And herein, I think, all the philosopher_The decline of the West, Volume 1

a serious criticism. What they do not possess is real standing in actual

life. Not one of them has intervened effectively, either in higher

politics, in the development of modern technics, in matters of

communication, in economics, or in any other _big_ actuality, with a

single act or a single compelling idea. Not one of them counts in

mathematics, in physics, in the science of government, even to the

extent that Kant counted. Let us glance at other times. Confucius was

several times a minister. Pythagoras was the organizer of an important

political movement[38] akin to the Cromwellian, the significance of

which is even now far underestimated by Classical researchers. Goethe,

besides being a model executive minister—though lacking, alas! the

operative sphere of a great state—was interested in the Suez and Panama

canals (the dates of which he foresaw with accuracy) and their effects

on the economy of the world, and he busied himself again and again with

the question of American economic life and its reactions on the Old

World, and with that of the dawning era of machine-industry. Hobbes was

one of the originators of the great plan of winning South America for

England, and although in execution the plan went no further than the

occupation of Jamaica, he has the glory of being one of the founders of

the British Colonial Empire. Leibniz, without doubt the greatest

intellect in Western philosophy, the founder of the differential

calculus and the _analysis situs_, conceived or co-operated in a number

of major political schemes, one of which was to relieve Germany by

drawing the attention of Louis XIV to the importance of Egypt as a

factor in French world-policy. The ideas of the memorandum on this

subject that he drew up for the Grand Monarch were so far in advance of

their time (1672) that it has been thought that Napoleon made use of

them for his Eastern venture. Even thus early, Leibniz laid down the

principle that Napoleon grasped more and more clearly after Wagram,

viz., that acquisitions on the Rhine and in Belgium would not

permanently better the position of France and that the neck of Suez

would one day be the key of world-dominance. Doubtless the King was not

equal to these deep political and strategic conceptions of the

Philosopher.

Turning from men of this mould to the “philosophers” of to-day, one is

dismayed and shamed. How poor their personalities, how commonplace their

political and practical outlook! Why is it that the mere idea of calling

upon one of them to prove his intellectual eminence in government,

diplomacy, large-scale organization, or direction of any big colonial,

commercial or transport concern is enough to evoke our pity? And this

insufficiency indicates, not that they possess inwardness, but simply

that they lack weight. I look round in vain for an instance in which a

modern “philosopher” has made a name by even one deep or far-seeing

pronouncement on an important question of the day. I see nothing but

provincial opinions of the same kind as anyone else’s. Whenever I take

up a work by a modern thinker, I find myself asking: has he any idea

whatever of the actualities of world-politics, world-city problems,

capitalism, the future of the state, the relation of technics to the

course of civilization, Russia, Science? Goethe would have understood

all this and revelled in it, but there is not one living philosopher

capable of taking it in. This sense of actualities is of course not the

same thing as the content of a philosophy but, I repeat, it is an

infallible symptom of its inward necessity, its fruitfulness and its

symbolic importance.

We must allow ourselves no illusions as to the gravity of this negative

result. It is palpable that we have lost sight of the final significance

of effective philosophy. We confuse philosophy with preaching, with

agitation, with novel-writing, with lecture-room jargon. We have

descended from the perspective of the bird to that of the frog. It has

come to this, that the very _possibility_ of a real philosophy of to-day

and to-morrow is in question. If not, it were far better to become a

colonist or an engineer, to do something, no matter what, that is true

and real, than to chew over once more the old dried-up themes under

cover of an alleged “new wave of philosophic thought”—far better to

construct an aero-engine than a new theory of apperception that is not

wanted. Truly it is a poor life’s work to restate once more, in slightly

different terms, views of a hundred predecessors on the Will or on

psycho-physical parallelism. This may be a profession, but a philosophy

it emphatically is not. A doctrine that does not attack and affect the

life of the period in its inmost depths is no doctrine and had better

not be taught. And what was possible even yesterday is, to-day, at least

not indispensable.

To me, the depths and refinement of mathematical and physical theories

are a joy; by comparison, the æsthete and the physiologist are fumblers.

I would sooner have the fine mind-begotten forms of a fast steamer, a

steel structure, a precision-lathe, the subtlety and elegance of many

chemical and optical processes, than all the pickings and stealings of

present-day “arts and crafts,” architecture and painting included. I

prefer one Roman aqueduct to all Roman temples and statues. I love the

Colosseum and the giant vault of the Palatine, for they display for me

to-day in the brown massiveness of their brick construction the _real_

Rome and the grand practical sense of her engineers, but it is a matter

of indifference to me whether the empty and pretentious marblery of the

Cæsars—their rows of statuary, their friezes, their overloaded

architraves—is preserved or not. Glance at some reconstruction of the

Imperial Fora—do we not find them the true counterpart of a modern

International Exhibition, obtrusive, bulky, empty, a boasting in

materials and dimensions wholly alien to Periclean Greece and the Rococo

alike, but exactly paralleled in the Egyptian modernism that is

displayed in the ruins of Rameses II (1300 B.C.) at Luxor and Karnak? It

was not for nothing that the genuine Roman despised the _Græculus

histrio_, the kind of “artist” and the kind of “philosopher” to be found

on the soil of Roman Civilization. The time for art and philosophy had

passed; they were exhausted, used up, superfluous, and his instinct for

the realities of life told him so. _One_ Roman law weighed more than all

the lyrics and school-metaphysics of the time together. And I maintain

that to-day many an inventor, many a diplomat, many a financier is a

sounder philosopher than all those who practise the dull craft of

experimental psychology. This is a situation which regularly repeats

itself at a certain historical level. It would have been absurd in a

Roman of intellectual eminence, who might as Consul or Prætor lead

armies, organize provinces, build cities and roads, or even be the

Princeps in Rome, to want to hatch out some new variant of post-Platonic

school philosophy at Athens or Rhodes. Consequently no one did so. It

was not in harmony with the tendency of the age, and therefore it only

attracted third-class men of the kind that always advances as far as the

_Zeitgeist_ of the day before yesterday. It is a very grave question

whether this stage has or has not set in for us already.

A century of purely extensive effectiveness, excluding big artistic and

metaphysical production—let us say frankly an irreligious time which

coincides exactly with the idea of the world-city—is a time of decline.

True. But we have not _chosen_ this time. We cannot help it if we are

born as men of the early winter of full Civilization, instead of on the

golden summit of a ripe Culture, in a Phidias or a Mozart time.

Everything depends on our seeing our own position, our _destiny_,

clearly, on our realizing that though we may lie to ourselves about it

we cannot evade it. He who does not acknowledge this in his heart,

ceases to be counted among the men of his generation, and remains either

a simpleton, a charlatan, or a pedant.

Therefore, in approaching a problem of the present, one must begin by

asking one’s self—a question answered in advance by instinct in the case

of the genuine adept—what to-day is possible and what he must forbid

himself. Only a very few of the problems of metaphysics are, so to say,

allocated for solution to any epoch of thought. Even thus soon, a whole

world separates Nietzsche’s time, in which a last trace of romanticism

was still operative, from our own, which has shed every vestige of it.

Systematic philosophy closes with the end of the 18th Century. Kant put

its utmost possibilities in forms both grand in themselves and—as a

rule—final for the Western soul. He is followed, as Plato and Aristotle

were followed, by a specifically megalopolitan philosophy that was not

speculative but practical, irreligious, social-ethical. This philosophy—

paralleled in the Chinese civilization by the schools of the “Epicurean”

Yang-chu, the “Socialist” Mo-ti, the “Pessimist” Chuang-tsü, the

“Positivist” Mencius, and in the Classical by the Cynics, the Cyrenaics,

the Stoics and the Epicureans—begins in the West with Schopenhauer, who

is the first to make the _Will to life_ (“creative life-force”) the

centre of gravity of his thought, although the deeper tendency of his

doctrine is obscured by his having, under the influence of a great

tradition, maintained the obsolete distinctions of phenomena and things-

in-themselves and suchlike. It is the same creative will-to-life that

was Schopenhauer-wise denied in “Tristan” and Darwin-wise asserted in

“Siegfried”; that was brilliantly and theatrically formulated by

Nietzsche in “Zarathustra”; that led the Hegelian Marx to an economic

and the Malthusian Darwin to a biological hypothesis which together have

subtly transformed the world-outlook of the Western megalopolis; and

that produced a homogeneous series of tragedy-conceptions extending from

Hebbel’s “Judith” to Ibsen’s “Epilogue.” It has embraced, therefore, all

the possibilities of a true philosophy—and at the same time it has

exhausted them.

Systematic philosophy, then, lies immensely far behind us, and ethical

has been wound up. _But a third possibility, corresponding to the

Classical Scepticism, still remains to the soul-world_ of the present-

day West, and it can be brought to light by the hitherto unknown methods

of historical morphology. That which is a possibility is a necessity.

The Classical scepticism is ahistoric, it doubts by denying outright.

But that of the West, if it is an inward necessity, a symbol of the

autumn of our spirituality, is obliged to be historical through and

through. Its solutions are got by treating everything as relative, as a

historical phenomenon, and its procedure is psychological. Whereas the

Sceptic philosophy arose within Hellenism as the negation of philosophy—

declaring philosophy to be purposeless—we, on the contrary, regard the

_history of philosophy_ as, in the last resort, philosophy’s gravest

theme. This _is_ “skepsis,” in the true sense, for whereas the Greek is

led to renounce absolute standpoints by contempt for the intellectual

past, we are led to do so by comprehension of that past as an organism.

In this work it will be our task to sketch out this unphilosophical

philosophy—the last that West Europe will know. Scepticism is the

expression of a pure Civilization; and it dissipates the world-picture

of the Culture that has gone before. For us, its success will lie in

resolving all the older problems into one, the genetic. The conviction

that what _is_ also _has become_, that the natural and cognizable is

rooted in the historic, that the World as the actual is founded on an

Ego as the potential actualized, that the “when” and the “how long” hold

as deep a secret as the “what,” leads directly to the fact that

everything, whatever else it may be, must at any rate be _the expression

of something living_. Cognitions and judgments too are acts of living

men. The thinkers of the past conceived external actuality as produced

by cognition and motiving ethical judgments, but to the thought of the

future they are above all _expressions and symbols. The Morphology of

world-history becomes inevitably a universal symbolism._

With that, the claim of higher thought to possess general and eternal

truths falls to the ground. Truths are truths only in relation to a

particular mankind. Thus, my own philosophy is able to express and

reflect _only_ the Western (as distinct from the Classical, Indian, or

other) soul, and that soul _only_ in its present civilized phase by

which its conception of the world, its practical range and its sphere of

effect are specified.

-----

Footnote 38:

See Vol. II, 373 ff.

-----

XVI

In concluding this Introduction, I may be permitted to add a personal

note. In 1911, I proposed to myself to put together some broad

considerations on the political phenomena of the day and their possible

developments. At that time the World-War appeared to me both as imminent

and also as the inevitable outward manifestation of the historical

crisis, and my endeavour was to comprehend it from an examination of the

spirit of the preceding centuries—not years. In the course of this

originally small task,[39] the conviction forced itself on me that for

an effective understanding of the epoch the area to be taken into the

foundation-plan must be very greatly enlarged, and that in an

investigation of this sort, if the results were to be fundamentally

conclusive and necessary results, it was impossible to restrict one’s

self to a single epoch and its political actualities, or to confine

one’s self to a pragmatical framework, or even to do without purely

metaphysical and highly transcendental methods of treatment. It became

evident that a political problem could not be comprehended by means of

politics themselves and that, frequently, important factors at work in

the depths could only be grasped through their artistic manifestations

or even distantly seen in the form of scientific or purely philosophical

ideas. Even the politico-social analysis of the last decades of the 19th

century—a period of tense quiet between two immense and outstanding

events: the one which, expressed in the Revolution and Napoleon, had

fixed the picture of West-European actuality for a century and another

of at least equal significance that was visibly and ever more rapidly

approaching—was found in the last resort to be impossible without

bringing in _all_ the great problems of Being in all their aspects. For,

in the historical as in the natural world-picture, there is found

nothing, however small, that does not embody in itself the entire sum of

fundamental tendencies. And thus the original theme came to be immensely

widened. A vast number of unexpected (and in the main entirely novel)

questions and interrelations presented themselves. And finally it became

perfectly clear that no single fragment of history could be thoroughly

illuminated unless and until the secret of world-history itself, to wit

the story of higher mankind as an organism of regular structure, had

been cleared up. And hitherto this has not been done, even in the least

degree.

From this moment on, relations and connexions—previously often

suspected, sometimes touched on but never comprehended—presented

themselves in ever-increasing volume. The forms of the arts linked

themselves to the forms of war and state-policy. Deep relations were

revealed between political and mathematical aspects of the same Culture,

between religious and technical conceptions, between mathematics, music

and sculpture, between economics and cognition-forms. Clearly and

unmistakably there appeared the fundamental dependence of the most

modern physical and chemical theories on the mythological concepts of

our Germanic ancestors, the style-congruence of tragedy and power-

technics and up-to-date finance, and the fact (bizarre at first but soon

self-evident) that oil-painting perspective, printing, the credit

system, longrange weapons, and contrapuntal music in one case, and the

nude statue, the city-state and coin-currency (discovered by the Greeks)

in another were identical expressions of one and the same spiritual

principle. And, beyond and above all, there stood out the fact that

these great _groups of morphological relations_, each one of which

symbolically represents a particular sort of mankind in the whole

picture of world-history, are strictly symmetrical in structure. It is

this perspective that first opens out for us the true style of history.

Belonging itself as symbol and expression to one time and therefore

inwardly possible and necessary only for present-day Western man, it can

but be compared—distantly—to certain ideas of ultra-modern mathematics

in the domain of the Theory of Groups. These were thoughts that had

occupied me for many years, though dark and undefined until enabled by

this method to emerge in tangible form.

Thereafter I saw the present—the approaching World-War—in a quite other

light. It was no longer a momentary constellation of casual facts due to

national sentiments, personal influences, or economic tendencies endowed

with an appearance of unity and necessity by some historian’s scheme of

political or social cause-and-effect, but the type of _a historical

change of phase_ occurring within a great historical organism of

definable compass at the point preordained for it hundreds of years ago.

The mark of the great crisis is its innumerable passionate questionings

and probings. In our own case there were books and ideas by the

thousand; but, scattered, disconnected, limited by the horizons of

specialisms as they were, they incited, depressed and confounded but

could not free. Hence, though these questions are seen, their identity

is missed. Consider those art-problems that (though never comprehended

in their depths) were evinced in the disputes between form and content,

line and space, drawing and colour, in the notion of style, in the idea

of Impressionism and the music of Wagner. Consider the decline of art

and the failing authority of science; the grave problems arising out of

the victory of the megalopolis over the country-side, such as

childlessness and land-depopulation; the place in society of a

fluctuating Fourth Estate; the crisis in materialism, in Socialism, in

parliamentary government; the position of the individual _vis-à-vis_ the

State; the problem of private property with its pendant the problem of

marriage. Consider at the same time one fact taken from what is

apparently an entirely different field, the voluminous work that was

being done in the domain of folk-psychology on the origins of myths,

arts, religions and thought—and done, moreover, no longer from an ideal

but from a strictly morphological standpoint. It is my belief that every

one of these questions was really aimed in the same direction as every

other, viz., towards that _one_ Riddle of History that had never yet

emerged with sufficient distinctness in the human consciousness. The

tasks before men were not, as supposed, infinitely numerous—they were

one and the same task. Everyone had an inkling that this was so, but no

one from his own narrow standpoint had seen the single and comprehensive

solution. And yet it had been in the air since Nietzsche, and Nietzsche

himself had gripped all the decisive problems although, being a

romantic, he had not dared to look strict reality in the face.

But herein precisely lies the inward necessity of the _stock-taking_

doctrine, so to call it. It had to come, and it could only come at this

time. Our scepticism is not an attack upon, but rather the verification

of, our stock of thoughts and works. It _confirms_ all that has been

sought and achieved for generations past, in that it integrates all the

truly living tendencies which it finds in the special spheres, no matter

what their aim may be.

Above all, there discovered itself the _opposition of History and

Nature_ through which alone it is possible to grasp the essence of the

former. As I have already said, man as an element and representative of

the World is a member, not only of nature, but also of history—which is

a second Cosmos different in structure and complexion, entirely

neglected by Metaphysics in favour of the first. I was originally

brought to reflect on this _fundamental_ question of our world-

consciousness through noticing how present-day historians as they fumble

round tangible events, things-become, believe themselves to have already

grasped History, the happening, the becoming itself. This is a prejudice

common to all who proceed by reason and cognition, as against intuitive

perception.[40] And it had long ago been a source of perplexity to the

great Eleatics with their doctrine that through cognition there could be

no becoming, but only a being (or having-become). In other words,

History was seen as Nature (in the objective sense of the physicist) and

treated accordingly, and it is to this that we must ascribe the baneful

mistake of applying the principles of causality, of law, of system—that

is, the structure of rigid being—to the picture of happenings. It was

assumed that a human culture existed just as electricity or gravitation

existed, and that it was capable of analysis in much the same way as

these. The habits of the scientific researcher were eagerly taken as a

model, and if, from time to time, some student asked what Gothic, or

Islam, or the Polis _was_, no one inquired why such symbols of something

living _inevitably_ appeared just _then, and there, in that form, and

for that space of time_. Historians were content, whenever they met one

of the innumerable similarities between widely discrete historical

phenomena, simply to register it, adding some clever remarks as to the

marvels of coincidence, dubbing Rhodes the “Venice of Antiquity” and

Napoleon the “modern Alexander,” or the like; yet it was just these

cases, in which the _destiny-problem_ came to the fore as the true

problem of history (viz., the problem of time), that needed to be

treated with all possible seriousness and scientifically regulated

_physiognomic_ in order to find out what strangely-constituted

necessity, so completely alien to the causal, was at work. That every

phenomenon _ipso facto_ propounds a metaphysical riddle, that the time

of its occurrence is _never_ irrelevant; that it still remained to be

discovered what kind of a _living_ interdependence (apart from the

inorganic, natural-law interdependence) subsists within the world-

picture, which radiates from nothing less than the whole man and not

merely (as Kant thought) from the cognizing part of him; that a

phenomenon is not only a fact for the understanding but also an

expression of the spiritual, not only an object but a symbol as well, be

it one of the highest creations of religion or art or a mere trifle of

everyday life—all this was, philosophically, something new.

And thus in the end I came to see the solution clearly before me in

immense outlines, possessed of full inward necessity, a solution derived

from one single principle that though discoverable had never been

discovered, that from my youth had haunted and attracted me, tormenting

me with the sense that it was there and must be attacked and yet defying

me to seize it. Thus, from an almost accidental occasion of beginning,

there has arisen the present work, which is put forward as the

provisional expression of a new world-picture. The book is laden, as I

know, with all the defects of a first attempt, incomplete, and certainly

not free from inconsistencies. Nevertheless I am convinced that it

contains the incontrovertible formulation of an idea which, once

enunciated clearly, will (I repeat) be accepted without dispute.

If, then, the narrower theme is an analysis of the Decline of that West-

European Culture which is now spread over the entire globe, yet the

object in view is the development of a philosophy and of the operative

method peculiar to it, which is now to be tried, viz., the method of

comparative morphology in world-history. The work falls naturally into

two parts. The first, “Form and Actuality,” starts from the form-

language of the great Cultures, attempts to penetrate to the deepest

roots of their origin and so provides itself with the basis for a

science of Symbolic. The second part, “World-historical Perspectives,”

starts from the _facts of actual life_, and from the historical practice

of higher mankind seeks to obtain a quintessence of historical

experience that we can set to work upon the formation of our own future.

The accompanying tables[41] present a general view of what has resulted

from the investigation. They may at the same time give some notion both

of the fruitfulness and of the scope of the new methods.

------------------------------------------------------------------------

CHAPTER II

THE MEANING OF NUMBERS

CHAPTER II

THE MEANING OF NUMBERS

It is necessary to begin by drawing attention to certain basic terms

which, as used in this work, carry strict and in some cases novel

connotations. Though the metaphysical content of these terms would

gradually become evident in following the course of the reasoning,

nevertheless, the exact significance to be attached to them ought to be

made clear beyond misunderstanding from the very outset.

The popular distinction—current also in philosophy—between “being” and

“becoming” seems to miss the essential point in the contrast it is meant

to express. An endless becoming—“action,” “actuality”—will always be

thought of also as a condition (as it is, for example, in physical

notions such as uniform velocity and the condition of motion, and in the

basic hypothesis of the kinetic theory of gases) and therefore ranked in

the category of “being.” On the other hand, out of the results that we

do in fact obtain by and in consciousness, we may, with Goethe,

distinguish as final elements “becoming” and “the become” (_Das Werden_,

_das Gewordne_). In all cases, though the atom of human-ness may lie

beyond the grasp of our powers of abstract conception, the very clear

and definite _feeling_ of this contrast—fundamental and diffused

throughout consciousness—is the most elemental something that we reach.

It necessarily follows therefore that “the become” is always founded on

a “becoming” and not the other way round.

I distinguish further, by the words “proper” and “alien” (_das Eigne_,

_das Fremde_), those two basic facts of consciousness which for all men

in the waking (not in the dreaming) state are established with an

immediate inward certainty, without the necessity or possibility of more

precise definition. The element called “alien” is always related in some

way to the basic fact expressed by the word “perception,” i.e., the

outer world, the life of sensation. Great thinkers have bent all their

powers of image-forming to the task of expressing this relation, more

and more rigorously, by the aid of half-intuitive dichotomies such as

“phenomena and things-in-themselves,” “world-as-will and world-as-idea,”

“ego and non-ego,” although human powers of exact knowing are surely

inadequate for the task.

Similarly, the element “proper” is involved with the basic fact known as

feeling, i.e., the inner life, in some intimate and invariable way that

equally defies analysis by the methods of abstract thought.

I distinguish, again, “soul” and “world.” _The existence of this

opposition is identical with the fact of purely human waking

consciousness (Wachsein)._ There are degrees of clearness and sharpness

in the opposition and therefore grades of the consciousness, of the

spirituality, of life. These grades range from the feeling-knowledge

that, unalert yet sometimes suffused through and through by an inward

light, is characteristic of the primitive and of the child (and also of

those moments of religious and artistic inspiration that occur ever less

and less often as a Culture grows older) right to the extremity of

waking and reasoning sharpness that we find, for instance, in the

thought of Kant and Napoleon, for whom soul and world have become

subject and object. This elementary structure of consciousness, as a

fact of immediate inner knowledge, is not susceptible of conceptual

subdivision. Nor, indeed, are the two factors distinguishable at all

except verbally and more or less artificially, since they are always

associated, always intertwined, and present themselves as a unit, a

totality. The epistemological starting-point of the born idealist and

the born realist alike, the assumption that soul is to world (or world

to soul, as the case may be) as foundation is to building, as primary to

derivative, as “cause” to “effect,” has no basis whatever in the pure

fact of consciousness, and when a philosophic system lays stress on the

one or the other, it only thereby informs us as to the personality of

the philosopher, a fact of purely biographical significance.

Thus, by regarding waking-consciousness structurally as a tension of

contraries, and applying to it the notions of “becoming” and “the thing-

become,” we find for the word _Life_ a perfectly definite meaning that

is closely allied to that of “becoming.” We may describe becomings and

the things-become as the form in which respectively the facts and the

results of life exist in the waking consciousness. To man in the waking

state his proper life, progressive and constantly self-fulfilling, is

presented through the element of Becoming in his consciousness—_this

fact we call “the present”_—and it possesses that mysterious property of

_Direction_ which in all the higher languages men have sought to impound

and—vainly—to rationalize by means of the enigmatic word _time_. It

follows necessarily from the above that there is a fundamental connexion

between _the become_ (the _hard-set_) and _Death_.

If, now, we designate the Soul—that is, the Soul as it is felt, not as

it is reasonably pictured—as the _possible_ and the World on the other

hand as the _actual_ (the meaning of these expressions is unmistakable

to man’s inner sense), we see life as _the form in which the actualizing

of the possible is accomplished_. With respect to the property of

Direction, the possible is called the _Future_ and the actualized the

_Past_. The actualizing itself, the centre-of-gravity and the centre-of-

meaning of life, we call the _Present_. “Soul” is the still-to-be-

accomplished, “World” the accomplished, “life” the accomplishing. In

this way we are enabled to assign to expressions like moment, duration,

development, life-content, vocation, scope, aim, fullness and emptiness

of life, the definite meanings which we shall need for all that follows

and especially for the understanding of historical phenomena.

Lastly, the words _History_ and _Nature_ are here employed, as the

reader will have observed already, in a quite definite and hitherto

unusual sense. These words comprise _possible_ modes of understanding,

of comprehending the totality of knowledge—becoming as well as things-

become, life as well as things-lived—as a homogeneous, spiritualized,

well-ordered _world-picture_ fashioned out of an indivisible mass-

impression in this way or in that according as the becoming or the

become, direction (“time”) or extension (“space”) is the dominant

factor. And it is not a question of one factor being alternative to the

other. The possibilities that we have of possessing an “outer world”

that reflects and attests our proper existence are infinitely numerous

and exceedingly heterogeneous, and the purely organic and the purely

mechanical world-view (in the precise literal sense of that familiar

term[42]) are only the extreme members of the series. Primitive man (so

far as we can imagine his waking-consciousness) and the child (as we can

remember) cannot fully see or grasp these possibilities. One condition

of this higher world-consciousness is the possession of _language_,

meaning thereby not mere human utterance but a culture-language, and

such is non-existent for primitive man and existent but not accessible

in the case of the child. In other words, neither possesses any clear

and distinct notion of the world. They have an inkling but no real

knowledge of history and nature, being too intimately incorporated with

the ensemble of these. _They have no Culture._

And therewith that important word is given a positive meaning of the

highest significance which henceforward will be assumed in using it. In

the same way as we have elected to distinguish the Soul as the possible

and the World as the actual, we can now differentiate between _possible_

and _actual_ culture, i.e., culture as _an idea in the_ (general or

individual) _existence_ and culture as the _body_ of that idea, as the

total of its visible, tangible and comprehensible expressions—acts and

opinions, religion and state, arts and sciences, peoples and cities,

economic and social forms, speech, laws, customs, characters, facial

lines and costumes. _Higher history_, intimately related to life and to

becoming, _is the actualizing of possible Culture_.[43]

We must not omit to add that these basic determinations of meaning are

largely incommunicable by specification, definition or proof, and in

their deeper import must be reached by feeling, experience and

intuition. There is a distinction, rarely appreciated as it should be,

between experience as lived and experience as learned (zwischen Erleben

und Erkennen), between the immediate certainty given by the various

kinds of intuition—such as illumination, inspiration, artistic flair,

experience of life, the power of “sizing men up” (Goethe’s

“exact percipient fancy”)—and the product of rational procedure and

technical experiment.

The first are imparted by means of analogy, picture, symbol, the second

by formula, law, scheme. The become is experienced by learning—indeed,

as we shall see, the having-become is for the human mind identical with

the completed act of cognition. A becoming, on the other hand, can only

be experienced by living, felt with a deep wordless understanding. It is

on this that what we call “knowledge of men” is based; in fact the

understanding of history implies a superlative knowledge of men. The eye

which can see into the depths of an alien soul—owes nothing to the

cognition-methods investigated in the “Critique of Pure Reason,” yet the

purer the historical picture is, the less accessible it becomes to any

other eye. The mechanism of a pure nature-picture, such as the world of

Newton and Kant, is cognized, grasped, dissected in laws and equations

and finally reduced to system: the organism of a pure history-picture,

like the world of Plotinus, Dante and Giordano Bruno, is intuitively

seen, inwardly experienced, grasped as a form or symbol and finally

rendered in poetical and artistic conceptions. Goethe’s “living nature”

is a _historical_ world-picture.[44]

-----

Footnote 39:

The work referred to is embodied in Vol. II (pp. 521 et seq., 562 et

seq., 631 et seq.).

Footnote 40:

The philosophy of this book I owe to the philosophy of Goethe, which

is practically unknown to-day, and also (but in a far less degree) to

that of Nietzsche. The position of Goethe in West-European metaphysics

is still not understood in the least; when philosophy is being

discussed he is not even named. For unfortunately he did not set down

his doctrines in a rigid system, and so the systematic philosophy has

overlooked him. Nevertheless he was a philosopher. His place _vis-à-

vis_ Kant is the same as that of Plato—who similarly eludes the would-

be-systematizer—_vis-à-vis_ Aristotle. Plato and Goethe stand for the

philosophy of Becoming, Aristotle and Kant the philosophy of Being.

Here we have intuition opposed to analysis. Something that it is

practically impossible to convey by the methods of reason is found in

individual sayings and poems of Goethe, e.g., in the Orphische

Urworte, and stanzas like “Wenn im Unendlichen” and “Sagt es Niemand,”

which must be regarded as the expression of a _perfectly definite_

metaphysical doctrine. I would not have one single word changed in

this: "The Godhead is effective in the living and not in the dead, in

the becoming and the changing, not in the become and the set-fast; and

therefore, similarly, the reason (_Vernunft_) is concerned only to

strive towards the divine through the becoming and the living, and the

understanding (_Verstand_) only to make use of the become and the set-

fast" (to Eckermann). This sentence comprises my entire philosophy.

Footnote 41:

At the end of the volume.

Footnote 42:

Weltanschauung im wörtlichen Sinne; Anschauung der Welt.

Footnote 43:

The case of mankind in the historyless state is discussed in Vol. II,

pp. 58 et seq.

Footnote 44:

With, moreover, a “biological horizon.” See Vol. II, p. 34.

-----

In order to exemplify the way in which a soul seeks to actualize itself

in the picture of its outer world—to show, that is, in how far Culture

in the “become” state can express or portray an idea of human existence—

I have chosen _number_, the primary element on which all mathematics

rests. I have done so because mathematics, accessible in its full depth

only to the very few, holds a quite peculiar position amongst the

creations of the mind. It is a science of the most rigorous kind, like

logic but more comprehensive and very much fuller; it is a true art,

along with sculpture and music, as needing the guidance of inspiration

and as developing under great conventions of form; it is, lastly, a

metaphysic of the highest rank, as Plato and above all Leibniz show us.

Every philosophy has hitherto grown up in conjunction with a mathematic

_belonging_ to it. Number is the symbol of causal necessity. Like the

conception of God, it contains the ultimate meaning of the world-as-

nature. The existence of numbers may therefore be called a mystery, and

the religious thought of every Culture has felt their impress.[45]

Just as all becoming possesses the original property of _direction_

(irreversibility), all things-become possess the property of

_extension_. But these two words seem unsatisfactory in that only an

artificial distinction can be made between them. The real secret of all

things-become, which are _ipso facto_ things extended (spatially and

materially), is embodied in mathematical number as contrasted with

chronological number. Mathematical number contains in its very essence

the notion of a _mechanical demarcation_, number being in that respect

akin to _word_, which, in the very fact of its comprising and denoting,

fences off world-impressions. The deepest depths, it is true, are here

both incomprehensible and inexpressible. But the actual number with

which the mathematician works, the figure, formula, sign, diagram, in

short the _number-sign which he thinks, speaks or writes exactly_, is

(like the exactly-used word) from the first a symbol of these depths,

something imaginable, communicable, comprehensible to the inner and the

outer eye, which can be accepted as representing the demarcation. The

origin of numbers resembles that of the myth. Primitive man elevates

indefinable nature-impressions (the “alien,” in our terminology) into

deities, _numina_, at the same time capturing and impounding them by a

_name_ which limits them. So also numbers are something that marks off

and captures nature-impressions, and it is by means of names and numbers

that the human understanding obtains power over the world. In the last

analysis, the number-language of a mathematic and the grammar of a

tongue are structurally alike. Logic is always a kind of mathematic and

vice versa. Consequently, in all acts of the intellect germane to

mathematical number—measuring, counting, drawing, weighing, arranging

and dividing[46]—men strive to delimit the extended in words as well,

i.e., to set it forth in the form of proofs, conclusions, theorems and

systems; and it is only through acts of this kind (which may be more or

less unintentioned) that waking man begins to be able to use numbers,

normatively, to specify objects and properties, relations and

differentiæ, unities and pluralities—briefly, that structure of the

world-picture which he feels as necessary and unshakable, calls “Nature”

and “cognizes.” _Nature is the numerable_, while History, on the other

hand, is the aggregate of that which has no relation to mathematics—

hence the mathematical certainty of the laws of Nature, the astounding

rightness of Galileo’s saying that Nature is “written in mathematical

language,” and the fact, emphasized by Kant, that exact natural science

reaches just as far as the possibilities of applied mathematics allow it

to reach. In number, then, as the _sign of completed demarcation_, lies

the _essence_ of everything actual, which is cognized, is delimited, and

has become all at once—as Pythagoras and certain others have been able

to see with complete inward certitude by a mighty and truly religious

intuition. Nevertheless, mathematics—meaning thereby the capacity to

think practically in figures—must not be confused with the far narrower

scientific mathematics, that is, the _theory_ of numbers as developed in

lecture and treatise. The mathematical vision and thought that a Culture

possesses within itself is as inadequately represented by its written

mathematic as its philosophical vision and thought by its philosophical

treatises. Number springs from a source that has also quite other

outlets. Thus at the beginning of every Culture we find an archaic

style, which might fairly have been called geometrical in other cases as

well as the Early Hellenic. There is a common factor which is expressly

mathematical in this early Classical style of the 10th Century B.C., in

the temple style of the Egyptian Fourth Dynasty with its absolutism of

straight line and right angle, in the Early Christian sarcophagus-

relief, and in Romanesque construction and ornament. Here every line,

every deliberately non-imitative figure of man and beast, reveals a

mystic number-thought in direct connexion with the mystery of death (the

hard-set).

Gothic cathedrals and Doric temples are _mathematics in stone_.

Doubtless Pythagoras was the first in the Classical Culture to conceive

number scientifically as the principle of a world-order of

comprehensible things—as _standard_ and as _magnitude_—but even before

him it had found expression, as a noble arraying of sensuous-material

units, in the strict canon of the statue and the Doric order of columns.

The great arts are, one and all, modes of interpretation by means of

limits based on number (consider, for example, the problem of space-

representation in oil painting). A high mathematical endowment may,

without any mathematical science whatsoever, come to fruition and full

self-knowledge in _technical_ spheres.

In the presence of so powerful a number-sense as that evidenced, even in

the Old Kingdom,[47] in the dimensioning of pyramid temples and in the

technique of building, water-control and public administration (not to

mention the calendar), no one surely would maintain that the valueless

arithmetic of Ahmes belonging to the New Empire represents the level of

Egyptian mathematics. The Australian natives, who rank intellectually as

thorough primitives, possess a mathematical instinct (or, what comes to

the same thing, a power of thinking in numbers which is not yet

communicable by signs or words) that as regards the interpretation of

pure space is far superior to that of the Greeks. Their discovery of the

boomerang can only be attributed to their having a sure feeling for

numbers of a class that we should refer to the higher geometry.

_Accordingly_—we shall justify the adverb later—they possess an

extraordinarily complicated ceremonial and, for expressing degrees of

affinity, such fine shades of language as not even the higher Cultures

themselves can show.

There is analogy, again, between the Euclidean mathematic and the

absence, in the Greek of the mature Periclean age, of any feeling either

for ceremonial public life or for loneliness, while the Baroque,

differing sharply from the Classical, presents us with a mathematic of

spatial analysis, a court of Versailles and a state system resting on

dynastic relations.

It is the style of a Soul that comes out in the world of numbers, and

the world of numbers includes something more than the science thereof.

-----

Footnote 45:

See Vol. II, pp. 327 et seq.

Footnote 46:

Also “thinking in money.” See Vol. II, pp. 603 et seq.

Footnote 47:

Dynasties I-VIII, or, effectively, I-VI. The Pyramid period coincides

with Dynasties IV-VI. Cheops, Chephren and Mycerinus belong to the IV

dynasty, under which also great water-control works were carried out

between Abydos and the Fayum.—_Tr._

-----

III

From this there follows a fact of decisive importance which has hitherto

been hidden from the mathematicians themselves.

_There is not, and cannot be, number as such._ There are several number-

worlds as there are several Cultures. We find an Indian, an Arabian, a

Classical, a Western type of mathematical thought and, corresponding

with each, a type of number—each type fundamentally peculiar and unique,

an expression of a specific world-feeling, a symbol having a specific

validity which is even capable of scientific definition, a principle of

ordering the Become which reflects the central essence of one and only

one soul, viz., the soul of that particular Culture. Consequently, there

are more mathematics than one. For indubitably the inner structure of

the Euclidean geometry is something quite different from that of the

Cartesian, the analysis of Archimedes is something other than the

analysis of Gauss, and not merely in matters of form, intuition and

method but above all in essence, in the intrinsic and obligatory meaning

of number which they respectively develop and set forth. This number,

the horizon within which it has been able to make phenomena self-

explanatory, and therefore the whole of the “nature” or world-extended

that is confined in the given limits and amenable to its particular sort

of mathematic, are not common to all mankind, but specific in each case

to one definite sort of mankind.

The style of any mathematic which comes into being, then, depends wholly

on the Culture in which it is rooted, the sort of mankind it is that

ponders it. The soul can bring its inherent possibilities to scientific

development, can manage them practically, can attain the highest levels

in its treatment of them—but is quite impotent to alter them. The idea

of the Euclidean geometry is actualized in the earliest forms of

Classical ornament, and that of the Infinitesimal Calculus in the

earliest forms of Gothic architecture, centuries before the first

learned mathematicians of the respective Cultures were born.

A deep inward experience, the genuine _awakening of the ego_, which

turns the child into the higher man and initiates him into community of

his Culture, marks the beginning of number-sense as it does that of

language-sense. It is only after this that objects come to exist for the

waking consciousness as things limitable and distinguishable as to

number and kind; only after this that properties, concepts, causal

necessity, system in the world-around, _a form of the world_, and _world

laws_ (for that which is set and settled is _ipso facto_ bounded,

hardened, number-governed) are susceptible of exact definition. And

therewith comes too a sudden, almost metaphysical, feeling of anxiety

and awe regarding the deeper meaning of measuring and counting, drawing

and form.

Now, Kant has classified the sum of human knowledge according to

syntheses _a priori_ (necessary and universally valid) and _a

posteriori_ (experiential and variable from case to case) and in the

former class has included mathematical knowledge. Thereby, doubtless, he

was enabled to reduce a strong inward feeling to abstract form. But,

quite apart from the fact (amply evidenced in modern mathematics and

mechanics) that there is no such sharp distinction between the two as is

originally and unconditionally implied in the principle, the _a priori_

itself, though certainly one of the most inspired conceptions of

philosophy, is a notion that seems to involve enormous difficulties.

With it Kant postulates—without attempting to prove what is quite

incapable of proof—both _unalterableness of form_ in all intellectual

activity and _identity of form for all men_ in the same. And, in

consequence, a factor of incalculable importance is—thanks to the

intellectual prepossessions of his period, not to mention his own—simply

ignored. This factor is the _varying degree_ of this alleged “universal

validity.” There are doubtless certain characters of very wide-ranging

validity which are (seemingly at any rate) independent of the Culture

and century to which the cognizing individual may belong, but along with

these there is a quite particular necessity of form which underlies all

his thought as axiomatic and to which he is subject by virtue of

belonging to his own Culture and no other. Here, then, we have two very

different kinds of _a priori_ thought-content, and the definition of a

frontier between them, or even the demonstration that such exists, is a

problem that lies beyond all possibilities of knowing and will never be

solved. So far, no one has dared to assume that the supposed constant

structure of the intellect is an illusion and that the history spread

out before us contains more than one _style of knowing_. But we must not

forget that unanimity about things that have not yet become problems may

just as well imply universal error as universal truth. True, there has

always been a certain sense of doubt and obscurity—so much so, that the

correct guess might have been made from that non-agreement of the

philosophers which every glance at the history of philosophy shows us.

But that this non-agreement is not due to imperfections of the human

intellect or present gaps in a perfectible knowledge, in a word, is not

due to defect, but to destiny and historical necessity—this is a

_discovery_. Conclusions on the deep and final things are to be reached

not by predicating constants but by studying differentiæ and developing

the _organic logic_ of differences. The _comparative morphology of

knowledge forms_ is a domain which Western thought has still to attack.

If mathematics were a mere science like astronomy or mineralogy, it

would be possible to define their object. This man is not and never has

been able to do. We West-Europeans may put our own scientific notion of

number to perform the same tasks as those with which the mathematicians

of Athens and Baghdad busied themselves, but the fact remains that the

theme, the intention and the methods of the like-named science in Athens

and in Baghdad were quite different from those of our own. _There is no

mathematic but only mathematics._ What we call “the history of

mathematics”—implying merely the progressive actualizing of a single

invariable ideal—is in fact, below the deceptive surface of history, a

complex of self-contained and independent developments, an ever-repeated

process of bringing to birth new form-worlds and appropriating,

transforming and sloughing alien form-worlds, a purely organic story of

blossoming, ripening, wilting and dying within the set period. The

student must not let himself be deceived. The mathematic of the

Classical soul sprouted almost out of nothingness, the historically-

constituted Western soul, already possessing the Classical science (not

inwardly, but outwardly as a thing learnt), had to win its own by

apparently altering and perfecting, but in reality destroying the

essentially alien Euclidean system. In the first case, the agent was

Pythagoras, in the second Descartes. In both cases the act is, at

bottom, the same.

The relationship between the form-language of a mathematic and that of

the cognate major arts,[48] is in this way put beyond doubt. The

temperament of the thinker and that of the artist differ widely indeed,

but the expression-methods of the waking consciousness are inwardly the

same for each. The sense of form of the sculptor, the painter, the

composer is essentially mathematical in its nature. The same inspired

ordering of an infinite world which manifested itself in the geometrical

analysis and projective geometry of the 17th Century, could vivify,

energize, and suffuse contemporary music with the harmony that it

developed out of the art of thoroughbass, (which is the geometry of the

sound-world) and contemporary painting with the principle of perspective

(the felt geometry of the space-world that only the West knows). This

inspired ordering is that which Goethe called “_The Idea, of which the

form is immediately apprehended in the domain of intuition_, whereas

pure science does not apprehend but observes and dissects.” The

Mathematic goes beyond observation and dissection, and in its highest

moments finds the way by vision, not abstraction. To Goethe again we owe

the profound saying: “the mathematician is only complete in so far as he

feels within himself the _beauty_ of the true.” Here we feel how nearly

the secret of number is related to the secret of artistic creation. And

so the born mathematician takes his place by the side of the great

masters of the fugue, the chisel and the brush; he and they alike

strive, and must strive, to actualize the grand order of all things by

clothing it in symbol and so to communicate it to the plain fellow-man

who hears that order within himself but cannot effectively possess it;

the domain of number, like the domains of tone, line and colour, becomes

an image of the world-form. For this reason the word “creative” means

more in the mathematical sphere than it does in the pure sciences—

Newton, Gauss, and Riemann were artist-natures, and we know with what

suddenness their great conceptions came upon them.[49] “A

mathematician,” said old Weierstrass “who is not at the same time a bit

of a poet will never be a full mathematician.”

The mathematic, then, is an art. As such it has its styles and style-

periods. It is not, as the layman and the philosopher (who is in this

matter a layman too) imagine, substantially unalterable, but subject

like every art to unnoticed changes from epoch to epoch. The development

of the great arts ought never to be treated without an (assuredly not

unprofitable) side-glance at contemporary mathematics. In the very deep

relation between changes of musical theory and the analysis of the

infinite, the details have never yet been investigated, although

æsthetics might have learned a great deal more from these than from all

so-called “psychology.” Still more revealing would be a history of

musical instruments written, not (as it always is) from the technical

standpoint of tone-production, but as a study of the deep spiritual

bases of the tone-colours and tone-effects aimed at. For it was the

wish, intensified to the point of a longing, to fill a spatial infinity

with sound which produced—in contrast to the Classical lyre and reed

(lyra, kithara; aulos, syrinx) and the Arabian lute—the two great

families of keyboard instruments (organ, pianoforte, etc.) and bow

instruments, and that as early as the Gothic time. The development of

both these families belongs spiritually (and possibly also in point of

technical origin) to the Celtic-Germanic North lying between Ireland,

the Weser and the Seine. The organ and clavichord belong certainly to

England, the bow instruments reached their definite forms in Upper Italy

between 1480 and 1530, while it was principally in Germany that the

organ was developed into the _space-commanding_ giant that we know, an

instrument the like of which does not exist in all musical history. The

free organ-playing of Bach and his time was nothing if it was not

analysis—analysis of a strange and vast tone-world. And, similarly, it

is in conformity with the Western number-thinking, and in opposition to

the Classical, that our string and wind instruments have been developed

not singly but in great groups (strings, woodwind, brass), ordered

within themselves according to the compass of the four human voices; the

history of the modern orchestra, with all its discoveries of new and

modification of old instruments, is in reality the self-contained

history of one tone-world—a world, moreover, that is quite capable of

being expressed in the forms of the higher analysis.

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 ε^{ix} which we constantly use, or even the

fractional index (e.g., 5^½) 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.

The Greek mathematic, as a science of perceivable magnitudes,

deliberately confines itself to facts of the comprehensibly present, and

limits its researches and their validity to the near and the small. As

compared with this impeccable consistency, the position of the Western

mathematic is seen to be, practically, somewhat illogical, though it is

only since the discovery of Non-Euclidean Geometry that the fact has

been really recognized. Numbers are images of the perfectly

desensualized understanding, of pure thought, and contain their abstract

validity within themselves.[52] Their exact application to the actuality

of conscious experience is therefore a problem in itself—a problem which

is always being posed anew and never solved—and the congruence of

mathematical system with empirical observation is at present anything

but self-evident. Although the lay idea—as found in Schopenhauer—is that

mathematics rest upon the direct evidences of the senses, Euclidean

geometry, superficially identical though it is with the popular geometry

of all ages, is only in agreement with the phenomenal world

approximately and within very narrow limits—in fact, the limits of a

drawing-board. Extend these limits, and what becomes, for instance, of

Euclidean parallels? They meet at the line of the horizon—a simple fact

upon which all our art-perspective is grounded.

Now, it is unpardonable that Kant, a Western thinker, should have evaded

the mathematic of distance, and appealed to a set of figure-examples

that their mere pettiness excludes from treatment by the specifically

Western infinitesimal methods. But Euclid, as a thinker of the Classical

age, was entirely consistent with its spirit when he refrained from

proving the phenomenal truth of his axioms by referring to, say, the

triangle formed by an observer and two infinitely distant fixed stars.

For these can neither be drawn nor “intuitively apprehended” and his

feeling was precisely the feeling which shrank from the irrationals,

which did not dare to give nothingness a value as zero (i.e., a number)

and even in the contemplation of cosmic relations shut its eyes to the

Infinite and held to its symbol of Proportion.

Aristarchus of Samos, who in 288-277 belonged to a circle of astronomers

at Alexandria that doubtless had relations with Chaldaeo-Persian

schools, projected the elements of a heliocentric world-system.[53]

Rediscovered by Copernicus, it was to shake the metaphysical passions of

the West to their foundations—witness Giordano Bruno[54]—to become the

fulfilment of mighty premonitions, and to justify that Faustian, Gothic

world-feeling which had already professed its faith in infinity through

the forms of its cathedrals. But the world of Aristarchus received his

work with entire indifference and in a brief space of time it was

forgotten—designedly, we may surmise. His few followers were nearly all

natives of Asia Minor, his most prominent supporter Seleucus (about 150)

being from the Persian Seleucia on Tigris. In fact, the Aristarchian

system had no spiritual appeal to the Classical Culture and might indeed

have become dangerous to it. And yet it was differentiated from the

Copernican (a point always missed) by something which made it perfectly

conformable to the Classical world-feeling, viz., the assumption that

the cosmos is _contained_ in a materially finite and optically

appreciable _hollow sphere_, in the middle of which the planetary

system, arranged as such on Copernican lines, moved. In the Classical

astronomy, the earth and the heavenly bodies are consistently regarded

as entities of two different kinds, however variously their movements in

detail might be interpreted. Equally, the opposite idea that the earth

is _only a star among stars_[55] is not inconsistent in itself with

either the Ptolemaic or the Copernican systems and in fact was pioneered

by Nicolaus Cusanus and Leonardo da Vinci. But by this device of a

celestial sphere the principle of infinity which would have endangered

the sensuous-Classical notion of bounds was smothered. One would have

supposed that the infinity-conception was inevitably implied by the

system of Aristarchus—long before his time, the Babylonian thinkers had

reached it. But no such thought emerges. On the contrary, in the famous

treatise on the grains of sand[56] Archimedes proves that the filling of

this stereometric body (for that is what Aristarchus’s Cosmos is, after

all) with atoms of sand leads to very high, but _not_ to infinite,

figure-results. This proposition, quoted though it may be, time and

again, as being a first step towards the Integral Calculus, amounts to a

denial (implicit indeed in the very title) of everything that we mean by

the word analysis. Whereas in our physics, the constantly-surging

hypotheses of a material (i.e., directly cognizable) æther, break

themselves one after the other against our refusal to acknowledge

material limitations of any kind, Eudoxus, Apollonius and Archimedes,

certainly the keenest and boldest of the Classical mathematicians,

completely worked out, in the main with rule and compass, a _purely

optical_ analysis of things-become on the basis of sculptural-Classical

bounds. They used deeply-thought-out (and for us hardly understandable)

methods of integration, but these possess only a superficial resemblance

even to Leibniz’s definite-integral method. They employed geometrical

loci and co-ordinates, but these are always specified lengths and units

of measurement and never, as in Fermat and above all in Descartes,

unspecified spatial relations, values of points in terms of their

positions in space. With these methods also should be classed the

exhaustion-method of Archimedes,[57] given by him in his recently

discovered letter to Eratosthenes on such subjects as the quadrature of

the parabola section by means of inscribed rectangles (instead of

through similar polygons). But the very subtlety and extreme

complication of his methods, which are grounded in certain of Plato’s

geometrical ideas, make us realize, in spite of superficial analogies,

what an enormous difference separates him from Pascal. Apart altogether

from the idea of Riemann’s integral, what sharper contrast could there

be to these ideas than the so-called quadratures of to-day? The name

itself is now no more than an unfortunate survival, the “surface” is

indicated by a bounding function, and the _drawing_ as such, has

vanished. Nowhere else did the two mathematical minds approach each

other more closely than in this instance, and nowhere is it more evident

that the gulf between the two souls thus expressing themselves is

impassable.

In the cubic style of their early architecture the Egyptians, so to say,

concealed pure numbers, fearful of stumbling upon their secret, and for

the Hellenes too they were the key to the meaning of the become, the

stiffened, the mortal. The stone statue and the scientific system deny

life. Mathematical number, the formal principle of an extension-world of

which the phenomenal existence is only the derivative and servant of

waking human consciousness, bears the hall-mark of causal necessity and

so is linked with _death_ as chronological number is with becoming, with

_life_, with the necessity of destiny. This connexion of strict

mathematical form with the _end_ of organic being, with the phenomenon

of its organic remainder the corpse, we shall see more and more clearly

to be the origin of all great art. We have already noticed the

development of early ornament on funerary equipments and receptacles.

_Numbers are symbols of the mortal._ Stiff forms are the negation of

life, formulas and laws spread rigidity over the face of nature, numbers

make dead—and the “Mothers” of Faust II sit enthroned, majestic and

withdrawn, in

“The realms of Image unconfined.

... Formation, transformation,

Eternal play of the eternal mind

With semblances of all things in creation

For ever and for ever sweeping round.”[58]

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.

-----

Footnote 48:

As also those of law and of money. See Vol. II, pp. 68 et seq., pp.

616 et seq.

Footnote 49:

Poincaré in his _Science et Méthode_ (Ch. III), searchingly analyses

the “becoming” of one of his own mathematical discoveries. Each

decisive stage in it bears “_les mêmes caractères de brièveté, de

soudaineté et de certitude absolue_” and in most cases this

“_certitude_” was such that he merely registered the discovery and put

off its working-out to any convenient season.—_Tr._

Footnote 50:

One may be permitted to add that according to legend, both Hippasus

who took to himself public credit for the discovery of a sphere of

twelve pentagons, viz., the regular dodecahedron (regarded by the

Pythagoreans as the quintessence—or æther—of a world of real

tetrahedrons, octahedrons, icosahedrons and cubes), and Archytas the

eighth successor of the Founder are reputed to have been drowned at

sea. The pentagon from which this dodecahedron is derived, itself

involves incommensurable numbers. The “pentagram” was the recognition

badge of Pythagoreans and the ἄλογον (incommensurable) their special

secret. It would be noted, too, that Pythagoreanism was popular till

its initiates were found to be dealing in these alarming and

subversive doctrines, and then they were suppressed and lynched—a

persecution which suggests more than one deep analogy with certain

heresy-suppressions of Western history. The English student may be

referred to G. J. Allman, _Greek Geometry from Thales to Euclid_

(Cambridge, 1889), and to his articles “Pythagoras,” “Philolaus” and

“Archytas” in the _Ency. Brit._, XI Edition.—_Tr._

Footnote 51:

Horace’s words (Odes I xi): “Tu ne quæsieris, scire nefas, quem mihi

quem tibi finem di dederint, Leuconoë, _nec Babylonios temptaris

numeros_ ... carpe diem, _quam minimum credula postero_.”—_Tr._

Footnote 52:

See Vol. II, pp. 11 et seq.

Footnote 53:

In the only writing of his that survives, indeed, Aristarchus

maintains the geocentric view; it may be presumed therefore that it

was only temporarily that he let himself be captivated by a hypothesis

of the Chaldaean learning.

Footnote 54:

Giordano Bruno (born 1548, burned for heresy 1600). His whole life

might be expressed as a crusade on behalf of God and the Copernican

universe against a degenerated orthodoxy and an Aristotelian world-

idea long coagulated in death.—_Tr._

Footnote 55:

F. Strunz, _Gesch. d. Naturwiss. im Mittelalter_ (1910), p. 90.

Footnote 56:

In the “Psammites,” or “Arenarius,” Archimedes framed a numerical

notation which was to be capable of expressing the number of grains of

sand _in a sphere of the size of our universe_.—_Tr._

Footnote 57:

This, for which the ground had been prepared by Eudoxus, was employed

for calculating the volume of pyramids and cones: “the means whereby

the Greeks were able _to evade_ the forbidden notion of infinity”

(Heiberg, _Naturwiss. u. Math. i. Klass. Alter._ [1912], p. 27).

Footnote 58:

Dr. Anster’s translation.—_Tr._

Footnote 59:

See Vol. II, 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.

-----

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.

-----

Footnote 63:

Born 1601, died 1665. See Ency. Brit., XI Ed., article _Fermat_, and

references therein.—_Tr._

-----

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]

-----

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.).

-----

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.

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.

-----

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.

-----

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]

-----

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.

-----

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.

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]

-----

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:

And herein, I think, all the philosophers of the newest age are open to