, 1977.

Vern S. Poythress. Creation and mathematics; or what does God have to do with numbers? The Journal of Christian Reconstruction, l, l, 1974.

Stephen Toulmin. Human Understanding: the Collective Use and Evolution of Concepts. Princeton: University Press, 1972.