renunciation alone can we build a temple for the worship of our own
ideals. Haunting foreshadowings of the temple appear in the realm of
imagination, in music, in architecture, in the untroubled kingdom of
reason, and in the golden sunset magic of lyrics, where beauty shines
and glows, remote from the touch of sorrow, remote from the fear of
change, remote from the failures and disenchantments of the world of
fact. In the contemplation of these things the vision of heaven will
shape itself in our hearts, giving at once a touchstone to judge the
world about us, and an inspiration by which to fashion to our needs
whatever is not incapable of serving as a stone in the sacred temple.
Except for those rare spirits that are born without sin, there is a
cavern of darkness to be traversed before that temple can be entered.
The gate of the cavern is despair, and its floor is paved with the
gravestones of abandoned hopes. There Self must die; there the
eagerness, the greed of untamed desire must be slain, for only so can
the soul be freed from the empire of Fate. But out of the cavern the
Gate of Renunciation leads again to the daylight of wisdom, by whose
radiance a new insight, a new joy, a new tenderness, shine forth to
gladden the pilgrim's heart.
When, without the bitterness of impotent rebellion, we have learnt
both to resign ourselves to the outward rule of Fate and to recognise
that the non-human world is unworthy of our worship, it becomes
possible at last so to transform and refashion the unconscious
universe, so to transmute it in the crucible of imagination, that a
new image of shining gold replaces the old idol of clay. In all the
multiform facts of the world--in the visual shapes of trees and
mountains and clouds, in the events of the life of man, even in the
very omnipotence of Death--the insight of creative idealism can find
the reflection of a beauty which its own thoughts first made. In this
way mind asserts its subtle mastery over the thoughtless forces of
Nature. The more evil the material with which it deals, the more
thwarting to untrained desire, the greater is its achievement in
inducing the reluctant rock to yield up its hidden treasures, the
prouder its victory in compelling the opposing forces to swell the
pageant of its triumph. Of all the arts, Tragedy is the proudest, the
most triumphant; for it builds its shining citadel in the very centre
of the enemy's country, on the very summit of his highest mountain;
from its impregnable watchtowers, his camps and arsenals, his columns
and forts, are all revealed; within its walls the free life continues,
while the legions of Death and Pain and Despair, and all the servile
captains of tyrant Fate, afford the burghers of that dauntless city
new spectacles of beauty. Happy those sacred ramparts, thrice happy
the dwellers on that all-seeing eminence. Honour to those brave
warriors who, through countless ages of warfare, have preserved for us
the priceless heritage of liberty, and have kept undefiled by
sacrilegious invaders the home of the unsubdued.
But the beauty of Tragedy does but make visible a quality which, in
more or less obvious shapes, is present always and everywhere in life.
In the spectacle of Death, in the endurance of intolerable pain, and
in the irrevocableness of a vanished past, there is a sacredness, an
overpowering awe, a feeling of the vastness, the depth, the
inexhaustible mystery of existence, in which, as by some strange
marriage of pain, the sufferer is bound to the world by bonds of
sorrow. In these moments of insight, we lose all eagerness of
temporary desire, all struggling and striving for petty ends, all care
for the little trivial things that, to a superficial view, make up the
common life of day by day; we see, surrounding the narrow raft
illumined by the flickering light of human comradeship, the dark ocean
on whose rolling waves we toss for a brief hour; from the great night
without, a chill blast breaks in upon our refuge; all the loneliness
of humanity amid hostile forces is concentrated upon the individual
soul, which must struggle alone, with what of courage it can command,
against the whole weight of a universe that cares nothing for its
hopes and fears. Victory, in this struggle with the powers of
darkness, is the true baptism into the glorious company of heroes, the
true initiation into the overmastering beauty of human existence. From
that awful encounter of the soul with the outer world, renunciation,
wisdom, and charity are born; and with their birth a new life begins.
To take into the inmost shrine of the soul the irresistible forces
whose puppets we seem to be--Death and change, the irrevocableness of
the past, and the powerlessness of man before the blind hurry of the
universe from vanity to vanity--to feel these things and know them is
to conquer them.
This is the reason why the Past has such magical power. The beauty of
its motionless and silent pictures is like the enchanted purity of
late autumn, when the leaves, though one breath would make them fall,
still glow against the sky in golden glory. The Past does not change
or strive; like Duncan, after life's fitful fever it sleeps well; what
was eager and grasping, what was petty and transitory, has faded away,
the things that were beautiful and eternal shine out of it like stars
in the night. Its beauty, to a soul not worthy of it, is unendurable;
but to a soul which has conquered Fate it is the key of religion.
The life of Man, viewed outwardly, is but a small thing in comparison
with the forces of Nature. The slave is doomed to worship Time and
Fate and Death, because they are greater than anything he finds in
himself, and because all his thoughts are of things which they devour.
But, great as they are, to think of them greatly, to feel their
passionless splendour, is greater still. And such thought makes us
free men; we no longer bow before the inevitable in Oriental
subjection, but we absorb it, and make it a part of ourselves. To
abandon the struggle for private happiness, to expel all eagerness of
temporary desire, to burn with passion for eternal things--this is
emancipation, and this is the free man's worship. And this liberation
is effected by a contemplation of Fate; for Fate itself is subdued by
the mind which leaves nothing to be purged by the purifying fire of
Time.
United with his fellow-men by the strongest of all ties, the tie of a
common doom, the free man finds that a new vision is with him always,
shedding over every daily task the light of love. The life of Man is a
long march through the night, surrounded by invisible foes, tortured
by weariness and pain, towards a goal that few can hope to reach, and
where none may tarry long. One by one, as they march, our comrades
vanish from our sight, seized by the silent orders of omnipotent
Death. Very brief is the time in which we can help them, in which
their happiness or misery is decided. Be it ours to shed sunshine on
their path, to lighten their sorrows by the balm of sympathy, to give
them the pure joy of a never-tiring affection, to strengthen failing
courage, to instil faith in hours of despair. Let us not weigh in
grudging scales their merits and demerits, but let us think only of
their need--of the sorrows, the difficulties, perhaps the blindnesses,
that make the misery of their lives; let us remember that they are
fellow-sufferers in the same darkness, actors in the same tragedy with
ourselves. And so, when their day is over, when their good and their
evil have become eternal by the immortality of the past, be it ours to
feel that, where they suffered, where they failed, no deed of ours was
the cause; but wherever a spark of the divine fire kindled in their
hearts, we were ready with encouragement, with sympathy, with brave
words in which high courage glowed.
Brief and powerless is Man's life; on him and all his race the slow,
sure doom falls pitiless and dark. Blind to good and evil, reckless of
destruction, omnipotent matter rolls on its relentless way; for Man,
condemned to-day to lose his dearest, to-morrow himself to pass
through the gate of darkness, it remains only to cherish, ere yet the
blow falls, the lofty thoughts that ennoble his little day; disdaining
the coward terrors of the slave of Fate, to worship at the shrine that
his own hands have built; undismayed by the empire of chance, to
preserve a mind free from the wanton tyranny that rules his outward
life; proudly defiant of the irresistible forces that tolerate, for a
moment, his knowledge and his condemnation, to sustain alone, a weary
but unyielding Atlas, the world that his own ideals have fashioned
despite the trampling march of unconscious power.
FOOTNOTES:
[9] Reprinted from the _Independent Review_, December, 1903.
IV
THE STUDY OF MATHEMATICS
In regard to every form of human activity it is necessary that the
question should be asked from time to time, What is its purpose and
ideal? In what way does it contribute to the beauty of human
existence? As respects those pursuits which contribute only remotely,
by providing the mechanism of life, it is well to be reminded that not
the mere fact of living is to be desired, but the art of living in the
contemplation of great things. Still more in regard to those
avocations which have no end outside themselves, which are to be
justified, if at all, as actually adding to the sum of the world's
permanent possessions, it is necessary to keep alive a knowledge of
their aims, a clear prefiguring vision of the temple in which creative
imagination is to be embodied.
The fulfilment of this need, in what concerns the studies forming the
material upon which custom has decided to train the youthful mind, is
indeed sadly remote--so remote as to make the mere statement of such a
claim appear preposterous. Great men, fully alive to the beauty of the
contemplations to whose service their lives are devoted, desiring that
others may share in their joys, persuade mankind to impart to the
successive generations the mechanical knowledge without which it is
impossible to cross the threshold. Dry pedants possess themselves of
the privilege of instilling this knowledge: they forget that it is to
serve but as a key to open the doors of the temple; though they spend
their lives on the steps leading up to those sacred doors, they turn
their backs upon the temple so resolutely that its very existence is
forgotten, and the eager youth, who would press forward to be
initiated to its domes and arches, is bidden to turn back and count
the steps.
Mathematics, perhaps more even than the study of Greece and Rome, has
suffered from this oblivion of its due place in civilisation. Although
tradition has decreed that the great bulk of educated men shall know
at least the elements of the subject, the reasons for which the
tradition arose are forgotten, buried beneath a great rubbish-heap of
pedantries and trivialities. To those who inquire as to the purpose of
mathematics, the usual answer will be that it facilitates the making
of machines, the travelling from place to place, and the victory over
foreign nations, whether in war or commerce. If it be objected that
these ends--all of which are of doubtful value--are not furthered by
the merely elementary study imposed upon those who do not become
expert mathematicians, the reply, it is true, will probably be that
mathematics trains the reasoning faculties. Yet the very men who make
this reply are, for the most part, unwilling to abandon the teaching
of definite fallacies, known to be such, and instinctively rejected by
the unsophisticated mind of every intelligent learner. And the
reasoning faculty itself is generally conceived, by those who urge its
cultivation, as merely a means for the avoidance of pitfalls and a
help in the discovery of rules for the guidance of practical life. All
these are undeniably important achievements to the credit of
mathematics; yet it is none of these that entitles mathematics to a
place in every liberal education. Plato, we know, regarded the
contemplation of mathematical truths as worthy of the Deity; and
Plato realised, more perhaps than any other single man, what those
elements are in human life which merit a place in heaven. There is in
mathematics, he says, "something which is _necessary_ and cannot be
set aside ... and, if I mistake not, of divine necessity; for as to
the human necessities of which the Many talk in this connection,
nothing can be more ridiculous than such an application of the words.
_Cleinias._ And what are these necessities of knowledge, Stranger,
which are divine and not human? _Athenian._ Those things without some
use or knowledge of which a man cannot become a God to the world, nor
a spirit, nor yet a hero, nor able earnestly to think and care for
man" (_Laws_, p. 818).[10] Such was Plato's judgment of mathematics;
but the mathematicians do not read Plato, while those who read him
know no mathematics, and regard his opinion upon this question as
merely a curious aberration.
Mathematics, rightly viewed, possesses not only truth, but supreme
beauty--a beauty cold and austere, like that of sculpture, without
appeal to any part of our weaker nature, without the gorgeous
trappings of painting or music, yet sublimely pure, and capable of a
stern perfection such as only the greatest art can show. The true
spirit of delight, the exaltation, the sense of being more than man,
which is the touchstone of the highest excellence, is to be found in
mathematics as surely as in poetry. What is best in mathematics
deserves not merely to be learnt as a task, but to be assimilated as a
part of daily thought, and brought again and again before the mind
with ever-renewed encouragement. Real life is, to most men, a long
second-best, a perpetual compromise between the ideal and the
possible; but the world of pure reason knows no compromise, no
practical limitations, no barrier to the creative activity embodying
in splendid edifices the passionate aspiration after the perfect from
which all great work springs. Remote from human passions, remote even
from the pitiful facts of nature, the generations have gradually
created an ordered cosmos, where pure thought can dwell as in its
natural home, and where one, at least, of our nobler impulses can
escape from the dreary exile of the actual world.
So little, however, have mathematicians aimed at beauty, that hardly
anything in their work has had this conscious purpose. Much, owing to
irrepressible instincts, which were better than avowed beliefs, has
been moulded by an unconscious taste; but much also has been spoilt by
false notions of what was fitting. The characteristic excellence of
mathematics is only to be found where the reasoning is rigidly
logical: the rules of logic are to mathematics what those of structure
are to architecture. In the most beautiful work, a chain of argument
is presented in which every link is important on its own account, in
which there is an air of ease and lucidity throughout, and the
premises achieve more than would have been thought possible, by means
which appear natural and inevitable. Literature embodies what is
general in particular circumstances whose universal significance
shines through their individual dress; but mathematics endeavours to
present whatever is most general in its purity, without any irrelevant
trappings.
How should the teaching of mathematics be conducted so as to
communicate to the learner as much as possible of this high ideal?
Here experience must, in a great measure, be our guide; but some
maxims may result from our consideration of the ultimate purpose to be
achieved.
One of the chief ends served by mathematics, when rightly taught, is
to awaken the learner's belief in reason, his confidence in the truth
of what has been demonstrated, and in the value of demonstration. This
purpose is not served by existing instruction; but it is easy to see
ways in which it might be served. At present, in what concerns
arithmetic, the boy or girl is given a set of rules, which present
themselves as neither true nor false, but as merely the will of the
teacher, the way in which, for some unfathomable reason, the teacher
prefers to have the game played. To some degree, in a study of such
definite practical utility, this is no doubt unavoidable; but as soon
as possible, the reasons of rules should be set forth by whatever
means most readily appeal to the childish mind. In geometry, instead
of the tedious apparatus of fallacious proofs for obvious truisms
which constitutes the beginning of Euclid, the learner should be
allowed at first to assume the truth of everything obvious, and should
be instructed in the demonstrations of theorems which are at once
startling and easily verifiable by actual drawing, such as those in
which it is shown that three or more lines meet in a point. In this
way belief is generated; it is seen that reasoning may lead to
startling conclusions, which nevertheless the facts will verify; and
thus the instinctive distrust of whatever is abstract or rational is
gradually overcome. Where theorems are difficult, they should be first
taught as exercises in geometrical drawing, until the figure has
become thoroughly familiar; it will then be an agreeable advance to be
taught the logical connections of the various lines or circles that
occur. It is desirable also that the figure illustrating a theorem
should be drawn in all possible cases and shapes, that so the abstract
relations with which geometry is concerned may of themselves emerge
as the residue of similarity amid such great apparent diversity. In
this way the abstract demonstrations should form but a small part of
the instruction, and should be given when, by familiarity with
concrete illustrations, they have come to be felt as the natural
embodiment of visible fact. In this early stage proofs should not be
given with pedantic fullness; definitely fallacious methods, such as
that of superposition, should be rigidly excluded from the first, but
where, without such methods, the proof would be very difficult, the
result should be rendered acceptable by arguments and illustrations
which are explicitly contrasted with demonstrations.
In the beginning of algebra, even the most intelligent child finds, as
a rule, very great difficulty. The use of letters is a mystery, which
seems to have no purpose except mystification. It is almost
impossible, at first, not to think that every letter stands for some
particular number, if only the teacher would reveal _what_ number it
stands for. The fact is, that in algebra the mind is first taught to
consider general truths, truths which are not asserted to hold only of
this or that particular thing, but of any one of a whole group of
things. It is in the power of understanding and discovering such
truths that the mastery of the intellect over the whole world of
things actual and possible resides; and ability to deal with the
general as such is one of the gifts that a mathematical education
should bestow. But how little, as a rule, is the teacher of algebra
able to explain the chasm which divides it from arithmetic, and how
little is the learner assisted in his groping efforts at
comprehension! Usually the method that has been adopted in arithmetic
is continued: rules are set forth, with no adequate explanation of
their grounds; the pupil learns to use the rules blindly, and
presently, when he is able to obtain the answer that the teacher
desires, he feels that he has mastered the difficulties of the
subject. But of inner comprehension of the processes employed he has
probably acquired almost nothing.
When algebra has been learnt, all goes smoothly until we reach those
studies in which the notion of infinity is employed--the infinitesimal
calculus and the whole of higher mathematics. The solution of the
difficulties which formerly surrounded the mathematical infinite is
probably the greatest achievement of which our own age has to boast.
Since the beginnings of Greek thought these difficulties have been
known; in every age the finest intellects have vainly endeavoured to
answer the apparently unanswerable questions that had been asked by
Zeno the Eleatic. At last Georg Cantor has found the answer, and has
conquered for the intellect a new and vast province which had been
given over to Chaos and old Night. It was assumed as self-evident,
until Cantor and Dedekind established the opposite, that if, from any
collection of things, some were taken away, the number of things left
must always be less than the original number of things. This
assumption, as a matter of fact, holds only of finite collections; and
the rejection of it, where the infinite is concerned, has been shown
to remove all the difficulties that had hitherto baffled human reason
in this matter, and to render possible the creation of an exact
science of the infinite. This stupendous fact ought to produce a
revolution in the higher teaching of mathematics; it has itself added
immeasurably to the educational value of the subject, and it has at
last given the means of treating with logical precision many studies
which, until lately, were wrapped in fallacy and obscurity. By those
who were educated on the old lines, the new work is considered to be
appallingly difficult, abstruse, and obscure; and it must be confessed
that the discoverer, as is so often the case, has hardly himself
emerged from the mists which the light of his intellect is dispelling.
But inherently, the new doctrine of the infinite, to all candid and
inquiring minds, has facilitated the mastery of higher mathematics;
for hitherto, it has been necessary to learn, by a long process of
sophistication, to give assent to arguments which, on first
acquaintance, were rightly judged to be confused and erroneous. So far
from producing a fearless belief in reason, a bold rejection of
whatever failed to fulfil the strictest requirements of logic, a
mathematical training, during the past two centuries, encouraged the
belief that many things, which a rigid inquiry would reject as
fallacious, must yet be accepted because they work in what the
mathematician calls "practice." By this means, a timid, compromising
spirit, or else a sacerdotal belief in mysteries not intelligible to
the profane, has been bred where reason alone should have ruled. All
this it is now time to sweep away; let those who wish to penetrate
into the arcana of mathematics be taught at once the true theory in
all its logical purity, and in the concatenation established by the
very essence of the entities concerned.
If we are considering mathematics as an end in itself, and not as a
technical training for engineers, it is very desirable to preserve the
purity and strictness of its reasoning. Accordingly those who have
attained a sufficient familiarity with its easier portions should be
led backward from propositions to which they have assented as
self-evident to more and more fundamental principles from which what
had previously appeared as premises can be deduced. They should be
taught--what the theory of infinity very aptly illustrates--that many
propositions seem self-evident to the untrained mind which,
nevertheless, a nearer scrutiny shows to be false. By this means they
will be led to a sceptical inquiry into first principles, an
examination of the foundations upon which the whole edifice of
reasoning is built, or, to take perhaps a more fitting metaphor, the
great trunk from which the spreading branches spring. At this stage,
it is well to study afresh the elementary portions of mathematics,
asking no longer merely whether a given proposition is true, but also
how it grows out of the central principles of logic. Questions of this
nature can now be answered with a precision and certainty which were
formerly quite impossible; and in the chains of reasoning that the
answer requires the unity of all mathematical studies at last unfolds
itself.