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
mathematics.
FOOTNOTES:
[10] This passage was pointed out to me by Professor Gilbert Murray.
V
MATHEMATICS AND THE METAPHYSICIANS
The nineteenth century, which prided itself upon the invention of
steam and evolution, might have derived a more legitimate title to
fame from the discovery of pure mathematics. This science, like most
others, was baptised long before it was born; and thus we find writers
before the nineteenth century alluding to what they called pure
mathematics. But if they had been asked what this subject was, they
would only have been able to say that it consisted of Arithmetic,
Algebra, Geometry, and so on. As to what these studies had in common,
and as to what distinguished them from applied mathematics, our
ancestors were completely in the dark.
Pure mathematics was discovered by Boole, in a work which he called
the _Laws of Thought_ (1854). This work abounds in asseverations that
it is not mathematical, the fact being that Boole was too modest to
suppose his book the first ever written on mathematics. He was also
mistaken in supposing that he was dealing with the laws of thought:
the question how people actually think was quite irrelevant to him,
and if his book had really contained the laws of thought, it was
curious that no one should ever have thought in such a way before. His
book was in fact concerned with formal logic, and this is the same
thing as mathematics.
Pure mathematics consists entirely of assertions to the effect that,
if such and such a proposition is true of _anything_, then such and
such another proposition is true of that thing. It is essential not to
discuss whether the first proposition is really true, and not to
mention what the anything is, of which it is supposed to be true. Both
these points would belong to applied mathematics. We start, in pure
mathematics, from certain rules of inference, by which we can infer
that _if_ one proposition is true, then so is some other proposition.
These rules of inference constitute the major part of the principles
of formal logic. We then take any hypothesis that seems amusing, and
deduce its consequences. _If_ our hypothesis is about _anything_, and
not about some one or more particular things, then our deductions
constitute mathematics. Thus mathematics may be defined as the subject
in which we never know what we are talking about, nor whether what we
are saying is true. People who have been puzzled by the beginnings of
mathematics will, I hope, find comfort in this definition, and will
probably agree that it is accurate.
As one of the chief triumphs of modern mathematics consists in having
discovered what mathematics really is, a few more words on this
subject may not be amiss. It is common to start any branch of
mathematics--for instance, Geometry--with a certain number of
primitive ideas, supposed incapable of definition, and a certain
number of primitive propositions or axioms, supposed incapable of
proof. Now the fact is that, though there are indefinables and
indemonstrables in every branch of applied mathematics, there are none
in pure mathematics except such as belong to general logic. Logic,
broadly speaking, is distinguished by the fact that its propositions
can be put into a form in which they apply to anything whatever. All
pure mathematics--Arithmetic, Analysis, and Geometry--is built up by
combinations of the primitive ideas of logic, and its propositions are
deduced from the general axioms of logic, such as the syllogism and
the other rules of inference. And this is no longer a dream or an
aspiration. On the contrary, over the greater and more difficult part
of the domain of mathematics, it has been already accomplished; in the
few remaining cases, there is no special difficulty, and it is now
being rapidly achieved. Philosophers have disputed for ages whether
such deduction was possible; mathematicians have sat down and made the
deduction. For the philosophers there is now nothing left but graceful
acknowledgments.
The subject of formal logic, which has thus at last shown itself to be
identical with mathematics, was, as every one knows, invented by
Aristotle, and formed the chief study (other than theology) of the
Middle Ages. But Aristotle never got beyond the syllogism, which is a
very small part of the subject, and the schoolmen never got beyond
Aristotle. If any proof were required of our superiority to the
mediæval doctors, it might be found in this. Throughout the Middle
Ages, almost all the best intellects devoted themselves to formal
logic, whereas in the nineteenth century only an infinitesimal
proportion of the world's thought went into this subject.
Nevertheless, in each decade since 1850 more has been done to advance
the subject than in the whole period from Aristotle to Leibniz. People
have discovered how to make reasoning symbolic, as it is in Algebra,
so that deductions are effected by mathematical rules. They have
discovered many rules besides the syllogism, and a new branch of
logic, called the Logic of Relatives,[11] has been invented to deal
with topics that wholly surpassed the powers of the old logic, though
they form the chief contents of mathematics.
It is not easy for the lay mind to realise the importance of symbolism
in discussing the foundations of mathematics, and the explanation may
perhaps seem strangely paradoxical. The fact is that symbolism is
useful because it makes things difficult. (This is not true of the
advanced parts of mathematics, but only of the beginnings.) What we
wish to know is, what can be deduced from what. Now, in the
beginnings, everything is self-evident; and it is very hard to see
whether one self-evident proposition follows from another or not.
Obviousness is always the enemy to correctness. Hence we invent some
new and difficult symbolism, in which nothing seems obvious. Then we
set up certain rules for operating on the symbols, and the whole thing
becomes mechanical. In this way we find out what must be taken as
premiss and what can be demonstrated or defined. For instance, the
whole of Arithmetic and Algebra has been shown to require three
indefinable notions and five indemonstrable propositions. But without
a symbolism it would have been very hard to find this out. It is so
obvious that two and two are four, that we can hardly make ourselves
sufficiently sceptical to doubt whether it can be proved. And the same
holds in other cases where self-evident things are to be proved.
But the proof of self-evident propositions may seem, to the
uninitiated, a somewhat frivolous occupation. To this we might reply
that it is often by no means self-evident that one obvious proposition
follows from another obvious proposition; so that we are really
discovering new truths when we prove what is evident by a method which
is not evident. But a more interesting retort is, that since people
have tried to prove obvious propositions, they have found that many of
them are false. Self-evidence is often a mere will-o'-the-wisp, which
is sure to lead us astray if we take it as our guide. For instance,
nothing is plainer than that a whole always has more terms than a
part, or that a number is increased by adding one to it. But these
propositions are now known to be usually false. Most numbers are
infinite, and if a number is infinite you may add ones to it as long
as you like without disturbing it in the least. One of the merits of a
proof is that it instils a certain doubt as to the result proved; and
when what is obvious can be proved in some cases, but not in others,
it becomes possible to suppose that in these other cases it is false.
The great master of the art of formal reasoning, among the men of our
own day, is an Italian, Professor Peano, of the University of
Turin.[12] He has reduced the greater part of mathematics (and he or
his followers will, in time, have reduced the whole) to strict
symbolic form, in which there are no words at all. In the ordinary
mathematical books, there are no doubt fewer words than most readers
would wish. Still, little phrases occur, such as _therefore, let us
assume, consider_, or _hence it follows_. All these, however, are a
concession, and are swept away by Professor Peano. For instance, if we
wish to learn the whole of Arithmetic, Algebra, the Calculus, and
indeed all that is usually called pure mathematics (except Geometry),
we must start with a dictionary of three words. One symbol stands for
_zero_, another for _number_, and a third for _next after_. What these
ideas mean, it is necessary to know if you wish to become an
arithmetician. But after symbols have been invented for these three
ideas, not another word is required in the whole development. All
future symbols are symbolically explained by means of these three.
Even these three can be explained by means of the notions of
_relation_ and _class_; but this requires the Logic of Relations,
which Professor Peano has never taken up. It must be admitted that
what a mathematician has to know to begin with is not much. There are
at most a dozen notions out of which all the notions in all pure
mathematics (including Geometry) are compounded. Professor Peano, who
is assisted by a very able school of young Italian disciples, has
shown how this may be done; and although the method which he has
invented is capable of being carried a good deal further than he has
carried it, the honour of the pioneer must belong to him.
Two hundred years ago, Leibniz foresaw the science which Peano has
perfected, and endeavoured to create it. He was prevented from
succeeding by respect for the authority of Aristotle, whom he could
not believe guilty of definite, formal fallacies; but the subject
which he desired to create now exists, in spite of the patronising
contempt with which his schemes have been treated by all superior
persons. From this "Universal Characteristic," as he called it, he
hoped for a solution of all problems, and an end to all disputes. "If
controversies were to arise," he says, "there would be no more need of
disputation between two philosophers than between two accountants. For
it would suffice to take their pens in their hands, to sit down to
their desks, and to say to each other (with a friend as witness, if
they liked), 'Let us calculate.'" This optimism has now appeared to be
somewhat excessive; there still are problems whose solution is
doubtful, and disputes which calculation cannot decide. But over an
enormous field of what was formerly controversial, Leibniz's dream has
become sober fact. In the whole philosophy of mathematics, which used
to be at least as full of doubt as any other part of philosophy, order
and certainty have replaced the confusion and hesitation which
formerly reigned. Philosophers, of course, have not yet discovered
this fact, and continue to write on such subjects in the old way. But
mathematicians, at least in Italy, have now the power of treating the
principles of mathematics in an exact and masterly manner, by means of
which the certainty of mathematics extends also to mathematical
philosophy. Hence many of the topics which used to be placed among the
great mysteries--for example, the natures of infinity, of continuity,
of space, time and motion--are now no longer in any degree open to
doubt or discussion. Those who wish to know the nature of these things
need only read the works of such men as Peano or Georg Cantor; they
will there find exact and indubitable expositions of all these quondam
mysteries.