We must not omit to add that these basic determinations of meaning are largely incommunicable by specification, definition or proof, and in their deeper import must be reached by feeling, experience and intuition. There is a distinction, rarely appreciated as it should be, between experience as lived and experience as learned (zwischen Erleben und Erkennen), between the immediate certainty given by the various kinds of intuition—such as illumination, inspiration, artistic flair, experience of life, the power of “sizing men up” (Goethe’s “exact percipient fancy”)—and the product of rational procedure and technical experiment.
The first are imparted by means of analogy, picture, symbol, the second by formula, law, scheme. The become is experienced by learning—indeed, as we shall see, the having-become is for the human mind identical with the completed act of cognition. A becoming, on the other hand, can only be experienced by living, felt with a deep wordless understanding. It is on this that what we call “knowledge of men” is based; in fact the understanding of history implies a superlative knowledge of men. The eye which can see into the depths of an alien soul—owes nothing to the cognition-methods investigated in the “Critique of Pure Reason,” yet the purer the historical picture is, the less accessible it becomes to any other eye. The mechanism of a pure nature-picture, such as the world of Newton and Kant, is cognized, grasped, dissected in laws and equations and finally reduced to system: the organism of a pure history-picture, like the world of Plotinus, Dante and Giordano Bruno, is intuitively seen, inwardly experienced, grasped as a form or symbol and finally rendered in poetical and artistic conceptions. Goethe’s “living nature” is a historical world-picture.[[44]]
II
In order to exemplify the way in which a soul seeks to actualize itself in the picture of its outer world—to show, that is, in how far Culture in the “become” state can express or portray an idea of human existence—I have chosen number, the primary element on which all mathematics rests. I have done so because mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creations of the mind. It is a science of the most rigorous kind, like logic but more comprehensive and very much fuller; it is a true art, along with sculpture and music, as needing the guidance of inspiration and as developing under great conventions of form; it is, lastly, a metaphysic of the highest rank, as Plato and above all Leibniz show us. Every philosophy has hitherto grown up in conjunction with a mathematic belonging to it. Number is the symbol of causal necessity. Like the conception of God, it contains the ultimate meaning of the world-as-nature. The existence of numbers may therefore be called a mystery, and the religious thought of every Culture has felt their impress.[[45]]
Just as all becoming possesses the original property of direction (irreversibility), all things-become possess the property of extension. But these two words seem unsatisfactory in that only an artificial distinction can be made between them. The real secret of all things-become, which are ipso facto things extended (spatially and materially), is embodied in mathematical number as contrasted with chronological number. Mathematical number contains in its very essence the notion of a mechanical demarcation, number being in that respect akin to word, which, in the very fact of its comprising and denoting, fences off world-impressions. The deepest depths, it is true, are here both incomprehensible and inexpressible. But the actual number with which the mathematician works, the figure, formula, sign, diagram, in short the number-sign which he thinks, speaks or writes exactly, is (like the exactly-used word) from the first a symbol of these depths, something imaginable, communicable, comprehensible to the inner and the outer eye, which can be accepted as representing the demarcation. The origin of numbers resembles that of the myth. Primitive man elevates indefinable nature-impressions (the “alien,” in our terminology) into deities, numina, at the same time capturing and impounding them by a name which limits them. So also numbers are something that marks off and captures nature-impressions, and it is by means of names and numbers that the human understanding obtains power over the world. In the last analysis, the number-language of a mathematic and the grammar of a tongue are structurally alike. Logic is always a kind of mathematic and vice versa. Consequently, in all acts of the intellect germane to mathematical number—measuring, counting, drawing, weighing, arranging and dividing[[46]]—men strive to delimit the extended in words as well, i.e., to set it forth in the form of proofs, conclusions, theorems and systems; and it is only through acts of this kind (which may be more or less unintentioned) that waking man begins to be able to use numbers, normatively, to specify objects and properties, relations and differentiæ, unities and pluralities—briefly, that structure of the world-picture which he feels as necessary and unshakable, calls “Nature” and “cognizes.” Nature is the numerable, while History, on the other hand, is the aggregate of that which has no relation to mathematics—hence the mathematical certainty of the laws of Nature, the astounding rightness of Galileo’s saying that Nature is “written in mathematical language,” and the fact, emphasized by Kant, that exact natural science reaches just as far as the possibilities of applied mathematics allow it to reach. In number, then, as the sign of completed demarcation, lies the essence of everything actual, which is cognized, is delimited, and has become all at once—as Pythagoras and certain others have been able to see with complete inward certitude by a mighty and truly religious intuition. Nevertheless, mathematics—meaning thereby the capacity to think practically in figures—must not be confused with the far narrower scientific mathematics, that is, the theory of numbers as developed in lecture and treatise. The mathematical vision and thought that a Culture possesses within itself is as inadequately represented by its written mathematic as its philosophical vision and thought by its philosophical treatises. Number springs from a source that has also quite other outlets. Thus at the beginning of every Culture we find an archaic style, which might fairly have been called geometrical in other cases as well as the Early Hellenic. There is a common factor which is expressly mathematical in this early Classical style of the 10th Century B.C., in the temple style of the Egyptian Fourth Dynasty with its absolutism of straight line and right angle, in the Early Christian sarcophagus-relief, and in Romanesque construction and ornament. Here every line, every deliberately non-imitative figure of man and beast, reveals a mystic number-thought in direct connexion with the mystery of death (the hard-set).
Gothic cathedrals and Doric temples are mathematics in stone. Doubtless Pythagoras was the first in the Classical Culture to conceive number scientifically as the principle of a world-order of comprehensible things—as standard and as magnitude—but even before him it had found expression, as a noble arraying of sensuous-material units, in the strict canon of the statue and the Doric order of columns. The great arts are, one and all, modes of interpretation by means of limits based on number (consider, for example, the problem of space-representation in oil painting). A high mathematical endowment may, without any mathematical science whatsoever, come to fruition and full self-knowledge in technical spheres.
In the presence of so powerful a number-sense as that evidenced, even in the Old Kingdom,[[47]] in the dimensioning of pyramid temples and in the technique of building, water-control and public administration (not to mention the calendar), no one surely would maintain that the valueless arithmetic of Ahmes belonging to the New Empire represents the level of Egyptian mathematics. The Australian natives, who rank intellectually as thorough primitives, possess a mathematical instinct (or, what comes to the same thing, a power of thinking in numbers which is not yet communicable by signs or words) that as regards the interpretation of pure space is far superior to that of the Greeks. Their discovery of the boomerang can only be attributed to their having a sure feeling for numbers of a class that we should refer to the higher geometry. Accordingly—we shall justify the adverb later—they possess an extraordinarily complicated ceremonial and, for expressing degrees of affinity, such fine shades of language as not even the higher Cultures themselves can show.
There is analogy, again, between the Euclidean mathematic and the absence, in the Greek of the mature Periclean age, of any feeling either for ceremonial public life or for loneliness, while the Baroque, differing sharply from the Classical, presents us with a mathematic of spatial analysis, a court of Versailles and a state system resting on dynastic relations.
It is the style of a Soul that comes out in the world of numbers, and the world of numbers includes something more than the science thereof.