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.”[[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.