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.

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.

Mysticism and Logic and Other Essays by Bertrand Russell

In what way does it contribute to the beauty of human

existence? As respects those pursuits which contribute only remotely,

by providing the mechanism of life, it is well to be reminded that not

the mere fact of living is to be desired, but the art of living in the

contemplation of great things. Still more in regard to those

avocations which have no end outside themselves, which are to be

justified, if at all, as actually adding to the sum of the world's

permanent possessions, it is necessary to keep alive a knowledge of

their aims, a clear prefiguring vision of the temple in which creative

imagination is to be embodied.

The fulfilment of this need, in what concerns the studies forming the

material upon which custom has decided to train the youthful mind, is

indeed sadly remote--so remote as to make the mere statement of such a

claim appear preposterous. Great men, fully alive to the beauty of the

contemplations to whose service their lives are devoted, desiring that

others may share in their joys, persuade mankind to impart to the

successive generations the mechanical knowledge without which it is

impossible to cross the threshold. Dry pedants possess themselves of

the privilege of instilling this knowledge: they forget that it is to

serve but as a key to open the doors of the temple; though they spend

their lives on the steps leading up to those sacred doors, they turn

their backs upon the temple so resolutely that its very existence is

forgotten, and the eager youth, who would press forward to be

initiated to its domes and arches, is bidden to turn back and count

the steps.

Mathematics, perhaps more even than the study of Greece and Rome, has

suffered from this oblivion of its due place in civilisation. Although

tradition has decreed that the great bulk of educated men shall know

at least the elements of the subject, the reasons for which the

tradition arose are forgotten, buried beneath a great rubbish-heap of

pedantries and trivialities. To those who inquire as to the purpose of

mathematics, the usual answer will be that it facilitates the making

of machines, the travelling from place to place, and the victory over

foreign nations, whether in war or commerce. If it be objected that

these ends--all of which are of doubtful value--are not furthered by

the merely elementary study imposed upon those who do not become

expert mathematicians, the reply, it is true, will probably be that

mathematics trains the reasoning faculties. Yet the very men who make

this reply are, for the most part, unwilling to abandon the teaching

of definite fallacies, known to be such, and instinctively rejected by

the unsophisticated mind of every intelligent learner. And the

reasoning faculty itself is generally conceived, by those who urge its

cultivation, as merely a means for the avoidance of pitfalls and a

help in the discovery of rules for the guidance of practical life. All

these are undeniably important achievements to the credit of

mathematics; yet it is none of these that entitles mathematics to a

place in every liberal education. Plato, we know, regarded the

contemplation of mathematical truths as worthy of the Deity; and

Plato realised, more perhaps than any other single man, what those

elements are in human life which merit a place in heaven. There is in

mathematics, he says, "something which is _necessary_ and cannot be

set aside . . . and, if I mistake not, of divine necessity; for as to

the human necessities of which the Many talk in this connection,

nothing can be more ridiculous than such an application of the words.

_Cleinias. _ And what are these necessities of knowledge, Stranger,

which are divine and not human? _Athenian. _ Those things without some

use or knowledge of which a man cannot become a God to the world, nor

a spirit, nor yet a hero, nor able earnestly to think and care for

man" (_Laws_, p. 818). [10] Such was Plato's judgment of mathematics;

but the mathematicians do not read Plato, while those who read him

know no mathematics, and regard his opinion upon this question as

merely a curious aberration.

Mathematics, rightly viewed, possesses not only truth, but supreme

beauty--a beauty cold and austere, like that of sculpture, without

appeal to any part of our weaker nature, without the gorgeous

trappings of painting or music, yet sublimely pure, and capable of a

stern perfection such as only the greatest art can show. The true

spirit of delight, the exaltation, the sense of being more than man,

which is the touchstone of the highest excellence, is to be found in

mathematics as surely as in poetry. What is best in mathematics

deserves not merely to be learnt as a task, but to be assimilated as a

part of daily thought, and brought again and again before the mind

with ever-renewed encouragement. Real life is, to most men, a long

second-best, a perpetual compromise between the ideal and the

possible; but the world of pure reason knows no compromise, no

practical limitations, no barrier to the creative activity embodying

in splendid edifices the passionate aspiration after the perfect from

which all great work springs. Remote from human passions, remote even

from the pitiful facts of nature, the generations have gradually

created an ordered cosmos, where pure thought can dwell as in its

natural home, and where one, at least, of our nobler impulses can

escape from the dreary exile of the actual world.

So little, however, have mathematicians aimed at beauty, that hardly

anything in their work has had this conscious purpose. Much, owing to

irrepressible instincts, which were better than avowed beliefs, has

been moulded by an unconscious taste; but much also has been spoilt by

false notions of what was fitting. The characteristic excellence of

mathematics is only to be found where the reasoning is rigidly

logical: the rules of logic are to mathematics what those of structure

are to architecture. In the most beautiful work, a chain of argument

is presented in which every link is important on its own account, in

which there is an air of ease and lucidity throughout, and the

premises achieve more than would have been thought possible, by means

which appear natural and inevitable. Literature embodies what is

general in particular circumstances whose universal significance

shines through their individual dress; but mathematics endeavours to

present whatever is most general in its purity, without any irrelevant

trappings.

How should the teaching of mathematics be conducted so as to

communicate to the learner as much as possible of this high ideal?

Here experience must, in a great measure, be our guide; but some

maxims may result from our consideration of the ultimate purpose to be

achieved.

One of the chief ends served by mathematics, when rightly taught, is

to awaken the learner's belief in reason, his confidence in the truth

of what has been demonstrated, and in the value of demonstration. This

purpose is not served by existing instruction; but it is easy to see

ways in which it might be served. At present, in what concerns

arithmetic, the boy or girl is given a set of rules, which present

themselves as neither true nor false, but as merely the will of the

teacher, the way in which, for some unfathomable reason, the teacher

prefers to have the game played. To some degree, in a study of such

definite practical utility, this is no doubt unavoidable; but as soon

as possible, the reasons of rules should be set forth by whatever

means most readily appeal to the childish mind. In geometry, instead

of the tedious apparatus of fallacious proofs for obvious truisms

which constitutes the beginning of Euclid, the learner should be

allowed at first to assume the truth of everything obvious, and should

be instructed in the demonstrations of theorems which are at once

startling and easily verifiable by actual drawing, such as those in

which it is shown that three or more lines meet in a point. In this

way belief is generated; it is seen that reasoning may lead to

startling conclusions, which nevertheless the facts will verify; and

thus the instinctive distrust of whatever is abstract or rational is

gradually overcome. Where theorems are difficult, they should be first

taught as exercises in geometrical drawing, until the figure has

become thoroughly familiar; it will then be an agreeable advance to be

taught the logical connections of the various lines or circles that

occur. It is desirable also that the figure illustrating a theorem

should be drawn in all possible cases and shapes, that so the abstract

relations with which geometry is concerned may of themselves emerge

as the residue of similarity amid such great apparent diversity. In

this way the abstract demonstrations should form but a small part of

the instruction, and should be given when, by familiarity with

concrete illustrations, they have come to be felt as the natural

embodiment of visible fact. In this early stage proofs should not be

given with pedantic fullness; definitely fallacious methods, such as

that of superposition, should be rigidly excluded from the first, but

where, without such methods, the proof would be very difficult, the

result should be rendered acceptable by arguments and illustrations

which are explicitly contrasted with demonstrations.

In the beginning of algebra, even the most intelligent child finds, as

a rule, very great difficulty. The use of letters is a mystery, which

seems to have no purpose except mystification. It is almost

impossible, at first, not to think that every letter stands for some

particular number, if only the teacher would reveal _what_ number it

stands for. The fact is, that in algebra the mind is first taught to

consider general truths, truths which are not asserted to hold only of

this or that particular thing, but of any one of a whole group of

things. It is in the power of understanding and discovering such

truths that the mastery of the intellect over the whole world of

things actual and possible resides; and ability to deal with the

general as such is one of the gifts that a mathematical education

should bestow. But how little, as a rule, is the teacher of algebra

able to explain the chasm which divides it from arithmetic, and how

little is the learner assisted in his groping efforts at

comprehension! Usually the method that has been adopted in arithmetic

is continued: rules are set forth, with no adequate explanation of

their grounds; the pupil learns to use the rules blindly, and

presently, when he is able to obtain the answer that the teacher

desires, he feels that he has mastered the difficulties of the

subject. But of inner comprehension of the processes employed he has

probably acquired almost nothing.

When algebra has been learnt, all goes smoothly until we reach those

studies in which the notion of infinity is employed--the infinitesimal

calculus and the whole of higher mathematics. The solution of the

difficulties which formerly surrounded the mathematical infinite is

probably the greatest achievement of which our own age has to boast.

Since the beginnings of Greek thought these difficulties have been

known; in every age the finest intellects have vainly endeavoured to

answer the apparently unanswerable questions that had been asked by

Zeno the Eleatic. At last Georg Cantor has found the answer, and has

conquered for the intellect a new and vast province which had been

given over to Chaos and old Night. It was assumed as self-evident,

until Cantor and Dedekind established the opposite, that if, from any

collection of things, some were taken away, the number of things left

must always be less than the original number of things. This

assumption, as a matter of fact, holds only of finite collections; and

the rejection of it, where the infinite is concerned, has been shown

to remove all the difficulties that had hitherto baffled human reason

in this matter, and to render possible the creation of an exact

science of the infinite. This stupendous fact ought to produce a

revolution in the higher teaching of mathematics; it has itself added

immeasurably to the educational value of the subject, and it has at

last given the means of treating with logical precision many studies

which, until lately, were wrapped in fallacy and obscurity. By those

who were educated on the old lines, the new work is considered to be

appallingly difficult, abstruse, and obscure; and it must be confessed

that the discoverer, as is so often the case, has hardly himself

emerged from the mists which the light of his intellect is dispelling.

But inherently, the new doctrine of the infinite, to all candid and

inquiring minds, has facilitated the mastery of higher mathematics;

for hitherto, it has been necessary to learn, by a long process of

sophistication, to give assent to arguments which, on first

acquaintance, were rightly judged to be confused and erroneous. So far

from producing a fearless belief in reason, a bold rejection of

whatever failed to fulfil the strictest requirements of logic, a

mathematical training, during the past two centuries, encouraged the

belief that many things, which a rigid inquiry would reject as

fallacious, must yet be accepted because they work in what the

mathematician calls "practice. " By this means, a timid, compromising

spirit, or else a sacerdotal belief in mysteries not intelligible to

the profane, has been bred where reason alone should have ruled. All

this it is now time to sweep away; let those who wish to penetrate

into the arcana of mathematics be taught at once the true theory in

all its logical purity, and in the concatenation established by the

very essence of the entities concerned.

If we are considering mathematics as an end in itself, and not as a

technical training for engineers, it is very desirable to preserve the

purity and strictness of its reasoning. Accordingly those who have

attained a sufficient familiarity with its easier portions should be

led backward from propositions to which they have assented as

self-evident to more and more fundamental principles from which what

had previously appeared as premises can be deduced. They should be

taught--what the theory of infinity very aptly illustrates--that many

propositions seem self-evident to the untrained mind which,

nevertheless, a nearer scrutiny shows to be false. By this means they

will be led to a sceptical inquiry into first principles, an

examination of the foundations upon which the whole edifice of

reasoning is built, or, to take perhaps a more fitting metaphor, the

great trunk from which the spreading branches spring. At this stage,

it is well to study afresh the elementary portions of mathematics,

asking no longer merely whether a given proposition is true, but also

how it grows out of the central principles of logic. Questions of this

nature can now be answered with a precision and certainty which were

formerly quite impossible; and in the chains of reasoning that the

answer requires the unity of all mathematical studies at last unfolds

itself.

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.

existence? As respects those pursuits which contribute only remotely,

by providing the mechanism of life, it is well to be reminded that not

the mere fact of living is to be desired, but the art of living in the

contemplation of great things. Still more in regard to those

avocations which have no end outside themselves, which are to be

justified, if at all, as actually adding to the sum of the world's

permanent possessions, it is necessary to keep alive a knowledge of

their aims, a clear prefiguring vision of the temple in which creative

imagination is to be embodied.

The fulfilment of this need, in what concerns the studies forming the

material upon which custom has decided to train the youthful mind, is

indeed sadly remote--so remote as to make the mere statement of such a

claim appear preposterous. Great men, fully alive to the beauty of the

contemplations to whose service their lives are devoted, desiring that

others may share in their joys, persuade mankind to impart to the

successive generations the mechanical knowledge without which it is

impossible to cross the threshold. Dry pedants possess themselves of

the privilege of instilling this knowledge: they forget that it is to

serve but as a key to open the doors of the temple; though they spend

their lives on the steps leading up to those sacred doors, they turn

their backs upon the temple so resolutely that its very existence is

forgotten, and the eager youth, who would press forward to be

initiated to its domes and arches, is bidden to turn back and count

the steps.

Mathematics, perhaps more even than the study of Greece and Rome, has

suffered from this oblivion of its due place in civilisation. Although

tradition has decreed that the great bulk of educated men shall know

at least the elements of the subject, the reasons for which the

tradition arose are forgotten, buried beneath a great rubbish-heap of

pedantries and trivialities. To those who inquire as to the purpose of

mathematics, the usual answer will be that it facilitates the making

of machines, the travelling from place to place, and the victory over

foreign nations, whether in war or commerce. If it be objected that

these ends--all of which are of doubtful value--are not furthered by

the merely elementary study imposed upon those who do not become

expert mathematicians, the reply, it is true, will probably be that

mathematics trains the reasoning faculties. Yet the very men who make

this reply are, for the most part, unwilling to abandon the teaching

of definite fallacies, known to be such, and instinctively rejected by

the unsophisticated mind of every intelligent learner. And the

reasoning faculty itself is generally conceived, by those who urge its

cultivation, as merely a means for the avoidance of pitfalls and a

help in the discovery of rules for the guidance of practical life. All

these are undeniably important achievements to the credit of

mathematics; yet it is none of these that entitles mathematics to a

place in every liberal education. Plato, we know, regarded the

contemplation of mathematical truths as worthy of the Deity; and

Plato realised, more perhaps than any other single man, what those

elements are in human life which merit a place in heaven. There is in

mathematics, he says, "something which is _necessary_ and cannot be

set aside . . . and, if I mistake not, of divine necessity; for as to

the human necessities of which the Many talk in this connection,

nothing can be more ridiculous than such an application of the words.

_Cleinias. _ And what are these necessities of knowledge, Stranger,

which are divine and not human? _Athenian. _ Those things without some

use or knowledge of which a man cannot become a God to the world, nor

a spirit, nor yet a hero, nor able earnestly to think and care for

man" (_Laws_, p. 818). [10] Such was Plato's judgment of mathematics;

but the mathematicians do not read Plato, while those who read him

know no mathematics, and regard his opinion upon this question as

merely a curious aberration.

Mathematics, rightly viewed, possesses not only truth, but supreme

beauty--a beauty cold and austere, like that of sculpture, without

appeal to any part of our weaker nature, without the gorgeous

trappings of painting or music, yet sublimely pure, and capable of a

stern perfection such as only the greatest art can show. The true

spirit of delight, the exaltation, the sense of being more than man,

which is the touchstone of the highest excellence, is to be found in

mathematics as surely as in poetry. What is best in mathematics

deserves not merely to be learnt as a task, but to be assimilated as a

part of daily thought, and brought again and again before the mind

with ever-renewed encouragement. Real life is, to most men, a long

second-best, a perpetual compromise between the ideal and the

possible; but the world of pure reason knows no compromise, no

practical limitations, no barrier to the creative activity embodying

in splendid edifices the passionate aspiration after the perfect from

which all great work springs. Remote from human passions, remote even

from the pitiful facts of nature, the generations have gradually

created an ordered cosmos, where pure thought can dwell as in its

natural home, and where one, at least, of our nobler impulses can

escape from the dreary exile of the actual world.

So little, however, have mathematicians aimed at beauty, that hardly

anything in their work has had this conscious purpose. Much, owing to

irrepressible instincts, which were better than avowed beliefs, has

been moulded by an unconscious taste; but much also has been spoilt by

false notions of what was fitting. The characteristic excellence of

mathematics is only to be found where the reasoning is rigidly

logical: the rules of logic are to mathematics what those of structure

are to architecture. In the most beautiful work, a chain of argument

is presented in which every link is important on its own account, in

which there is an air of ease and lucidity throughout, and the

premises achieve more than would have been thought possible, by means

which appear natural and inevitable. Literature embodies what is

general in particular circumstances whose universal significance

shines through their individual dress; but mathematics endeavours to

present whatever is most general in its purity, without any irrelevant

trappings.

How should the teaching of mathematics be conducted so as to

communicate to the learner as much as possible of this high ideal?

Here experience must, in a great measure, be our guide; but some

maxims may result from our consideration of the ultimate purpose to be

achieved.

One of the chief ends served by mathematics, when rightly taught, is

to awaken the learner's belief in reason, his confidence in the truth

of what has been demonstrated, and in the value of demonstration. This

purpose is not served by existing instruction; but it is easy to see

ways in which it might be served. At present, in what concerns

arithmetic, the boy or girl is given a set of rules, which present

themselves as neither true nor false, but as merely the will of the

teacher, the way in which, for some unfathomable reason, the teacher

prefers to have the game played. To some degree, in a study of such

definite practical utility, this is no doubt unavoidable; but as soon

as possible, the reasons of rules should be set forth by whatever

means most readily appeal to the childish mind. In geometry, instead

of the tedious apparatus of fallacious proofs for obvious truisms

which constitutes the beginning of Euclid, the learner should be

allowed at first to assume the truth of everything obvious, and should

be instructed in the demonstrations of theorems which are at once

startling and easily verifiable by actual drawing, such as those in

which it is shown that three or more lines meet in a point. In this

way belief is generated; it is seen that reasoning may lead to

startling conclusions, which nevertheless the facts will verify; and

thus the instinctive distrust of whatever is abstract or rational is

gradually overcome. Where theorems are difficult, they should be first

taught as exercises in geometrical drawing, until the figure has

become thoroughly familiar; it will then be an agreeable advance to be

taught the logical connections of the various lines or circles that

occur. It is desirable also that the figure illustrating a theorem

should be drawn in all possible cases and shapes, that so the abstract

relations with which geometry is concerned may of themselves emerge

as the residue of similarity amid such great apparent diversity. In

this way the abstract demonstrations should form but a small part of

the instruction, and should be given when, by familiarity with

concrete illustrations, they have come to be felt as the natural

embodiment of visible fact. In this early stage proofs should not be

given with pedantic fullness; definitely fallacious methods, such as

that of superposition, should be rigidly excluded from the first, but

where, without such methods, the proof would be very difficult, the

result should be rendered acceptable by arguments and illustrations

which are explicitly contrasted with demonstrations.

In the beginning of algebra, even the most intelligent child finds, as

a rule, very great difficulty. The use of letters is a mystery, which

seems to have no purpose except mystification. It is almost

impossible, at first, not to think that every letter stands for some

particular number, if only the teacher would reveal _what_ number it

stands for. The fact is, that in algebra the mind is first taught to

consider general truths, truths which are not asserted to hold only of

this or that particular thing, but of any one of a whole group of

things. It is in the power of understanding and discovering such

truths that the mastery of the intellect over the whole world of

things actual and possible resides; and ability to deal with the

general as such is one of the gifts that a mathematical education

should bestow. But how little, as a rule, is the teacher of algebra

able to explain the chasm which divides it from arithmetic, and how

little is the learner assisted in his groping efforts at

comprehension! Usually the method that has been adopted in arithmetic

is continued: rules are set forth, with no adequate explanation of

their grounds; the pupil learns to use the rules blindly, and

presently, when he is able to obtain the answer that the teacher

desires, he feels that he has mastered the difficulties of the

subject. But of inner comprehension of the processes employed he has

probably acquired almost nothing.

When algebra has been learnt, all goes smoothly until we reach those

studies in which the notion of infinity is employed--the infinitesimal

calculus and the whole of higher mathematics. The solution of the

difficulties which formerly surrounded the mathematical infinite is

probably the greatest achievement of which our own age has to boast.

Since the beginnings of Greek thought these difficulties have been

known; in every age the finest intellects have vainly endeavoured to

answer the apparently unanswerable questions that had been asked by

Zeno the Eleatic. At last Georg Cantor has found the answer, and has

conquered for the intellect a new and vast province which had been

given over to Chaos and old Night. It was assumed as self-evident,

until Cantor and Dedekind established the opposite, that if, from any

collection of things, some were taken away, the number of things left

must always be less than the original number of things. This

assumption, as a matter of fact, holds only of finite collections; and

the rejection of it, where the infinite is concerned, has been shown

to remove all the difficulties that had hitherto baffled human reason

in this matter, and to render possible the creation of an exact

science of the infinite. This stupendous fact ought to produce a

revolution in the higher teaching of mathematics; it has itself added

immeasurably to the educational value of the subject, and it has at

last given the means of treating with logical precision many studies

which, until lately, were wrapped in fallacy and obscurity. By those

who were educated on the old lines, the new work is considered to be

appallingly difficult, abstruse, and obscure; and it must be confessed

that the discoverer, as is so often the case, has hardly himself

emerged from the mists which the light of his intellect is dispelling.

But inherently, the new doctrine of the infinite, to all candid and

inquiring minds, has facilitated the mastery of higher mathematics;

for hitherto, it has been necessary to learn, by a long process of

sophistication, to give assent to arguments which, on first

acquaintance, were rightly judged to be confused and erroneous. So far

from producing a fearless belief in reason, a bold rejection of

whatever failed to fulfil the strictest requirements of logic, a

mathematical training, during the past two centuries, encouraged the

belief that many things, which a rigid inquiry would reject as

fallacious, must yet be accepted because they work in what the

mathematician calls "practice. " By this means, a timid, compromising

spirit, or else a sacerdotal belief in mysteries not intelligible to

the profane, has been bred where reason alone should have ruled. All

this it is now time to sweep away; let those who wish to penetrate

into the arcana of mathematics be taught at once the true theory in

all its logical purity, and in the concatenation established by the

very essence of the entities concerned.

If we are considering mathematics as an end in itself, and not as a

technical training for engineers, it is very desirable to preserve the

purity and strictness of its reasoning. Accordingly those who have

attained a sufficient familiarity with its easier portions should be

led backward from propositions to which they have assented as

self-evident to more and more fundamental principles from which what

had previously appeared as premises can be deduced. They should be

taught--what the theory of infinity very aptly illustrates--that many

propositions seem self-evident to the untrained mind which,

nevertheless, a nearer scrutiny shows to be false. By this means they

will be led to a sceptical inquiry into first principles, an

examination of the foundations upon which the whole edifice of

reasoning is built, or, to take perhaps a more fitting metaphor, the

great trunk from which the spreading branches spring. At this stage,

it is well to study afresh the elementary portions of mathematics,

asking no longer merely whether a given proposition is true, but also

how it grows out of the central principles of logic. Questions of this

nature can now be answered with a precision and certainty which were

formerly quite impossible; and in the chains of reasoning that the

answer requires the unity of all mathematical studies at last unfolds

itself.

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.