Chapter III
.
Footnote 60:
Oresme was, equally, prelate, church reformer, scholar, scientist and economist—the very type of the philosopher-leader.—_Tr._
Footnote 61:
Oresme in his _Latitudines Formarum_ used ordinate and abscissa, not indeed to specify numerically, but certainly to describe, change, i.e., fundamentally, to express functions.—_Tr._
Footnote 62:
Alexandria ceased to be a world-city in the second century A.D. and became a collection of houses left over from the Classical civilization which harboured a primitive population of quite different spiritual constitution. See Vol. II, pp. 122 et seq.
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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.
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Footnote 63:
Born 1601, died 1665. See Ency. Brit., XI Ed., article _Fermat_, and references therein.—_Tr._
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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, α^{1⁄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]
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Footnote 64:
Similarly, coinage and double-entry book-keeping play analogous parts in the money-thinking of the Classical and the Western Cultures respectively. See Vol. II, pp. 610 et seq.
Footnote 65:
The same may be said in the matter of Roman Law (see Vol. II, pp. 96 et seq.) and of coinage (see Vol. II, pp. 616 et seq.).
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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 3^x + 4^x = 5^x and x^n + y^n = z^n (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.
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Footnote 66:
That is, “it is impossible to part a cube into two cubes, a biquadrate into two biquadrates, and generally any power above the square into two powers having the same exponent.” Fermat claimed to possess a proof of the proposition, but this has not been preserved, and no general proof has hitherto been obtained.—_Tr._
Footnote 67:
Thus Bishop Berkeley’s _Discourse addressed to an infidel mathematician_ (1735) shrewdly asked whether the mathematician were in a position to criticize the divine for proceeding on the basis of faith.—_Tr._
Footnote 68:
From the savage conjuror with his naming-magic to the modern scientist who subjects things by attaching technical labels to them, the form has in no wise changed. See Vol. II, pp. 116 et seq., 322 et seq.
Footnote 69:
See Vol. II, pp. 137 et seq.
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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 mathematicians ever use the word σῶμα for their entities, just as the Greek lawyers used it for persons as distinct from things (σώματα καὶ πράγματα: _personæ et res_).
Classical number, integral and corporeal, therefore inevitably seeks to relate itself with the birth of bodily man, the σῶμα. The number 1 is hardly yet conceived of as actual number but rather as ἀρχή, the prime stuff of the number-series, the origin of all true numbers and therefore all magnitudes, measures and materiality (Dinglichkeit). In the group of the Pythagoreans (the date does not matter) its figured-sign was also the symbol of the mother-womb, the origin of all life. The digit 2, the first _true_ number, which doubles the 1, was therefore correlated with the male principle and given the sign of the phallus. And, finally, 3, the “holy number” of the Pythagoreans, denoted the act of union between man and woman, the act of propagation—the erotic suggestion in adding and multiplying (the only two processes of increasing, of _propagating_, magnitude useful to Classical man) is easily seen—and its sign was the combination of the two first. Now, all this throws quite a new light upon the legends previously alluded to, concerning the sacrilege of disclosing the irrational. The irrational—in our language the employment of unending decimal fractions—implied the destruction of an organic and corporeal and reproductive order that the gods had laid down. There is no doubt that the Pythagorean reforms of the Classical religion were themselves based upon the immemorial Demeter-cult. Demeter, Gæa, is akin to Mother Earth. There is a deep relation between the honour paid to her and this exalted conception of the numbers.
Thus, inevitably, the Classical became by degrees the Culture of the _small_. The Apollinian soul had tried to tie down the meaning of things-become by means of the principle of _visible limits_; its taboo was focused upon the immediately-present and proximate alien. What was far away, invisible, was _ipso facto_ “not there.” The Greek and the Roman alike sacrificed to the gods of the place in which he happened to stay or reside; all other deities were outside the range of vision. Just as the Greek tongue—again and again we shall note the mighty symbolism of such language-phenomena—possessed _no word for space_, so the Greek himself was destitute of our feeling of landscape, horizons, outlooks, distances, clouds, and of the idea of the far-spread fatherland embracing the great nation. _Home_, for Classical man, is what he can see from the citadel of his native town and no more. All that lay beyond the visual range of this political atom was alien, and hostile to boot; beyond that narrow range, fear set in at once, and hence the appalling bitterness with which these petty towns strove to destroy one another. The Polis is the smallest of all conceivable state-forms, and its policy is frankly short-range, therein differing in the extreme from our own cabinet-diplomacy which is the policy of the unlimited. Similarly, the Classical temple, which can be taken in in one glance, is the smallest of all first-rate architectural forms. Classical geometry from Archytas to Euclid—like the school geometry of to-day which is still dominated by it—concerned itself with small, manageable figures and bodies, and therefore remained unaware of the difficulties that arise in establishing figures of astronomical dimensions, which in many cases are not amenable to Euclidean geometry.[70] Otherwise the subtle Attic spirit would almost surely have arrived at some notion of the problems of non-Euclidean geometry, for its criticism of the well-known “parallel” axiom,[71] the doubtfulness of which soon aroused opposition yet could not in any way be elucidated, brought it very close indeed to the decisive discovery. The Classical mind as unquestioningly devoted and limited itself to the study of the small and the near as ours has to that of the infinite and ultra-visual. All the mathematical ideas that the West found for itself or borrowed from others were automatically subjected to the form-language of the Infinitesimal—and that long before the actual Differential Calculus was discovered. Arabian algebra, Indian trigonometry, Classical mechanics were incorporated as a matter of course in analysis. Even the most “self-evident” propositions of elementary arithmetic such as 2 × 2 = 4 become, when considered analytically, problems, and the solution of these problems was only made possible by deductions from the Theory of Aggregates, and is in many points still unaccomplished. Plato and his age would have looked upon this sort of thing not only as a hallucination but also as evidence of an utterly nonmathematical mind. In a certain measure, geometry may be treated algebraically and algebra geometrically, that is, the eye may be switched off or it may be allowed to govern. We take the first alternative, the Greeks the second. Archimedes, in his beautiful management of spirals, touches upon certain general facts that are also fundamentals in Leibniz’s method of the definite integral; but his processes, for all their superficial appearance of modernity, are subordinated to stereometric principles; in like case, an Indian mathematician would naturally have found some trigonometrical formulation.[72]
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Footnote 70:
A beginning is now being made with the application of non-Euclidean geometries to astronomy. The hypothesis of curved space, closed but without limits, filled by the system of fixed stars on a radius of about 470,000,000 earth-distances, would lead to the hypothesis of a counter-image of the sun which to us appears as a star of medium brilliancy. (See translator’s footnote, p. 332.)
Footnote 71:
That only one parallel to a given straight line is possible through a given point—a proposition that is incapable of proof.
Footnote 72:
It is impossible to say, with certainty, how much of the Indian mathematics that we possess is old, i.e., before Buddha.
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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.
XV
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]
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Footnote 73:
The technical difference (in German usage) between _Grenz_ and _Grenzwert_ is in most cases ignored in this translation as it is only the underlying conception of “number” common to both that concerns us. _Grenz_ is the “limit” strictly speaking, i.e., the number _a_ to which the terms _a__{1}₁, _a__{2}₂, _a_₃ ... of a particular _series_ approximate more and more closely, till nearer to _a_ than any assignable number whatever. The _Grenzwert_ of a _function_, on the other hand, is the “limit” of the value which the function takes for a given value _a_ of the variable _x_. These methods of reasoning and their derivatives enable solutions to be obtained for series such as (1⁄_m_¹,) (1⁄_m_²,) (1⁄_m_³,) ... (1⁄_m_^{_x_}) or functions such as
_x_(2_x_ - 1) _y_ = ——————————————— (_x_+ 2)(_x_- 3)
where _x_ is _infinite_ or _indefinite_.—_Tr._
Footnote 74:
“Function, rightly understood, is existence considered as an activity” (Goethe). Cf. Vol. II, p. 618, for functional money.
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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.
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Footnote 75:
Built for August II, in 1711, as barbican or fore-building for a projected palace.—_Tr._
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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 a^{_n_}, 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.
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Footnote 76:
From the standpoint of the theory of “aggregates” (or “sets of points”), a well-ordered set of points, irrespective of the dimension figure, is called a corpus; and thus an aggregate of _n_ - 1 dimensions is considered, _relatively_ to one of _n_ dimensions, as a surface. Thus the limit (wall, edge) of an “aggregate” represents an aggregate of lower “potentiality.”
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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.
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## CHAPTER III THE PROBLEM OF WORLD-HISTORY
I PHYSIOGNOMIC AND SYSTEMATIC
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