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,”[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 elastically[61] 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.
VIII
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.
IX
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]
X
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 problem[66]) 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.
XI
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.
XII
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 mathematiciansmathematicians 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]
XIII
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.
XIV
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.