The Meaning of Numbers

It is necessary to begin by drawing attention to certain basic terms which, as used in this work, carry strict and in some cases novel connotations. Though the metaphysical content of these terms would gradually become evident in following the course of the reasoning, nevertheless, the exact significance to be attached to them ought to be made clear beyond misunderstanding from the very outset.

The popular distinction—current also in philosophy—between “being” and “becoming” seems to miss the essential point in the contrast it is meant to express. An endless becoming—“action,” “actuality”—will always be thought of also as a condition (as it is, for example, in physical notions such as uniform velocity and the condition of motion, and in the basic hypothesis of the kinetic theory of gases) and therefore ranked in the category of “being.” On the other hand, out of the results that we do in fact obtain by and in consciousness, we may, with Goethe, distinguish as final elements “becoming” and “the become” (Das Werden, das Gewordne). In all cases, though the atom of human-ness may lie beyond the grasp of our powers of abstract conception, the very clear and definite feeling of this contrast—fundamental and diffused throughout consciousness—is the most elemental something that we reach. It necessarily follows therefore that “the become” is always founded on a “becoming” and not the other way round.

I distinguish further, by the words “proper” and “alien” (das Eigne, das Fremde), those two basic facts of consciousness which for all men in the waking (not in the dreaming) state are established with an immediate inward certainty, without the necessity or possibility of more precise definition. The element called “alien” is always related in some way to the basic fact expressed by the word “perception,” i.e., the outer world, the life of sensation. Great thinkers have bent all their powers of image-forming to the task of expressing this relation, more and more rigorously, by the aid of half-intuitive dichotomies such as “phenomena and things-in-themselves,” “world-as-will and world-as-idea,” “ego and non-ego,” although human powers of exact knowing are surely inadequate for the task.

Similarly, the element “proper” is involved with the basic fact known as feeling, i.e., the inner life, in some intimate and invariable way that equally defies analysis by the methods of abstract thought.

I distinguish, again, “soul” and “world.” The existence of this opposition is identical with the fact of purely human waking consciousness (Wachsein). There are degrees of clearness and sharpness in the opposition and therefore grades of the consciousness, of the spirituality, of life. These grades range from the feeling-knowledge that, unalert yet sometimes suffused through and through by an inward light, is characteristic of the primitive and of the child (and also of those moments of religious and artistic inspiration that occur ever less and less often as a Culture grows older) right to the extremity of waking and reasoning sharpness that we find, for instance, in the thought of Kant and Napoleon, for whom soul and world have become subject and object. This elementary structure of consciousness, as a fact of immediate inner knowledge, is not susceptible of conceptual subdivision. Nor, indeed, are the two factors distinguishable at all except verbally and more or less artificially, since they are always associated, always intertwined, and present themselves as a unit, a totality. The epistemological starting-point of the born idealist and the born realist alike, the assumption that soul is to world (or world to soul, as the case may be) as foundation is to building, as primary to derivative, as “cause” to “effect,” has no basis whatever in the pure fact of consciousness, and when a philosophic system lays stress on the one or the other, it only thereby informs us as to the personality of the philosopher, a fact of purely biographical significance.

Thus, by regarding waking-consciousness structurally as a tension of contraries, and applying to it the notions of “becoming” and “the thing-become,” we find for the word Life a perfectly definite meaning that is closely allied to that of “becoming.” We may describe becomings and the things-become as the form in which respectively the facts and the results of life exist in the waking consciousness. To man in the waking state his proper life, progressive and constantly self-fulfilling, is presented through the element of Becoming in his consciousness—this fact we call “the present”—and it possesses that mysterious property of Direction which in all the higher languages men have sought to impound and—vainly—to rationalize by means of the enigmatic word time. It follows necessarily from the above that there is a fundamental connexion between the become (the hard-set) and Death.

If, now, we designate the Soul—that is, the Soul as it is felt, not as it is reasonably pictured—as the possible and the World on the other hand as the actual (the meaning of these expressions is unmistakable to man’s inner sense), we see life as the form in which the actualizing of the possible is accomplished. With respect to the property of Direction, the possible is called the Future and the actualized the Past. The actualizing itself, the centre-of-gravity and the centre-of-meaning of life, we call the Present. “Soul” is the still-to-be-accomplished, “World” the accomplished, “life” the accomplishing. In this way we are enabled to assign to expressions like moment, duration, development, life-content, vocation, scope, aim, fullness and emptiness of life, the definite meanings which we shall need for all that follows and especially for the understanding of historical phenomena.

Lastly, the words History and Nature are here employed, as the reader will have observed already, in a quite definite and hitherto unusual sense. These words comprise possible modes of understanding, of comprehending the totality of knowledge—becoming as well as things-become, life as well as things-lived—as a homogeneous, spiritualized, well-ordered world-picture fashioned out of an indivisible mass-impression in this way or in that according as the becoming or the become, direction (“time”) or extension (“space”) is the dominant factor. And it is not a question of one factor being alternative to the other. The possibilities that we have of possessing an “outer world” that reflects and attests our proper existence are infinitely numerous and exceedingly heterogeneous, and the purely organic and the purely mechanical world-view (in the precise literal sense of that familiar term⁴²) are only the extreme members of the series. Primitive man (so far as we can imagine his waking-consciousness) and the child (as we can remember) cannot fully see or grasp these possibilities. One condition of this higher world-consciousness is the possession of language, meaning thereby not mere human utterance but a culture-language, and such is non-existent for primitive man and existent but not accessible in the case of the child. In other words, neither possesses any clear and distinct notion of the world. They have an inkling but no real knowledge of history and nature, being too intimately incorporated with the ensemble of these. They have no Culture.

And therewith that important word is given a positive meaning of the highest significance which henceforward will be assumed in using it. In the same way as we have elected to distinguish the Soul as the possible and the World as the actual, we can now differentiate between possible and actual culture, i.e., culture as an idea in the (general or individual) existence and culture as the body of that idea, as the total of its visible, tangible and comprehensible expressions—acts and opinions, religion and state, arts and sciences, peoples and cities, economic and social forms, speech, laws, customs, characters, facial lines and costumes. Higher history, intimately related to life and to becoming, is the actualizing of possible Culture.⁴³

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” 56(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

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 dividing⁴⁶—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.

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From this there follows a fact of decisive importance which has hitherto been hidden from the mathematicians themselves.

There is not, and cannot be, number as such. There are several number-worlds as there are several Cultures. We find an Indian, an Arabian, a Classical, a Western type of mathematical thought and, corresponding with each, a type of number—each type fundamentally peculiar and unique, an expression of a specific world-feeling, a symbol having a specific validity which is even capable of scientific definition, a principle of ordering the Become which reflects the central essence of one and only one soul, viz., the soul of that particular Culture. Consequently, there are more mathematics than one. For indubitably the inner structure of the Euclidean geometry is something quite different from that of the Cartesian, the analysis of Archimedes is something other than the analysis of Gauss, and not merely in matters of form, intuition and method but above all in essence, in the intrinsic and obligatory meaning of number which they respectively develop and set forth. This number, the horizon within which it has been able to make phenomena self-explanatory, and therefore the whole of the “nature” or world-extended that is confined in the given limits and amenable to its particular sort of mathematic, are not common to all mankind, but specific in each case to one definite sort of mankind.

The style of any mathematic which comes into being, then, depends wholly on the Culture in which it is rooted, the sort of mankind it is that ponders it. The soul can bring its inherent possibilities to scientific development, can manage them practically, can attain the highest levels in its treatment of them—but is quite impotent to alter them. The idea of the Euclidean geometry is actualized in the earliest forms of Classical ornament, and that of the Infinitesimal Calculus in the earliest forms of Gothic architecture, centuries before the first learned mathematicians of the respective Cultures were born.

A deep inward experience, the genuine awakening of the ego, which turns the child into the higher man and initiates him into community of his Culture, marks the beginning of number-sense as it does that of language-sense. It is only after this that objects come to exist for the waking consciousness as things limitable and distinguishable as to number and kind; only after this that properties, concepts, causal necessity, system in the world-around, a form of the world, and world laws (for that which is set and settled is ipso facto bounded, hardened, number-governed) are susceptible of exact definition. And therewith comes too a sudden, almost metaphysical, feeling of anxiety and awe regarding the deeper meaning of measuring and counting, drawing and form.

Now, Kant has classified the sum of human knowledge according to syntheses a priori (necessary and universally valid) and a posteriori (experiential and variable from case to case) and in the former class has included mathematical knowledge. Thereby, doubtless, he was enabled to reduce a strong inward feeling to abstract form. But, quite apart from the fact (amply evidenced in modern mathematics and mechanics) that there is no such sharp distinction between the two as is originally and unconditionally implied in the principle, the a priori itself, though certainly one of the most inspired conceptions of philosophy, is a notion that seems to involve enormous difficulties. With it Kant postulates—without attempting to prove what is quite incapable of proof—both unalterableness of form in all intellectual activity and identity of form for all men in the same. And, in consequence, a factor of incalculable importance is—thanks to the intellectual prepossessions of his period, not to mention his own—simply ignored. This factor is the varying degree of this alleged “universal validity.” There are doubtless certain characters of very wide-ranging validity which are (seemingly at any rate) independent of the Culture and century to which the cognizing individual may belong, but along with these there is a quite particular necessity of form which underlies all his thought as axiomatic and to which he is subject by virtue of belonging to his own Culture and no other. Here, then, we have two very different kinds of a priori thought-content, and the definition of a frontier between them, or even the demonstration that such exists, is a problem that lies beyond all possibilities of knowing and will never be solved. So far, no one has dared to assume that the supposed constant structure of the intellect is an illusion and that the history spread out before us contains more than one style of knowing. But we must not forget that unanimity about things that have not yet become problems may just as well imply universal error as universal truth. True, there has always been a certain sense of doubt and obscurity—so much so, that the correct guess might have been made from that non-agreement of the philosophers which every glance at the history of philosophy shows us. But that this non-agreement is not due to imperfections of the human intellect or present gaps in a perfectible knowledge, in a word, is not due to defect, but to destiny and historical necessity—this is a discovery. Conclusions on the deep and final things are to be reached not by predicating constants but by studying differentiæ and developing the organic logic of differences. The comparative morphology of knowledge forms is a domain which Western thought has still to attack.

If mathematics were a mere science like astronomy or mineralogy, it would be possible to define their object. This man is not and never has been able to do. We West-Europeans may put our own scientific notion of number to perform the same tasks as those with which the mathematicians of Athens and Baghdad busied themselves, but the fact remains that the theme, the intention and the methods of the like-named science in Athens and in Baghdad were quite different from those of our own. There is no mathematic but only mathematics. What we call “the history of mathematics”—implying merely the progressive actualizing of a single invariable ideal—is in fact, below the deceptive surface of history, a complex of self-contained and independent developments, an ever-repeated process of bringing to birth new form-worlds and appropriating, transforming and sloughing alien form-worlds, a purely organic story of blossoming, ripening, wilting and dying within the set period. The student must not let himself be deceived. The mathematic of the Classical soul sprouted almost out of nothingness, the historically-constituted Western soul, already possessing the Classical science (not inwardly, but outwardly as a thing learnt), had to win its own by apparently altering and perfecting, but in reality destroying the essentially alien Euclidean system. In the first case, the agent was Pythagoras, in the second Descartes. In both cases the act is, at bottom, the same.

The relationship between the form-language of a mathematic and that of the cognate major arts,⁴⁸ is in this way put beyond doubt. The temperament of the thinker and that of the artist differ widely indeed, but the expression-methods of the waking consciousness are inwardly the same for each. The sense of form of the sculptor, the painter, the composer is essentially mathematical in its nature. The same inspired ordering of an infinite world which manifested itself in the geometrical analysis and projective geometry of the 17th Century, could vivify, energize, and suffuse contemporary music with the harmony that it developed out of the art of thoroughbass, (which is the geometry of the sound-world) and contemporary painting with the principle of perspective (the felt geometry of the space-world that only the West knows). This inspired ordering is that which Goethe called “The Idea, of which the form is immediately apprehended in the domain of intuition, whereas pure science does not apprehend but observes and dissects.” The Mathematic goes beyond observation and dissection, and in its highest moments finds the way by vision, not abstraction. To Goethe again we owe the profound saying: “the mathematician is only complete in so far as he feels within himself the beauty of the true.” Here we feel how nearly the secret of number is related to the secret of artistic creation. And so the born mathematician takes his place by the side of the great masters of the fugue, the chisel and the brush; he and they alike strive, and must strive, to actualize the grand order of all things by clothing it in symbol and so to communicate it to the plain fellow-man who hears that order within himself but cannot effectively possess it; the domain of number, like the domains of tone, line and colour, becomes an image of the world-form. For this reason the word “creative” means more in the mathematical sphere than it does in the pure sciences—Newton, Gauss, and Riemann were artist-natures, and we know with what suddenness their great conceptions came upon them.⁴⁹ “A mathematician,” said old Weierstrass “who is not at the same time a bit of a poet will never be a full mathematician.”

The mathematic, then, is an art. As such it has its styles and style-periods. It is not, as the layman and the philosopher (who is in this matter a layman too) imagine, substantially unalterable, but subject like every art to unnoticed changes from epoch to epoch. The development of the great arts ought never to be treated without an (assuredly not unprofitable) side-glance at contemporary mathematics. In the very deep relation between changes of musical theory and the analysis of the infinite, the details have never yet been investigated, although æsthetics might have learned a great deal more from these than from all so-called “psychology.” Still more revealing would be a history of musical instruments written, not (as it always is) from the technical standpoint of tone-production, but as a study of the deep spiritual bases of the tone-colours and tone-effects aimed at. For it was the wish, intensified to the point of a longing, to fill a spatial infinity with sound which produced—in contrast to the Classical lyre and reed (lyra, kithara; aulos, syrinx) and the Arabian lute—the two great families of keyboard instruments (organ, pianoforte, etc.) and bow instruments, and that as early as the Gothic time. The development of both these families belongs spiritually (and possibly also in point of technical origin) to the Celtic-Germanic North lying between Ireland, the Weser and the Seine. The organ and clavichord belong certainly to England, the bow instruments reached their definite forms in Upper Italy between 1480 and 1530, while it was principally in Germany that the organ was developed into the space-commanding giant that we know, an instrument the like of which does not exist in all musical history. The free organ-playing of Bach and his time was nothing if it was not analysis—analysis of a strange and vast tone-world. And, similarly, it is in conformity with the Western number-thinking, and in opposition to the Classical, that our string and wind instruments have been developed not singly but in great groups (strings, woodwind, brass), ordered within themselves according to the compass of the four human voices; the history of the modern orchestra, with all its discoveries of new and modification of old instruments, is in reality the self-contained history of one tone-world—a world, moreover, that is quite capable of being expressed in the forms of the higher analysis.

When, about 540 B.C., the circle of the Pythagoreans arrived at the idea that number is the essence of all things, it was not “a step in the development of mathematics” that was made, but a wholly new mathematic that was born. Long heralded by metaphysical problem-posings and artistic form-tendencies, now it came forth from the depths of the Classical soul as a formulated theory, a mathematic born in one act at one great historical moment—just as the mathematic of the Egyptians had been, and the algebra-astronomy of the Babylonian Culture with its ecliptic co-ordinate system—and new—for these older mathematics had long been extinguished and the Egyptian was never written down. Fulfilled by the 2nd century A.D., the Classical mathematic vanished in its turn (for though it seemingly exists even to-day, it is only as a convenience of notation that it does so), and gave place to the Arabian. From what we know of the Alexandrian mathematic, it is a necessary presumption that there was a great movement within the Middle East, of which the centre of gravity must have lain in the Persian-Babylonian schools (such as Edessa, Gundisapora and Ctesiphon) and of which only details found their way into the regions of Classical speech. In spite of their Greek names, the Alexandrian mathematicians—Zenodorus who dealt with figures of equal perimeter, Serenus who worked on the properties of a harmonic pencil in space, Hypsicles who introduced the Chaldean circle-division, Diophantus above all—were all without doubt Aramæans, and their works only a small part of a literature which was written principally in Syriac. This mathematic found its completion in the investigations of the Arabian-Islamic thinkers, and after these there was again a long interval. And then a perfectly new mathematic was born, the Western, our own, which in our infatuation we regard as “Mathematics,” as the culmination and the implicit purpose of two thousand years’ evolution, though in reality its centuries are (strictly) numbered and to-day almost spent.

The most valuable thing in the Classical mathematic is its proposition that number is the essence of all things perceptible to the senses. Defining number as a measure, it contains the whole world-feeling of a soul passionately devoted to the “here” and the “now.” Measurement in this sense means the measurement of something near and corporeal. Consider the content of the Classical art-work, say the free-standing statue of a naked man; here every essential and important element of Being, its whole rhythm, is exhaustively rendered by surfaces, dimensions and the sensuous relations of the parts. The Pythagorean notion of the harmony of numbers, although it was probably deduced from music—a music, be it noted, that knew not polyphony or harmony, and formed its instruments to render single plump, almost fleshy, tones—seems to be the very mould for a sculpture that has this ideal. The worked stone is only a something in so far as it has considered limits and measured form; what it is is what it has become under the sculptor’s chisel. Apart from this it is a chaos, something not yet actualized, in fact for the time being a null. The same feeling transferred to the grander stage produces, as an opposite to the state of chaos, that of cosmos, which for the Classical soul implies a cleared-up situation of the external world, a harmonic order which includes each separate thing as a well-defined, comprehensible and present entity. The sum of such things constitutes neither more nor less than the whole world, and the interspaces between them, which for us are filled with the impressive symbol of the Universe of Space, are for them the nonent (τὸ μὴ ὅν).

Extension means, for Classical mankind body, and for us space, and it is as a function of space that, to us, things “appear.” And, looking backward from this standpoint, we may perhaps see into the deepest concept of the Classical metaphysics, Anaximander’s ἄπειρον—a word that is quite untranslatable into any Western tongue. It is that which possesses no “number” in the Pythagorean sense of the word, no measurable dimensions or definable limits, and therefore no being; the measureless, the negation of form, the statue not yet carved out of the block; the ἀρχὴ optically boundless and formless, which only becomes a something (namely, the world) after being split up by the senses. It is the underlying form a priori of Classical cognition, bodiliness as such, which is replaced exactly in the Kantian world-picture by that Space out of which Kant maintained that all things could be “thought forth.”

We can now understand what it is that divides one mathematic from another, and in particular the Classical from the Western. The whole world-feeling of the matured Classical world led it to see mathematics only as the theory of relations of magnitude, dimension and form between bodies. When, from out of this feeling, Pythagoras evolved and expressed the decisive formula, number had come, for him, to be an optical symbol—not a measure of form generally, an abstract relation, but a frontier-post of the domain of the Become, or rather of that part of it which the senses were able to split up and pass under review. By the whole Classical world without exception numbers are conceived as units of measure, as magnitude, lengths, or surfaces, and for it no other sort of extension is imaginable. The whole Classical mathematic is at bottom Stereometry (solid geometry). To Euclid, who rounded off its system in the third century, the triangle is of deep necessity the bounding surface of a body, never a system of three intersecting straight lines or a group of three points in three-dimensional space. He defines a line as “length without breadth” (μῆκος ἀπλατές). In our mouths such a definition would be pitiful—in the Classical mathematic it was brilliant.

The Western number, too, is not, as Kant and even Helmholtz thought, something proceeding out of Time as an a priori form of conception, but is something specifically spatial, in that it is an order (or ordering) of like units. Actual time (as we shall see more and more clearly in the sequel) has not the slightest relation with mathematical things. Numbers belong exclusively to the domain of extension. But there are precisely as many possibilities—and therefore necessities—of ordered presentation of the extended as there are Cultures. Classical number is a thought-process dealing not with spatial relations but with visibly limitable and tangible units, and it follows naturally and necessarily that the Classical knows only the “natural” (positive and whole) numbers, which on the contrary play in our Western mathematics a quite undistinguished part in the midst of complex, hypercomplex, non-Archimedean and other number-systems.

On this account, the idea of irrational numbers—the unending decimal fractions of our notation—was unrealizable within the Greek spirit. Euclid says—and he ought to have been better understood—that incommensurable lines are “not related to one another like numbers.” In fact, it is the idea of irrational number that, once achieved, separates the notion of number from that of magnitude, for the magnitude of such a number (π, for example) can never be defined or exactly represented by any straight line. Moreover, it follows from this that in considering the relation, say, between diagonal and side in a square the Greek would be brought up suddenly against a quite other sort of number, which was fundamentally alien to the Classical soul, and was consequently feared as a secret of its proper existence too dangerous to be unveiled. There is a singular and significant late-Greek legend, according to which the man who first published the hidden mystery of the irrational perished by shipwreck, “for the unspeakable and the formless must be left hidden for ever.”⁵⁰

The fear that underlies this legend is the selfsame notion that prevented even the ripest Greeks from extending their tiny city-states so as to organize the country-side politically, from laying out their streets to end in prospects and their alleys to give vistas, that made them recoil time and again from the Babylonian astronomy with its penetration of endless starry space,⁵¹ and refuse to venture out of the Mediterranean along sea-paths long before dared by the Phœnicians and the Egyptians. It is the deep metaphysical fear that the sense-comprehensible and present in which the Classical existence had entrenched itself would collapse and precipitate its cosmos (largely created and sustained by art) into unknown primitive abysses. And to understand this fear is to understand the final significance of Classical number—that is, measure in contrast to the immeasurable—and to grasp the high ethical significance of its limitation. Goethe too, as a nature-student, felt it—hence his almost terrified aversion to mathematics, which as we can now see was really an involuntary reaction against the non-Classical mathematic, the Infinitesimal Calculus which underlay the natural philosophy of his time.

Religious feeling in Classical man focused itself ever more and more intensely upon physically present, localized cults which alone expressed a college of Euclidean deities. Abstractions, dogmas floating homeless in the space of thought, were ever alien to it. A cult of this kind has as much in common with a Roman Catholic dogma as the statue has with the cathedral organ. There is no doubt that something of cult was comprised in the Euclidean mathematic—consider, for instance, the secret doctrines of the Pythagoreans and the Theorems of regular polyhedrons with their esoteric significance in the circle of Plato. Just so, there is a deep relation between Descartes’ analysis of the infinite and contemporary dogmatic theology as it progressed from the final decisions of the Reformation and the Counter-Reformation to entirely desensualized deism. Descartes and Pascal were mathematicians and Jansenists, Leibniz a mathematician and pietist. Voltaire, Lagrange and D’Alembert were contemporaries. Now, the Classical soul felt the principle of the irrational, which overturned the statuesquely-ordered array of whole numbers and the complete and self-sufficing world-order for which these stood, as an impiety against the Divine itself. In Plato’s “Timæus” this feeling is unmistakable. For the transformation of a series of discrete numbers into a continuum challenged not merely the Classical notion of number but the Classical world-idea itself, and so it is understandable that even negative numbers, which to us offer no conceptual difficulty, were impossible in the Classical mathematic, let alone zero as a number, that refined creation of a wonderful abstractive power which, for the Indian soul that conceived it as base for a positional numeration, was nothing more nor less than the key to the meaning of existence. Negative magnitudes have no existence. The expression -2×-3=+6 is neither something perceivable nor a representation of magnitude. The series of magnitudes ends with +1, and in graphic representation of negative numbers

( + 3 + 2 + 1 0 - 1 - 2 - 3 )

— ･ — ･ — ･ ･ — ･ — ･ — ･

we have suddenly, from zero onwards, positive symbols of something negative; they mean something, but they no longer are. But the fulfilment of this act did not lie within the direction of Classical number-thinking.

Every product of the waking consciousness of the Classical world, then, is elevated to the rank of actuality by way of sculptural definition. That which cannot be drawn is not “number.” Archytas and Eudoxus use the terms surface- and volume-numbers to mean what we call second and third powers, and it is easy to understand that the notion of higher integral powers did not exist for them, for a fourth power would predicate at once, for the mind based on the plastic feeling, an extension in four dimensions, and four material dimensions into the bargain, “which is absurd.” Expressions like εix which we constantly use, or even the fractional index (e.g., 5½) which is employed in the Western mathematics as early as Oresme (14th Century), would have been to them utter nonsense. Euclid calls the factors of a product its sides πλευραί and fractions (finite of course) were treated as whole-number relationships between two lines. Clearly, out of this no conception of zero as a number could possibly come, for from the point of view of a draughtsman it is meaningless. We, having minds differently constituted, must not argue from our habits to theirs and treat their mathematic as a “first stage” in the development of “Mathematics.” Within and for the purposes of the world that Classical man evolved for himself, the Classical mathematic was a complete thing—it is merely not so for us. Babylonian and Indian mathematics had long contained, as essential elements of their number-worlds, things which the Classical number-feeling regarded as nonsense—and not from ignorance either, since many a Greek thinker was acquainted with them. It must be repeated, “Mathematics” is an illusion. A mathematical, and, generally, a scientific way of thinking is right, convincing, a “necessity of thought,” when it completely expresses the life-feeling proper to it. Otherwise it is either impossible, futile and senseless, or else, as we in the arrogance of our historical soul like to say, “primitive.” The modern mathematic, though “true” only for the Western spirit, is undeniably a master-work of that spirit; and yet to Plato it would have seemed a ridiculous and painful aberration from the path leading to the “true”—to wit, the Classical—mathematic. And so with ourselves. Plainly, we have almost no notion of the multitude of great ideas belonging to other Cultures that we have suffered to lapse because our thought with its limitations has not permitted us to assimilate them, or (which comes to the same thing) has led us to reject them as false, superfluous, and nonsensical.

The Greek mathematic, as a science of perceivable magnitudes, deliberately confines itself to facts of the comprehensibly present, and limits its researches and their validity to the near and the small. As compared with this impeccable consistency, the position of the Western mathematic is seen to be, practically, somewhat illogical, though it is only since the discovery of Non-Euclidean Geometry that the fact has been really recognized. Numbers are images of the perfectly desensualized understanding, of pure thought, and contain their abstract validity within themselves.⁵² Their exact application to the actuality of conscious experience is therefore a problem in itself—a problem which is always being posed anew and never solved—and the congruence of mathematical system with empirical observation is at present anything but self-evident. Although the lay idea—as found in Schopenhauer—is that mathematics rest upon the direct evidences of the senses, Euclidean geometry, superficially identical though it is with the popular geometry of all ages, is only in agreement with the phenomenal world approximately and within very narrow limits—in fact, the limits of a drawing-board. Extend these limits, and what becomes, for instance, of Euclidean parallels? They meet at the line of the horizon—a simple fact upon which all our art-perspective is grounded.

Now, it is unpardonable that Kant, a Western thinker, should have evaded the mathematic of distance, and appealed to a set of figure-examples that their mere pettiness excludes from treatment by the specifically Western infinitesimal methods. But Euclid, as a thinker of the Classical age, was entirely consistent with its spirit when he refrained from proving the phenomenal truth of his axioms by referring to, say, the triangle formed by an observer and two infinitely distant fixed stars. For these can neither be drawn nor “intuitively apprehended” and his feeling was precisely the feeling which shrank from the irrationals, which did not dare to give nothingness a value as zero (i.e., a number) and even in the contemplation of cosmic relations shut its eyes to the Infinite and held to its symbol of Proportion.

Aristarchus of Samos, who in 288-277 belonged to a circle of astronomers at Alexandria that doubtless had relations with Chaldaeo-Persian schools, projected the elements of a heliocentric world-system.⁵³ Rediscovered by Copernicus, it was to shake the metaphysical passions of the West to their foundations—witness Giordano Bruno⁵⁴—to become the fulfilment of mighty premonitions, and to justify that Faustian, Gothic world-feeling which had already professed its faith in infinity through the forms of its cathedrals. But the world of Aristarchus received his work with entire indifference and in a brief space of time it was forgotten—designedly, we may surmise. His few followers were nearly all natives of Asia Minor, his most prominent supporter Seleucus (about 150) being from the Persian Seleucia on Tigris. In fact, the Aristarchian system had no spiritual appeal to the Classical Culture and might indeed have become dangerous to it. And yet it was differentiated from the Copernican (a point always missed) by something which made it perfectly conformable to the Classical world-feeling, viz., the assumption that the cosmos is contained in a materially finite and optically appreciable hollow sphere, in the middle of which the planetary system, arranged as such on Copernican lines, moved. In the Classical astronomy, the earth and the heavenly bodies are consistently regarded as entities of two different kinds, however variously their movements in detail might be interpreted. Equally, the opposite idea that the earth is only a star among stars⁵⁵ is not inconsistent in itself with either the Ptolemaic or the Copernican systems and in fact was pioneered by Nicolaus Cusanus and Leonardo da Vinci. But by this device of a celestial sphere the principle of infinity which would have endangered the sensuous-Classical notion of bounds was smothered. One would have supposed that the infinity-conception was inevitably implied by the system of Aristarchus—long before his time, the Babylonian thinkers had reached it. But no such thought emerges. On the contrary, in the famous treatise on the grains of sand⁵⁶ Archimedes proves that the filling of this stereometric body (for that is what Aristarchus’s Cosmos is, after all) with atoms of sand leads to very high, but not to infinite, figure-results. This proposition, quoted though it may be, time and again, as being a first step towards the Integral Calculus, amounts to a denial (implicit indeed in the very title) of everything that we mean by the word analysis. Whereas in our physics, the constantly-surging hypotheses of a material (i.e., directly cognizable) æther, break themselves one after the other against our refusal to acknowledge material limitations of any kind, Eudoxus, Apollonius and Archimedes, certainly the keenest and boldest of the Classical mathematicians, completely worked out, in the main with rule and compass, a purely optical analysis of things-become on the basis of sculptural-Classical bounds. They used deeply-thought-out (and for us hardly understandable) methods of integration, but these possess only a superficial resemblance even to Leibniz’s definite-integral method. They employed geometrical loci and co-ordinates, but these are always specified lengths and units of measurement and never, as in Fermat and above all in Descartes, unspecified spatial relations, values of points in terms of their positions in space. With these methods also should be classed the exhaustion-method of Archimedes,⁵⁷ given by him in his recently discovered letter to Eratosthenes on such subjects as the quadrature of the parabola section by means of inscribed rectangles (instead of through similar polygons). But the very subtlety and extreme complication of his methods, which are grounded in certain of Plato’s geometrical ideas, make us realize, in spite of superficial analogies, what an enormous difference separates him from Pascal. Apart altogether from the idea of Riemann’s integral, what sharper contrast could there be to these ideas than the so-called quadratures of to-day? The name itself is now no more than an unfortunate survival, the “surface” is indicated by a bounding function, and the drawing as such, has vanished. Nowhere else did the two mathematical minds approach each other more closely than in this instance, and nowhere is it more evident that the gulf between the two souls thus expressing themselves is impassable.

In the cubic style of their early architecture the Egyptians, so to say, concealed pure numbers, fearful of stumbling upon their secret, and for the Hellenes too they were the key to the meaning of the become, the stiffened, the mortal. The stone statue and the scientific system deny life. Mathematical number, the formal principle of an extension-world of which the phenomenal existence is only the derivative and servant of waking human consciousness, bears the hall-mark of causal necessity and so is linked with death as chronological number is with becoming, with life, with the necessity of destiny. This connexion of strict mathematical form with the end of organic being, with the phenomenon of its organic remainder the corpse, we shall see more and more clearly to be the origin of all great art. We have already noticed the development of early ornament on funerary equipments and receptacles. Numbers are symbols of the mortal. Stiff forms are the negation of life, formulas and laws spread rigidity over the face of nature, numbers make dead—and the “Mothers” of Faust II sit enthroned, majestic and withdrawn, in

“The realms of Image unconfined.

... Formation, transformation,

Eternal play of the eternal mind

With semblances of all things in creation

For ever and for ever sweeping round.”⁵⁸

Goethe draws very near to Plato in this divination of one of the final secrets. For his unapproachable Mothers are Plato’s Ideas—the possibilities of a spirituality, the unborn forms to be realized as active and purposed Culture, as art, thought, polity and religion, in a world ordered and determined by that spirituality. And so the number-thought and the world-idea of a Culture are related, and by this relation, the former is elevated above mere knowledge and experience and becomes a view of the universe, there being consequently as many mathematics—as many number-worlds—as there are higher Cultures. Only so can we understand, as something necessary, the fact that the greatest mathematical thinkers, the creative artists of the realm of numbers, have been brought to the decisive mathematical discoveries of their several Cultures by a deep religious intuition.

Classical, Apollinian number we must regard as the creation of Pythagoras—who founded a religion. It was an instinct that guided Nicolaus Cusanus, the great Bishop of Brixen (about 1450), from the idea of the unendingness of God in nature to the elements of the Infinitesimal Calculus. Leibniz himself, who two centuries later definitely settled the methods and notation of the Calculus, was led by purely metaphysical speculations about the divine principle and its relation to infinite extent to conceive and develop the notion of an analysis situs—probably the most inspired of all interpretations of pure and emancipated space—the possibilities of which were to be developed later by Grassmann in his Ausdehnungslehre and above all by Riemann, their real creator, in his symbolism of two-sided planes representative of the nature of equations. And Kepler and Newton, strictly religious natures both, were and remained convinced, like Plato, that it was precisely through the medium of number that they had been able to apprehend intuitively the essence of the divine world-order.

The Classical arithmetic, we are always told, was first liberated from its sense-bondage, widened and extended by Diophantus, who did not indeed create algebra (the science of undefined magnitudes) but brought it to expression within the framework of the Classical mathematic that we know—and so suddenly that we have to assume that there was a pre-existent stock of ideas which he worked out. But this amounts, not to an enrichment of, but a complete victory over, the Classical world-feeling, and the mere fact should have sufficed in itself to show that, inwardly, Diophantus does not belong to the Classical Culture at all. What is active in him is a new number-feeling, or let us say a new limit-feeling with respect to the actual and become, and no longer that Hellenic feeling of sensuously-present limits which had produced the Euclidean geometry, the nude statue and the coin. Details of the formation of this new mathematic we do not know—Diophantus stands so completely by himself in the history of so-called late-Classical mathematics that an Indian influence has been presumed. But here also the influence it must really have been that of those early-Arabian schools whose studies (apart from the dogmatic) have hitherto been so imperfectly investigated. In Diophantus, unconscious though he may be of his own essential antagonism to the Classical foundations on which he attempted to build, there emerges from under the surface of Euclidean intention the new limit-feeling which I designate the “Magian.” He did not widen the idea of number as magnitude, but (unwittingly) eliminated it. No Greek could have stated anything about an undefined number a or an undenominated number 3—which are neither magnitudes nor lines—whereas the new limit-feeling sensibly expressed by numbers of this sort at least underlay, if it did not constitute, Diophantine treatment; and the letter-notation which we employ to clothe our own (again transvalued) algebra was first introduced by Vieta in 1591, an unmistakable, if unintended, protest against the classicizing tendency of Renaissance mathematics.

Diophantus lived about 250 A.D., that is, in the third century of that Arabian Culture whose organic history, till now smothered under the surface-forms of the Roman Empire and the “Middle Ages,”⁵⁹ comprises everything that happened after the beginning of our era in the region that was later to be Islam’s. It was precisely in the time of Diophantus that the last shadow of the Attic statuary art paled before the new space-sense of cupola, mosaic and sarcophagus-relief that we have in the Early-Christian-Syrian style. In that time there was once more archaic art and strictly geometrical ornament; and at that time too Diocletian completed the transformation of the now merely sham Empire into a Caliphate. The four centuries that separate Euclid and Diophantus, separate also Plato and Plotinus—the last and conclusive thinker, the Kant, of a fulfilled Culture and the first schoolman, the Duns Scotus, of a Culture just awakened.

It is here that we are made aware for the first time of the existence of those higher individualities whose coming, growth and decay constitute the real substance of history underlying the myriad colours and changes of the surface. The Classical spirituality, which reached its final phase in the cold intelligence of the Romans and of which the whole Classical Culture with all its works, thoughts, deeds and ruins forms the “body,” had been born about 1100 B.C. in the country about the Ægean Sea. The Arabian Culture, which, under cover of the Classical Civilization, had been germinating in the East since Augustus, came wholly out of the region between Armenia and Southern Arabia, Alexandria and Ctesiphon, and we have to consider as expressions of this new soul almost the whole “late-Classical” art of the Empire, all the young ardent religions of the East—Mandæanism, Manichæism, Christianity, Neo-Platonism, and in Rome itself, as well as the Imperial Fora, that Pantheon which is the first of all mosques.

That Alexandria and Antioch still wrote in Greek and imagined that they were thinking in Greek is a fact of no more importance than the facts that Latin was the scientific language of the West right up to the time of Kant and that Charlemagne “renewed” the Roman Empire.

In Diophantus, number has ceased to be the measure and essence of plastic things. In the Ravennate mosaics man has ceased to be a body. Unnoticed, Greek designations have lost their original connotations. We have left the realm of Attic καλοκάγαθία the Stoic ἀταραξία and γαλήνη. Diophantus does not yet know zero and negative numbers, it is true, but he has ceased to know Pythagorean numbers. And this Arabian indeterminateness of number is, in its turn, something quite different from the controlled variability of the later Western mathematics, the variability of the function.

The Magian mathematic—we can see the outline, though we are ignorant of the details—advanced through Diophantus (who is obviously not a starting-point) boldly and logically to a culmination in the Abbassid period (9th century) that we can appreciate in Al-Khwarizmi and Alsidzshi. And as Euclidean geometry is to Attic statuary (the same expression-form in a different medium) and the analysis of space to polyphonic music, so this algebra is to the Magian art with its mosaic, its arabesque (which the Sassanid Empire and later Byzantium produced with an ever-increasing profusion and luxury of tangible-intangible organic motives) and its Constantinian high-relief in which uncertain deep-darks divide the freely-handled figures of the foreground. As algebra is to Classical arithmetic and Western analysis, so is the cupola-church to the Doric temple and the Gothic cathedral. It is not as though Diophantus were one of the great mathematicians. On the contrary, much of what we have been accustomed to associate with his name is not his work alone. His accidental importance lies in the fact that, so far as our knowledge goes, he was the first mathematician in whom the new number-feeling is unmistakably present. In comparison with the masters who conclude the development of a mathematic—with Apollonius and Archimedes, with Gauss, Cauchy, Riemann—Diophantus has, in his form-language especially, something primitive. This something, which till now we have been pleased to refer to “late-Classical” decadence, we shall presently learn to understand and value, just as we are revising our ideas as to the despised “late-Classical” art and beginning to see in it the tentative expression of the nascent Early Arabian Culture. Similarly archaic, primitive, and groping was the mathematic of Nicolas Oresme, Bishop of Lisieux (1323-1382),⁶⁰ who was the first Western who used co-ordinates so to say elastically⁶¹ and, more important still, to employ fractional powers—both of which presuppose a number-feeling, obscure it may be but quite unmistakable, which is completely non-Classical and also non-Arabic. But if, further, we think of Diophantus together with the early-Christian sarcophagi of the Roman collections, and of Oresme together with the Gothic wall-statuary of the German cathedrals, we see that the mathematicians as well as the artists have something in common, which is, that they stand in their respective Cultures at the same (viz., the primitive) level of abstract understanding. In the world and age of Diophantus the stereometric sense of bounds, which had long ago reached in Archimedes the last stages of refinement and elegance proper to the megalopolitan intelligence, had passed away. Throughout that world men were unclear, longing, mystic, and no longer bright and free in the Attic way; they were men rooted in the earth of a young country-side, not megalopolitans like Euclid and D’Alembert.⁶² They no longer understood the deep and complicated forms of the Classical thought, and their own were confused and new, far as yet from urban clarity and tidiness. Their Culture was in the Gothic condition, as all Cultures have been in their youth—as even the Classical was in the early Doric period which is known to us now only by its Dipylon pottery. Only in Baghdad and in the 9th and 10th Centuries were the young ideas of the age of Diophantus carried through to completion by ripe masters of the calibre of Plato and Gauss.

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The decisive act of Descartes, whose geometry appeared in 1637, consisted not in the introduction of a new method or idea in the domain of traditional geometry (as we are so frequently told), but in the definitive conception of a new number-idea, which conception was expressed in the emancipation of geometry from servitude to optically-realizable constructions and to measured and measurable lines generally. With that, the analysis of the infinite became a fact. The rigid, so-called Cartesian, system of co-ordinates—a semi-Euclidean method of ideally representing measurable magnitudes—had long been known (witness Oresme) and regarded as of high importance, and when we get to the bottom of Descartes’ thought we find that what he did was not to round off the system but to overcome it. Its last historic representative was Descartes’ contemporary Fermat.⁶³

In place of the sensuous element of concrete lines and planes—the specific character of the Classical feeling of bounds—there emerged the abstract, spatial, un-Classical element of the point which from then on was regarded as a group of co-ordered pure numbers. The idea of magnitude and of perceivable dimension derived from Classical texts and Arabian traditions was destroyed and replaced by that of variable relation-values between positions in space. It is not in general realized that this amounted to the supersession of geometry, which thenceforward enjoyed only a fictitious existence behind a façade of Classical tradition. The word “geometry” has an inextensible Apollinian meaning, and from the time of Descartes what is called the “new geometry” is made up in part of synthetic work upon the position of points in a space which is no longer necessarily three-dimensional (a “manifold of points”), and in part of analysis, in which numbers are defined through point-positions in space. And this replacement of lengths by positions carries with it a purely spatial, and no longer a material, conception of extension.

The clearest example of this destruction of the inherited optical-finite geometry seems to me to be the conversion of angular functions—which in the Indian mathematic had been numbers (in a sense of the word that is hardly accessible to our minds)—into periodic functions, and their passage thence into an infinite number-realm, in which they become series and not the smallest trace remains of the Euclidean figure. In all parts of that realm the circle-number π, like the Napierian base ε, generates relations of all sorts which obliterate all the old distinctions of geometry, trigonometry and algebra, which are neither arithmetical nor geometrical in their nature, and in which no one any longer dreams of actually drawing circles or working out powers.

75

At the moment exactly corresponding to that at which (c. 540) the Classical Soul in the person of Pythagoras discovered its own proper Apollinian number, the measurable magnitude, the Western soul in the persons of Descartes and his generation (Pascal, Fermat, Desargues) discovered a notion of number that was the child of a passionate Faustian tendency towards the infinite. Number as pure magnitude inherent in the material presentness of things is paralleled by numbers as pure relation,⁶⁴ and if we may characterize the Classical “world,” the cosmos, as being based on a deep need of visible limits and composed accordingly as a sum of material things, so we may say that our world-picture is an actualizing of an infinite space in which things visible appear very nearly as realities of a lower order, limited in the presence of the illimitable. The symbol of the West is an idea of which no other Culture gives even a hint, the idea of Function. The function is anything rather than an expansion of, it is complete emancipation from, any pre-existent idea of number. With the function, not only the Euclidean geometry (and with it the common human geometry of children and laymen, based on everyday experience) but also the Archimedean arithmetic, ceased to have any value for the really significant mathematic of Western Europe. Henceforward, this consisted solely in abstract analysis. For Classical man geometry and arithmetic were self-contained and complete sciences of the highest rank, both phenomenal and both concerned with magnitudes that could be drawn or numbered. For us, on the contrary, those things are only practical auxiliaries of daily life. Addition and multiplication, the two Classical methods of reckoning magnitudes, have, like their sister geometrical-drawing, utterly vanished in the infinity of functional processes. Even the power, which in the beginning denotes numerically a set of multiplications (products of equal magnitudes), is, through the exponential idea (logarithm) and its employment in complex, negative and fractional forms, dissociated from all connexion with magnitude and transferred to a transcendent relational world which the Greeks, knowing only the two positive whole-number powers that represent areas and volumes, were unable to approach. Think, for instance, of expressions like ε-x, π√x, α1⁄i.

Every one of the significant creations which succeeded one another so rapidly from the Renaissance onward—imaginary and complex numbers, introduced by Cardanus as early as 1550; infinite series, established theoretically by Newton’s great discovery of the binomial theorem in 1666; the differential geometry, the definite integral of Leibniz; the aggregate as a new number-unit, hinted at even by Descartes; new processes like those of general integrals; the expansion of functions into series and even into infinite series of other functions—is a victory over the popular and sensuous number-feeling in us, a victory which the new mathematic had to win in order to make the new world-feeling actual.

In all history, so far, there is no second example of one Culture paying to another Culture long extinguished such reverence and submission in matters of science as ours has paid to the Classical. It was very long before we found courage to think our proper thought. But though the wish to emulate the Classical was constantly present, every step of the attempt took us in reality further away from the imagined ideal. The history of Western knowledge is thus one of progressive emancipation from Classical thought, an emancipation never willed but enforced in the depths of the unconscious. And so the development of the new mathematic consists of a long, secret and finally victorious battle against the notion of magnitude.⁶⁵

One result of this Classicizing tendency has been to prevent us from finding the new notation proper to our Western number as such. The present-day sign-language of mathematics perverts its real content. It is principally owing to that tendency that the belief in numbers as magnitudes still rules to-day even amongst mathematicians, for is it not the base of all our written notation?

But it is not the separate signs (e.g., χ, π, ς) serving to express the functions but the function itself as unit, as element, the variable relation no longer capable of being optically defined, that constitutes the new number; and this new number should have demanded a new notation built up with entire disregard of Classical influences. Consider the difference between two equations (if the same word can be used of two such dissimilar things) such as 3x + 4x = 5x and xn + yn = zn (the equation of Fermat’s theorem). The first consists of several Classical numbers—i.e., magnitudes—but the second is one number of a different sort, veiled by being written down according to Euclidean-Archimedean tradition in the identical form of the first. In the first case, the sign = establishes a rigid connexion between definite and tangible magnitudes, but in the second it states that within a domain of variable images there exists a relation such that from certain alterations certain other alterations necessarily follow. The first equation has as its aim the specification by measurement of a concrete magnitude, viz., a “result,” while the second has, in general, no result but is simply the picture and sign of a relation which for n>2 (this is the famous Fermat problem⁶⁶) can probably be shown to exclude integers. A Greek mathematician would have found it quite impossible to understand the purport of an operation like this, which was not meant to be “worked out.”

As applied to the letters in Fermat’s equation, the notion of the unknown is completely misleading. In the first equation x is a magnitude, defined and measurable, which it is our business to compute. In the second, the word “defined” has no meaning at all for x, y, z, n, and consequently we do not attempt to compute their “values.” Hence they are not numbers at all in the plastic sense but signs representing a connexion that is destitute of the hallmarks of magnitude, shape and unique meaning, an infinity of possible positions of like character, an ensemble unified and so attaining existence as a number. The whole equation, though written in our unfortunate notation as a plurality of terms, is actually one single number, x, y, z being no more numbers than + and = are.

In fact, directly the essentially anti-Hellenic idea of the irrationals is introduced, the foundations of the idea of number as concrete and definite collapse. Thenceforward, the series of such numbers is no longer a visible row of increasing, discrete, numbers capable of plastic embodiment but a unidimensional continuum in which each “cut” (in Dedekind’s sense) represents a number. Such a number is already difficult to reconcile with Classical number, for the Classical mathematic knows only one number between 1 and 3, whereas for the Western the totality of such numbers is an infinite aggregate. But when we introduce further the imaginary (√-1 or i) and finally the complex numbers (general form a + bi), the linear continuum is broadened into the highly transcendent form of a number-body, i.e., the content of an aggregate of homogeneous elements in which a “cut” now stands for a number-surface containing an infinite aggregate of numbers of a lower “potency” (for instance, all the real numbers), and there remains not a trace of number in the Classical and popular sense. These number-surfaces, which since Cauchy and Riemann have played an important part in the theory of functions, are pure thought-pictures. Even positive irrational number (e.g., √2) could be conceived in a sort of negative fashion by Classical minds; they had, in fact, enough idea of it to ban it as ἄῤῥητος and ἄλογος. But expressions of the form x + yi lie beyond every possibility of comprehension by Classical thought, whereas it is on the extension of the mathematical laws over the whole region of the complex numbers, within which these laws remain operative, that we have built up the function theory which has at last exhibited the Western mathematic in all purity and unity. Not until that point was reached could this mathematic be unreservedly brought to bear in the parallel sphere of our dynamic Western physics; for the Classical mathematic was fitted precisely to its own stereometric world of individual objects and to static mechanics as developed from Leucippus to Archimedes.

The brilliant period of the Baroque mathematic—the counterpart of the Ionian—lies substantially in the 18th Century and extends from the decisive discoveries of Newton and Leibniz through Euler, Lagrange, Laplace and D’Alembert to Gauss. Once this immense creation found wings, its rise was miraculous. Men hardly dared believe their senses. The age of refined scepticism witnessed the emergence of one seemingly impossible truth after another.⁶⁷ Regarding the theory of the differential coefficient, D’Alembert had to say: “Go forward, and faith will come to you.” Logic itself seemed to raise objections and to prove foundations fallacious. But the goal was reached.

This century was a very carnival of abstract and immaterial thinking, in which the great masters of analysis and, with them, Bach, Gluck, Haydn and Mozart—a small group of rare and deep intellects—revelled in the most refined discoveries and speculations, from which Goethe and Kant remained aloof; and in point of content it is exactly paralleled by the ripest century of the Ionic, the century of Eudoxus and Archytas (440-350) and, we may add, of Phidias, Polycletus, Alcamenes and the Acropolis buildings—in which the form-world of Classical mathematic and sculpture displayed the whole fullness of its possibilities, and so ended.

And now for the first time it is possible to comprehend in full the elemental opposition of the Classical and the Western souls. In the whole panorama of history, innumerable and intense as historical relations are, we find no two things so fundamentally alien to one another as these. And it is because extremes meet—because it may be there is some deep common origin behind their divergence—that we find in the Western Faustian soul this yearning effort towards the Apollinian ideal, the only alien ideal which we have loved and, for its power of intensely living in the pure sensuous present, have envied.

We have already observed that, like a child, a primitive mankind acquires (as part of the inward experience that is the birth of the ego) an understanding of number and ipso facto possession of an external world referred to the ego. As soon as the primitive’s astonished eye perceives the dawning world of ordered extension, and the significant emerges in great outlines from the welter of mere impressions, and the irrevocable parting of the outer world from his proper, his inner, world gives form and direction to his waking life, there arises in the soul—instantly conscious of its loneliness—the root-feeling of longing (Sehnsucht). It is this that urges “becoming” towards its goal, that motives the fulfilment and actualizing of every inward possibility, that unfolds the idea of individual being. It is the child’s longing, which will presently come into the consciousness more and more clearly as a feeling of constant direction and finally stand before the mature spirit as the enigma of Time—queer, tempting, insoluble. Suddenly, the words “past” and “future” have acquired a fateful meaning.

But this longing which wells out of the bliss of the inner life is also, in the intimate essence of every soul, a dread as well. As all becoming moves towards a having-become wherein it ends, so the prime feeling of becoming—the longing—touches the prime feeling of having-become, the dread. In the present we feel a trickling-away, the past implies a passing. Here is the root of our eternal dread of the irrevocable, the attained, the final—our dread of mortality, of the world itself as a thing-become, where death is set as a frontier like birth—our dread in the moment when the possible is actualized, the life is inwardly fulfilled and consciousness stands at its goal. It is the deep world-fear of the child—which never leaves the higher man, the believer, the poet, the artist—that makes him so infinitely lonely in the presence of the alien powers that loom, threatening in the dawn, behind the screen of sense-phenomena. The element of direction, too, which is inherent in all “becoming,” is felt owing to its inexorable irreversibility to be something alien and hostile, and the human will-to-understanding ever seeks to bind the inscrutable by the spell of a name. It is something beyond comprehension, this transformation of future into past, and thus time, in its contrast with space, has always a queer, baffling, oppressive ambiguity from which no serious man can wholly protect himself.

This world-fear is assuredly the most creative of all prime feelings. Man owes to it the ripest and deepest forms and images, not only of his conscious inward life, but also of the infinitely-varied external culture which reflects this life. Like a secret melody that not every ear can perceive, it runs through the form-language of every true art-work, every inward philosophy, every important deed, and, although those who can perceive it in that domain are the very few, it lies at the root of the great problems of mathematics. Only the spiritually dead man of the autumnal cities—Hammurabi’s Babylon, Ptolemaic Alexandria, Islamic Baghdad, Paris and Berlin to-day—only the pure intellectual, the sophist, the sensualist, the Darwinian, loses it or is able to evade it by setting up a secretless “scientific world-view” between himself and the alien. As the longing attaches itself to that impalpable something whose thousand-formed elusive manifestations are comprised in, rather than denoted by, the word “time,” so the other prime feeling, dread, finds its expression in the intellectual, understandable, outlinable symbols of extension; and thus we find that every Culture is aware (each in its own special way) of an opposition of time and space, of direction and extension, the former underlying the latter as becoming precedes having-become. It is the longing that underlies the dread, becomes the dread, and not vice versa. The one is not subject to the intellect, the other is its servant. The rôle of the one is purely to experience, that of the other purely to know (erleben, erkennen). In the Christian language, the opposition of the two world-feelings is expressed by: “Fear God and love Him.”

In the soul of all primitive mankind, just as in that of earliest childhood, there is something which impels it to find means of dealing with the alien powers of the extension-world that assert themselves, inexorable, in and through space. To bind, to bridle, to placate, to “know” are all, in the last analysis, the same thing. In the mysticism of all primitive periods, to know God means to conjure him, to make him favourable, to appropriate him inwardly. This is achieved, principally, by means of a word, the Name—the “nomen” which designates and calls up the “numen”—and also by ritual practices of secret potency; and the subtlest, as well as the most powerful, form of this defence is causal and systematic knowledge, delimitation by label and number. In this respect man only becomes wholly man when he has acquired language. When cognition has ripened to the point of words, the original chaos of impressions necessarily transforms itself into a “Nature” that has laws and must obey them, and the world-in-itself becomes a world-for-us.⁶⁸

The world-fear is stilled when an intellectual form-language hammers out brazen vessels in which the mysterious is captured and made comprehensible. This is the idea of “taboo,”⁶⁹ which plays a decisive part in the spiritual life of all primitive men, though the original content of the word lies so far from us that it is incapable of translation into any ripe culture-language. Blind terror, religious awe, deep loneliness, melancholy, hate, obscure impulses to draw near, to be merged, to escape—all those formed feelings of mature souls are in the childish condition blurred in a monotonous indecision. The two senses of the word “conjure” (verschwören), meaning to bind and to implore at once, may serve to make clear the sense of the mystical process by which for primitive man the formidable alien becomes “taboo.” Reverent awe before that which is independent of one’s self, things ordained and fixed by law, the alien powers of the world, is the source from which the elementary formative acts, one and all, spring. In early times this feeling is actualized in ornament, in laborious ceremonies and rites, and the rigid laws of primitive intercourse. At the zeniths of the great Cultures those formations, though retaining inwardly the mark of their origin, the characteristic of binding and conjuring, have become the complete form-worlds of the various arts and of religious, scientific and, above all, mathematical thought. The method common to all—the only way of actualizing itself that the soul knows—is the symbolizing of extension, of space or of things; and we find it alike in the conceptions of absolute space that pervade Newtonian physics, Gothic cathedral-interiors and Moorish mosques, and the atmospheric infinity of Rembrandt’s paintings and again the dark tone-worlds of Beethoven’s quartets; in the regular polyhedrons of Euclid, the Parthenon sculptures and the pyramids of Old Egypt, the Nirvana of Buddha, the aloofness of court-customs under Sesostris, Justinian I and Louis XIV, in the God-idea of an Æschylus, a Plotinus, a Dante; and in the world-embracing spatial energy of modern technics.

To return to mathematics. In the Classical world the starting-point of every formative act was, as we have seen, the ordering of the “become,” in so far as this was present, visible, measurable and numerable. The Western, Gothic, form-feeling on the contrary is that of an unrestrained, strong-willed far-ranging soul, and its chosen badge is pure, imperceptible, unlimited space. But we must not be led into regarding such symbols as unconditional. On the contrary, they are strictly conditional, though apt to be taken as having identical essence and validity. Our universe of infinite space, whose existence, for us, goes without saying, simply does not exist for Classical man. It is not even capable of being presented to him. On the other hand, the Hellenic cosmos, which is (as we might have discovered long ago) entirely foreign to our way of thinking, was for the Hellene something self-evident. The fact is that the infinite space of our physics is a form of very numerous and extremely complicated elements tacitly assumed, which have come into being only as the copy and expression of our soul, and are actual, necessary and natural only for our type of waking life. The simple notions are always the most difficult. They are simple, in that they comprise a vast deal that not only is incapable of being exhibited in words but does not even need to be stated, because for men of the particular group it is anchored in the intuition; and they are difficult because for all alien men their real content is ipso facto quite inaccessible. Such a notion, at once simple and difficult, is our specifically Western meaning of the word “space.” The whole of our mathematic from Descartes onward is devoted to the theoretical interpretation of this great and wholly religious symbol. The aim of all our physics since Galileo is identical; but in the Classical mathematics and physics the content of this word is simply not known.

Here, too, Classical names, inherited from the literature of Greece and retained in use, have veiled the realities. Geometry means the art of measuring, arithmetic the art of numbering. The mathematic of the West has long ceased to have anything to do with both these forms of defining, but it has not managed to find new names for its own elements—for the word “analysis” is hopelessly inadequate.

The beginning and end of the Classical mathematic is consideration of the properties of individual bodies and their boundary-surfaces; thus indirectly taking in conic sections and higher curves. We, on the other hand, at bottom know only the abstract space-element of the point, which can neither be seen, nor measured, nor yet named, but represents simply a centre of reference. The straight line, for the Greeks a measurable edge, is for us an infinite continuum of points. Leibniz illustrates his infinitesimal principle by presenting the straight line as one limiting case and the point as the other limiting case of a circle having infinitely great or infinitely little radius. But for the Greek the circle is a plane and the problem that interested him was that of bringing it into a commensurable condition. Thus the squaring of the circle became for the Classical intellect the supreme problem of the finite. The deepest problem of world-form seemed to it to be to alter surfaces bounded by curved lines, without change of magnitude, into rectangles and so to render them measureable. For us, on the other hand, it has become the usual, and not specially significant, practice to represent the number π by algebraic means, regardless of any geometrical image.

The Classical mathematician knows only what he sees and grasps. Where definite and defining visibility—the domain of his thought—ceases, his science comes to an end. The Western mathematician, as soon as he has quite shaken off the trammels of Classical prejudice, goes off into a wholly abstract region of infinitely numerous “manifolds” of n (no longer 3) dimensions, in which his so-called geometry always can and generally must do without every commonplace aid. When Classical man turns to artistic expressions of his form-feeling, he tries with marble and bronze to give the dancing or the wrestling human form that pose and attitude in which surfaces and contours have all attainable proportion and meaning. But the true artist of the West shuts his eyes and loses himself in the realm of bodiless music, in which harmony and polyphony bring him to images of utter “beyondness” that transcend all possibilities of visual definition. One need only think of the meanings of the word “figure” as used respectively by the Greek sculptor and the Northern contrapuntist, and the opposition of the two worlds, the two mathematics, is immediately presented. The Greek mathematicians ever use the word σῶμα for their entities, just as the Greek lawyers used it for persons as distinct from things (σώματα καὶ πράγματα: personæ et res).

Classical number, integral and corporeal, therefore inevitably seeks to relate itself with the birth of bodily man, the σῶμα. The number 1 is hardly yet conceived of as actual number but rather as ἀρχή, the prime stuff of the number-series, the origin of all true numbers and therefore all magnitudes, measures and materiality (Dinglichkeit). In the group of the Pythagoreans (the date does not matter) its figured-sign was also the symbol of the mother-womb, the origin of all life. The digit 2, the first true number, which doubles the 1, was therefore correlated with the male principle and given the sign of the phallus. And, finally, 3, the “holy number” of the Pythagoreans, denoted the act of union between man and woman, the act of propagation—the erotic suggestion in adding and multiplying (the only two processes of increasing, of propagating, magnitude useful to Classical man) is easily seen—and its sign was the combination of the two first. Now, all this throws quite a new light upon the legends previously alluded to, concerning the sacrilege of disclosing the irrational. The irrational—in our language the employment of unending decimal fractions—implied the destruction of an organic and corporeal and reproductive order that the gods had laid down. There is no doubt that the Pythagorean reforms of the Classical religion were themselves based upon the immemorial Demeter-cult. Demeter, Gæa, is akin to Mother Earth. There is a deep relation between the honour paid to her and this exalted conception of the numbers.

Thus, inevitably, the Classical became by degrees the Culture of the small. The Apollinian soul had tried to tie down the meaning of things-become by means of the principle of visible limits; its taboo was focused upon the immediately-present and proximate alien. What was far away, invisible, was ipso facto “not there.” The Greek and the Roman alike sacrificed to the gods of the place in which he happened to stay or reside; all other deities were outside the range of vision. Just as the Greek tongue—again and again we shall note the mighty symbolism of such language-phenomena—possessed no word for space, so the Greek himself was destitute of our feeling of landscape, horizons, outlooks, distances, clouds, and of the idea of the far-spread fatherland embracing the great nation. Home, for Classical man, is what he can see from the citadel of his native town and no more. All that lay beyond the visual range of this political atom was alien, and hostile to boot; beyond that narrow range, fear set in at once, and hence the appalling bitterness with which these petty towns strove to destroy one another. The Polis is the smallest of all conceivable state-forms, and its policy is frankly short-range, therein differing in the extreme from our own cabinet-diplomacy which is the policy of the unlimited. Similarly, the Classical temple, which can be taken in in one glance, is the smallest of all first-rate architectural forms. Classical geometry from Archytas to Euclid—like the school geometry of to-day which is still dominated by it—concerned itself with small, manageable figures and bodies, and therefore remained unaware of the difficulties that arise in establishing figures of astronomical dimensions, which in many cases are not amenable to Euclidean geometry.⁷⁰ Otherwise the subtle Attic spirit would almost surely have arrived at some notion of the problems of non-Euclidean geometry, for its criticism of the well-known “parallel” axiom,⁷¹ the doubtfulness of which soon aroused opposition yet could not in any way be elucidated, brought it very close indeed to the decisive discovery. The Classical mind as unquestioningly devoted and limited itself to the study of the small and the near as ours has to that of the infinite and ultra-visual. All the mathematical ideas that the West found for itself or borrowed from others were automatically subjected to the form-language of the Infinitesimal—and that long before the actual Differential Calculus was discovered. Arabian algebra, Indian trigonometry, Classical mechanics were incorporated as a matter of course in analysis. Even the most “self-evident” propositions of elementary arithmetic such as 2 × 2 = 4 become, when considered analytically, problems, and the solution of these problems was only made possible by deductions from the Theory of Aggregates, and is in many points still unaccomplished. Plato and his age would have looked upon this sort of thing not only as a hallucination but also as evidence of an utterly nonmathematical mind. In a certain measure, geometry may be treated algebraically and algebra geometrically, that is, the eye may be switched off or it may be allowed to govern. We take the first alternative, the Greeks the second. Archimedes, in his beautiful management of spirals, touches upon certain general facts that are also fundamentals in Leibniz’s method of the definite integral; but his processes, for all their superficial appearance of modernity, are subordinated to stereometric principles; in like case, an Indian mathematician would naturally have found some trigonometrical formulation.⁷²

From this fundamental opposition of Classical and Western numbers there arises an equally radical difference in the relationship of element to element in each of these number-worlds. The nexus of magnitudes is called proportion, that of relations is comprised in the notion of function. The significance of these two words is not confined to mathematics proper; they are of high importance also in the allied arts of sculpture and music. Quite apart from the rôle of proportion in ordering the parts of the individual statue, the typically Classical artforms of the statue, the relief, and the fresco, admit enlargements and reductions of scale—words that in music have no meaning at all—as we see in the art of the gems, in which the subjects are essentially reductions from life-sized originals. In the domain of Function, on the contrary, it is the idea of transformation of groups that is of decisive importance, and the musician will readily agree that similar ideas play an essential part in modern composition-theory. I need only allude to one of the most elegant orchestral forms of the 18th Century, the Tema con Variazioni.

All proportion assumes the constancy, all transformation the variability of the constituents. Compare, for instance, the congruence theorems of Euclid, the proof of which depends in fact on the assumed ratio 1 : 1, with the modern deduction of the same by means of angular functions.

The Alpha and Omega of the Classical mathematic is construction (which in the broad sense includes elementary arithmetic), that is, the production of a single visually-present figure. The chisel, in this second sculptural art, is the compass. On the other hand, in function-research, where the object is not a result of the magnitude sort but a discussion of general formal possibilities, the way of working is best described as a sort of composition-procedure closely analogous to the musical; and in fact, a great number of the ideas met with in the theory of music (key, phrasing, chromatics, for instance) can be directly employed in physics, and it is at least arguable that many relations would be clarified by so doing.

Every construction affirms, and every operation denies appearances, in that the one works out that which is optically given and the other dissolves it. And so we meet with yet another contrast between the two kinds of mathematic; the Classical mathematic of small things deals with the concrete individual instance and produces a once-for-all construction, while the mathematic of the infinite handles whole classes of formal possibilities, groups of functions, operations, equations, curves, and does so with an eye, not to any result they may have, but to their course. And so for the last two centuries—though present-day mathematicians hardly realize the fact—there has been growing up the idea of a general morphology of mathematical operations, which we are justified in regarding as the real meaning of modern mathematics as a whole. All this, as we shall perceive more and more clearly, is one of the manifestations of a general tendency inherent in the Western intellect, proper to the Faustian spirit and Culture and found in no other. The great majority of the problems which occupy our mathematic, and are regarded as “our” problems in the same sense as the squaring of the circle was the Greeks’,—e.g., the investigation of convergence in infinite series (Cauchy) and the transformation of elliptic and algebraic integrals into multiply-periodic functions (Abel, Gauss)—would probably have seemed to the Ancients, who strove for simple and definite quantitative results, to be an exhibition of rather abstruse virtuosity. And so indeed the popular mind regards them even to-day. There is nothing less “popular” than the modern mathematic, and it too contains its symbolism of the infinitely far, of distance. All the great works of the West, from the “Divina Commedia” to “Parsifal,” are unpopular, whereas everything Classical from Homer to the Altar of Pergamum was popular in the highest degree.

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Thus, finally, the whole content of Western number-thought centres itself upon the historic limit-problem of the Faustian mathematic, the key which opens the way to the Infinite, that Faustian infinite which is so different from the infinity of Arabian and Indian world-ideas. Whatever the guise—infinite series, curves or functions—in which number appears in the particular case, the essence of it is the theory of the limit.⁷³ This limit is the absolute opposite of the limit which (without being so called) figures in the Classical problem of the quadrature of the circle. Right into the 18th Century, Euclidean popular prepossessions obscured the real meaning of the differential principle. The idea of infinitely small quantities lay, so to say, ready to hand, and however skilfully they were handled, there was bound to remain a trace of the Classical constancy, the semblance of magnitude, about them, though Euclid would never have known them or admitted them as such. Thus, zero is a constant, a whole number in the linear continuum between +1 and -1; and it was a great hindrance to Euler in his analytical researches that, like many after him, he treated the differentials as zero. Only in the 19th Century was this relic of Classical number-feeling finally removed and the Infinitesimal Calculus made logically secure by Cauchy’s definitive elucidation of the limit-idea; only the intellectual step from the “infinitely small quantity” to the “lower limit of every possible finite magnitude” brought out the conception of a variable number which oscillates beneath any assignable number that is not zero. A number of this sort has ceased to possess any character of magnitude whatever: the limit, as thus finally presented by theory, is no longer that which is approximated to, but the approximation, the process, the operation itself. It is not a state, but a relation. And so in this decisive problem of our mathematic, we are suddenly made to see how historical is the constitution of the Western soul.⁷⁴

The liberation of geometry from the visual, and of algebra from the notion of magnitude, and the union of both, beyond all elementary limitations of drawing and counting, in the great structure of function-theory—this was the grand course of Western number-thought. The constant number of the Classical mathematic was dissolved into the variable. Geometry became analytical and dissolved all concrete forms, replacing the mathematical bodies from which the rigid geometrical values had been obtained, by abstract spatial relations which in the end ceased to have any application at all to sense-present phenomena. It began by substituting for Euclid’s optical figures geometrical loci referred to a co-ordinate system of arbitrarily chosen “origin,” and reducing the postulated objectiveness of existence of the geometrical object to the one condition that during the operation (which itself was one of equating and not of measurement) the selected co-ordinate system should not be changed. But these co-ordinates immediately came to be regarded as values pure and simple, serving not so much to determine as to represent and replace the position of points as space-elements. Number, the boundary of things-become, was represented, not as before pictorially by a figure, but symbolically by an equation. “Geometry” altered its meaning; the co-ordinate system as a picturing disappeared and the point became an entirely abstract number-group. In architecture, we find this inward transformation of Renaissance into Baroque through the innovations of Michael Angelo and Vignola. Visually pure lines became, in palace and church façades as in mathematics, ineffectual. In place of the clear co-ordinates that we have in Romano-Florentine colonnading and storeying, the “infinitesimal” appears in the graceful flow of elements, the scrollwork, the cartouches. The constructive dissolves in the wealth of the decorative—in mathematical language, the functional. Columns and pilasters, assembled in groups and clusters, break up the façades, gather and disperse again restlessly. The flat surfaces of wall, roof, storey melt into a wealth of stucco work and ornaments, vanish and break into a play of light and shade. The light itself, as it is made to play upon the form-world of mature Baroque—viz., the period from Bernini (1650) to the Rococo of Dresden, Vienna and Paris—has become an essentially musical element. The Dresden Zwinger⁷⁵ is a sinfonia. Along with 18th Century mathematics, 18th Century architecture develops into a form-world of musical characters.

This mathematics of ours was bound in due course to reach the point at which not merely the limits of artificial geometrical form but the limits of the visual itself were felt by theory and by the soul alike as limits indeed, as obstacles to the unreserved expression of inward possibilities—in other words, the point at which the ideal of transcendent extension came into fundamental conflict with the limitations of immediate perception. The Classical soul, with the entire abdication of Platonic and Stoic ἀταραξία, submitted to the sensuous and (as the erotic under-meaning of the Pythagorean numbers shows) it rather felt than emitted its great symbols. Of transcending the corporeal here-and-now it was quite incapable. But whereas number, as conceived by a Pythagorean, exhibited the essence of individual and discrete data in “Nature” Descartes and his successors looked upon number as something to be conquered, to be wrung out, an abstract relation royally indifferent to all phenomenal support and capable of holding its own against “Nature” on all occasions. The will-to-power (to use Nietzsche’s great formula) that from the earliest Gothic of the Eddas, the Cathedrals and Crusades, and even from the old conquering Goths and Vikings, has distinguished the attitude of the Northern soul to its world, appears also in the sense-transcending energy, the dynamic of Western number. In the Apollinian mathematic the intellect is the servant of the eye, in the Faustian its master. Mathematical, “absolute” space, we see then, is utterly un-Classical, and from the first, although mathematicians with their reverence for the Hellenic tradition did not dare to observe the fact, it was something different from the indefinite spaciousness of daily experience and customary painting, the a priori space of Kant which seemed so unambiguous and sure a concept. It is a pure abstract, an ideal and unfulfillable postulate of a soul which is ever less and less satisfied with sensuous means of expression and in the end passionately brushes them aside. The inner eye has awakened.

And then, for the first time, those who thought deeply were obliged to see that the Euclidean geometry, which is the true and only geometry of the simple of all ages, is when regarded from the higher standpoint nothing but a hypothesis, the general validity of which, since Gauss, we know it to be quite impossible to prove in the face of other and perfectly non-perceptual geometries. The critical proposition of this geometry, Euclid’s axiom of parallels, is an assertion, for which we are quite at liberty to substitute another assertion. We may assert, in fact, that through a given point, no parallels, or two, or many parallels may be drawn to a given straight line, and all these assumptions lead to completely irreproachable geometries of three dimensions, which can be employed in physics and even in astronomy, and are in some cases preferable to the Euclidean.

Even the simple axiom that extension is boundless (boundlessness, since Riemann and the theory of curved space, is to be distinguished from endlessness) at once contradicts the essential character of all immediate perception, in that the latter depends upon the existence of light-resistances and ipso facto has material bounds. But abstract principles of boundary can be imagined which transcend, in an entirely new sense, the possibilities of optical definition. For the deep thinker, there exists even in the Cartesian geometry the tendency to get beyond the three dimensions of experiential space, regarded as an unnecessary restriction on the symbolism of number. And although it was not till about 1800 that the notion of multi-dimensional space (it is a pity that no better word was found) provided analysis with broader foundations, the real first step was taken at the moment when powers—that is, really, logarithms—were released from their original relation with sensually realizable surfaces and solids and, through the employment of irrational and complex exponents, brought within the realm of function as perfectly general relation-values. It will be admitted by everyone who understands anything of mathematical reasoning that directly we passed from the notion of a³ as a natural maximum to that of an, the unconditional necessity of three-dimensional space was done away with.

Once the space-element or point had lost its last persistent relic of visualness and, instead of being represented to the eye as a cut in co-ordinate lines, was defined as a group of three independent numbers, there was no longer any inherent objection to replacing the number 3 by the general number n. The notion of dimension was radically changed. It was no longer a matter of treating the properties of a point metrically with reference to its position in a visible system, but of representing the entirely abstract properties of a number-group by means of any dimensions that we please. The number-group—consisting of n independent ordered elements—is an image of the point and it is called a point. Similarly, an equation logically arrived therefrom is called a plane and is the image of a plane. And the aggregate of all points of n dimensions is called an n-dimensional space.⁷⁶ In these transcendent space-worlds, which are remote from every sort of sensualism, lie the relations which it is the business of analysis to investigate and which are found to be consistently in agreement with the data of experimental physics. This space of higher degree is a symbol which is through-and-through the peculiar property of the Western mind. That mind alone has attempted, and successfully too, to capture the “become” and the extended in these forms, to conjure and bind—to “know”—the alien by this kind of appropriation or taboo. Not until such spheres of number-thought are reached, and not for any men but the few who have reached them, do such imaginings as systems of hypercomplex numbers (e.g., the quaternions of the calculus of vectors) and apparently quite meaningless symbols like ∞n acquire the character of something actual. And here if anywhere it must be understood that actuality is not only sensual actuality. The spiritual is in no wise limited to perception-forms for the actualizing of its idea.

From this grand intuition of symbolic space-worlds came the last and conclusive creation of Western mathematic—the expansion and subtilizing of the function theory in that of groups. Groups are aggregates or sets of homogeneous mathematical images—e.g., the totality of all differential equations of a certain type—which in structure and ordering are analogous to the Dedekind number-bodies. Here are worlds, we feel, of perfectly new numbers, which are nevertheless not utterly sense-transcendent for the inner eye of the adept; and the problem now is to discover in those vast abstract form-systems certain elements which, relatively to a particular group of operations (viz., of transformations of the system), remain unaffected thereby, that is, possess invariance. In mathematical language, the problem, as stated generally by Klein, is—given an n-dimensional manifold (“space”) and a group of transformations, it is required to examine the forms belonging to the manifold in respect of such properties as are not altered by transformation of the group.

And with this culmination our Western mathematic, having exhausted every inward possibility and fulfilled its destiny as the copy and purest expression of the idea of the Faustian soul, closes its development in the same way as the mathematic of the Classical Culture concluded in the third century. Both those sciences (the only ones of which the organic structure can even to-day be examined historically) arose out of a wholly new idea of number, in the one case Pythagoras’s, in the other Descartes’. Both, expanding in all beauty, reached their maturity one hundred years later; and both, after flourishing for three centuries, completed the structure of their ideas at the same moment as the Cultures to which they respectively belonged passed over into the phase of megalopolitan Civilization. The deep significance of this interdependence will be made clear in due course. It is enough for the moment that for us the time of the great mathematicians is past. Our tasks to-day are those of preserving, rounding off, refining, selection—in place of big dynamic creation, the same clever detail-work which characterized the Alexandrian mathematic of late Hellenism.

A historical paradigm will make this clearer.

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