The decline of the West

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

XVI

The liberation of geometry from the visual, and of algebra from the notion of magnitude, and the union of both, beyond all elementary limitations of drawing and counting, in the great structure of function-theory—this was the grand course of Western number-thought. The constant number of the Classical mathematic was dissolved into the variable. Geometry became analytical and dissolved all concrete forms, replacing the mathematical bodies from which the rigid geometrical values had been obtained, by abstract spatial relations which in the end ceased to have any application at all to sense-present phenomena. It began by substituting for Euclid’s optical figures geometrical loci referred to a co-ordinate system of arbitrarily chosen “origin,” and reducing the postulated objectiveness of existence of the geometrical object to the one condition that during the operation (which itself was one of equating and not of measurement) the selected co-ordinate system should not be changed. But these co-ordinates immediately came to be regarded as values pure and simple, serving not so much to determine as to represent and replace the position of points as space-elements. Number, the boundary of things-become, was represented, not as before pictorially by a figure, but symbolically by an equation. “Geometry” altered its meaning; the co-ordinate system as a picturing disappeared and the point became an entirely abstract number-group. In architecture, we find this inward transformation of Renaissance into Baroque through the innovations of Michael Angelo and Vignola. Visually pure lines became, in palace and church façades as in mathematics, ineffectual. In place of the clear co-ordinates that we have in Romano-Florentine colonnading and storeying, the “infinitesimal” appears in the graceful flow of elements, the scrollwork, the cartouches. The constructive dissolves in the wealth of the decorative—in mathematical language, the functional. Columns and pilasters, assembled in groups and clusters, break up the façades, gather and disperse again restlessly. The flat surfaces of wall, roof, storey melt into a wealth of stucco work and ornaments, vanish and break into a play of light and shade. The light itself, as it is made to play upon the form-world of mature Baroque—viz., the period from Bernini (1650) to the Rococo of Dresden, Vienna and Paris—has become an essentially musical element. The Dresden Zwinger^[75] is a sinfonia. Along with 18th Century mathematics, 18th Century architecture develops into a form-world of musical characters.

XVII

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

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

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

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

XVIII

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

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

A historical paradigm will make this clearer.