The Two-Fold Nature of Mathematics
The Writings declare that all things in God's creation are wrought by the conjunction or marriage of good and truth.
An angel teaching in the spiritual world is quoted:
It is important that we come to understand more clearly what the conjunction of good and truth is like, as this quotation emphasizes (italics added):
The purpose of this paper is to show that mathematics has a twofold nature that exemplifies the marriage of good and truth. I hope that this example may serve to increase our knowledge of how good and truth are conjoined. An interesting passage in Arcana Coelestia says
My hope concerning this paper is that the things which are of heaven (the marriage of good and truth) may appear as in a mirror, or in a certain representative image, in the things which are of natural light (the two-fold nature of mathematics).
The study and development of mathematics involve the interplay of two distinct kinds of mental processes. One of these I shall refer to as the intuitive process or intuition. Its function is to admit into the imagination some relationship or concept that is as yet tenuous, vague, amorphous and undefined. This mental process is, in mathematical creation, the analog of good.
The analog of truth is the mental activity that I shall call the formal process. It entails the creation of distinct, precise thought patterns that cause the intuitive concept to stand forth with greater clarity in our understanding, and it provides the rigorous logic necessary to demonstrate the validity of a newly discovered relationship. Thus it gives form to what had been amorphous and brings it forth to view. It makes the intuitive entity visible to the understanding, as truth makes good visible.
The work of the mathematician is to discover new relationships between mathematical concepts and to invent new concepts that will enhance mathematical knowledge. This work involves both the intuitive and the formal process. The thought of a new relationship enters the mind by an intuitive act, but it must then be proved by a rigorous logical demonstration so that no doubt remains about its correctness and so that others can also see its validity. Then, when the intuitively perceived relationship is made manifest by its proof, it attains the status of a theorem. Similarly, an entirely new concept may come to the mind of a mathematician by an act of intuition, but then the formal process must furnish this concept with a precise definition so that it can be treated with mathematical rigor.
The hard labor of mathematics is making the formalism fit the intuition; that is, devising the precise definitions and the rigorous proofs that are required to validate the concepts and theorems. This labor may be analogous to the temptations by which good and truth are conjoined. In Arcana Coelestia 4572 we read, "All conjunction of good with truth is effected by means of temptations." And just as temptations are followed by consolations, so the labor of mathematics is followed by a joyful sense of satisfaction. When his concept has been clearly defined or his theorem elegantly proved, the mathematician is elated. "All conjunction of good and truth has joy in it, since it is the heavenly marriage in which is the Divine" (AC 4572).
Although each thing in the universe has relation to both good and truth, certain things have a special relation to one or the other of these. For example, light has a special relation to truth and heat to good, and similarly with wine and bread. The case is the same with mathematics and its two fundamental processes every part of mathematics entails both of them, but still certain parts have a special relation to one or the other. Algebra is specially related to the formal process, and geometry (using the term in a broad sense to include topology) is specially related to the intuitive process. The area of mathematics known as analysis is a union, or marriage, of algebra and geometry, and thus it enjoys a certain fullness that has kept it in the mainstream of mathematical thought for centuries. In the present century algebra and geometry are being conjoined at a higher level of abstraction to create such subjects as algebraic topology.
In regard to the various levels of mathematical abstraction, it is interesting to view the historical development of mathematics from the point of view of its two-fold nature. The table shown divides the subject matter of mathematics into two parts: one part bearing a special relation to the formal process and the other part bearing a special relation to the intuitive process. It also separates the subject matter into four levels of abstraction, the lowest level being at the bottom and the highest at the top.
The boxes in the table are numbered 1 through 8. Proceeding through the table in this order gives one a roughly chronological outline of the historical development of mathematics. Boxes 1, 2 and. 3 refer to developments that are pre-historic, but commonly accepted theories : of the primitive origins of mathematics agree chronologically with the numerical order of these boxes. It seems likely that tangible collections, such as fruits in a basket, sheep in a flock, or people at a gathering could be conceptualized (box 1) before there was a conscious conception of such continuous magnitudes as length or duration (box 2). And it was probably still later that numerical concepts were totally abstracted from tangible objects to obtain numbers and the counting process (box 3), which entails a higher level of abstract thought. However that may be, the earliest civilizations from which there is extensive archeological evidence had numbers for counting and at least some capability for computing with fractions. Thus they were at the mathematical stage represented by box 3.
The content of box 4 is the mathematics that originated in ancient Greece at the dawn of Western culture.. Thales is perhaps the earliest historical name that can be associated with it. It culminated in the definitive work Elements by Euclid about 300 B.C. This mathematics lives today as high school geometry.
Box 5 refers to mathematics developed during the Middle Ages in different parts of the civilized world-especially the Arabic parts. Notation was invented for writing and solving abstract algebraic equations, which led to the extension of the number system. This type of mathematics is studied today as high school algebra.
The mathematics of box 6 originated in the early nineteenth century with the publication of the revolutionary geometrical ideas of Lobaschevsky and Bolyai. They led to the realization that geometry need not be tied to the space of the physical universe and that the mathematician can invent all manner of different abstract "spaces" in which to work. This box includes a flood of new geometrical abstractions that is still rising today.
Box 7 can hardly be separated chronologically from box 6 because their contents have been developing simultaneously. If we regard Hamilton's discovery of quaternions (the first noncommutative algebra) as the initial impetus for the mathematics of box 7, then it trailed the start of box 6 by less than a quarter century. In any case, box 7 is related to box 5 in the same way that box 6 is related to box 4: both are sweeping abstractions of the concepts of their lower-level predecessors. The contents of these two boxes form a large part of current university mathematics.
Finally we arrive at box 8. Its content, category theory, is quite different from the content of any previous box. Category theory is a kind of super abstraction from mathematics as a whole. It can be thought of as a mathematics of mathematics. The entities with which it deals are whole classes (or categories) of mathematical structures and mappings (called functors) from one such class to another. Algebraic topology, for example, from the viewpoint of category theory, is the study of functors from the category of topological spaces to the category of groups. Historically category theory originated about thirty years ago. It has served to organize and unify the mathematics of boxes 6 and 7.
Having finished this brief outline of the history of mathematics, we must now return to the table to consider the significance of the three double-headed arrows that link boxes 2 and 3, 4 and 5, 6 and 7. This triple pairing of an intuitive box with a formal box is the most important feature of the table. Each of these pairs -denotes a conjunction of, mathematical concepts that has been or, in the case of the last pair, will be of extreme importance in the intellectual history of mankind, and each is analogous to the conjunction of good and truth.
Let us consider first the 2-3 pair. The conjunction of the intuitive concept of continuous magnitude (box 2) with the formal concept of number (box 3) resulted in the new and tremendously fruitful concept of measurement. This concept was already well formulated prior to 500 B.C. by the Pythagoreans, who eagerly sought to apply it to every aspect of learning. Although their endeavors encountered theoretical difficulties because of the inadequacy of their number system (they had only the rational numbers), the concept of measurement was much too useful to be discarded. It thrived in practical applications until such time as an. adequate number system was developed. For the more sophisticated mathematicians, of course, the soundness of the concept was assured by the work of Eudoxus, and Archimedes brought it to a pinnacle of success. It continues to play such a basic role in our culture that what we call civilization would be impossible without it.
The next pair of conjoined boxes, 4 and 5, contain respectively the idealizations of Euclidean geometry and the formalism of high school algebra. Perhaps the most direct symbol of the conjunction of these two kinds of mathematics (again; of course, one is intuitive and the other formal) is the Cartesian coordinate system of analytic geometry, a device that enables the student to translate from algebra to geometry and vice versa. But the major strand in the connecting bond was the work of Newton and those who followed him in combining the two kinds of mathematics into the rate-of-change concept of calculus. The resulting mathematics was capable of expressing the "laws of motion," the "law of gravitation," and all the other scientific "laws" on which our science and technology are founded. Its effect on the- world has been enormous, and it is still growing.
The last two paired boxes, 6 and 7, are presently in the process of being conjoined. It is too early to tell what will be the nature of their union, what will be their most significant offspring, or what will be their effect on the history of mankind. Some consequences of their union are already at hand: for example algebraic topology, which we have alluded to before, and algebraic geometry. Also there are some hints about the way in which this mathematics will affect our conception of physical reality. The modern formulation of differential geometry, a product of the union of boxes 6 and 7, provides the mathematical' expression of relativity theory, an idea that has fundamentally changed our conception of space and time. Another product of this union is the recently devised branch of mathematics called catastrophe theory, which permits mathematical analysis of sudden drastic changes in contrast to the gradual changes dealt with by calculus and. classical analysis. This mathematical theory may also alter our conception of the natural world. But the general consequences of the union of boxes 6 and 7 are not yet foreseeable.
It is an obvious characteristic of each of the three pairs of conjoined boxes that the box on the formal side of the table is on a higher level of abstraction than its mate on the intuitive side. This seems to be an accurate reflection of the corresponding characteristic of the marriage of good and truth.
The mathematics that is produced by the union of the contents of two paired boxes I shall call, for lack of a better term, conjunctive mathematics. Now; if we tilt the table slightly so that the arrows become horizontal, as shown in the diagram, we can regard the three arrows as denoting three discrete degrees of conjunctive mathematics. Boxes 1 and 8 then lie respectively below find above the intermediate body of mathematics, which seems entirely appropriate in view of their contents. For conceptualizing tangible collections (box 1) is a submathematical activity, and category theory (box 8) is a supramathematical activity.
The appearance of three discrete degrees of anything is of course suggestive to any student of the Writings. I shall refrain from speculation in this paper about the significance of the three degrees of conjunctive mathematics. I hope that the reader will be enticed to think about them, and that he will come to see that spiritual truths are reflected in the natural organization of mathematics.
-The New Philosophy 1977;80:63-69