III

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.

IV

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, 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. “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.

V

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.”

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, 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.

VI

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. 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. Rediscovered by Copernicus, it was to shake the metaphysical passions of the West to their foundations—witness Giordano Bruno—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 stars 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 sand 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, 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.”

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.

VII

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,” 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), who was the first Western who used co-ordinates so to say elastically 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. 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.