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.
In the great majority of mathematical text-books there is a total lack
of unity in method and of systematic development of a central theme.
Propositions of very diverse kinds are proved by whatever means are
thought most easily intelligible, and much space is devoted to mere
curiosities which in no way contribute to the main argument. But in
the greatest works, unity and inevitability are felt as in the
unfolding of a drama; in the premisses a subject is proposed for
consideration, and in every subsequent step some definite advance is
made towards mastery of its nature. The love of system, of
interconnection, which is perhaps the inmost essence of the
intellectual impulse, can find free play in mathematics as nowhere
else. The learner who feels this impulse must not be repelled by an
array of meaningless examples or distracted by amusing oddities, but
must be encouraged to dwell upon central principles, to become
familiar with the structure of the various subjects which are put
before him, to travel easily over the steps of the more important
deductions. In this way a good tone of mind is cultivated, and
selective attention is taught to dwell by preference upon what is
weighty and essential.
When the separate studies into which mathematics is divided have each
been viewed as a logical whole, as a natural growth from the
propositions which constitute their principles, the learner will be
able to understand the fundamental science which unifies and
systematises the whole of deductive reasoning. This is symbolic
logic--a study which, though it owes its inception to Aristotle, is
yet, in its wider developments, a product, almost wholly, of the
nineteenth century, and is indeed, in the present day, still growing
with great rapidity. The true method of discovery in symbolic logic,
and probably also the best method for introducing the study to a
learner acquainted with other parts of mathematics, is the analysis of
actual examples of deductive reasoning, with a view to the discovery
of the principles employed. These principles, for the most part, are
so embedded in our ratiocinative instincts, that they are employed
quite unconsciously, and can be dragged to light only by much patient
effort. But when at last they have been found, they are seen to be few
in number, and to be the sole source of everything in pure
mathematics. The discovery that all mathematics follows inevitably
from a small collection of fundamental laws is one which immeasurably
enhances the intellectual beauty of the whole; to those who have been
oppressed by the fragmentary and incomplete nature of most existing
chains of deduction this discovery comes with all the overwhelming
force of a revelation; like a palace emerging from the autumn mist as
the traveller ascends an Italian hill-side, the stately storeys of the
mathematical edifice appear in their due order and proportion, with a
new perfection in every part.
Until symbolic logic had acquired its present development, the
principles upon which mathematics depends were always supposed to be
philosophical, and discoverable only by the uncertain, unprogressive
methods hitherto employed by philosophers. So long as this was
thought, mathematics seemed to be not autonomous, but dependent upon a
study which had quite other methods than its own.
Moreover, since the
nature of the postulates from which arithmetic, analysis, and geometry
are to be deduced was wrapped in all the traditional obscurities of
metaphysical discussion, the edifice built upon such dubious
foundations began to be viewed as no better than a castle in the air.
In this respect, the discovery that the true principles are as much a
part of mathematics as any of their consequences has very greatly
increased the intellectual satisfaction to be obtained.
This
satisfaction ought not to be refused to learners capable of enjoying
it, for it is of a kind to increase our respect for human powers and
our knowledge of the beauties belonging to the abstract world.
Philosophers have commonly held that the laws of logic, which underlie
mathematics, are laws of thought, laws regulating the operations of
our minds. By this opinion the true dignity of reason is very greatly
lowered: it ceases to be an investigation into the very heart and
immutable essence of all things actual and possible, becoming,
instead, an inquiry into something more or less human and subject to
our limitations. The contemplation of what is non-human, the discovery
that our minds are capable of dealing with material not created by
them, above all, the realisation that beauty belongs to the outer
world as to the inner, are the chief means of overcoming the terrible
sense of impotence, of weakness, of exile amid hostile powers, which
is too apt to result from acknowledging the all-but omnipotence of
alien forces. To reconcile us, by the exhibition of its awful beauty,
to the reign of Fate--which is merely the literary personification of
these forces--is the task of tragedy. But mathematics takes us still
further from what is human, into the region of absolute necessity, to
which not only the actual world, but every possible world, must
conform; and even here it builds a habitation, or rather finds a
habitation eternally standing, where our ideals are fully satisfied
and our best hopes are not thwarted. It is only when we thoroughly
understand the entire independence of ourselves, which belongs to this
world that reason finds, that we can adequately realise the profound
importance of its beauty.
Not only is mathematics independent of us and our thoughts, but in
another sense we and the whole universe of existing things are
independent of mathematics. The apprehension of this purely ideal
character is indispensable, if we are to understand rightly the place
of mathematics as one among the arts. It was formerly supposed that
pure reason could decide, in some respects, as to the nature of the
actual world: geometry, at least, was thought to deal with the space
in which we live. But we now know that pure mathematics can never
pronounce upon questions of actual existence: the world of reason, in
a sense, controls the world of fact, but it is not at any point
creative of fact, and in the application of its results to the world
in time and space, its certainty and precision are lost among
approximations and working hypotheses. The objects considered by
mathematicians have, in the past, been mainly of a kind suggested by
phenomena; but from such restrictions the abstract imagination should
be wholly free. A reciprocal liberty must thus be accorded: reason
cannot dictate to the world of facts, but the facts cannot restrict
reason's privilege of dealing with whatever objects its love of beauty
may cause to seem worthy of consideration. Here, as elsewhere, we
build up our own ideals out of the fragments to be found in the world;
and in the end it is hard to say whether the result is a creation or a
discovery.
It is very desirable, in instruction, not merely to persuade the
student of the accuracy of important theorems, but to persuade him in
the way which itself has, of all possible ways, the most beauty. The
true interest of a demonstration is not, as traditional modes of
exposition suggest, concentrated wholly in the result; where this does
occur, it must be viewed as a defect, to be remedied, if possible, by
so generalising the steps of the proof that each becomes important in
and for itself. An argument which serves only to prove a conclusion is
like a story subordinated to some moral which it is meant to teach:
for æsthetic perfection no part of the whole should be merely a means.
A certain practical spirit, a desire for rapid progress, for conquest
of new realms, is responsible for the undue emphasis upon results
which prevails in mathematical instruction. The better way is to
propose some theme for consideration--in geometry, a figure having
important properties; in analysis, a function of which the study is
illuminating, and so on. Whenever proofs depend upon some only of the
marks by which we define the object to be studied, these marks should
be isolated and investigated on their own account. For it is a defect,
in an argument, to employ more premisses than the conclusion demands:
what mathematicians call elegance results from employing only the
essential principles in virtue of which the thesis is true. It is a
merit in Euclid that he advances as far as he is able to go without
employing the axiom of parallels--not, as is often said, because this
axiom is inherently objectionable, but because, in mathematics, every
new axiom diminishes the generality of the resulting theorems, and the
greatest possible generality is before all things to be sought.
Of the effects of mathematics outside its own sphere more has been
written than on the subject of its own proper ideal. The effect upon
philosophy has, in the past, been most notable, but most varied; in
the seventeenth century, idealism and rationalism, in the eighteenth,
materialism and sensationalism, seemed equally its offspring. Of the
effect which it is likely to have in the future it would be very rash
to say much; but in one respect a good result appears probable.
Against that kind of scepticism which abandons the pursuit of ideals
because the road is arduous and the goal not certainly attainable,
mathematics, within its own sphere, is a complete answer. Too often it
is said that there is no absolute truth, but only opinion and private
judgment; that each of us is conditioned, in his view of the world, by
his own peculiarities, his own taste and bias; that there is no
external kingdom of truth to which, by patience and discipline, we may
at last obtain admittance, but only truth for me, for you, for every
separate person. By this habit of mind one of the chief ends of human
effort is denied, and the supreme virtue of candour, of fearless
acknowledgment of what is, disappears from our moral vision. Of such
scepticism mathematics is a perpetual reproof; for its edifice of
truths stands unshakable and inexpungable to all the weapons of
doubting cynicism.
The effects of mathematics upon practical life, though they should not
be regarded as the motive of our studies, may be used to answer a
doubt to which the solitary student must always be liable. In a world
so full of evil and suffering, retirement into the cloister of
contemplation, to the enjoyment of delights which, however noble, must
always be for the few only, cannot but appear as a somewhat selfish
refusal to share the burden imposed upon others by accidents in which
justice plays no part. Have any of us the right, we ask, to withdraw
from present evils, to leave our fellow-men unaided, while we live a
life which, though arduous and austere, is yet plainly good in its own
nature? When these questions arise, the true answer is, no doubt, that
some must keep alive the sacred fire, some must preserve, in every
generation, the haunting vision which shadows forth the goal of so
much striving. But when, as must sometimes occur, this answer seems
too cold, when we are almost maddened by the spectacle of sorrows to
which we bring no help, then we may reflect that indirectly the
mathematician often does more for human happiness than any of his more
practically active contemporaries. The history of science abundantly
proves that a body of abstract propositions--even if, as in the case
of conic sections, it remains two thousand years without effect upon
daily life--may yet, at any moment, be used to cause a revolution in
the habitual thoughts and occupations of every citizen.
The use of
steam and electricity--to take striking instances--is rendered
possible only by mathematics. In the results of abstract thought the
world possesses a capital of which the employment in enriching the
common round has no hitherto discoverable limits. Nor does experience
give any means of deciding what parts of mathematics will be found
useful. Utility, therefore, can be only a consolation in moments of
discouragement, not a guide in directing our studies.
For the health of the moral life, for ennobling the tone of an age or
a nation, the austerer virtues have a strange power, exceeding the
power of those not informed and purified by thought. Of these austerer
virtues the love of truth is the chief, and in mathematics, more than
elsewhere, the love of truth may find encouragement for waning faith.
Every great study is not only an end in itself, but also a means of
creating and sustaining a lofty habit of mind; and this purpose should
be kept always in view throughout the teaching and learning of