No More Learning

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 aesthetic 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
mediaeval 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.

In this capricious world, nothing is more capricious than posthumous
fame. One of the most notable examples of posterity's lack of judgment
is the Eleatic Zeno. This man, who may be regarded as the founder of
the philosophy of infinity, appears in Plato's Parmenides in the
privileged position of instructor to Socrates. He invented four
arguments, all immeasurably subtle and profound, to prove that motion
is impossible, that Achilles can never overtake the tortoise, and that
an arrow in flight is really at rest. After being refuted by
Aristotle, and by every subsequent philosopher from that day to our
own, these arguments were reinstated, and made the basis of a
mathematical renaissance, by a German professor, who probably never
dreamed of any connection between himself and Zeno. Weierstrass,[13]
by strictly banishing from mathematics the use of infinitesimals, has
at last shown that we live in an unchanging world, and that the arrow
in its flight is truly at rest. Zeno's only error lay in inferring (if
he did infer) that, because there is no such thing as a state of
change, therefore the world is in the same state at any one time as at
any other. This is a consequence which by no means follows; and in
this respect, the German mathematician is more constructive than the
ingenious Greek. Weierstrass has been able, by embodying his views in
mathematics, where familiarity with truth eliminates the vulgar
prejudices of common sense, to invest Zeno's paradoxes with the
respectable air of platitudes; and if the result is less delightful to
the lover of reason than Zeno's bold defiance, it is at any rate more
calculated to appease the mass of academic mankind.

Zeno was concerned, as a matter of fact, with three problems, each
presented by motion, but each more abstract than motion, and capable
of a purely arithmetical treatment. These are the problems of the
infinitesimal, the infinite, and continuity. To state clearly the
difficulties involved, was to accomplish perhaps the hardest part of
the philosopher's task. This was done by Zeno. From him to our own
day, the finest intellects of each generation in turn attacked the
problems, but achieved, broadly speaking, nothing. In our own time,
however, three men--Weierstrass, Dedekind, and Cantor--have not merely
advanced the three problems, but have completely solved them. The
solutions, for those acquainted with mathematics, are so clear as to
leave no longer the slightest doubt or difficulty. This achievement is
probably the greatest of which our age has to boast; and I know of no
age (except perhaps the golden age of Greece) which has a more
convincing proof to offer of the transcendent genius of its great men.
Of the three problems, that of the infinitesimal was solved by
Weierstrass; the solution of the other two was begun by Dedekind, and
definitively accomplished by Cantor.

The infinitesimal played formerly a great part in mathematics. It was
introduced by the Greeks, who regarded a circle as differing
infinitesimally from a polygon with a very large number of very small
equal sides. It gradually grew in importance, until, when Leibniz
invented the Infinitesimal Calculus, it seemed to become the
fundamental notion of all higher mathematics. Carlyle tells, in his
_Frederick the Great_, how Leibniz used to discourse to Queen Sophia
Charlotte of Prussia concerning the infinitely little, and how she
would reply that on that subject she needed no instruction--the
behaviour of courtiers had made her thoroughly familiar with it. But
philosophers and mathematicians--who for the most part had less
acquaintance with courts--continued to discuss this topic, though
without making any advance. The Calculus required continuity, and
continuity was supposed to require the infinitely little; but nobody
could discover what the infinitely little might be. It was plainly not
quite zero, because a sufficiently large number of infinitesimals,
added together, were seen to make up a finite whole. But nobody could
point out any fraction which was not zero, and yet not finite. Thus
there was a deadlock. But at last Weierstrass discovered that the
infinitesimal was not needed at all, and that everything could be
accomplished without it. Thus there was no longer any need to suppose
that there was such a thing. Nowadays, therefore, mathematicians are
more dignified than Leibniz: instead of talking about the infinitely
small, they talk about the infinitely great--a subject which, however
appropriate to monarchs, seems, unfortunately, to interest them even
less than the infinitely little interested the monarchs to whom
Leibniz discoursed.

The banishment of the infinitesimal has all sorts of odd consequences,
to which one has to become gradually accustomed. For example, there is
no such thing as the next moment. The interval between one moment and
the next would have to be infinitesimal, since, if we take two moments
with a finite interval between them, there are always other moments in
the interval. Thus if there are to be no infinitesimals, no two
moments are quite consecutive, but there are always other moments
between any two. Hence there must be an infinite number of moments
between any two; because if there were a finite number one would be
nearest the first of the two moments, and therefore next to it. This
might be thought to be a difficulty; but, as a matter of fact, it is
here that the philosophy of the infinite comes in, and makes all
straight.

The same sort of thing happens in space.
If any piece of matter be cut
in two, and then each part be halved, and so on, the bits will become
smaller and smaller, and can theoretically be made as small as we
please. However small they may be, they can still be cut up and made
smaller still. But they will always have _some_ finite size, however
small they may be. We never reach the infinitesimal in this way, and
no finite number of divisions will bring us to points. Nevertheless
there _are_ points, only these are not to be reached by successive
divisions. Here again, the philosophy of the infinite shows us how
this is possible, and why points are not infinitesimal lengths.

As regards motion and change, we get similarly curious results. People
used to think that when a thing changes, it must be in a state of
change, and that when a thing moves, it is in a state of motion. This
is now known to be a mistake. When a body moves, all that can be said
is that it is in one place at one time and in another at another. We
must not say that it will be in a neighbouring place at the next
instant, since there is no next instant. Philosophers often tell us
that when a body is in motion, it changes its position within the
instant. To this view Zeno long ago made the fatal retort that every
body always is where it is; but a retort so simple and brief was not
of the kind to which philosophers are accustomed to give weight, and
they have continued down to our own day to repeat the same phrases
which roused the Eleatic's destructive ardour. It was only recently
that it became possible to explain motion in detail in accordance with
Zeno's platitude, and in opposition to the philosopher's paradox. We
may now at last indulge the comfortable belief that a body in motion
is just as truly where it is as a body at rest. Motion consists merely
in the fact that bodies are sometimes in one place and sometimes in
another, and that they are at intermediate places at intermediate
times. Only those who have waded through the quagmire of philosophic
speculation on this subject can realise what a liberation from antique
prejudices is involved in this simple and straightforward commonplace.

The philosophy of the infinitesimal, as we have just seen, is mainly
negative. People used to believe in it, and now they have found out
their mistake. The philosophy of the infinite, on the other hand, is
wholly positive. It was formerly supposed that infinite numbers, and
the mathematical infinite generally, were self-contradictory. But as
it was obvious that there were infinities--for example, the number of
numbers--the contradictions of infinity seemed unavoidable, and
philosophy seemed to have wandered into a "cul-de-sac. " This
difficulty led to Kant's antinomies, and hence, more or less
indirectly, to much of Hegel's dialectic method. Almost all current
philosophy is upset by the fact (of which very few philosophers are as
yet aware) that all the ancient and respectable contradictions in the
notion of the infinite have been once for all disposed of. The method
by which this has been done is most interesting and instructive. In
the first place, though people had talked glibly about infinity ever
since the beginnings of Greek thought, nobody had ever thought of
asking, What is infinity? If any philosopher had been asked for a
definition of infinity, he might have produced some unintelligible
rigmarole, but he would certainly not have been able to give a
definition that had any meaning at all. Twenty years ago, roughly
speaking, Dedekind and Cantor asked this question, and, what is more
remarkable, they answered it. They found, that is to say, a perfectly
precise definition of an infinite number or an infinite collection of
things. This was the first and perhaps the greatest step. It then
remained to examine the supposed contradictions in this notion. Here
Cantor proceeded in the only proper way. He took pairs of
contradictory propositions, in which both sides of the contradiction
would be usually regarded as demonstrable, and he strictly examined
the supposed proofs. He found that all proofs adverse to infinity
involved a certain principle, at first sight obviously true, but
destructive, in its consequences, of almost all mathematics. The
proofs favourable to infinity, on the other hand, involved no
principle that had evil consequences. It thus appeared that common
sense had allowed itself to be taken in by a specious maxim, and that,
when once this maxim was rejected, all went well.

The maxim in question is, that if one collection is part of another,
the one which is a part has fewer terms than the one of which it is a
part. This maxim is true of finite numbers. For example, Englishmen
are only some among Europeans, and there are fewer Englishmen than
Europeans. But when we come to infinite numbers, this is no longer
true. This breakdown of the maxim gives us the precise definition of
infinity. A collection of terms is infinite when it contains as parts
other collections which have just as many terms as it has. If you can
take away some of the terms of a collection, without diminishing the
number of terms, then there are an infinite number of terms in the
collection. For example, there are just as many even numbers as there
are numbers altogether, since every number can be doubled. This may be
seen by putting odd and even numbers together in one row, and even
numbers alone in a row below:--

1, 2, 3, 4, 5, _ad infinitum_.
2, 4, 6, 8, 10, _ad infinitum_.

There are obviously just as many numbers in the row below as in the
row above, because there is one below for each one above. This
property, which was formerly thought to be a contradiction, is now
transformed into a harmless definition of infinity, and shows, in the
above case, that the number of finite numbers is infinite.

But the uninitiated may wonder how it is possible to deal with a
number which cannot be counted. It is impossible to count up _all_ the
numbers, one by one, because, however many we may count, there are
always more to follow. The fact is that counting is a very vulgar and
elementary way of finding out how many terms there are in a
collection. And in any case, counting gives us what mathematicians
call the _ordinal_ number of our terms; that is to say, it arranges
our terms in an order or series, and its result tells us what type of
series results from this arrangement. In other words, it is impossible
to count things without counting some first and others afterwards, so
that counting always has to do with order. Now when there are only a
finite number of terms, we can count them in any order we like; but
when there are an infinite number, what corresponds to counting will
give us quite different results according to the way in which we carry
out the operation. Thus the ordinal number, which results from what,
in a general sense may be called counting, depends not only upon how
many terms we have, but also (where the number of terms is infinite)
upon the way in which the terms are arranged.

The fundamental infinite numbers are not ordinal, but are what is
called _cardinal_. They are not obtained by putting our terms in order
and counting them, but by a different method, which tells us, to begin
with, whether two collections have the same number of terms, or, if
not, which is the greater. [14] It does not tell us, in the way in
which counting does, _what_ number of terms a collection has; but if
we define a number as the number of terms in such and such a
collection, then this method enables us to discover whether some other
collection that may be mentioned has more or fewer terms. An
illustration will show how this is done. If there existed some country
in which, for one reason or another, it was impossible to take a
census, but in which it was known that every man had a wife and every
woman a husband, then (provided polygamy was not a national
institution) we should know, without counting, that there were exactly
as many men as there were women in that country, neither more nor
less. This method can be applied generally. If there is some relation
which, like marriage, connects the things in one collection each with
one of the things in another collection, and vice versa, then the two
collections have the same number of terms. This was the way in which
we found that there are as many even numbers as there are numbers.
Every number can be doubled, and every even number can be halved, and
each process gives just one number corresponding to the one that is
doubled or halved. And in this way we can find any number of
collections each of which has just as many terms as there are finite
numbers. If every term of a collection can be hooked on to a number,
and all the finite numbers are used once, and only once, in the
process, then our collection must have just as many terms as there are
finite numbers. This is the general method by which the numbers of
infinite collections are defined.

But it must not be supposed that all infinite numbers are equal. On
the contrary, there are infinitely more infinite numbers than finite
ones. There are more ways of arranging the finite numbers in different
types of series than there are finite numbers. There are probably more
points in space and more moments in time than there are finite
numbers. There are exactly as many fractions as whole numbers,
although there are an infinite number of fractions between any two
whole numbers. But there are more irrational numbers than there are
whole numbers or fractions. There are probably exactly as many points
in space as there are irrational numbers, and exactly as many points
on a line a millionth of an inch long as in the whole of infinite
space. There is a greatest of all infinite numbers, which is the
number of things altogether, of every sort and kind. It is obvious
that there cannot be a greater number than this, because, if
everything has been taken, there is nothing left to add. Cantor has a
proof that there is no greatest number, and if this proof were valid,
the contradictions of infinity would reappear in a sublimated form.
But in this one point, the master has been guilty of a very subtle
fallacy, which I hope to explain in some future work. [15]

We can now understand why Zeno believed that Achilles cannot overtake
the tortoise and why as a matter of fact he can overtake it. We shall
see that all the people who disagreed with Zeno had no right to do so,
because they all accepted premises from which his conclusion followed.
The argument is this: Let Achilles and the tortoise start along a road
at the same time, the tortoise (as is only fair) being allowed a
handicap. Let Achilles go twice as fast as the tortoise, or ten times
or a hundred times as fast. Then he will never reach the tortoise. For
at every moment the tortoise is somewhere and Achilles is somewhere;
and neither is ever twice in the same place while the race is going
on. Thus the tortoise goes to just as many places as Achilles does,
because each is in one place at one moment, and in another at any
other moment. But if Achilles were to catch up with the tortoise, the
places where the tortoise would have been would be only part of the
places where Achilles would have been. Here, we must suppose, Zeno
appealed to the maxim that the whole has more terms than the part. [16]
Thus if Achilles were to overtake the tortoise, he would have been in
more places than the tortoise; but we saw that he must, in any period,
be in exactly as many places as the tortoise. Hence we infer that he
can never catch the tortoise. This argument is strictly correct, if we
allow the axiom that the whole has more terms than the part. As the
conclusion is absurd, the axiom must be rejected, and then all goes
well. But there is no good word to be said for the philosophers of the
past two thousand years and more, who have all allowed the axiom and
denied the conclusion.

The retention of this axiom leads to absolute contradictions, while
its rejection leads only to oddities. Some of these oddities, it must
be confessed, are very odd. One of them, which I call the paradox of
Tristram Shandy, is the converse of the Achilles, and shows that the
tortoise, if you give him time, will go just as far as Achilles.
Tristram Shandy, as we know, employed two years in chronicling the
first two days of his life, and lamented that, at this rate, material
would accumulate faster than he could deal with it, so that, as years
went by, he would be farther and farther from the end of his history.
Now I maintain that, if he had lived for ever, and had not wearied of
his task, then, even if his life had continued as event fully as it
began, no part of his biography would have remained unwritten. For
consider: the hundredth day will be described in the hundredth year,
the thousandth in the thousandth year, and so on. Whatever day we may
choose as so far on that he cannot hope to reach it, that day will be
described in the corresponding year. Thus any day that may be
mentioned will be written up sooner or later, and therefore no part of
the biography will remain permanently unwritten. This paradoxical but
perfectly true proposition depends upon the fact that the number of
days in all time is no greater than the number of years.

Thus on the subject of infinity it is impossible to avoid conclusions
which at first sight appear paradoxical, and this is the reason why so
many philosophers have supposed that there were inherent
contradictions in the infinite. But a little practice enables one to
grasp the true principles of Cantor's doctrine, and to acquire new and
better instincts as to the true and the false. The oddities then
become no odder than the people at the antipodes, who used to be
thought impossible because they would find it so inconvenient to stand
on their heads.

The solution of the problems concerning infinity has enabled Cantor to
solve also the problems of continuity. Of this, as of infinity, he has
given a perfectly precise definition, and has shown that there are no
contradictions in the notion so defined. But this subject is so
technical that it is impossible to give any account of it here.

The notion of continuity depends upon that of _order_, since
continuity is merely a particular type of order. Mathematics has, in
modern times, brought order into greater and greater prominence. In
former days, it was supposed (and philosophers are still apt to
suppose) that quantity was the fundamental notion of mathematics. But
nowadays, quantity is banished altogether, except from one little
corner of Geometry, while order more and more reigns supreme. The
investigation of different kinds of series and their relations is now
a very large part of mathematics, and it has been found that this
investigation can be conducted without any reference to quantity, and,
for the most part, without any reference to number. All types of
series are capable of formal definition, and their properties can be
deduced from the principles of symbolic logic by means of the Algebra
of Relatives. The notion of a limit, which is fundamental in the
greater part of higher mathematics, used to be defined by means of
quantity, as a term to which the terms of some series approximate as
nearly as we please. But nowadays the limit is defined quite
differently, and the series which it limits may not approximate to it
at all. This improvement also is due to Cantor, and it is one which
has revolutionised mathematics. Only order is now relevant to limits.
Thus, for instance, the smallest of the infinite integers is the limit
of the finite integers, though all finite integers are at an infinite
distance from it. The study of different types of series is a general
subject of which the study of ordinal numbers (mentioned above) is a
special and very interesting branch. But the unavoidable
technicalities of this subject render it impossible to explain to any
but professed mathematicians.

Geometry, like Arithmetic, has been subsumed, in recent times, under
the general study of order. It was formerly supposed that Geometry was
the study of the nature of the space in which we live, and accordingly
it was urged, by those who held that what exists can only be known
empirically, that Geometry should really be regarded as belonging to
applied mathematics. But it has gradually appeared, by the increase of
non-Euclidean systems, that Geometry throws no more light upon the
nature of space than Arithmetic throws upon the population of the
United States. Geometry is a whole collection of deductive sciences
based on a corresponding collection of sets of axioms. One set of
axioms is Euclid's; other equally good sets of axioms lead to other
results. Whether Euclid's axioms are true, is a question as to which
the pure mathematician is indifferent; and, what is more, it is a
question which it is theoretically impossible to answer with certainty
in the affirmative. It might possibly be shown, by very careful
measurements, that Euclid's axioms are false; but no measurements
could ever assure us (owing to the errors of observation) that they
are exactly true. Thus the geometer leaves to the man of science to
decide, as best he may, what axioms are most nearly true in the actual
world. The geometer takes any set of axioms that seem interesting, and
deduces their consequences. What defines Geometry, in this sense, is
that the axioms must give rise to a series of more than one dimension.
And it is thus that Geometry becomes a department in the study of
order.

In Geometry, as in other parts of mathematics, Peano and his disciples
have done work of the very greatest merit as regards principles.
Formerly, it was held by philosophers and mathematicians alike that
the proofs in Geometry depended on the figure; nowadays, this is known
to be false. In the best books there are no figures at all. The
reasoning proceeds by the strict rules of formal logic from a set of
axioms laid down to begin with. If a figure is used, all sorts of
things seem obviously to follow, which no formal reasoning can prove
from the explicit axioms, and which, as a matter of fact, are only
accepted because they are obvious. By banishing the figure, it becomes
possible to discover _all_ the axioms that are needed; and in this way
all sorts of possibilities, which would have otherwise remained
undetected, are brought to light.

One great advance, from the point of view of correctness, has been
made by introducing points as they are required, and not starting, as
was formerly done, by assuming the whole of space. This method is due
partly to Peano, partly to another Italian named Fano. To those
unaccustomed to it, it has an air of somewhat wilful pedantry. In this
way, we begin with the following axioms: (1) There is a class of
entities called _points_. (2) There is at least one point. (3) If _a_
be a point, there is at least one other point besides _a_. Then we
bring in the straight line joining two points, and begin again with
(4), namely, on the straight line joining _a_ and _b_, there is at
least one other point besides _a_ and _b_. (5) There is at least one
point not on the line _ab_. And so we go on, till we have the means of
obtaining as many points as we require. But the word _space_, as Peano
humorously remarks, is one for which Geometry has no use at all.

The rigid methods employed by modern geometers have deposed Euclid
from his pinnacle of correctness. It was thought, until recent times,
that, as Sir Henry Savile remarked in 1621, there were only two
blemishes in Euclid, the theory of parallels and the theory of
proportion. It is now known that these are almost the only points in
which Euclid is free from blemish. Countless errors are involved in
his first eight propositions. That is to say, not only is it doubtful
whether his axioms are true, which is a comparatively trivial matter,
but it is certain that his propositions do not follow from the axioms
which he enunciates. A vastly greater number of axioms, which Euclid
unconsciously employs, are required for the proof of his propositions.
Even in the first proposition of all, where he constructs an
equilateral triangle on a given base, he uses two circles which are
assumed to intersect. But no explicit axiom assures us that they do
so, and in some kinds of spaces they do not always intersect. It is
quite doubtful whether our space belongs to one of these kinds or not.
Thus Euclid fails entirely to prove his point in the very first
proposition. As he is certainly not an easy author, and is terribly
long-winded, he has no longer any but an historical interest. Under
these circumstances, it is nothing less than a scandal that he should
still be taught to boys in England. [17] A book should have either
intelligibility or correctness; to combine the two is impossible, but
to lack both is to be unworthy of such a place as Euclid has occupied
in education.

The most remarkable result of modern methods in mathematics is the
importance of symbolic logic and of rigid formalism. Mathematicians,
under the influence of Weierstrass, have shown in modern times a care
for accuracy, and an aversion to slipshod reasoning, such as had not
been known among them previously since the time of the Greeks. The
great inventions of the seventeenth century--Analytical Geometry and
the Infinitesimal Calculus--were so fruitful in new results that
mathematicians had neither time nor inclination to examine their
foundations. Philosophers, who should have taken up the task, had too
little mathematical ability to invent the new branches of mathematics
which have now been found necessary for any adequate discussion. Thus
mathematicians were only awakened from their "dogmatic slumbers" when
Weierstrass and his followers showed that many of their most cherished
propositions are in general false. Macaulay, contrasting the certainty
of mathematics with the uncertainty of philosophy, asks who ever heard
of a reaction against Taylor's theorem? If he had lived now, he
himself might have heard of such a reaction, for this is precisely one
of the theorems which modern investigations have overthrown. Such rude
shocks to mathematical faith have produced that love of formalism
which appears, to those who are ignorant of its motive, to be mere
outrageous pedantry.

The proof that all pure mathematics, including Geometry, is nothing
but formal logic, is a fatal blow to the Kantian philosophy. Kant,
rightly perceiving that Euclid's propositions could not be deduced
from Euclid's axioms without the help of the figures, invented a
theory of knowledge to account for this fact; and it accounted so
successfully that, when the fact is shown to be a mere defect in
Euclid, and not a result of the nature of geometrical reasoning,
Kant's theory also has to be abandoned. The whole doctrine of _a
priori_ intuitions, by which Kant explained the possibility of pure
mathematics, is wholly inapplicable to mathematics in its present
form. The Aristotelian doctrines of the schoolmen come nearer in
spirit to the doctrines which modern mathematics inspire; but the
schoolmen were hampered by the fact that their formal logic was very
defective, and that the philosophical logic based upon the syllogism
showed a corresponding narrowness. What is now required is to give the
greatest possible development to mathematical logic, to allow to the
full the importance of relations, and then to found upon this secure
basis a new philosophical logic, which may hope to borrow some of the
exactitude and certainty of its mathematical foundation. If this can
be successfully accomplished, there is every reason to hope that the
near future will be as great an epoch in pure philosophy as the
immediate past has been in the principles of mathematics. Great
triumphs inspire great hopes; and pure thought may achieve, within our
generation, such results as will place our time, in this respect, on a
level with the greatest age of Greece. [18]

FOOTNOTES:

[11] This subject is due in the main to Mr. C. S. Peirce.

[12] I ought to have added Frege, but his writings were unknown to me
when this article was written. [Note added in 1917. ]

[13] Professor of Mathematics in the University of Berlin. He died in
1897.

[14] [Note added in 1917. ] Although some infinite numbers are greater
than some others, it cannot be proved that of any two infinite numbers
one must be the greater.

[15] Cantor was not guilty of a fallacy on this point. His proof that
there is no greatest number is valid. The solution of the puzzle is
complicated and depends upon the theory of types, which is explained
in _Principia Mathematica_, Vol. I (Camb. Univ. Press, 1910). [Note
added in 1917. ]

[16] This must not be regarded as a historically correct account of
what Zeno actually had in mind. It is a new argument for his
conclusion, not the argument which influenced him. On this point, see
e. g. C. D. Broad, "Note on Achilles and the Tortoise," _Mind_, N. S. ,
Vol.