Mysticism and Logic and Other Essays
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Title: Mysticism and Logic and Other Essays
Author: Bertrand Russell
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* * * * *
BERTRAND RUSSELL
MYSTICISM AND
LOGIC
AND OTHER ESSAYS
_LONDON_
GEORGE ALLEN & UNWIN LTD
RUSKIN HOUSE MUSEUM STREET
MYSTICISM AND LOGIC
AND OTHER ESSAYS
BY BERTRAND RUSSELL
_The ABC of Relativity_
_The Analysis of Matter_
_Human Society in Ethics and Politics_
_The Impact of Science on Society_
_New Hopes for a Changing World_
_Authority and the Individual_
_Human Knowledge_
_History of Western Philosophy_
_The Principles of Mathematics_
_Introduction to Mathematical Philosophy_
_The Analysis of Mind_
_Our Knowledge of the External World_
_An Outline of Philosophy_
_The Philosophy of Leibniz_
_An Inquiry into Meaning and Truth_
_Logic and Knowledge_
_The Problems of Philosophy_
_Principia Mathematica_
_Common Sense and Nuclear Warfare_
_Why I am Not a Christian_
_Portraits from Memory_
_My Philosophical Development_
_Unpopular Essays_
_Power_
_In Praise of Idleness_
_The Conquest of Happiness_
_Sceptical Essays_
_The Scientific Outlook_
_Marriage and Morals_
_Education and the Social Order_
_On Education_
_Freedom and Organization_
_Principles of Social Reconstruction_
_Roads to Freedom_
_Practice and Theory of Bolshevism_
_Satan in The Suburbs_
_Nightmares of Eminent Persons_
_First published as "Philosophical Essays"_ _October 1910_
_Second Edition as "Mysticism and Logic"_ _December 1917_
_Third Impression_ _April 1918_
_Fourth Impression_ _February 1919_
_Fifth Impression_ _October 1921_
_Sixth Impression_ _August 1925_
_Seventh Impression_ _January 1932_
_Eighth Impression_ _1949_
_Ninth Impression_ _1950_
_Tenth Impression_ _1951_
_Eleventh Impression_ _1959_
_This book is copyright under the Berne Convention. Apart from any
fair dealing for the purpose of private study, research, criticism
or review, as permitted under the Copyright Act, 1956, no portion
may be reproduced by any process without written permission.
Enquiry should be made to the publisher._
PRINTED IN GREAT BRITAIN
_by Taylor Garnett Evans & Co. Ltd.,
Watford, Herts._
PREFACE
The following essays have been written and published at various times,
and my thanks are due to the previous publishers for the permission to
reprint them.
The essay on "Mysticism and Logic" appeared in the _Hibbert Journal_
for July, 1914. "The Place of Science in a Liberal Education" appeared
in two numbers of _The New Statesman_, May 24 and 31, 1913. "The Free
Man's Worship" and "The Study of Mathematics" were included in a
former collection (now out of print), _Philosophical Essays_, also
published by Messrs. Longmans, Green & Co. Both were written in 1902;
the first appeared originally in the _Independent Review_ for 1903,
the second in the _New Quarterly_, November, 1907. In theoretical
Ethics, the position advocated in "The Free Man's Worship" is not
quite identical with that which I hold now: I feel less convinced than
I did then of the objectivity of good and evil. But the general
attitude towards life which is suggested in that essay still seems to
me, in the main, the one which must be adopted in times of stress and
difficulty by those who have no dogmatic religious beliefs, if inward
defeat is to be avoided.
The essay on "Mathematics and the Metaphysicians" was written in 1901,
and appeared in an American magazine, _The International Monthly_,
under the title "Recent Work in the Philosophy of Mathematics." Some
points in this essay require modification in view of later work.
These are indicated in footnotes. Its tone is partly explained by the
fact that the editor begged me to make the article "as romantic as
possible."
All the above essays are entirely popular, but those that follow are
somewhat more technical. "On Scientific Method in Philosophy" was the
Herbert Spencer lecture at Oxford in 1914, and was published by the
Clarendon Press, which has kindly allowed me to include it in this
collection. "The Ultimate Constituents of Matter" was an address to
the Manchester Philosophical Society, early in 1915, and was published
in the _Monist_ in July of that year. The essay on "The Relation of
Sense-data to Physics" was written in January, 1914, and first
appeared in No. 4 of that year's volume of _Scientia_, an
International Review of Scientific Synthesis, edited by M. Eugenio
Rignano, published monthly by Messrs. Williams and Norgate, London,
Nicola Zanichelli, Bologna, and Félix Alcan, Paris. The essay "On the
Notion of Cause" was the presidential address to the Aristotelian
Society in November, 1912, and was published in their _Proceedings_
for 1912-13. "Knowledge by Acquaintance and Knowledge by Description"
was also a paper read before the Aristotelian Society, and published
in their _Proceedings_ for 1910-11.
LONDON,
_September, 1917_
CONTENTS
_Chapter_ _Page_
I. _Mysticism and Logic_ 1
II. _The Place of Science in a Liberal Education_ 33
III. _A Free Man's Worship_ 46
IV. _The Study of Mathematics_ 58
V. _Mathematics and the Metaphysicians_ 74
VI. _On Scientific Method in Philosophy_ 97
VII. _The Ultimate Constituents of Matter_ 125
VIII. _The Relation of Sense-data to Physics_ 145
IX. _On the Notion of Cause_ 180
X. _Knowledge by Acquaintance and Knowledge by
Description_ 209
MYSTICISM AND LOGIC AND OTHER ESSAYS
I
MYSTICISM AND LOGIC
Metaphysics, or the attempt to conceive the world as a whole by means
of thought, has been developed, from the first, by the union and
conflict of two very different human impulses, the one urging men
towards mysticism, the other urging them towards science. Some men
have achieved greatness through one of these impulses alone, others
through the other alone: in Hume, for example, the scientific impulse
reigns quite unchecked, while in Blake a strong hostility to science
co-exists with profound mystic insight. But the greatest men who have
been philosophers have felt the need both of science and of mysticism:
the attempt to harmonise the two was what made their life, and what
always must, for all its arduous uncertainty, make philosophy, to some
minds, a greater thing than either science or religion.
Before attempting an explicit characterisation of the scientific and
the mystical impulses, I will illustrate them by examples from two
philosophers whose greatness lies in the very intimate blending which
they achieved. The two philosophers I mean are Heraclitus and Plato.
Heraclitus, as every one knows, was a believer in universal flux: time
builds and destroys all things. From the few fragments that remain, it
is not easy to discover how he arrived at his opinions, but there are
some sayings that strongly suggest scientific observation as the
source.
"The things that can be seen, heard, and learned," he says, "are what
I prize the most." This is the language of the empiricist, to whom
observation is the sole guarantee of truth. "The sun is new every
day," is another fragment; and this opinion, in spite of its
paradoxical character, is obviously inspired by scientific reflection,
and no doubt seemed to him to obviate the difficulty of understanding
how the sun can work its way underground from west to east during the
night. Actual observation must also have suggested to him his central
doctrine, that Fire is the one permanent substance, of which all
visible things are passing phases. In combustion we see things change
utterly, while their flame and heat rise up into the air and vanish.
"This world, which is the same for all," he says, "no one of gods or
men has made; but it was ever, is now, and ever shall be, an
ever-living Fire, with measures kindling, and measures going out."
"The transformations of Fire are, first of all, sea; and half of the
sea is earth, half whirlwind."
This theory, though no longer one which science can accept, is
nevertheless scientific in spirit. Science, too, might have inspired
the famous saying to which Plato alludes: "You cannot step twice into
the same rivers; for fresh waters are ever flowing in upon you." But
we find also another statement among the extant fragments: "We step
and do not step into the same rivers; we are and are not."
The comparison of this statement, which is mystical, with the one
quoted by Plato, which is scientific, shows how intimately the two
tendencies are blended in the system of Heraclitus. Mysticism is, in
essence, little more than a certain intensity and depth of feeling in
regard to what is believed about the universe; and this kind of
feeling leads Heraclitus, on the basis of his science, to strangely
poignant sayings concerning life and the world, such as:
"Time is a child playing draughts, the kingly power is a child's."
It is poetic imagination, not science, which presents Time as despotic
lord of the world, with all the irresponsible frivolity of a child. It
is mysticism, too, which leads Heraclitus to assert the identity of
opposites: "Good and ill are one," he says; and again: "To God all
things are fair and good and right, but men hold some things wrong and
some right."
Much of mysticism underlies the ethics of Heraclitus. It is true that
a scientific determinism alone might have inspired the statement:
"Man's character is his fate"; but only a mystic would have said:
"Every beast is driven to the pasture with blows"; and again:
"It is hard to fight with one's heart's desire. Whatever it wishes to
get, it purchases at the cost of soul"; and again:
"Wisdom is one thing. It is to know the thought by which all things
are steered through all things."[1]
Examples might be multiplied, but those that have been given are
enough to show the character of the man: the facts of science, as they
appeared to him, fed the flame in his soul, and in its light he saw
into the depths of the world by the reflection of his own dancing
swiftly penetrating fire. In such a nature we see the true union of
the mystic and the man of science--the highest eminence, as I think,
that it is possible to achieve in the world of thought.
In Plato, the same twofold impulse exists, though the mystic impulse
is distinctly the stronger of the two, and secures ultimate victory
whenever the conflict is sharp. His description of the cave is the
classical statement of belief in a knowledge and reality truer and
more real than that of the senses:
"Imagine[2] a number of men living in an underground cavernous
chamber, with an entrance open to the light, extending along the
entire length of the cavern, in which they have been confined, from
their childhood, with their legs and necks so shackled that they
are obliged to sit still and look straight forwards, because their
chains render it impossible for them to turn their heads round: and
imagine a bright fire burning some way off, above and behind them,
and an elevated roadway passing between the fire and the prisoners,
with a low wall built along it, like the screens which conjurors
put up in front of their audience, and above which they exhibit
their wonders.
I have it, he replied.
Also figure to yourself a number of persons walking behind this
wall, and carrying with them statues of men, and images of other
animals, wrought in wood and stone and all kinds of materials,
together with various other articles, which overtop the wall; and,
as you might expect, let some of the passers-by be talking, and
others silent.
You are describing a strange scene, and strange prisoners.
They resemble us, I replied.
Now consider what would happen if the course of nature brought them
a release from their fetters, and a remedy for their foolishness,
in the following manner. Let us suppose that one of them has been
released, and compelled suddenly to stand up, and turn his neck
round and walk with open eyes towards the light; and let us suppose
that he goes through all these actions with pain, and that the
dazzling splendour renders him incapable of discerning those
objects of which he used formerly to see the shadows. What answer
should you expect him to make, if some one were to tell him that in
those days he was watching foolish phantoms, but that now he is
somewhat nearer to reality, and is turned towards things more real,
and sees more correctly; above all, if he were to point out to him
the several objects that are passing by, and question him, and
compel him to answer what they are? Should you not expect him to be
puzzled, and to regard his old visions as truer than the objects
now forced upon his notice?
Yes, much truer....
Hence, I suppose, habit will be necessary to enable him to perceive
objects in that upper world. At first he will be most successful in
distinguishing shadows; then he will discern the reflections of men
and other things in water, and afterwards the realities; and after
this he will raise his eyes to encounter the light of the moon and
stars, finding it less difficult to study the heavenly bodies and
the heaven itself by night, than the sun and the sun's light by
day.
Doubtless.
Last of all, I imagine, he will be able to observe and contemplate
the nature of the sun, not as it _appears_ in water or on alien
ground, but as it is in itself in its own territory.
Of course.
His next step will be to draw the conclusion, that the sun is the
author of the seasons and the years, and the guardian of all things
in the visible world, and in a manner the cause of all those things
which he and his companions used to see.
Obviously, this will be his next step....
Now this imaginary case, my dear Glancon, you must apply in all its
parts to our former statements, by comparing the region which the
eye reveals to the prison house, and the light of the fire therein
to the power of the sun: and if, by the upward ascent and the
contemplation of the upper world, you understand the mounting of
the soul into the intellectual region, you will hit the tendency of
my own surmises, since you desire to be told what they are; though,
indeed, God only knows whether they are correct. But, be that as it
may, the view which I take of the subject is to the following
effect. In the world of knowledge, the essential Form of Good is
the limit of our enquiries, and can barely be perceived; but, when
perceived, we cannot help concluding that it is in every case the
source of all that is bright and beautiful,--in the visible world
giving birth to light and its master, and in the intellectual world
dispensing, immediately and with full authority, truth and
reason;--and that whosoever would act wisely, either in private or
in public, must set this Form of Good before his eyes."
But in this passage, as throughout most of Plato's teaching, there is
an identification of the good with the truly real, which became
embodied in the philosophical tradition, and is still largely
operative in our own day. In thus allowing a legislative function to
the good, Plato produced a divorce between philosophy and science,
from which, in my opinion, both have suffered ever since and are still
suffering. The man of science, whatever his hopes may be, must lay
them aside while he studies nature; and the philosopher, if he is to
achieve truth, must do the same. Ethical considerations can only
legitimately appear when the truth has been ascertained: they can and
should appear as determining our feeling towards the truth, and our
manner of ordering our lives in view of the truth, but not as
themselves dictating what the truth is to be.
There are passages in Plato--among those which illustrate the
scientific side of his mind--where he seems clearly aware of this. The
most noteworthy is the one in which Socrates, as a young man, is
explaining the theory of ideas to Parmenides.
After Socrates has explained that there is an idea of the good, but
not of such things as hair and mud and dirt, Parmenides advises him
"not to despise even the meanest things," and this advice shows the
genuine scientific temper. It is with this impartial temper that the
mystic's apparent insight into a higher reality and a hidden good has
to be combined if philosophy is to realise its greatest possibilities.
And it is failure in this respect that has made so much of idealistic
philosophy thin, lifeless, and insubstantial. It is only in marriage
with the world that our ideals can bear fruit: divorced from it, they
remain barren. But marriage with the world is not to be achieved by an
ideal which shrinks from fact, or demands in advance that the world
shall conform to its desires.
Parmenides himself is the source of a peculiarly interesting strain
of mysticism which pervades Plato's thought--the mysticism which may
be called "logical" because it is embodied in theories on logic. This
form of mysticism, which appears, so far as the West is concerned, to
have originated with Parmenides, dominates the reasonings of all the
great mystical metaphysicians from his day to that of Hegel and his
modern disciples. Reality, he says, is uncreated, indestructible,
unchanging, indivisible; it is "immovable in the bonds of mighty
chains, without beginning and without end; since coming into being and
passing away have been driven afar, and true belief has cast them
away." The fundamental principle of his inquiry is stated in a
sentence which would not be out of place in Hegel: "Thou canst not
know what is not--that is impossible--nor utter it; for it is the same
thing that can be thought and that can be." And again: "It needs must
be that what can be thought and spoken of is; for it is possible for
it to be, and it is not possible for what is nothing to be." The
impossibility of change follows from this principle; for what is past
can be spoken of, and therefore, by the principle, still is.
Mystical philosophy, in all ages and in all parts of the world, is
characterised by certain beliefs which are illustrated by the
doctrines we have been considering.
There is, first, the belief in insight as against discursive analytic
knowledge: the belief in a way of wisdom, sudden, penetrating,
coercive, which is contrasted with the slow and fallible study of
outward appearance by a science relying wholly upon the senses. All
who are capable of absorption in an inward passion must have
experienced at times the strange feeling of unreality in common
objects, the loss of contact with daily things, in which the solidity
of the outer world is lost, and the soul seems, in utter loneliness,
to bring forth, out of its own depths, the mad dance of fantastic
phantoms which have hitherto appeared as independently real and
living. This is the negative side of the mystic's initiation: the
doubt concerning common knowledge, preparing the way for the reception
of what seems a higher wisdom. Many men to whom this negative
experience is familiar do not pass beyond it, but for the mystic it is
merely the gateway to an ampler world.
The mystic insight begins with the sense of a mystery unveiled, of a
hidden wisdom now suddenly become certain beyond the possibility of a
doubt. The sense of certainty and revelation comes earlier than any
definite belief. The definite beliefs at which mystics arrive are the
result of reflection upon the inarticulate experience gained in the
moment of insight. Often, beliefs which have no real connection with
this moment become subsequently attracted into the central nucleus;
thus in addition to the convictions which all mystics share, we find,
in many of them, other convictions of a more local and temporary
character, which no doubt become amalgamated with what was essentially
mystical in virtue of their subjective certainty. We may ignore such
inessential accretions, and confine ourselves to the beliefs which all
mystics share.
The first and most direct outcome of the moment of illumination is
belief in the possibility of a way of knowledge which may be called
revelation or insight or intuition, as contrasted with sense, reason,
and analysis, which are regarded as blind guides leading to the morass
of illusion. Closely connected with this belief is the conception of a
Reality behind the world of appearance and utterly different from it.
This Reality is regarded with an admiration often amounting to
worship; it is felt to be always and everywhere close at hand, thinly
veiled by the shows of sense, ready, for the receptive mind, to shine
in its glory even through the apparent folly and wickedness of Man.
The poet, the artist, and the lover are seekers after that glory: the
haunting beauty that they pursue is the faint reflection of its sun.
But the mystic lives in the full light of the vision: what others
dimly seek he knows, with a knowledge beside which all other knowledge
is ignorance.
The second characteristic of mysticism is its belief in unity, and its
refusal to admit opposition or division anywhere. We found Heraclitus
saying "good and ill are one"; and again he says, "the way up and the
way down is one and the same." The same attitude appears in the
simultaneous assertion of contradictory propositions, such as: "We
step and do not step into the same rivers; we are and are not." The
assertion of Parmenides, that reality is one and indivisible, comes
from the same impulse towards unity. In Plato, this impulse is less
prominent, being held in check by his theory of ideas; but it
reappears, so far as his logic permits, in the doctrine of the primacy
of the Good.
A third mark of almost all mystical metaphysics is the denial of the
reality of Time. This is an outcome of the denial of division; if all
is one, the distinction of past and future must be illusory. We have
seen this doctrine prominent in Parmenides; and among moderns it is
fundamental in the systems of Spinoza and Hegel.
The last of the doctrines of mysticism which we have to consider is
its belief that all evil is mere appearance, an illusion produced by
the divisions and oppositions of the analytic intellect. Mysticism
does not maintain that such things as cruelty, for example, are good,
but it denies that they are real: they belong to that lower world of
phantoms from which we are to be liberated by the insight of the
vision. Sometimes--for example in Hegel, and at least verbally in
Spinoza--not only evil, but good also, is regarded as illusory, though
nevertheless the emotional attitude towards what is held to be Reality
is such as would naturally be associated with the belief that Reality
is good. What is, in all cases, ethically characteristic of mysticism
is absence of indignation or protest, acceptance with joy, disbelief
in the ultimate truth of the division into two hostile camps, the good
and the bad. This attitude is a direct outcome of the nature of the
mystical experience: with its sense of unity is associated a feeling
of infinite peace. Indeed it may be suspected that the feeling of
peace produces, as feelings do in dreams, the whole system of
associated beliefs which make up the body of mystic doctrine. But this
is a difficult question, and one on which it cannot be hoped that
mankind will reach agreement.
Four questions thus arise in considering the truth or falsehood of
mysticism, namely:
I. Are there two ways of knowing, which may be called respectively
reason and intuition? And if so, is either to be preferred to the
other?
II. Is all plurality and division illusory?
III. Is time unreal?
IV. What kind of reality belongs to good and evil?
On all four of these questions, while fully developed mysticism seems
to me mistaken, I yet believe that, by sufficient restraint, there is
an element of wisdom to be learned from the mystical way of feeling,
which does not seem to be attainable in any other manner. If this is
the truth, mysticism is to be commended as an attitude towards life,
not as a creed about the world. The meta-physical creed, I shall
maintain, is a mistaken outcome of the emotion, although this emotion,
as colouring and informing all other thoughts and feelings, is the
inspirer of whatever is best in Man. Even the cautious and patient
investigation of truth by science, which seems the very antithesis of
the mystic's swift certainty, may be fostered and nourished by that
very spirit of reverence in which mysticism lives and moves.
I. REASON AND INTUITION[3]
Of the reality or unreality of the mystic's world I know nothing. I
have no wish to deny it, nor even to declare that the insight which
reveals it is not a genuine insight. What I do wish to maintain--and
it is here that the scientific attitude becomes imperative--is that
insight, untested and unsupported, is an insufficient guarantee of
truth, in spite of the fact that much of the most important truth is
first suggested by its means. It is common to speak of an opposition
between instinct and reason; in the eighteenth century, the opposition
was drawn in favour of reason, but under the influence of Rousseau and
the romantic movement instinct was given the preference, first by
those who rebelled against artificial forms of government and thought,
and then, as the purely rationalistic defence of traditional theology
became increasingly difficult, by all who felt in science a menace to
creeds which they associated with a spiritual outlook on life and the
world. Bergson, under the name of "intuition," has raised instinct to
the position of sole arbiter of metaphysical truth. But in fact the
opposition of instinct and reason is mainly illusory. Instinct,
intuition, or insight is what first leads to the beliefs which
subsequent reason confirms or confutes; but the confirmation, where it
is possible, consists, in the last analysis, of agreement with other
beliefs no less instinctive. Reason is a harmonising, controlling
force rather than a creative one. Even in the most purely logical
realm, it is insight that first arrives at what is new.
Where instinct and reason do sometimes conflict is in regard to single
beliefs, held instinctively, and held with such determination that no
degree of inconsistency with other beliefs leads to their abandonment.
Instinct, like all human faculties, is liable to error. Those in whom
reason is weak are often unwilling to admit this as regards
themselves, though all admit it in regard to others. Where instinct is
least liable to error is in practical matters as to which right
judgment is a help to survival: friendship and hostility in others,
for instance, are often felt with extraordinary discrimination through
very careful disguises. But even in such matters a wrong impression
may be given by reserve or flattery; and in matters less directly
practical, such as philosophy deals with, very strong instinctive
beliefs are sometimes wholly mistaken, as we may come to know through
their perceived inconsistency with other equally strong beliefs. It is
such considerations that necessitate the harmonising mediation of
reason, which tests our beliefs by their mutual compatibility, and
examines, in doubtful cases, the possible sources of error on the one
side and on the other. In this there is no opposition to instinct as a
whole, but only to blind reliance upon some one interesting aspect of
instinct to the exclusion of other more commonplace but not less
trustworthy aspects. It is such one-sidedness, not instinct itself,
that reason aims at correcting.
These more or less trite maxims may be illustrated by application to
Bergson's advocacy of "intuition" as against "intellect." There are,
he says, "two profoundly different ways of knowing a thing. The first
implies that we move round the object: the second that we enter into
it. The first depends on the point of view at which we are placed and
on the symbols by which we express ourselves. The second neither
depends on a point of view nor relies on any symbol. The first kind of
knowledge may be said to stop at the _relative_; the second, in those
cases where it is possible, to attain the _absolute_."[4] The second
of these, which is intuition, is, he says, "the kind of _intellectual
sympathy_ by which one places oneself within an object in order to
coincide with what is unique in it and therefore inexpressible" (p.
6). In illustration, he mentions self-knowledge: "there is one
reality, at least, which we all seize from within, by intuition and
not by simple analysis. It is our own personality in its flowing
through time--our self which endures" (p. 8). The rest of Bergson's
philosophy consists in reporting, through the imperfect medium of
words, the knowledge gained by intuition, and the consequent complete
condemnation of all the pretended knowledge derived from science and
common sense.
This procedure, since it takes sides in a conflict of instinctive
beliefs, stands in need of justification by proving the greater
trustworthiness of the beliefs on one side than of those on the other.
Bergson attempts this justification in two ways, first by explaining
that intellect is a purely practical faculty to secure biological
success, secondly by mentioning remarkable feats of instinct in
animals and by pointing out characteristics of the world which, though
intuition can apprehend them, are baffling to intellect as he
interprets it.
Of Bergson's theory that intellect is a purely practical faculty,
developed in the struggle for survival, and not a source of true
beliefs, we may say, first, that it is only through intellect that we
know of the struggle for survival and of the biological ancestry of
man: if the intellect is misleading, the whole of this merely inferred
history is presumably untrue. If, on the other hand, we agree with him
in thinking that evolution took place as Darwin believed, then it is
not only intellect, but all our faculties, that have been developed
under the stress of practical utility. Intuition is seen at its best
where it is directly useful, for example in regard to other people's
characters and dispositions. Bergson apparently holds that capacity
for this kind of knowledge is less explicable by the struggle for
existence than, for example, capacity for pure mathematics. Yet the
savage deceived by false friendship is likely to pay for his mistake
with his life; whereas even in the most civilised societies men are
not put to death for mathematical incompetence. All the most striking
of his instances of intuition in animals have a very direct survival
value. The fact is, of course, that both intuition and intellect have
been developed because they are useful, and that, speaking broadly,
they are useful when they give truth and become harmful when they give
falsehood. Intellect, in civilised man, like artistic capacity, has
occasionally been developed beyond the point where it is useful to the
individual; intuition, on the other hand, seems on the whole to
diminish as civilisation increases. It is greater, as a rule, in
children than in adults, in the uneducated than in the educated.
Probably in dogs it exceeds anything to be found in human beings. But
those who see in these facts a recommendation of intuition ought to
return to running wild in the woods, dyeing themselves with woad and
living on hips and haws.
Let us next examine whether intuition possesses any such infallibility
as Bergson claims for it. The best instance of it, according to him,
is our acquaintance with ourselves; yet self-knowledge is proverbially
rare and difficult. Most men, for example, have in their nature
meannesses, vanities, and envies of which they are quite unconscious,
though even their best friends can perceive them without any
difficulty. It is true that intuition has a convincingness which is
lacking to intellect: while it is present, it is almost impossible to
doubt its truth. But if it should appear, on examination, to be at
least as fallible as intellect, its greater subjective certainty
becomes a demerit, making it only the more irresistibly deceptive.
Apart from self-knowledge, one of the most notable examples of
intuition is the knowledge people believe themselves to possess of
those with whom they are in love: the wall between different
personalities seems to become transparent, and people think they see
into another soul as into their own. Yet deception in such cases is
constantly practised with success; and even where there is no
intentional deception, experience gradually proves, as a rule, that
the supposed insight was illusory, and that the slower more groping
methods of the intellect are in the long run more reliable.
Bergson maintains that intellect can only deal with things in so far
as they resemble what has been experienced in the past, while
intuition has the power of apprehending the uniqueness and novelty
that always belong to each fresh moment. That there is something
unique and new at every moment, is certainly true; it is also true
that this cannot be fully expressed by means of intellectual concepts.
Only direct acquaintance can give knowledge of what is unique and new.
But direct acquaintance of this kind is given fully in sensation, and
does not require, so far as I can see, any special faculty of
intuition for its apprehension. It is neither intellect nor intuition,
but sensation, that supplies new data; but when the data are new in
any remarkable manner, intellect is much more capable of dealing with
them than intuition would be. The hen with a brood of ducklings no
doubt has intuition which seems to place her inside them, and not
merely to know them analytically; but when the ducklings take to the
water, the whole apparent intuition is seen to be illusory, and the
hen is left helpless on the shore. Intuition, in fact, is an aspect
and development of instinct, and, like all instinct, is admirable in
those customary surroundings which have moulded the habits of the
animal in question, but totally incompetent as soon as the
surroundings are changed in a way which demands some non-habitual mode
of action.
The theoretical understanding of the world, which is the aim of
philosophy, is not a matter of great practical importance to animals,
or to savages, or even to most civilised men. It is hardly to be
supposed, therefore, that the rapid, rough and ready methods of
instinct or intuition will find in this field a favourable ground for
their application. It is the older kinds of activity, which bring out
our kinship with remote generations of animal and semi-human
ancestors, that show intuition at its best. In such matters as
self-preservation and love, intuition will act sometimes (though not
always) with a swiftness and precision which are astonishing to the
critical intellect. But philosophy is not one of the pursuits which
illustrate our affinity with the past: it is a highly refined, highly
civilised pursuit, demanding, for its success, a certain liberation
from the life of instinct, and even, at times, a certain aloofness
from all mundane hopes and fears. It is not in philosophy, therefore,
that we can hope to see intuition at its best. On the contrary, since
the true objects of philosophy, and the habit of thought demanded for
their apprehension, are strange, unusual, and remote, it is here, more
almost than anywhere else, that intellect proves superior to
intuition, and that quick unanalysed convictions are least deserving
of uncritical acceptance.
In advocating the scientific restraint and balance, as against the
self-assertion of a confident reliance upon intuition, we are only
urging, in the sphere of knowledge, that largeness of contemplation,
that impersonal disinterestedness, and that freedom from practical
preoccupations which have been inculcated by all the great religions
of the world. Thus our conclusion, however it may conflict with the
explicit beliefs of many mystics, is, in essence, not contrary to the
spirit which inspires those beliefs, but rather the outcome of this
very spirit as applied in the realm of thought.
II. UNITY AND PLURALITY
One of the most convincing aspects of the mystic illumination is the
apparent revelation of the oneness of all things, giving rise to
pantheism in religion and to monism in philosophy. An elaborate logic,
beginning with Parmenides, and culminating in Hegel and his followers,
has been gradually developed, to prove that the universe is one
indivisible Whole, and that what seem to be its parts, if considered
as substantial and self-existing, are mere illusion. The conception
of a Reality quite other than the world of appearance, a reality one,
indivisible, and unchanging, was introduced into Western philosophy by
Parmenides, not, nominally at least, for mystical or religious
reasons, but on the basis of a logical argument as to the
impossibility of not-being, and most subsequent metaphysical systems
are the outcome of this fundamental idea.
The logic used in defence of mysticism seems to be faulty as logic,
and open to technical criticisms, which I have explained elsewhere. I
shall not here repeat these criticisms, since they are lengthy and
difficult, but shall instead attempt an analysis of the state of mind
from which mystical logic has arisen.
Belief in a reality quite different from what appears to the senses
arises with irresistible force in certain moods, which are the source
of most mysticism, and of most metaphysics. While such a mood is
dominant, the need of logic is not felt, and accordingly the more
thoroughgoing mystics do not employ logic, but appeal directly to the
immediate deliverance of their insight. But such fully developed
mysticism is rare in the West. When the intensity of emotional
conviction subsides, a man who is in the habit of reasoning will
search for logical grounds in favour of the belief which he finds in
himself. But since the belief already exists, he will be very
hospitable to any ground that suggests itself. The paradoxes
apparently proved by his logic are really the paradoxes of mysticism,
and are the goal which he feels his logic must reach if it is to be in
accordance with insight. The resulting logic has rendered most
philosophers incapable of giving any account of the world of science
and daily life. If they had been anxious to give such an account, they
would probably have discovered the errors of their logic; but most of
them were less anxious to understand the world of science and daily
life than to convict it of unreality in the interests of a
super-sensible "real" world.
It is in this way that logic has been pursued by those of the great
philosophers who were mystics. But since they usually took for granted
the supposed insight of the mystic emotion, their logical doctrines
were presented with a certain dryness, and were believed by their
disciples to be quite independent of the sudden illumination from
which they sprang. Nevertheless their origin clung to them, and they
remained--to borrow a useful word from Mr. Santayana--"malicious" in
regard to the world of science and common sense. It is only so that we
can account for the complacency with which philosophers have accepted
the inconsistency of their doctrines with all the common and
scientific facts which seem best established and most worthy of
belief.
The logic of mysticism shows, as is natural, the defects which are
inherent in anything malicious. The impulse to logic, not felt while
the mystic mood is dominant, reasserts itself as the mood fades, but
with a desire to retain the vanishing insight, or at least to prove
that it _was_ insight, and that what seems to contradict it is
illusion. The logic which thus arises is not quite disinterested or
candid, and is inspired by a certain hatred of the daily world to
which it is to be applied. Such an attitude naturally does not tend to
the best results. Everyone knows that to read an author simply in
order to refute him is not the way to understand him; and to read the
book of Nature with a conviction that it is all illusion is just as
unlikely to lead to understanding. If our logic is to find the common
world intelligible, it must not be hostile, but must be inspired by a
genuine acceptance such as is not usually to be found among
metaphysicians.
III. TIME
The unreality of time is a cardinal doctrine of many metaphysical
systems, often nominally based, as already by Parmenides, upon logical
arguments, but originally derived, at any rate in the founders of new
systems, from the certainty which is born in the moment of mystic
insight. As a Persian Sufi poet says:
"Past and future are what veil God from our sight.
Burn up both of them with fire! How long
Wilt thou be partitioned by these segments as a reed?"[5]
The belief that what is ultimately real must be immutable is a very
common one: it gave rise to the metaphysical notion of substance, and
finds, even now, a wholly illegitimate satisfaction in such scientific
doctrines as the conservation of energy and mass.
It is difficult to disentangle the truth and the error in this view.
The arguments for the contention that time is unreal and that the
world of sense is illusory must, I think, be regarded as fallacious.
Nevertheless there is some sense--easier to feel than to state--in
which time is an unimportant and superficial characteristic of
reality. Past and future must be acknowledged to be as real as the
present, and a certain emancipation from slavery to time is essential
to philosophic thought. The importance of time is rather practical
than theoretical, rather in relation to our desires than in relation
to truth. A truer image of the world, I think, is obtained by
picturing things as entering into the stream of time from an eternal
world outside, than from a view which regards time as the devouring
tyrant of all that is. Both in thought and in feeling, even though
time be real, to realise the unimportance of time is the gate of
wisdom.
That this is the case may be seen at once by asking ourselves why our
feelings towards the past are so different from our feelings towards
the future. The reason for this difference is wholly practical: our
wishes can affect the future but not the past, the future is to some
extent subject to our power, while the past is unalterably fixed. But
every future will some day be past: if we see the past truly now, it
must, when it was still future, have been just what we now see it to
be, and what is now future must be just what we shall see it to be
when it has become past. The felt difference of quality between past
and future, therefore, is not an intrinsic difference, but only a
difference in relation to us: to impartial contemplation, it ceases to
exist. And impartiality of contemplation is, in the intellectual
sphere, that very same virtue of disinterestedness which, in the
sphere of action, appears as justice and unselfishness. Whoever wishes
to see the world truly, to rise in thought above the tyranny of
practical desires, must learn to overcome the difference of attitude
towards past and future, and to survey the whole stream of time in one
comprehensive vision.
The kind of way in which, as it seems to me, time ought not to enter
into our theoretic philosophical thought, may be illustrated by the
philosophy which has become associated with the idea of evolution, and
which is exemplified by Nietzsche, pragmatism, and Bergson. This
philosophy, on the basis of the development which has led from the
lowest forms of life up to man, sees in _progress_ the fundamental law
of the universe, and thus admits the difference between _earlier_ and
_later_ into the very citadel of its contemplative outlook. With its
past and future history of the world, conjectural as it is, I do not
wish to quarrel. But I think that, in the intoxication of a quick
success, much that is required for a true understanding of the
universe has been forgotten. Something of Hellenism, something, too,
of Oriental resignation, must be combined with its hurrying Western
self-assertion before it can emerge from the ardour of youth into the
mature wisdom of manhood. In spite of its appeals to science, the true
scientific philosophy, I think, is something more arduous and more
aloof, appealing to less mundane hopes, and requiring a severer
discipline for its successful practice.
Darwin's _Origin of Species_ persuaded the world that the difference
between different species of animals and plants is not the fixed
immutable difference that it appears to be. The doctrine of natural
kinds, which had rendered classification easy and definite, which was
enshrined in the Aristotelian tradition, and protected by its supposed
necessity for orthodox dogma, was suddenly swept away for ever out of
the biological world. The difference between man and the lower
animals, which to our human conceit appears enormous, was shown to be
a gradual achievement, involving intermediate being who could not with
certainty be placed either within or without the human family. The sun
and the planets had already been shown by Laplace to be very probably
derived from a primitive more or less undifferentiated nebula. Thus
the old fixed landmarks became wavering and indistinct, and all sharp
outlines were blurred. Things and species lost their boundaries, and
none could say where they began or where they ended.
But if human conceit was staggered for a moment by its kinship with
the ape, it soon found a way to reassert itself, and that way is the
"philosophy" of evolution. A process which led from the amoeba to Man
appeared to the philosophers to be obviously a progress--though
whether the amoeba would agree with this opinion is not known. Hence
the cycle of changes which science had shown to be the probable
history of the past was welcomed as revealing a law of development
towards good in the universe--an evolution or unfolding of an idea
slowly embodying itself in the actual. But such a view, though it
might satisfy Spencer and those whom we may call Hegelian
evolutionists, could not be accepted as adequate by the more
whole-hearted votaries of change. An ideal to which the world
continuously approaches is, to these minds, too dead and static to be
inspiring. Not only the aspiration, but the ideal too, must change and
develop with the course of evolution: there must be no fixed goal, but
a continual fashioning of fresh needs by the impulse which is life and
which alone gives unity to the process.
Life, in this philosophy, is a continuous stream, in which all
divisions are artificial and unreal. Separate things, beginnings and
endings, are mere convenient fictions: there is only smooth unbroken
transition. The beliefs of to-day may count as true to-day, if they
carry us along the stream; but to-morrow they will be false, and must
be replaced by new beliefs to meet the new situation. All our thinking
consists of convenient fictions, imaginary congealings of the stream:
reality flows on in spite of all our fictions, and though it can be
lived, it cannot be conceived in thought. Somehow, without explicit
statement, the assurance is slipped in that the future, though we
cannot foresee it, will be better than the past or the present: the
reader is like the child which expects a sweet because it has been
told to open its mouth and shut its eyes. Logic, mathematics, physics
disappear in this philosophy, because they are too "static"; what is
real is no impulse and movement towards a goal which, like the
rainbow, recedes as we advance, and makes every place different when
it reaches it from what it appeared to be at a distance.
I do not propose to enter upon a technical examination of this
philosophy. I wish only to maintain that the motives and interests
which inspire it are so exclusively practical, and the problems with
which it deals are so special, that it can hardly be regarded as
touching any of the questions that, to my mind, constitute genuine
philosophy.
The predominant interest of evolutionism is in the question of human
destiny, or at least of the destiny of Life. It is more interested in
morality and happiness than in knowledge for its own sake. It must be
admitted that the same may be said of many other philosophies, and
that a desire for the kind of knowledge which philosophy can give is
very rare. But if philosophy is to attain truth, it is necessary first
and foremost that philosophers should acquire the disinterested
intellectual curiosity which characterises the genuine man of science.
Knowledge concerning the future--which is the kind of knowledge that
must be sought if we are to know about human destiny--is possible
within certain narrow limits. It is impossible to say how much the
limits may be enlarged with the progress of science. But what is
evident is that any proposition about the future belongs by its
subject-matter to some particular science, and is to be ascertained,
if at all, by the methods of that science. Philosophy is not a short
cut to the same kind of results as those of the other sciences: if it
is to be a genuine study, it must have a province of its own, and aim
at results which the other sciences can neither prove nor disprove.
Evolutionism, in basing itself upon the notion of _progress_, which is
change from the worse to the better, allows the notion of time, as it
seems to me, to become its tyrant rather than its servant, and thereby
loses that impartiality of contemplation which is the source of all
that is best in philosophic thought and feeling. Metaphysicians, as we
saw, have frequently denied altogether the reality of time. I do not
wish to do this; I wish only to preserve the mental outlook which
inspired the denial, the attitude which, in thought, regards the past
as having the same reality as the present and the same importance as
the future. "In so far," says Spinoza,[6] "as the mind conceives a
thing according to the dictate of reason, it will be equally affected
whether the idea is that of a future, past, or present thing." It is
this "conceiving according to the dictate of reason" that I find
lacking in the philosophy which is based on evolution.
IV. GOOD AND EVIL
Mysticism maintains that all evil is illusory, and sometimes maintains
the same view as regards good, but more often holds that all Reality
is good. Both views are to be found in Heraclitus: "Good and ill are
one," he says, but again, "To God all things are fair and good and
right, but men hold some things wrong and some right." A similar
twofold position is to be found in Spinoza, but he uses the word
"perfection" when he means to speak of the good that is not merely
human. "By reality and perfection I mean the same thing," he says;[7]
but elsewhere we find the definition: "By _good_ I shall mean that
which we certainly know to be useful to us."[8] Thus perfection
belongs to Reality in its own nature, but goodness is relative to
ourselves and our needs, and disappears in an impartial survey. Some
such distinction, I think, is necessary in order to understand the
ethical outlook of mysticism: there is a lower mundane kind of good
and evil, which divides the world of appearance into what seem to be
conflicting parts; but there is also a higher, mystical kind of good,
which belongs to Reality and is not opposed by any correlative kind of
evil.
It is difficult to give a logically tenable account of this position
without recognising that good and evil are subjective, that what is
good is merely that towards which we have one kind of feeling, and
what is evil is merely that towards which we have another kind of
feeling. In our active life, where we have to exercise choice, and to
prefer this to that of two possible acts, it is necessary to have a
distinction of good and evil, or at least of better and worse. But
this distinction, like everything pertaining to action, belongs to
what mysticism regards as the world of illusion, if only because it is
essentially concerned with time. In our contemplative life, where
action is not called for, it is possible to be impartial, and to
overcome the ethical dualism which action requires. So long as we
remain _merely_ impartial, we may be content to say that both the good
and the evil of action are illusions. But if, as we must do if we have
the mystic vision, we find the whole world worthy of love and worship,
if we see
"The earth, and every common sight....
Apparell'd in celestial light,"
we shall say that there is a higher good than that of action, and that
this higher good belongs to the whole world as it is in reality. In
this way the twofold attitude and the apparent vacillation of
mysticism are explained and justified.
The possibility of this universal love and joy in all that exists is
of supreme importance for the conduct and happiness of life, and gives
inestimable value to the mystic emotion, apart from any creeds which
may be built upon it. But if we are not to be led into false beliefs,
it is necessary to realise exactly _what_ the mystic emotion reveals.
It reveals a possibility of human nature--a possibility of a nobler,
happier, freer life than any that can be otherwise achieved. But it
does not reveal anything about the non-human, or about the nature of
the universe in general. Good and bad, and even the higher good that
mysticism finds everywhere, are the reflections of our own emotions on
other things, not part of the substance of things as they are in
themselves. And therefore an impartial contemplation, freed from all
pre-occupation with Self, will not judge things good or bad, although
it is very easily combined with that feeling of universal love which
leads the mystic to say that the whole world is good.
The philosophy of evolution, through the notion of progress, is bound
up with the ethical dualism of the worse and the better, and is thus
shut out, not only from the kind of survey which discards good and
evil altogether from its view, but also from the mystical belief in
the goodness of everything. In this way the distinction of good and
evil, like time, becomes a tyrant in this philosophy, and introduces
into thought the restless selectiveness of action. Good and evil, like
time, are, it would seem, not general or fundamental in the world of
thought, but late and highly specialised members of the intellectual
hierarchy.
Although, as we saw, mysticism can be interpreted so as to agree with
the view that good and evil are not intellectually fundamental, it
must be admitted that here we are no longer in verbal agreement with
most of the great philosophers and religious teachers of the past. I
believe, however, that the elimination of ethical considerations from
philosophy is both scientifically necessary and--though this may seem
a paradox--an ethical advance. Both these contentions must be briefly
defended.
The hope of satisfaction to our more human desires--the hope of
demonstrating that the world has this or that desirable ethical
characteristic--is not one which, so far as I can see, a scientific
philosophy can do anything whatever to satisfy. The difference between
a good world and a bad one is a difference in the particular
characteristics of the particular things that exist in these worlds:
it is not a sufficiently abstract difference to come within the
province of philosophy. Love and hate, for example, are ethical
opposites, but to philosophy they are closely analogous attitudes
towards objects. The general form and structure of those attitudes
towards objects which constitute mental phenomena is a problem for
philosophy, but the difference between love and hate is not a
difference of form or structure, and therefore belongs rather to the
special science of psychology than to philosophy. Thus the ethical
interests which have often inspired philosophers must remain in the
background: some kind of ethical interest may inspire the whole study,
but none must obtrude in the detail or be expected in the special
results which are sought.
If this view seems at first sight disappointing, we may remind
ourselves that a similar change has been found necessary in all the
other sciences. The physicist or chemist is not now required to prove
the ethical importance of his ions or atoms; the biologist is not
expected to prove the utility of the plants or animals which he
dissects. In pre-scientific ages this was not the case. Astronomy, for
example, was studied because men believed in astrology: it was thought
that the movements of the planets had the most direct and important
bearing upon the lives of human beings. Presumably, when this belief
decayed and the disinterested study of astronomy began, many who had
found astrology absorbingly interesting decided that astronomy had too
little human interest to be worthy of study. Physics, as it appears in
Plato's Timæus for example, is full of ethical notions: it is an
essential part of its purpose to show that the earth is worthy of
admiration. The modern physicist, on the contrary, though he has no
wish to deny that the earth is admirable, is not concerned, as
physicist, with its ethical attributes: he is merely concerned to find
out facts, not to consider whether they are good or bad. In
psychology, the scientific attitude is even more recent and more
difficult than in the physical sciences: it is natural to consider
that human nature is either good or bad, and to suppose that the
difference between good and bad, so all-important in practice, must be
important in theory also. It is only during the last century that an
ethically neutral psychology has grown up; and here too, ethical
neutrality has been essential to scientific success.
In philosophy, hitherto, ethical neutrality has been seldom sought and
hardly ever achieved. Men have remembered their wishes, and have
judged philosophies in relation to their wishes. Driven from the
particular sciences, the belief that the notions of good and evil must
afford a key to the understanding of the world has sought a refuge in
philosophy. But even from this last refuge, if philosophy is not to
remain a set of pleasing dreams, this belief must be driven forth. It
is a commonplace that happiness is not best achieved by those who
seek it directly; and it would seem that the same is true of the good.
In thought, at any rate, those who forget good and evil and seek only
to know the facts are more likely to achieve good than those who view
the world through the distorting medium of their own desires.
We are thus brought back to our seeming paradox, that a philosophy
which does not seek to impose upon the world its own conceptions of
good and evil is not only more likely to achieve truth, but is also
the outcome of a higher ethical standpoint than one which, like
evolutionism and most traditional systems, is perpetually appraising
the universe and seeking to find in it an embodiment of present
ideals. In religion, and in every deeply serious view of the world and
of human destiny, there is an element of submission, a realisation of
the limits of human power, which is somewhat lacking in the modern
world, with its quick material successes and its insolent belief in
the boundless possibilities of progress. "He that loveth his life
shall lose it"; and there is danger lest, through a too confident love
of life, life itself should lose much of what gives it its highest
worth. The submission which religion inculcates in action is
essentially the same in spirit as that which science teaches in
thought; and the ethical neutrality by which its victories have been
achieved is the outcome of that submission.
The good which it concerns us to remember is the good which it lies in
our power to create--the good in our own lives and in our attitude
towards the world. Insistence on belief in an external realisation of
the good is a form of self-assertion, which, while it cannot secure
the external good which it desires, can seriously impair the inward
good which lies within our power, and destroy that reverence towards
fact which constitutes both what is valuable in humility and what is
fruitful in the scientific temper.
Human beings cannot, of course, wholly transcend human nature;
something subjective, if only the interest that determines the
direction of our attention, must remain in all our thought. But
scientific philosophy comes nearer to objectivity than any other human
pursuit, and gives us, therefore, the closest constant and the most
intimate relation with the outer world that it is possible to achieve.
To the primitive mind, everything is either friendly or hostile; but
experience has shown that friendliness and hostility are not the
conceptions by which the world is to be understood. Scientific
philosophy thus represents, though as yet only in a nascent condition,
a higher form of thought than any pre-scientific belief or
imagination, and, like every approach to self-transcendence, it brings
with it a rich reward in increase of scope and breadth and
comprehension. Evolutionism, in spite of its appeals to particular
scientific facts, fails to be a truly scientific philosophy because of
its slavery to time, its ethical preoccupations, and its predominant
interest in our mundane concerns and destiny. A truly scientific
philosophy will be more humble, more piecemeal, more arduous, offering
less glitter of outward mirage to flatter fallacious hopes, but more
indifferent to fate, and more capable of accepting the world without
the tyrannous imposition of our human and temporary demands.
FOOTNOTES:
[1] All the above quotations are from Burnet's _Early Greek
Philosophy_, (2nd ed., 1908), pp. 146-156.
[2] _Republic_, 514, translated by Davies and Vaughan.
[3] This section, and also one or two pages in later sections, have
been printed in a course of Lowell lectures _On our knowledge of the
external world_, published by the Open Court Publishing Company. But I
have left them here, as this is the context for which they were
originally written.
[4] _Introduction to Metaphysics_, p. 1.
[5] Whinfield's translation of the _Masnavi_ (Trübner, 1887), p. 34.
[6] _Ethics_, Bk. IV, Prop. LXII.
[7] Ib., Pt. IV, Df. I.
[8] _Ethics_. Pt. II. Df. VI.
II
THE PLACE OF SCIENCE IN A LIBERAL EDUCATION
I
Science, to the ordinary reader of newspapers, is represented by a
varying selection of sensational triumphs, such as wireless telegraphy
and aeroplanes, radio-activity and the marvels of modern alchemy. It is
not of this aspect of science that I wish to speak. Science, in this
aspect, consists of detached up-to-date fragments, interesting only
until they are replaced by something newer and more up-to-date,
displaying nothing of the systems of patiently constructed knowledge
out of which, almost as a casual incident, have come the practically
useful results which interest the man in the street. The increased
command over the forces of nature which is derived from science is
undoubtedly an amply sufficient reason for encouraging scientific
research, but this reason has been so often urged and is so easily
appreciated that other reasons, to my mind quite as important, are apt
to be overlooked. It is with these other reasons, especially with the
intrinsic value of a scientific habit of mind in forming our outlook
on the world, that I shall be concerned in what follows.
The instance of wireless telegraphy will serve to illustrate the
difference between the two points of view. Almost all the serious
intellectual labour required for the possibility of this invention is
due to three men--Faraday, Maxwell, and Hertz. In alternating layers
of experiment and theory these three men built up the modern theory of
electromagnetism, and demonstrated the identity of light with
electromagnetic waves. The system which they discovered is one of
profound intellectual interest, bringing together and unifying an
endless variety of apparently detached phenomena, and displaying a
cumulative mental power which cannot but afford delight to every
generous spirit. The mechanical details which remained to be adjusted
in order to utilise their discoveries for a practical system of
telegraphy demanded, no doubt, very considerable ingenuity, but had
not that broad sweep and that universality which could give them
intrinsic interest as an object of disinterested contemplation.
From the point of view of training the mind, of giving that
well-informed, impersonal outlook which constitutes culture in the
good sense of this much-misused word, it seems to be generally held
indisputable that a literary education is superior to one based on
science. Even the warmest advocates of science are apt to rest their
claims on the contention that culture ought to be sacrificed to
utility. Those men of science who respect culture, when they associate
with men learned in the classics, are apt to admit, not merely
politely, but sincerely, a certain inferiority on their side,
compensated doubtless by the services which science renders to
humanity, but none the less real. And so long as this attitude exists
among men of science, it tends to verify itself: the intrinsically
valuable aspects of science tend to be sacrificed to the merely
useful, and little attempt is made to preserve that leisurely,
systematic survey by which the finer quality of mind is formed and
nourished.
But even if there be, in present fact, any such inferiority as is
supposed in the educational value of science, this is, I believe, not
the fault of science itself, but the fault of the spirit in which
science is taught. If its full possibilities were realised by those
who teach it, I believe that its capacity of producing those habits of
mind which constitute the highest mental excellence would be at least
as great as that of literature, and more particularly of Greek and
Latin literature. In saying this I have no wish whatever to disparage
a classical education. I have not myself enjoyed its benefits, and my
knowledge of Greek and Latin authors is derived almost wholly from
translations. But I am firmly persuaded that the Greeks fully deserve
all the admiration that is bestowed upon them, and that it is a very
great and serious loss to be unacquainted with their writings. It is
not by attacking them, but by drawing attention to neglected
excellences in science, that I wish to conduct my argument.
One defect, however, does seem inherent in a purely classical
education--namely, a too exclusive emphasis on the past. By the study
of what is absolutely ended and can never be renewed, a habit of
criticism towards the present and the future is engendered. The
qualities in which the present excels are qualities to which the study
of the past does not direct attention, and to which, therefore, the
student of Greek civilisation may easily become blind. In what is new
and growing there is apt to be something crude, insolent, even a
little vulgar, which is shocking to the man of sensitive taste;
quivering from the rough contact, he retires to the trim gardens of a
polished past, forgetting that they were reclaimed from the wilderness
by men as rough and earth-soiled as those from whom he shrinks in his
own day. The habit of being unable to recognise merit until it is
dead is too apt to be the result of a purely bookish life, and a
culture based wholly on the past will seldom be able to pierce through
everyday surroundings to the essential splendour of contemporary
things, or to the hope of still greater splendour in the future.
"My eyes saw not the men of old;
And now their age away has rolled.
I weep--to think I shall not see
The heroes of posterity."
So says the Chinese poet; but such impartiality is rare in the more
pugnacious atmosphere of the West, where the champions of past and
future fight a never-ending battle, instead of combining to seek out
the merits of both.
This consideration, which militates not only against the exclusive
study of the classics, but against every form of culture which has
become static, traditional, and academic, leads inevitably to the
fundamental question: What is the true end of education? But before
attempting to answer this question it will be well to define the sense
in which we are to use the word "education." For this purpose I shall
distinguish the sense in which I mean to use it from two others, both
perfectly legitimate, the one broader and the other narrower than the
sense in which I mean to use the word.
In the broader sense, education will include not only what we learn
through instruction, but all that we learn through personal
experience--the formation of character through the education of life.
Of this aspect of education, vitally important as it is, I will say
nothing, since its consideration would introduce topics quite foreign
to the question with which we are concerned.
In the narrower sense, education may be confined to instruction, the
imparting of definite information on various subjects, because such
information, in and for itself, is useful in daily life. Elementary
education--reading, writing, and arithmetic--is almost wholly of this
kind. But instruction, necessary as it is, does not _per se_
constitute education in the sense in which I wish to consider it.
Education, in the sense in which I mean it, may be defined as _the
formation, by means of instruction, of certain mental habits and a
certain outlook on life and the world_. It remains to ask ourselves,
what mental habits, and what sort of outlook, can be hoped for as the
result of instruction? When we have answered this question we can
attempt to decide what science has to contribute to the formation of
the habits and outlook which we desire.
Our whole life is built about a certain number--not a very small
number--of primary instincts and impulses. Only what is in some way
connected with these instincts and impulses appears to us desirable or
important; there is no faculty, whether "reason" or "virtue" or
whatever it may be called, that can take our active life and our hopes
and fears outside the region controlled by these first movers of all
desire. Each of them is like a queen-bee, aided by a hive of workers
gathering honey; but when the queen is gone the workers languish and
die, and the cells remain empty of their expected sweetness. So with
each primary impulse in civilised man: it is surrounded and protected
by a busy swarm of attendant derivative desires, which store up in its
service whatever honey the surrounding world affords. But if the
queen-impulse dies, the death-dealing influence, though retarded a
little by habit, spreads slowly through all the subsidiary impulses,
and a whole tract of life becomes inexplicably colourless. What was
formerly full of zest, and so obviously worth doing that it raised no
questions, has now grown dreary and purposeless: with a sense of
disillusion we inquire the meaning of life, and decide, perhaps, that
all is vanity. The search for an outside meaning that can _compel_ an
inner response must always be disappointed: all "meaning" must be at
bottom related to our primary desires, and when they are extinct no
miracle can restore to the world the value which they reflected upon
it.
The purpose of education, therefore, cannot be to create any primary
impulse which is lacking in the uneducated; the purpose can only be to
enlarge the scope of those that human nature provides, by increasing
the number and variety of attendant thoughts, and by showing where the
most permanent satisfaction is to be found. Under the impulse of a
Calvinistic horror of the "natural man," this obvious truth has been
too often misconceived in the training of the young; "nature" has been
falsely regarded as excluding all that is best in what is natural, and
the endeavour to teach virtue has led to the production of stunted and
contorted hypocrites instead of full-grown human beings. From such
mistakes in education a better psychology or a kinder heart is
beginning to preserve the present generation; we need, therefore,
waste no more words on the theory that the purpose of education is to
thwart or eradicate nature.
But although nature must supply the initial force of desire, nature is
not, in the civilised man, the spasmodic, fragmentary, and yet violent
set of impulses that it is in the savage. Each impulse has its
constitutional ministry of thought and knowledge and reflection,
through which possible conflicts of impulses are foreseen, and
temporary impulses are controlled by the unifying impulse which may be
called wisdom. In this way education destroys the crudity of
instinct, and increases through knowledge the wealth and variety of
the individual's contacts with the outside world, making him no longer
an isolated fighting unit, but a citizen of the universe, embracing
distant countries, remote regions of space, and vast stretches of past
and future within the circle of his interests. It is this simultaneous
softening in the insistence of desire and enlargement of its scope
that is the chief moral end of education.
Closely connected with this moral end is the more purely intellectual
aim of education, the endeavour to make us see and imagine the world
in an objective manner, as far as possible as it is in itself, and not
merely through the distorting medium of personal desire. The complete
attainment of such an objective view is no doubt an ideal,
indefinitely approachable, but not actually and fully realisable.
Education, considered as a process of forming our mental habits and
our outlook on the world, is to be judged successful in proportion as
its outcome approximates to this ideal; in proportion, that is to say,
as it gives us a true view of our place in society, of the relation of
the whole human society to its non-human environment, and of the
nature of the non-human world as it is in itself apart from our
desires and interests. If this standard is admitted, we can return to
the consideration of science, inquiring how far science contributes to
such an aim, and whether it is in any respect superior to its rivals
in educational practice.
II
Two opposite and at first sight conflicting merits belong to science
as against literature and art. The one, which is not inherently
necessary, but is certainly true at the present day, is hopefulness
as to the future of human achievement, and in particular as to the
useful work that may be accomplished by any intelligent student. This
merit and the cheerful outlook which it engenders prevent what might
otherwise be the depressing effect of another aspect of science, to my
mind also a merit, and perhaps its greatest merit--I mean the
irrelevance of human passions and of the whole subjective apparatus
where scientific truth is concerned. Each of these reasons for
preferring the study of science requires some amplification. Let us
begin with the first.
In the study of literature or art our attention is perpetually riveted
upon the past: the men of Greece or of the Renaissance did better than
any men do now; the triumphs of former ages, so far from facilitating
fresh triumphs in our own age, actually increase the difficulty of
fresh triumphs by rendering originality harder of attainment; not only
is artistic achievement not cumulative, but it seems even to depend
upon a certain freshness and _naïveté_ of impulse and vision which
civilisation tends to destroy. Hence comes, to those who have been
nourished on the literary and artistic productions of former ages, a
certain peevishness and undue fastidiousness towards the present, from
which there seems no escape except into the deliberate vandalism which
ignores tradition and in the search after originality achieves only
the eccentric. But in such vandalism there is none of the simplicity
and spontaneity out of which great art springs: theory is still the
canker in its core, and insincerity destroys the advantages of a
merely pretended ignorance.
The despair thus arising from an education which suggests no
pre-eminent mental activity except that of artistic creation is wholly
absent from an education which gives the knowledge of scientific
method. The discovery of scientific method, except in pure
mathematics, is a thing of yesterday; speaking broadly, we may say
that it dates from Galileo. Yet already it has transformed the world,
and its success proceeds with ever-accelerating velocity. In science
men have discovered an activity of the very highest value in which
they are no longer, as in art, dependent for progress upon the
appearance of continually greater genius, for in science the
successors stand upon the shoulders of their predecessors; where one
man of supreme genius has invented a method, a thousand lesser men can
apply it. No transcendent ability is required in order to make useful
discoveries in science; the edifice of science needs its masons,
bricklayers, and common labourers as well as its foremen,
master-builders, and architects. In art nothing worth doing can be
done without genius; in science even a very moderate capacity can
contribute to a supreme achievement.
In science the man of real genius is the man who invents a new method.
The notable discoveries are often made by his successors, who can
apply the method with fresh vigour, unimpaired by the previous labour
of perfecting it; but the mental calibre of the thought required for
their work, however brilliant, is not so great as that required by the
first inventor of the method. There are in science immense numbers of
different methods, appropriate to different classes of problems; but
over and above them all, there is something not easily definable,
which may be called _the_ method of science. It was formerly customary
to identify this with the inductive method, and to associate it with
the name of Bacon. But the true inductive method was not discovered by
Bacon, and the true method of science is something which includes
deduction as much as induction, logic and mathematics as much as
botany and geology. I shall not attempt the difficult task of stating
what the scientific method is, but I will try to indicate the temper
of mind out of which the scientific method grows, which is the second
of the two merits that were mentioned above as belonging to a
scientific education.
The kernel of the scientific outlook is a thing so simple, so obvious,
so seemingly trivial, that the mention of it may almost excite
derision. The kernel of the scientific outlook is the refusal to
regard our own desires, tastes, and interests as affording a key to
the understanding of the world. Stated thus baldly, this may seem no
more than a trite truism. But to remember it consistently in matters
arousing our passionate partisanship is by no means easy, especially
where the available evidence is uncertain and inconclusive. A few
illustrations will make this clear.
Aristotle, I understand, considered that the stars must move in
circles because the circle is the most perfect curve. In the absence
of evidence to the contrary, he allowed himself to decide a question
of fact by an appeal to æsthetico-moral considerations. In such a case
it is at once obvious to us that this appeal was unjustifiable. We
know now how to ascertain as a fact the way in which the heavenly
bodies move, and we know that they do not move in circles, or even in
accurate ellipses, or in any other kind of simply describable curve.
This may be painful to a certain hankering after simplicity of pattern
in the universe, but we know that in astronomy such feelings are
irrelevant. Easy as this knowledge seems now, we owe it to the courage
and insight of the first inventors of scientific method, and more
especially of Galileo.
We may take as another illustration Malthus's doctrine of population.
This illustration is all the better for the fact that his actual
doctrine is now known to be largely erroneous. It is not his
conclusions that are valuable, but the temper and method of his
inquiry. As everyone knows, it was to him that Darwin owed an
essential part of his theory of natural selection, and this was only
possible because Malthus's outlook was truly scientific. His great
merit lies in considering man not as the object of praise or blame,
but as a part of nature, a thing with a certain characteristic
behaviour from which certain consequences must follow. If the
behaviour is not quite what Malthus supposed, if the consequences are
not quite what he inferred, that may falsify his conclusions, but does
not impair the value of his method. The objections which were made
when his doctrine was new--that it was horrible and depressing, that
people ought not to act as he said they did, and so on--were all such
as implied an unscientific attitude of mind; as against all of them,
his calm determination to treat man as a natural phenomenon marks an
important advance over the reformers of the eighteenth century and the
Revolution.
Under the influence of Darwinism the scientific attitude towards man
has now become fairly common, and is to some people quite natural,
though to most it is still a difficult and artificial intellectual
contortion. There is however, one study which is as yet almost wholly
untouched by the scientific spirit--I mean the study of philosophy.
Philosophers and the public imagine that the scientific spirit must
pervade pages that bristle with allusions to ions, germ-plasms, and
the eyes of shell-fish. But as the devil can quote Scripture, so the
philosopher can quote science. The scientific spirit is not an affair
of quotation, of externally acquired information, any more than
manners are an affair of the etiquette-book. The scientific attitude
of mind involves a sweeping away of all other desires in the interests
of the desire to know--it involves suppression of hopes and fears,
loves and hates, and the whole subjective emotional life, until we
become subdued to the material, able to see it frankly, without
preconceptions, without bias, without any wish except to see it as it
is, and without any belief that what it is must be determined by some
relation, positive or negative, to what we should like it to be, or to
what we can easily imagine it to be.
Now in philosophy this attitude of mind has not as yet been achieved.
A certain self-absorption, not personal, but human, has marked almost
all attempts to conceive the universe as a whole. Mind, or some aspect
of it--thought or will or sentience--has been regarded as the pattern
after which the universe is to be conceived, for no better reason, at
bottom, than that such a universe would not seem strange, and would
give us the cosy feeling that every place is like home. To conceive
the universe as essentially progressive or essentially deteriorating,
for example, is to give to our hopes and fears a cosmic importance
which _may_, of course, be justified, but which we have as yet no
reason to suppose justified. Until we have learnt to think of it in
ethically neutral terms, we have not arrived at a scientific attitude
in philosophy; and until we have arrived at such an attitude, it is
hardly to be hoped that philosophy will achieve any solid results.
I have spoken so far largely of the negative aspect of the scientific
spirit, but it is from the positive aspect that its value is derived.
The instinct of constructiveness, which is one of the chief incentives
to artistic creation, can find in scientific systems a satisfaction
more massive than any epic poem. Disinterested curiosity, which is the
source of almost all intellectual effort, finds with astonished
delight that science can unveil secrets which might well have seemed
for ever undiscoverable. The desire for a larger life and wider
interests, for an escape from private circumstances, and even from the
whole recurring human cycle of birth and death, is fulfilled by the
impersonal cosmic outlook of science as by nothing else. To all these
must be added, as contributing to the happiness of the man of science,
the admiration of splendid achievement, and the consciousness of
inestimable utility to the human race. A life devoted to science is
therefore a happy life, and its happiness is derived from the very
best sources that are open to dwellers on this troubled and passionate
planet.
III
A FREE MAN'S WORSHIP[9]
To Dr. Faustus in his study Mephistopheles told the history of the
Creation, saying:
"The endless praises of the choirs of angels had begun to grow
wearisome; for, after all, did he not deserve their praise? Had he not
given them endless joy? Would it not be more amusing to obtain
undeserved praise, to be worshipped by beings whom he tortured? He
smiled inwardly, and resolved that the great drama should be
performed.
"For countless ages the hot nebula whirled aimlessly through space. At
length it began to take shape, the central mass threw off planets, the
planets cooled, boiling seas and burning mountains heaved and tossed,
from black masses of cloud hot sheets of rain deluged the barely solid
crust. And now the first germ of life grew in the depths of the ocean,
and developed rapidly in the fructifying warmth into vast forest
trees, huge ferns springing from the damp mould, sea monsters
breeding, fighting, devouring, and passing away. And from the
monsters, as the play unfolded itself, Man was born, with the power of
thought, the knowledge of good and evil, and the cruel thirst for
worship. And Man saw that all is passing in this mad, monstrous world,
that all is struggling to snatch, at any cost, a few brief moments of
life before Death's inexorable decree. And Man said: 'There is a
hidden purpose, could we but fathom it, and the purpose is good; for
we must reverence something, and in the visible world there is nothing
worthy of reverence.' And Man stood aside from the struggle, resolving
that God intended harmony to come out of chaos by human efforts. And
when he followed the instincts which God had transmitted to him from
his ancestry of beasts of prey, he called it Sin, and asked God to
forgive him. But he doubted whether he could be justly forgiven, until
he invented a divine Plan by which God's wrath was to have been
appeased. And seeing the present was bad, he made it yet worse, that
thereby the future might be better. And he gave God thanks for the
strength that enabled him to forgo even the joys that were possible.
And God smiled; and when he saw that Man had become perfect in
renunciation and worship, he sent another sun through the sky, which
crashed into Man's sun; and all returned again to nebula.
"'Yes,' he murmured, 'it was a good play; I will have it performed
again.'"
Such, in outline, but even more purposeless, more void of meaning, is
the world which Science presents for our belief. Amid such a world, if
anywhere, our ideals henceforward must find a home. That Man is the
product of causes which had no prevision of the end they were
achieving; that his origin, his growth, his hopes and fears, his loves
and his beliefs, are but the outcome of accidental collocations of
atoms; that no fire, no heroism, no intensity of thought and feeling,
can preserve an individual life beyond the grave; that all the labours
of the ages, all the devotion, all the inspiration, all the noonday
brightness of human genius, are destined to extinction in the vast
death of the solar system, and that the whole temple of Man's
achievement must inevitably be buried beneath the débris of a universe
in ruins--all these things, if not quite beyond dispute, are yet so
nearly certain, that no philosophy which rejects them can hope to
stand. Only within the scaffolding of these truths, only on the firm
foundation of unyielding despair, can the soul's habitation henceforth
be safely built.
How, in such an alien and inhuman world, can so powerless a creature
as Man preserve his aspirations untarnished? A strange mystery it is
that Nature, omnipotent but blind, in the revolutions of her secular
hurryings through the abysses of space, has brought forth at last a
child, subject still to her power, but gifted with sight, with
knowledge of good and evil, with the capacity of judging all the works
of his unthinking Mother. In spite of Death, the mark and seal of the
parental control, Man is yet free, during his brief years, to examine,
to criticise, to know, and in imagination to create. To him alone, in
the world with which he is acquainted, this freedom belongs; and in
this lies his superiority to the resistless forces that control his
outward life.
The savage, like ourselves, feels the oppression of his impotence
before the powers of Nature; but having in himself nothing that he
respects more than Power, he is willing to prostrate himself before
his gods, without inquiring whether they are worthy of his worship.
Pathetic and very terrible is the long history of cruelty and torture,
of degradation and human sacrifice, endured in the hope of placating
the jealous gods: surely, the trembling believer thinks, when what is
most precious has been freely given, their lust for blood must be
appeased, and more will not be required. The religion of Moloch--as
such creeds may be generically called--is in essence the cringing
submission of the slave, who dare not, even in his heart, allow the
thought that his master deserves no adulation. Since the independence
of ideals is not yet acknowledged, Power may be freely worshipped, and
receive an unlimited respect, despite its wanton infliction of pain.
But gradually, as morality grows bolder, the claim of the ideal world
begins to be felt; and worship, if it is not to cease, must be given
to gods of another kind than those created by the savage. Some, though
they feel the demands of the ideal, will still consciously reject
them, still urging that naked Power is worthy of worship. Such is the
attitude inculcated in God's answer to Job out of the whirlwind: the
divine power and knowledge are paraded, but of the divine goodness
there is no hint. Such also is the attitude of those who, in our own
day, base their morality upon the struggle for survival, maintaining
that the survivors are necessarily the fittest. But others, not
content with an answer so repugnant to the moral sense, will adopt the
position which we have become accustomed to regard as specially
religious, maintaining that, in some hidden manner, the world of fact
is really harmonious with the world of ideals. Thus Man creates God,
all-powerful and all-good, the mystic unity of what is and what should
be.
But the world of fact, after all, is not good; and, in submitting our
judgment to it, there is an element of slavishness from which our
thoughts must be purged. For in all things it is well to exalt the
dignity of Man, by freeing him as far as possible from the tyranny of
non-human Power. When we have realised that Power is largely bad, that
man, with his knowledge of good and evil, is but a helpless atom in a
world which has no such knowledge, the choice is again presented to
us: Shall we worship Force, or shall we worship Goodness? Shall our
God exist and be evil, or shall he be recognised as the creation of
our own conscience?
The answer to this question is very momentous, and affects profoundly
our whole morality. The worship of Force, to which Carlyle and
Nietzsche and the creed of Militarism have accustomed us, is the
result of failure to maintain our own ideals against a hostile
universe: it is itself a prostrate submission to evil, a sacrifice of
our best to Moloch. If strength indeed is to be respected, let us
respect rather the strength of those who refuse that false
"recognition of facts" which fails to recognise that facts are often
bad. Let us admit that, in the world we know, there are many things
that would be better otherwise, and that the ideals to which we do and
must adhere are not realised in the realm of matter. Let us preserve
our respect for truth, for beauty, for the ideal of perfection which
life does not permit us to attain, though none of these things meet
with the approval of the unconscious universe. If Power is bad, as it
seems to be, let us reject it from our hearts. In this lies Man's true
freedom: in determination to worship only the God created by our own
love of the good, to respect only the heaven which inspires the
insight of our best moments. In action, in desire, we must submit
perpetually to the tyranny of outside forces; but in thought, in
aspiration, we are free, free from our fellow-men, free from the petty
planet on which our bodies impotently crawl, free even, while we live,
from the tyranny of death. Let us learn, then, that energy of faith
which enables us to live constantly in the vision of the good; and let
us descend, in action, into the world of fact, with that vision always
before us.
When first the opposition of fact and ideal grows fully visible, a
spirit of fiery revolt, of fierce hatred of the gods, seems necessary
to the assertion of freedom. To defy with Promethean constancy a
hostile universe, to keep its evil always in view, always actively
hated, to refuse no pain that the malice of Power can invent, appears
to be the duty of all who will not bow before the inevitable. But
indignation is still a bondage, for it compels our thoughts to be
occupied with an evil world; and in the fierceness of desire from
which rebellion springs there is a kind of self-assertion which it is
necessary for the wise to overcome. Indignation is a submission of our
thoughts, but not of our desires; the Stoic freedom in which wisdom
consists is found in the submission of our desires, but not of our
thoughts. From the submission of our desires springs the virtue of
resignation; from the freedom of our thoughts springs the whole world
of art and philosophy, and the vision of beauty by which, at last, we
half reconquer the reluctant world. But the vision of beauty is
possible only to unfettered contemplation, to thoughts not weighted by
the load of eager wishes; and thus Freedom comes only to those who no
longer ask of life that it shall yield them any of those personal
goods that are subject to the mutations of Time.
Although the necessity of renunciation is evidence of the existence of
evil, yet Christianity, in preaching it, has shown a wisdom exceeding
that of the Promethean philosophy of rebellion. It must be admitted
that, of the things we desire, some, though they prove impossible, are
yet real goods; others, however, as ardently longed for, do not form
part of a fully purified ideal. The belief that what must be renounced
is bad, though sometimes false, is far less often false than untamed
passion supposes; and the creed of religion, by providing a reason
for proving that it is never false, has been the means of purifying
our hopes by the discovery of many austere truths.
But there is in resignation a further good element: even real goods,
when they are unattainable, ought not to be fretfully desired. To
every man comes, sooner or later, the great renunciation. For the
young, there is nothing unattainable; a good thing desired with the
whole force of a passionate will, and yet impossible, is to them not
credible. Yet, by death, by illness, by poverty, or by the voice of
duty, we must learn, each one of us, that the world was not made for
us, and that, however beautiful may be the things we crave, Fate may
nevertheless forbid them. It is the part of courage, when misfortune
comes, to bear without repining the ruin of our hopes, to turn away
our thoughts from vain regrets. This degree of submission to Power is
not only just and right: it is the very gate of wisdom.
But passive renunciation is not the whole of wisdom; for not by
renunciation alone can we build a temple for the worship of our own
ideals. Haunting foreshadowings of the temple appear in the realm of
imagination, in music, in architecture, in the untroubled kingdom of
reason, and in the golden sunset magic of lyrics, where beauty shines
and glows, remote from the touch of sorrow, remote from the fear of
change, remote from the failures and disenchantments of the world of
fact. In the contemplation of these things the vision of heaven will
shape itself in our hearts, giving at once a touchstone to judge the
world about us, and an inspiration by which to fashion to our needs
whatever is not incapable of serving as a stone in the sacred temple.
Except for those rare spirits that are born without sin, there is a
cavern of darkness to be traversed before that temple can be entered.
The gate of the cavern is despair, and its floor is paved with the
gravestones of abandoned hopes. There Self must die; there the
eagerness, the greed of untamed desire must be slain, for only so can
the soul be freed from the empire of Fate. But out of the cavern the
Gate of Renunciation leads again to the daylight of wisdom, by whose
radiance a new insight, a new joy, a new tenderness, shine forth to
gladden the pilgrim's heart.
When, without the bitterness of impotent rebellion, we have learnt
both to resign ourselves to the outward rule of Fate and to recognise
that the non-human world is unworthy of our worship, it becomes
possible at last so to transform and refashion the unconscious
universe, so to transmute it in the crucible of imagination, that a
new image of shining gold replaces the old idol of clay. In all the
multiform facts of the world--in the visual shapes of trees and
mountains and clouds, in the events of the life of man, even in the
very omnipotence of Death--the insight of creative idealism can find
the reflection of a beauty which its own thoughts first made. In this
way mind asserts its subtle mastery over the thoughtless forces of
Nature. The more evil the material with which it deals, the more
thwarting to untrained desire, the greater is its achievement in
inducing the reluctant rock to yield up its hidden treasures, the
prouder its victory in compelling the opposing forces to swell the
pageant of its triumph. Of all the arts, Tragedy is the proudest, the
most triumphant; for it builds its shining citadel in the very centre
of the enemy's country, on the very summit of his highest mountain;
from its impregnable watchtowers, his camps and arsenals, his columns
and forts, are all revealed; within its walls the free life continues,
while the legions of Death and Pain and Despair, and all the servile
captains of tyrant Fate, afford the burghers of that dauntless city
new spectacles of beauty. Happy those sacred ramparts, thrice happy
the dwellers on that all-seeing eminence. Honour to those brave
warriors who, through countless ages of warfare, have preserved for us
the priceless heritage of liberty, and have kept undefiled by
sacrilegious invaders the home of the unsubdued.
But the beauty of Tragedy does but make visible a quality which, in
more or less obvious shapes, is present always and everywhere in life.
In the spectacle of Death, in the endurance of intolerable pain, and
in the irrevocableness of a vanished past, there is a sacredness, an
overpowering awe, a feeling of the vastness, the depth, the
inexhaustible mystery of existence, in which, as by some strange
marriage of pain, the sufferer is bound to the world by bonds of
sorrow. In these moments of insight, we lose all eagerness of
temporary desire, all struggling and striving for petty ends, all care
for the little trivial things that, to a superficial view, make up the
common life of day by day; we see, surrounding the narrow raft
illumined by the flickering light of human comradeship, the dark ocean
on whose rolling waves we toss for a brief hour; from the great night
without, a chill blast breaks in upon our refuge; all the loneliness
of humanity amid hostile forces is concentrated upon the individual
soul, which must struggle alone, with what of courage it can command,
against the whole weight of a universe that cares nothing for its
hopes and fears. Victory, in this struggle with the powers of
darkness, is the true baptism into the glorious company of heroes, the
true initiation into the overmastering beauty of human existence. From
that awful encounter of the soul with the outer world, renunciation,
wisdom, and charity are born; and with their birth a new life begins.
To take into the inmost shrine of the soul the irresistible forces
whose puppets we seem to be--Death and change, the irrevocableness of
the past, and the powerlessness of man before the blind hurry of the
universe from vanity to vanity--to feel these things and know them is
to conquer them.
This is the reason why the Past has such magical power. The beauty of
its motionless and silent pictures is like the enchanted purity of
late autumn, when the leaves, though one breath would make them fall,
still glow against the sky in golden glory. The Past does not change
or strive; like Duncan, after life's fitful fever it sleeps well; what
was eager and grasping, what was petty and transitory, has faded away,
the things that were beautiful and eternal shine out of it like stars
in the night. Its beauty, to a soul not worthy of it, is unendurable;
but to a soul which has conquered Fate it is the key of religion.
The life of Man, viewed outwardly, is but a small thing in comparison
with the forces of Nature. The slave is doomed to worship Time and
Fate and Death, because they are greater than anything he finds in
himself, and because all his thoughts are of things which they devour.
But, great as they are, to think of them greatly, to feel their
passionless splendour, is greater still. And such thought makes us
free men; we no longer bow before the inevitable in Oriental
subjection, but we absorb it, and make it a part of ourselves. To
abandon the struggle for private happiness, to expel all eagerness of
temporary desire, to burn with passion for eternal things--this is
emancipation, and this is the free man's worship. And this liberation
is effected by a contemplation of Fate; for Fate itself is subdued by
the mind which leaves nothing to be purged by the purifying fire of
Time.
United with his fellow-men by the strongest of all ties, the tie of a
common doom, the free man finds that a new vision is with him always,
shedding over every daily task the light of love. The life of Man is a
long march through the night, surrounded by invisible foes, tortured
by weariness and pain, towards a goal that few can hope to reach, and
where none may tarry long. One by one, as they march, our comrades
vanish from our sight, seized by the silent orders of omnipotent
Death. Very brief is the time in which we can help them, in which
their happiness or misery is decided. Be it ours to shed sunshine on
their path, to lighten their sorrows by the balm of sympathy, to give
them the pure joy of a never-tiring affection, to strengthen failing
courage, to instil faith in hours of despair. Let us not weigh in
grudging scales their merits and demerits, but let us think only of
their need--of the sorrows, the difficulties, perhaps the blindnesses,
that make the misery of their lives; let us remember that they are
fellow-sufferers in the same darkness, actors in the same tragedy with
ourselves. And so, when their day is over, when their good and their
evil have become eternal by the immortality of the past, be it ours to
feel that, where they suffered, where they failed, no deed of ours was
the cause; but wherever a spark of the divine fire kindled in their
hearts, we were ready with encouragement, with sympathy, with brave
words in which high courage glowed.
Brief and powerless is Man's life; on him and all his race the slow,
sure doom falls pitiless and dark. Blind to good and evil, reckless of
destruction, omnipotent matter rolls on its relentless way; for Man,
condemned to-day to lose his dearest, to-morrow himself to pass
through the gate of darkness, it remains only to cherish, ere yet the
blow falls, the lofty thoughts that ennoble his little day; disdaining
the coward terrors of the slave of Fate, to worship at the shrine that
his own hands have built; undismayed by the empire of chance, to
preserve a mind free from the wanton tyranny that rules his outward
life; proudly defiant of the irresistible forces that tolerate, for a
moment, his knowledge and his condemnation, to sustain alone, a weary
but unyielding Atlas, the world that his own ideals have fashioned
despite the trampling march of unconscious power.
FOOTNOTES:
[9] Reprinted from the _Independent Review_, December, 1903.
IV
THE STUDY OF MATHEMATICS
In regard to every form of human activity it is necessary that the
question should be asked from time to time, What is its purpose and
ideal? In what way does it contribute to the beauty of human
existence? As respects those pursuits which contribute only remotely,
by providing the mechanism of life, it is well to be reminded that not
the mere fact of living is to be desired, but the art of living in the
contemplation of great things. Still more in regard to those
avocations which have no end outside themselves, which are to be
justified, if at all, as actually adding to the sum of the world's
permanent possessions, it is necessary to keep alive a knowledge of
their aims, a clear prefiguring vision of the temple in which creative
imagination is to be embodied.
The fulfilment of this need, in what concerns the studies forming the
material upon which custom has decided to train the youthful mind, is
indeed sadly remote--so remote as to make the mere statement of such a
claim appear preposterous. Great men, fully alive to the beauty of the
contemplations to whose service their lives are devoted, desiring that
others may share in their joys, persuade mankind to impart to the
successive generations the mechanical knowledge without which it is
impossible to cross the threshold. Dry pedants possess themselves of
the privilege of instilling this knowledge: they forget that it is to
serve but as a key to open the doors of the temple; though they spend
their lives on the steps leading up to those sacred doors, they turn
their backs upon the temple so resolutely that its very existence is
forgotten, and the eager youth, who would press forward to be
initiated to its domes and arches, is bidden to turn back and count
the steps.
Mathematics, perhaps more even than the study of Greece and Rome, has
suffered from this oblivion of its due place in civilisation. Although
tradition has decreed that the great bulk of educated men shall know
at least the elements of the subject, the reasons for which the
tradition arose are forgotten, buried beneath a great rubbish-heap of
pedantries and trivialities. To those who inquire as to the purpose of
mathematics, the usual answer will be that it facilitates the making
of machines, the travelling from place to place, and the victory over
foreign nations, whether in war or commerce. If it be objected that
these ends--all of which are of doubtful value--are not furthered by
the merely elementary study imposed upon those who do not become
expert mathematicians, the reply, it is true, will probably be that
mathematics trains the reasoning faculties. Yet the very men who make
this reply are, for the most part, unwilling to abandon the teaching
of definite fallacies, known to be such, and instinctively rejected by
the unsophisticated mind of every intelligent learner. And the
reasoning faculty itself is generally conceived, by those who urge its
cultivation, as merely a means for the avoidance of pitfalls and a
help in the discovery of rules for the guidance of practical life. All
these are undeniably important achievements to the credit of
mathematics; yet it is none of these that entitles mathematics to a
place in every liberal education. Plato, we know, regarded the
contemplation of mathematical truths as worthy of the Deity; and
Plato realised, more perhaps than any other single man, what those
elements are in human life which merit a place in heaven. There is in
mathematics, he says, "something which is _necessary_ and cannot be
set aside ... and, if I mistake not, of divine necessity; for as to
the human necessities of which the Many talk in this connection,
nothing can be more ridiculous than such an application of the words.
_Cleinias._ And what are these necessities of knowledge, Stranger,
which are divine and not human? _Athenian._ Those things without some
use or knowledge of which a man cannot become a God to the world, nor
a spirit, nor yet a hero, nor able earnestly to think and care for
man" (_Laws_, p. 818).[10] Such was Plato's judgment of mathematics;
but the mathematicians do not read Plato, while those who read him
know no mathematics, and regard his opinion upon this question as
merely a curious aberration.
Mathematics, rightly viewed, possesses not only truth, but supreme
beauty--a beauty cold and austere, like that of sculpture, without
appeal to any part of our weaker nature, without the gorgeous
trappings of painting or music, yet sublimely pure, and capable of a
stern perfection such as only the greatest art can show. The true
spirit of delight, the exaltation, the sense of being more than man,
which is the touchstone of the highest excellence, is to be found in
mathematics as surely as in poetry. What is best in mathematics
deserves not merely to be learnt as a task, but to be assimilated as a
part of daily thought, and brought again and again before the mind
with ever-renewed encouragement. Real life is, to most men, a long
second-best, a perpetual compromise between the ideal and the
possible; but the world of pure reason knows no compromise, no
practical limitations, no barrier to the creative activity embodying
in splendid edifices the passionate aspiration after the perfect from
which all great work springs. Remote from human passions, remote even
from the pitiful facts of nature, the generations have gradually
created an ordered cosmos, where pure thought can dwell as in its
natural home, and where one, at least, of our nobler impulses can
escape from the dreary exile of the actual world.
So little, however, have mathematicians aimed at beauty, that hardly
anything in their work has had this conscious purpose. Much, owing to
irrepressible instincts, which were better than avowed beliefs, has
been moulded by an unconscious taste; but much also has been spoilt by
false notions of what was fitting. The characteristic excellence of
mathematics is only to be found where the reasoning is rigidly
logical: the rules of logic are to mathematics what those of structure
are to architecture. In the most beautiful work, a chain of argument
is presented in which every link is important on its own account, in
which there is an air of ease and lucidity throughout, and the
premises achieve more than would have been thought possible, by means
which appear natural and inevitable. Literature embodies what is
general in particular circumstances whose universal significance
shines through their individual dress; but mathematics endeavours to
present whatever is most general in its purity, without any irrelevant
trappings.
How should the teaching of mathematics be conducted so as to
communicate to the learner as much as possible of this high ideal?
Here experience must, in a great measure, be our guide; but some
maxims may result from our consideration of the ultimate purpose to be
achieved.
One of the chief ends served by mathematics, when rightly taught, is
to awaken the learner's belief in reason, his confidence in the truth
of what has been demonstrated, and in the value of demonstration. This
purpose is not served by existing instruction; but it is easy to see
ways in which it might be served. At present, in what concerns
arithmetic, the boy or girl is given a set of rules, which present
themselves as neither true nor false, but as merely the will of the
teacher, the way in which, for some unfathomable reason, the teacher
prefers to have the game played. To some degree, in a study of such
definite practical utility, this is no doubt unavoidable; but as soon
as possible, the reasons of rules should be set forth by whatever
means most readily appeal to the childish mind. In geometry, instead
of the tedious apparatus of fallacious proofs for obvious truisms
which constitutes the beginning of Euclid, the learner should be
allowed at first to assume the truth of everything obvious, and should
be instructed in the demonstrations of theorems which are at once
startling and easily verifiable by actual drawing, such as those in
which it is shown that three or more lines meet in a point. In this
way belief is generated; it is seen that reasoning may lead to
startling conclusions, which nevertheless the facts will verify; and
thus the instinctive distrust of whatever is abstract or rational is
gradually overcome. Where theorems are difficult, they should be first
taught as exercises in geometrical drawing, until the figure has
become thoroughly familiar; it will then be an agreeable advance to be
taught the logical connections of the various lines or circles that
occur. It is desirable also that the figure illustrating a theorem
should be drawn in all possible cases and shapes, that so the abstract
relations with which geometry is concerned may of themselves emerge
as the residue of similarity amid such great apparent diversity. In
this way the abstract demonstrations should form but a small part of
the instruction, and should be given when, by familiarity with
concrete illustrations, they have come to be felt as the natural
embodiment of visible fact. In this early stage proofs should not be
given with pedantic fullness; definitely fallacious methods, such as
that of superposition, should be rigidly excluded from the first, but
where, without such methods, the proof would be very difficult, the
result should be rendered acceptable by arguments and illustrations
which are explicitly contrasted with demonstrations.
In the beginning of algebra, even the most intelligent child finds, as
a rule, very great difficulty. The use of letters is a mystery, which
seems to have no purpose except mystification. It is almost
impossible, at first, not to think that every letter stands for some
particular number, if only the teacher would reveal _what_ number it
stands for. The fact is, that in algebra the mind is first taught to
consider general truths, truths which are not asserted to hold only of
this or that particular thing, but of any one of a whole group of
things. It is in the power of understanding and discovering such
truths that the mastery of the intellect over the whole world of
things actual and possible resides; and ability to deal with the
general as such is one of the gifts that a mathematical education
should bestow. But how little, as a rule, is the teacher of algebra
able to explain the chasm which divides it from arithmetic, and how
little is the learner assisted in his groping efforts at
comprehension! Usually the method that has been adopted in arithmetic
is continued: rules are set forth, with no adequate explanation of
their grounds; the pupil learns to use the rules blindly, and
presently, when he is able to obtain the answer that the teacher
desires, he feels that he has mastered the difficulties of the
subject. But of inner comprehension of the processes employed he has
probably acquired almost nothing.
When algebra has been learnt, all goes smoothly until we reach those
studies in which the notion of infinity is employed--the infinitesimal
calculus and the whole of higher mathematics. The solution of the
difficulties which formerly surrounded the mathematical infinite is
probably the greatest achievement of which our own age has to boast.
Since the beginnings of Greek thought these difficulties have been
known; in every age the finest intellects have vainly endeavoured to
answer the apparently unanswerable questions that had been asked by
Zeno the Eleatic. At last Georg Cantor has found the answer, and has
conquered for the intellect a new and vast province which had been
given over to Chaos and old Night. It was assumed as self-evident,
until Cantor and Dedekind established the opposite, that if, from any
collection of things, some were taken away, the number of things left
must always be less than the original number of things. This
assumption, as a matter of fact, holds only of finite collections; and
the rejection of it, where the infinite is concerned, has been shown
to remove all the difficulties that had hitherto baffled human reason
in this matter, and to render possible the creation of an exact
science of the infinite. This stupendous fact ought to produce a
revolution in the higher teaching of mathematics; it has itself added
immeasurably to the educational value of the subject, and it has at
last given the means of treating with logical precision many studies
which, until lately, were wrapped in fallacy and obscurity. By those
who were educated on the old lines, the new work is considered to be
appallingly difficult, abstruse, and obscure; and it must be confessed
that the discoverer, as is so often the case, has hardly himself
emerged from the mists which the light of his intellect is dispelling.
But inherently, the new doctrine of the infinite, to all candid and
inquiring minds, has facilitated the mastery of higher mathematics;
for hitherto, it has been necessary to learn, by a long process of
sophistication, to give assent to arguments which, on first
acquaintance, were rightly judged to be confused and erroneous. So far
from producing a fearless belief in reason, a bold rejection of
whatever failed to fulfil the strictest requirements of logic, a
mathematical training, during the past two centuries, encouraged the
belief that many things, which a rigid inquiry would reject as
fallacious, must yet be accepted because they work in what the
mathematician calls "practice." By this means, a timid, compromising
spirit, or else a sacerdotal belief in mysteries not intelligible to
the profane, has been bred where reason alone should have ruled. All
this it is now time to sweep away; let those who wish to penetrate
into the arcana of mathematics be taught at once the true theory in
all its logical purity, and in the concatenation established by the
very essence of the entities concerned.
If we are considering mathematics as an end in itself, and not as a
technical training for engineers, it is very desirable to preserve the
purity and strictness of its reasoning. Accordingly those who have
attained a sufficient familiarity with its easier portions should be
led backward from propositions to which they have assented as
self-evident to more and more fundamental principles from which what
had previously appeared as premises can be deduced. They should be
taught--what the theory of infinity very aptly illustrates--that many
propositions seem self-evident to the untrained mind which,
nevertheless, a nearer scrutiny shows to be false. By this means they
will be led to a sceptical inquiry into first principles, an
examination of the foundations upon which the whole edifice of
reasoning is built, or, to take perhaps a more fitting metaphor, the
great trunk from which the spreading branches spring. At this stage,
it is well to study afresh the elementary portions of mathematics,
asking no longer merely whether a given proposition is true, but also
how it grows out of the central principles of logic. Questions of this
nature can now be answered with a precision and certainty which were
formerly quite impossible; and in the chains of reasoning that the
answer requires the unity of all mathematical studies at last unfolds
itself.
In the great majority of mathematical text-books there is a total lack
of unity in method and of systematic development of a central theme.
Propositions of very diverse kinds are proved by whatever means are
thought most easily intelligible, and much space is devoted to mere
curiosities which in no way contribute to the main argument. But in
the greatest works, unity and inevitability are felt as in the
unfolding of a drama; in the premisses a subject is proposed for
consideration, and in every subsequent step some definite advance is
made towards mastery of its nature. The love of system, of
interconnection, which is perhaps the inmost essence of the
intellectual impulse, can find free play in mathematics as nowhere
else. The learner who feels this impulse must not be repelled by an
array of meaningless examples or distracted by amusing oddities, but
must be encouraged to dwell upon central principles, to become
familiar with the structure of the various subjects which are put
before him, to travel easily over the steps of the more important
deductions. In this way a good tone of mind is cultivated, and
selective attention is taught to dwell by preference upon what is
weighty and essential.
When the separate studies into which mathematics is divided have each
been viewed as a logical whole, as a natural growth from the
propositions which constitute their principles, the learner will be
able to understand the fundamental science which unifies and
systematises the whole of deductive reasoning. This is symbolic
logic--a study which, though it owes its inception to Aristotle, is
yet, in its wider developments, a product, almost wholly, of the
nineteenth century, and is indeed, in the present day, still growing
with great rapidity. The true method of discovery in symbolic logic,
and probably also the best method for introducing the study to a
learner acquainted with other parts of mathematics, is the analysis of
actual examples of deductive reasoning, with a view to the discovery
of the principles employed. These principles, for the most part, are
so embedded in our ratiocinative instincts, that they are employed
quite unconsciously, and can be dragged to light only by much patient
effort. But when at last they have been found, they are seen to be few
in number, and to be the sole source of everything in pure
mathematics. The discovery that all mathematics follows inevitably
from a small collection of fundamental laws is one which immeasurably
enhances the intellectual beauty of the whole; to those who have been
oppressed by the fragmentary and incomplete nature of most existing
chains of deduction this discovery comes with all the overwhelming
force of a revelation; like a palace emerging from the autumn mist as
the traveller ascends an Italian hill-side, the stately storeys of the
mathematical edifice appear in their due order and proportion, with a
new perfection in every part.
Until symbolic logic had acquired its present development, the
principles upon which mathematics depends were always supposed to be
philosophical, and discoverable only by the uncertain, unprogressive
methods hitherto employed by philosophers. So long as this was
thought, mathematics seemed to be not autonomous, but dependent upon a
study which had quite other methods than its own. Moreover, since the
nature of the postulates from which arithmetic, analysis, and geometry
are to be deduced was wrapped in all the traditional obscurities of
metaphysical discussion, the edifice built upon such dubious
foundations began to be viewed as no better than a castle in the air.
In this respect, the discovery that the true principles are as much a
part of mathematics as any of their consequences has very greatly
increased the intellectual satisfaction to be obtained. This
satisfaction ought not to be refused to learners capable of enjoying
it, for it is of a kind to increase our respect for human powers and
our knowledge of the beauties belonging to the abstract world.
Philosophers have commonly held that the laws of logic, which underlie
mathematics, are laws of thought, laws regulating the operations of
our minds. By this opinion the true dignity of reason is very greatly
lowered: it ceases to be an investigation into the very heart and
immutable essence of all things actual and possible, becoming,
instead, an inquiry into something more or less human and subject to
our limitations. The contemplation of what is non-human, the discovery
that our minds are capable of dealing with material not created by
them, above all, the realisation that beauty belongs to the outer
world as to the inner, are the chief means of overcoming the terrible
sense of impotence, of weakness, of exile amid hostile powers, which
is too apt to result from acknowledging the all-but omnipotence of
alien forces. To reconcile us, by the exhibition of its awful beauty,
to the reign of Fate--which is merely the literary personification of
these forces--is the task of tragedy. But mathematics takes us still
further from what is human, into the region of absolute necessity, to
which not only the actual world, but every possible world, must
conform; and even here it builds a habitation, or rather finds a
habitation eternally standing, where our ideals are fully satisfied
and our best hopes are not thwarted. It is only when we thoroughly
understand the entire independence of ourselves, which belongs to this
world that reason finds, that we can adequately realise the profound
importance of its beauty.
Not only is mathematics independent of us and our thoughts, but in
another sense we and the whole universe of existing things are
independent of mathematics. The apprehension of this purely ideal
character is indispensable, if we are to understand rightly the place
of mathematics as one among the arts. It was formerly supposed that
pure reason could decide, in some respects, as to the nature of the
actual world: geometry, at least, was thought to deal with the space
in which we live. But we now know that pure mathematics can never
pronounce upon questions of actual existence: the world of reason, in
a sense, controls the world of fact, but it is not at any point
creative of fact, and in the application of its results to the world
in time and space, its certainty and precision are lost among
approximations and working hypotheses. The objects considered by
mathematicians have, in the past, been mainly of a kind suggested by
phenomena; but from such restrictions the abstract imagination should
be wholly free. A reciprocal liberty must thus be accorded: reason
cannot dictate to the world of facts, but the facts cannot restrict
reason's privilege of dealing with whatever objects its love of beauty
may cause to seem worthy of consideration. Here, as elsewhere, we
build up our own ideals out of the fragments to be found in the world;
and in the end it is hard to say whether the result is a creation or a
discovery.
It is very desirable, in instruction, not merely to persuade the
student of the accuracy of important theorems, but to persuade him in
the way which itself has, of all possible ways, the most beauty. The
true interest of a demonstration is not, as traditional modes of
exposition suggest, concentrated wholly in the result; where this does
occur, it must be viewed as a defect, to be remedied, if possible, by
so generalising the steps of the proof that each becomes important in
and for itself. An argument which serves only to prove a conclusion is
like a story subordinated to some moral which it is meant to teach:
for æsthetic perfection no part of the whole should be merely a means.
A certain practical spirit, a desire for rapid progress, for conquest
of new realms, is responsible for the undue emphasis upon results
which prevails in mathematical instruction. The better way is to
propose some theme for consideration--in geometry, a figure having
important properties; in analysis, a function of which the study is
illuminating, and so on. Whenever proofs depend upon some only of the
marks by which we define the object to be studied, these marks should
be isolated and investigated on their own account. For it is a defect,
in an argument, to employ more premisses than the conclusion demands:
what mathematicians call elegance results from employing only the
essential principles in virtue of which the thesis is true. It is a
merit in Euclid that he advances as far as he is able to go without
employing the axiom of parallels--not, as is often said, because this
axiom is inherently objectionable, but because, in mathematics, every
new axiom diminishes the generality of the resulting theorems, and the
greatest possible generality is before all things to be sought.
Of the effects of mathematics outside its own sphere more has been
written than on the subject of its own proper ideal. The effect upon
philosophy has, in the past, been most notable, but most varied; in
the seventeenth century, idealism and rationalism, in the eighteenth,
materialism and sensationalism, seemed equally its offspring. Of the
effect which it is likely to have in the future it would be very rash
to say much; but in one respect a good result appears probable.
Against that kind of scepticism which abandons the pursuit of ideals
because the road is arduous and the goal not certainly attainable,
mathematics, within its own sphere, is a complete answer. Too often it
is said that there is no absolute truth, but only opinion and private
judgment; that each of us is conditioned, in his view of the world, by
his own peculiarities, his own taste and bias; that there is no
external kingdom of truth to which, by patience and discipline, we may
at last obtain admittance, but only truth for me, for you, for every
separate person. By this habit of mind one of the chief ends of human
effort is denied, and the supreme virtue of candour, of fearless
acknowledgment of what is, disappears from our moral vision. Of such
scepticism mathematics is a perpetual reproof; for its edifice of
truths stands unshakable and inexpungable to all the weapons of
doubting cynicism.
The effects of mathematics upon practical life, though they should not
be regarded as the motive of our studies, may be used to answer a
doubt to which the solitary student must always be liable. In a world
so full of evil and suffering, retirement into the cloister of
contemplation, to the enjoyment of delights which, however noble, must
always be for the few only, cannot but appear as a somewhat selfish
refusal to share the burden imposed upon others by accidents in which
justice plays no part. Have any of us the right, we ask, to withdraw
from present evils, to leave our fellow-men unaided, while we live a
life which, though arduous and austere, is yet plainly good in its own
nature? When these questions arise, the true answer is, no doubt, that
some must keep alive the sacred fire, some must preserve, in every
generation, the haunting vision which shadows forth the goal of so
much striving. But when, as must sometimes occur, this answer seems
too cold, when we are almost maddened by the spectacle of sorrows to
which we bring no help, then we may reflect that indirectly the
mathematician often does more for human happiness than any of his more
practically active contemporaries. The history of science abundantly
proves that a body of abstract propositions--even if, as in the case
of conic sections, it remains two thousand years without effect upon
daily life--may yet, at any moment, be used to cause a revolution in
the habitual thoughts and occupations of every citizen. The use of
steam and electricity--to take striking instances--is rendered
possible only by mathematics. In the results of abstract thought the
world possesses a capital of which the employment in enriching the
common round has no hitherto discoverable limits. Nor does experience
give any means of deciding what parts of mathematics will be found
useful. Utility, therefore, can be only a consolation in moments of
discouragement, not a guide in directing our studies.
For the health of the moral life, for ennobling the tone of an age or
a nation, the austerer virtues have a strange power, exceeding the
power of those not informed and purified by thought. Of these austerer
virtues the love of truth is the chief, and in mathematics, more than
elsewhere, the love of truth may find encouragement for waning faith.
Every great study is not only an end in itself, but also a means of
creating and sustaining a lofty habit of mind; and this purpose should
be kept always in view throughout the teaching and learning of
mathematics.
FOOTNOTES:
[10] This passage was pointed out to me by Professor Gilbert Murray.
V
MATHEMATICS AND THE METAPHYSICIANS
The nineteenth century, which prided itself upon the invention of
steam and evolution, might have derived a more legitimate title to
fame from the discovery of pure mathematics. This science, like most
others, was baptised long before it was born; and thus we find writers
before the nineteenth century alluding to what they called pure
mathematics. But if they had been asked what this subject was, they
would only have been able to say that it consisted of Arithmetic,
Algebra, Geometry, and so on. As to what these studies had in common,
and as to what distinguished them from applied mathematics, our
ancestors were completely in the dark.
Pure mathematics was discovered by Boole, in a work which he called
the _Laws of Thought_ (1854). This work abounds in asseverations that
it is not mathematical, the fact being that Boole was too modest to
suppose his book the first ever written on mathematics. He was also
mistaken in supposing that he was dealing with the laws of thought:
the question how people actually think was quite irrelevant to him,
and if his book had really contained the laws of thought, it was
curious that no one should ever have thought in such a way before. His
book was in fact concerned with formal logic, and this is the same
thing as mathematics.
Pure mathematics consists entirely of assertions to the effect that,
if such and such a proposition is true of _anything_, then such and
such another proposition is true of that thing. It is essential not to
discuss whether the first proposition is really true, and not to
mention what the anything is, of which it is supposed to be true. Both
these points would belong to applied mathematics. We start, in pure
mathematics, from certain rules of inference, by which we can infer
that _if_ one proposition is true, then so is some other proposition.
These rules of inference constitute the major part of the principles
of formal logic. We then take any hypothesis that seems amusing, and
deduce its consequences. _If_ our hypothesis is about _anything_, and
not about some one or more particular things, then our deductions
constitute mathematics. Thus mathematics may be defined as the subject
in which we never know what we are talking about, nor whether what we
are saying is true. People who have been puzzled by the beginnings of
mathematics will, I hope, find comfort in this definition, and will
probably agree that it is accurate.
As one of the chief triumphs of modern mathematics consists in having
discovered what mathematics really is, a few more words on this
subject may not be amiss. It is common to start any branch of
mathematics--for instance, Geometry--with a certain number of
primitive ideas, supposed incapable of definition, and a certain
number of primitive propositions or axioms, supposed incapable of
proof. Now the fact is that, though there are indefinables and
indemonstrables in every branch of applied mathematics, there are none
in pure mathematics except such as belong to general logic. Logic,
broadly speaking, is distinguished by the fact that its propositions
can be put into a form in which they apply to anything whatever. All
pure mathematics--Arithmetic, Analysis, and Geometry--is built up by
combinations of the primitive ideas of logic, and its propositions are
deduced from the general axioms of logic, such as the syllogism and
the other rules of inference. And this is no longer a dream or an
aspiration. On the contrary, over the greater and more difficult part
of the domain of mathematics, it has been already accomplished; in the
few remaining cases, there is no special difficulty, and it is now
being rapidly achieved. Philosophers have disputed for ages whether
such deduction was possible; mathematicians have sat down and made the
deduction. For the philosophers there is now nothing left but graceful
acknowledgments.
The subject of formal logic, which has thus at last shown itself to be
identical with mathematics, was, as every one knows, invented by
Aristotle, and formed the chief study (other than theology) of the
Middle Ages. But Aristotle never got beyond the syllogism, which is a
very small part of the subject, and the schoolmen never got beyond
Aristotle. If any proof were required of our superiority to the
mediæval doctors, it might be found in this. Throughout the Middle
Ages, almost all the best intellects devoted themselves to formal
logic, whereas in the nineteenth century only an infinitesimal
proportion of the world's thought went into this subject.
Nevertheless, in each decade since 1850 more has been done to advance
the subject than in the whole period from Aristotle to Leibniz. People
have discovered how to make reasoning symbolic, as it is in Algebra,
so that deductions are effected by mathematical rules. They have
discovered many rules besides the syllogism, and a new branch of
logic, called the Logic of Relatives,[11] has been invented to deal
with topics that wholly surpassed the powers of the old logic, though
they form the chief contents of mathematics.
It is not easy for the lay mind to realise the importance of symbolism
in discussing the foundations of mathematics, and the explanation may
perhaps seem strangely paradoxical. The fact is that symbolism is
useful because it makes things difficult. (This is not true of the
advanced parts of mathematics, but only of the beginnings.) What we
wish to know is, what can be deduced from what. Now, in the
beginnings, everything is self-evident; and it is very hard to see
whether one self-evident proposition follows from another or not.
Obviousness is always the enemy to correctness. Hence we invent some
new and difficult symbolism, in which nothing seems obvious. Then we
set up certain rules for operating on the symbols, and the whole thing
becomes mechanical. In this way we find out what must be taken as
premiss and what can be demonstrated or defined. For instance, the
whole of Arithmetic and Algebra has been shown to require three
indefinable notions and five indemonstrable propositions. But without
a symbolism it would have been very hard to find this out. It is so
obvious that two and two are four, that we can hardly make ourselves
sufficiently sceptical to doubt whether it can be proved. And the same
holds in other cases where self-evident things are to be proved.
But the proof of self-evident propositions may seem, to the
uninitiated, a somewhat frivolous occupation. To this we might reply
that it is often by no means self-evident that one obvious proposition
follows from another obvious proposition; so that we are really
discovering new truths when we prove what is evident by a method which
is not evident. But a more interesting retort is, that since people
have tried to prove obvious propositions, they have found that many of
them are false. Self-evidence is often a mere will-o'-the-wisp, which
is sure to lead us astray if we take it as our guide. For instance,
nothing is plainer than that a whole always has more terms than a
part, or that a number is increased by adding one to it. But these
propositions are now known to be usually false. Most numbers are
infinite, and if a number is infinite you may add ones to it as long
as you like without disturbing it in the least. One of the merits of a
proof is that it instils a certain doubt as to the result proved; and
when what is obvious can be proved in some cases, but not in others,
it becomes possible to suppose that in these other cases it is false.
The great master of the art of formal reasoning, among the men of our
own day, is an Italian, Professor Peano, of the University of
Turin.[12] He has reduced the greater part of mathematics (and he or
his followers will, in time, have reduced the whole) to strict
symbolic form, in which there are no words at all. In the ordinary
mathematical books, there are no doubt fewer words than most readers
would wish. Still, little phrases occur, such as _therefore, let us
assume, consider_, or _hence it follows_. All these, however, are a
concession, and are swept away by Professor Peano. For instance, if we
wish to learn the whole of Arithmetic, Algebra, the Calculus, and
indeed all that is usually called pure mathematics (except Geometry),
we must start with a dictionary of three words. One symbol stands for
_zero_, another for _number_, and a third for _next after_. What these
ideas mean, it is necessary to know if you wish to become an
arithmetician. But after symbols have been invented for these three
ideas, not another word is required in the whole development. All
future symbols are symbolically explained by means of these three.
Even these three can be explained by means of the notions of
_relation_ and _class_; but this requires the Logic of Relations,
which Professor Peano has never taken up. It must be admitted that
what a mathematician has to know to begin with is not much. There are
at most a dozen notions out of which all the notions in all pure
mathematics (including Geometry) are compounded. Professor Peano, who
is assisted by a very able school of young Italian disciples, has
shown how this may be done; and although the method which he has
invented is capable of being carried a good deal further than he has
carried it, the honour of the pioneer must belong to him.
Two hundred years ago, Leibniz foresaw the science which Peano has
perfected, and endeavoured to create it. He was prevented from
succeeding by respect for the authority of Aristotle, whom he could
not believe guilty of definite, formal fallacies; but the subject
which he desired to create now exists, in spite of the patronising
contempt with which his schemes have been treated by all superior
persons. From this "Universal Characteristic," as he called it, he
hoped for a solution of all problems, and an end to all disputes. "If
controversies were to arise," he says, "there would be no more need of
disputation between two philosophers than between two accountants. For
it would suffice to take their pens in their hands, to sit down to
their desks, and to say to each other (with a friend as witness, if
they liked), 'Let us calculate.'" This optimism has now appeared to be
somewhat excessive; there still are problems whose solution is
doubtful, and disputes which calculation cannot decide. But over an
enormous field of what was formerly controversial, Leibniz's dream has
become sober fact. In the whole philosophy of mathematics, which used
to be at least as full of doubt as any other part of philosophy, order
and certainty have replaced the confusion and hesitation which
formerly reigned. Philosophers, of course, have not yet discovered
this fact, and continue to write on such subjects in the old way. But
mathematicians, at least in Italy, have now the power of treating the
principles of mathematics in an exact and masterly manner, by means of
which the certainty of mathematics extends also to mathematical
philosophy. Hence many of the topics which used to be placed among the
great mysteries--for example, the natures of infinity, of continuity,
of space, time and motion--are now no longer in any degree open to
doubt or discussion. Those who wish to know the nature of these things
need only read the works of such men as Peano or Georg Cantor; they
will there find exact and indubitable expositions of all these quondam
mysteries.
In this capricious world, nothing is more capricious than posthumous
fame. One of the most notable examples of posterity's lack of judgment
is the Eleatic Zeno. This man, who may be regarded as the founder of
the philosophy of infinity, appears in Plato's Parmenides in the
privileged position of instructor to Socrates. He invented four
arguments, all immeasurably subtle and profound, to prove that motion
is impossible, that Achilles can never overtake the tortoise, and that
an arrow in flight is really at rest. After being refuted by
Aristotle, and by every subsequent philosopher from that day to our
own, these arguments were reinstated, and made the basis of a
mathematical renaissance, by a German professor, who probably never
dreamed of any connection between himself and Zeno. Weierstrass,[13]
by strictly banishing from mathematics the use of infinitesimals, has
at last shown that we live in an unchanging world, and that the arrow
in its flight is truly at rest. Zeno's only error lay in inferring (if
he did infer) that, because there is no such thing as a state of
change, therefore the world is in the same state at any one time as at
any other. This is a consequence which by no means follows; and in
this respect, the German mathematician is more constructive than the
ingenious Greek. Weierstrass has been able, by embodying his views in
mathematics, where familiarity with truth eliminates the vulgar
prejudices of common sense, to invest Zeno's paradoxes with the
respectable air of platitudes; and if the result is less delightful to
the lover of reason than Zeno's bold defiance, it is at any rate more
calculated to appease the mass of academic mankind.
Zeno was concerned, as a matter of fact, with three problems, each
presented by motion, but each more abstract than motion, and capable
of a purely arithmetical treatment. These are the problems of the
infinitesimal, the infinite, and continuity. To state clearly the
difficulties involved, was to accomplish perhaps the hardest part of
the philosopher's task. This was done by Zeno. From him to our own
day, the finest intellects of each generation in turn attacked the
problems, but achieved, broadly speaking, nothing. In our own time,
however, three men--Weierstrass, Dedekind, and Cantor--have not merely
advanced the three problems, but have completely solved them. The
solutions, for those acquainted with mathematics, are so clear as to
leave no longer the slightest doubt or difficulty. This achievement is
probably the greatest of which our age has to boast; and I know of no
age (except perhaps the golden age of Greece) which has a more
convincing proof to offer of the transcendent genius of its great men.
Of the three problems, that of the infinitesimal was solved by
Weierstrass; the solution of the other two was begun by Dedekind, and
definitively accomplished by Cantor.
The infinitesimal played formerly a great part in mathematics. It was
introduced by the Greeks, who regarded a circle as differing
infinitesimally from a polygon with a very large number of very small
equal sides. It gradually grew in importance, until, when Leibniz
invented the Infinitesimal Calculus, it seemed to become the
fundamental notion of all higher mathematics. Carlyle tells, in his
_Frederick the Great_, how Leibniz used to discourse to Queen Sophia
Charlotte of Prussia concerning the infinitely little, and how she
would reply that on that subject she needed no instruction--the
behaviour of courtiers had made her thoroughly familiar with it. But
philosophers and mathematicians--who for the most part had less
acquaintance with courts--continued to discuss this topic, though
without making any advance. The Calculus required continuity, and
continuity was supposed to require the infinitely little; but nobody
could discover what the infinitely little might be. It was plainly not
quite zero, because a sufficiently large number of infinitesimals,
added together, were seen to make up a finite whole. But nobody could
point out any fraction which was not zero, and yet not finite. Thus
there was a deadlock. But at last Weierstrass discovered that the
infinitesimal was not needed at all, and that everything could be
accomplished without it. Thus there was no longer any need to suppose
that there was such a thing. Nowadays, therefore, mathematicians are
more dignified than Leibniz: instead of talking about the infinitely
small, they talk about the infinitely great--a subject which, however
appropriate to monarchs, seems, unfortunately, to interest them even
less than the infinitely little interested the monarchs to whom
Leibniz discoursed.
The banishment of the infinitesimal has all sorts of odd consequences,
to which one has to become gradually accustomed. For example, there is
no such thing as the next moment. The interval between one moment and
the next would have to be infinitesimal, since, if we take two moments
with a finite interval between them, there are always other moments in
the interval. Thus if there are to be no infinitesimals, no two
moments are quite consecutive, but there are always other moments
between any two. Hence there must be an infinite number of moments
between any two; because if there were a finite number one would be
nearest the first of the two moments, and therefore next to it. This
might be thought to be a difficulty; but, as a matter of fact, it is
here that the philosophy of the infinite comes in, and makes all
straight.
The same sort of thing happens in space. If any piece of matter be cut
in two, and then each part be halved, and so on, the bits will become
smaller and smaller, and can theoretically be made as small as we
please. However small they may be, they can still be cut up and made
smaller still. But they will always have _some_ finite size, however
small they may be. We never reach the infinitesimal in this way, and
no finite number of divisions will bring us to points. Nevertheless
there _are_ points, only these are not to be reached by successive
divisions. Here again, the philosophy of the infinite shows us how
this is possible, and why points are not infinitesimal lengths.
As regards motion and change, we get similarly curious results. People
used to think that when a thing changes, it must be in a state of
change, and that when a thing moves, it is in a state of motion. This
is now known to be a mistake. When a body moves, all that can be said
is that it is in one place at one time and in another at another. We
must not say that it will be in a neighbouring place at the next
instant, since there is no next instant. Philosophers often tell us
that when a body is in motion, it changes its position within the
instant. To this view Zeno long ago made the fatal retort that every
body always is where it is; but a retort so simple and brief was not
of the kind to which philosophers are accustomed to give weight, and
they have continued down to our own day to repeat the same phrases
which roused the Eleatic's destructive ardour. It was only recently
that it became possible to explain motion in detail in accordance with
Zeno's platitude, and in opposition to the philosopher's paradox. We
may now at last indulge the comfortable belief that a body in motion
is just as truly where it is as a body at rest. Motion consists merely
in the fact that bodies are sometimes in one place and sometimes in
another, and that they are at intermediate places at intermediate
times. Only those who have waded through the quagmire of philosophic
speculation on this subject can realise what a liberation from antique
prejudices is involved in this simple and straightforward commonplace.
The philosophy of the infinitesimal, as we have just seen, is mainly
negative. People used to believe in it, and now they have found out
their mistake. The philosophy of the infinite, on the other hand, is
wholly positive. It was formerly supposed that infinite numbers, and
the mathematical infinite generally, were self-contradictory. But as
it was obvious that there were infinities--for example, the number of
numbers--the contradictions of infinity seemed unavoidable, and
philosophy seemed to have wandered into a "cul-de-sac." This
difficulty led to Kant's antinomies, and hence, more or less
indirectly, to much of Hegel's dialectic method. Almost all current
philosophy is upset by the fact (of which very few philosophers are as
yet aware) that all the ancient and respectable contradictions in the
notion of the infinite have been once for all disposed of. The method
by which this has been done is most interesting and instructive. In
the first place, though people had talked glibly about infinity ever
since the beginnings of Greek thought, nobody had ever thought of
asking, What is infinity? If any philosopher had been asked for a
definition of infinity, he might have produced some unintelligible
rigmarole, but he would certainly not have been able to give a
definition that had any meaning at all. Twenty years ago, roughly
speaking, Dedekind and Cantor asked this question, and, what is more
remarkable, they answered it. They found, that is to say, a perfectly
precise definition of an infinite number or an infinite collection of
things. This was the first and perhaps the greatest step. It then
remained to examine the supposed contradictions in this notion. Here
Cantor proceeded in the only proper way. He took pairs of
contradictory propositions, in which both sides of the contradiction
would be usually regarded as demonstrable, and he strictly examined
the supposed proofs. He found that all proofs adverse to infinity
involved a certain principle, at first sight obviously true, but
destructive, in its consequences, of almost all mathematics. The
proofs favourable to infinity, on the other hand, involved no
principle that had evil consequences. It thus appeared that common
sense had allowed itself to be taken in by a specious maxim, and that,
when once this maxim was rejected, all went well.
The maxim in question is, that if one collection is part of another,
the one which is a part has fewer terms than the one of which it is a
part. This maxim is true of finite numbers. For example, Englishmen
are only some among Europeans, and there are fewer Englishmen than
Europeans. But when we come to infinite numbers, this is no longer
true. This breakdown of the maxim gives us the precise definition of
infinity. A collection of terms is infinite when it contains as parts
other collections which have just as many terms as it has. If you can
take away some of the terms of a collection, without diminishing the
number of terms, then there are an infinite number of terms in the
collection. For example, there are just as many even numbers as there
are numbers altogether, since every number can be doubled. This may be
seen by putting odd and even numbers together in one row, and even
numbers alone in a row below:--
1, 2, 3, 4, 5, _ad infinitum_.
2, 4, 6, 8, 10, _ad infinitum_.
There are obviously just as many numbers in the row below as in the
row above, because there is one below for each one above. This
property, which was formerly thought to be a contradiction, is now
transformed into a harmless definition of infinity, and shows, in the
above case, that the number of finite numbers is infinite.
But the uninitiated may wonder how it is possible to deal with a
number which cannot be counted. It is impossible to count up _all_ the
numbers, one by one, because, however many we may count, there are
always more to follow. The fact is that counting is a very vulgar and
elementary way of finding out how many terms there are in a
collection. And in any case, counting gives us what mathematicians
call the _ordinal_ number of our terms; that is to say, it arranges
our terms in an order or series, and its result tells us what type of
series results from this arrangement. In other words, it is impossible
to count things without counting some first and others afterwards, so
that counting always has to do with order. Now when there are only a
finite number of terms, we can count them in any order we like; but
when there are an infinite number, what corresponds to counting will
give us quite different results according to the way in which we carry
out the operation. Thus the ordinal number, which results from what,
in a general sense may be called counting, depends not only upon how
many terms we have, but also (where the number of terms is infinite)
upon the way in which the terms are arranged.
The fundamental infinite numbers are not ordinal, but are what is
called _cardinal_. They are not obtained by putting our terms in order
and counting them, but by a different method, which tells us, to begin
with, whether two collections have the same number of terms, or, if
not, which is the greater.[14] It does not tell us, in the way in
which counting does, _what_ number of terms a collection has; but if
we define a number as the number of terms in such and such a
collection, then this method enables us to discover whether some other
collection that may be mentioned has more or fewer terms. An
illustration will show how this is done. If there existed some country
in which, for one reason or another, it was impossible to take a
census, but in which it was known that every man had a wife and every
woman a husband, then (provided polygamy was not a national
institution) we should know, without counting, that there were exactly
as many men as there were women in that country, neither more nor
less. This method can be applied generally. If there is some relation
which, like marriage, connects the things in one collection each with
one of the things in another collection, and vice versa, then the two
collections have the same number of terms. This was the way in which
we found that there are as many even numbers as there are numbers.
Every number can be doubled, and every even number can be halved, and
each process gives just one number corresponding to the one that is
doubled or halved. And in this way we can find any number of
collections each of which has just as many terms as there are finite
numbers. If every term of a collection can be hooked on to a number,
and all the finite numbers are used once, and only once, in the
process, then our collection must have just as many terms as there are
finite numbers. This is the general method by which the numbers of
infinite collections are defined.
But it must not be supposed that all infinite numbers are equal. On
the contrary, there are infinitely more infinite numbers than finite
ones. There are more ways of arranging the finite numbers in different
types of series than there are finite numbers. There are probably more
points in space and more moments in time than there are finite
numbers. There are exactly as many fractions as whole numbers,
although there are an infinite number of fractions between any two
whole numbers. But there are more irrational numbers than there are
whole numbers or fractions. There are probably exactly as many points
in space as there are irrational numbers, and exactly as many points
on a line a millionth of an inch long as in the whole of infinite
space. There is a greatest of all infinite numbers, which is the
number of things altogether, of every sort and kind. It is obvious
that there cannot be a greater number than this, because, if
everything has been taken, there is nothing left to add. Cantor has a
proof that there is no greatest number, and if this proof were valid,
the contradictions of infinity would reappear in a sublimated form.
But in this one point, the master has been guilty of a very subtle
fallacy, which I hope to explain in some future work.[15]
We can now understand why Zeno believed that Achilles cannot overtake
the tortoise and why as a matter of fact he can overtake it. We shall
see that all the people who disagreed with Zeno had no right to do so,
because they all accepted premises from which his conclusion followed.
The argument is this: Let Achilles and the tortoise start along a road
at the same time, the tortoise (as is only fair) being allowed a
handicap. Let Achilles go twice as fast as the tortoise, or ten times
or a hundred times as fast. Then he will never reach the tortoise. For
at every moment the tortoise is somewhere and Achilles is somewhere;
and neither is ever twice in the same place while the race is going
on. Thus the tortoise goes to just as many places as Achilles does,
because each is in one place at one moment, and in another at any
other moment. But if Achilles were to catch up with the tortoise, the
places where the tortoise would have been would be only part of the
places where Achilles would have been. Here, we must suppose, Zeno
appealed to the maxim that the whole has more terms than the part.[16]
Thus if Achilles were to overtake the tortoise, he would have been in
more places than the tortoise; but we saw that he must, in any period,
be in exactly as many places as the tortoise. Hence we infer that he
can never catch the tortoise. This argument is strictly correct, if we
allow the axiom that the whole has more terms than the part. As the
conclusion is absurd, the axiom must be rejected, and then all goes
well. But there is no good word to be said for the philosophers of the
past two thousand years and more, who have all allowed the axiom and
denied the conclusion.
The retention of this axiom leads to absolute contradictions, while
its rejection leads only to oddities. Some of these oddities, it must
be confessed, are very odd. One of them, which I call the paradox of
Tristram Shandy, is the converse of the Achilles, and shows that the
tortoise, if you give him time, will go just as far as Achilles.
Tristram Shandy, as we know, employed two years in chronicling the
first two days of his life, and lamented that, at this rate, material
would accumulate faster than he could deal with it, so that, as years
went by, he would be farther and farther from the end of his history.
Now I maintain that, if he had lived for ever, and had not wearied of
his task, then, even if his life had continued as event fully as it
began, no part of his biography would have remained unwritten. For
consider: the hundredth day will be described in the hundredth year,
the thousandth in the thousandth year, and so on. Whatever day we may
choose as so far on that he cannot hope to reach it, that day will be
described in the corresponding year. Thus any day that may be
mentioned will be written up sooner or later, and therefore no part of
the biography will remain permanently unwritten. This paradoxical but
perfectly true proposition depends upon the fact that the number of
days in all time is no greater than the number of years.
Thus on the subject of infinity it is impossible to avoid conclusions
which at first sight appear paradoxical, and this is the reason why so
many philosophers have supposed that there were inherent
contradictions in the infinite. But a little practice enables one to
grasp the true principles of Cantor's doctrine, and to acquire new and
better instincts as to the true and the false. The oddities then
become no odder than the people at the antipodes, who used to be
thought impossible because they would find it so inconvenient to stand
on their heads.
The solution of the problems concerning infinity has enabled Cantor to
solve also the problems of continuity. Of this, as of infinity, he has
given a perfectly precise definition, and has shown that there are no
contradictions in the notion so defined. But this subject is so
technical that it is impossible to give any account of it here.
The notion of continuity depends upon that of _order_, since
continuity is merely a particular type of order. Mathematics has, in
modern times, brought order into greater and greater prominence. In
former days, it was supposed (and philosophers are still apt to
suppose) that quantity was the fundamental notion of mathematics. But
nowadays, quantity is banished altogether, except from one little
corner of Geometry, while order more and more reigns supreme. The
investigation of different kinds of series and their relations is now
a very large part of mathematics, and it has been found that this
investigation can be conducted without any reference to quantity, and,
for the most part, without any reference to number. All types of
series are capable of formal definition, and their properties can be
deduced from the principles of symbolic logic by means of the Algebra
of Relatives. The notion of a limit, which is fundamental in the
greater part of higher mathematics, used to be defined by means of
quantity, as a term to which the terms of some series approximate as
nearly as we please. But nowadays the limit is defined quite
differently, and the series which it limits may not approximate to it
at all. This improvement also is due to Cantor, and it is one which
has revolutionised mathematics. Only order is now relevant to limits.
Thus, for instance, the smallest of the infinite integers is the limit
of the finite integers, though all finite integers are at an infinite
distance from it. The study of different types of series is a general
subject of which the study of ordinal numbers (mentioned above) is a
special and very interesting branch. But the unavoidable
technicalities of this subject render it impossible to explain to any
but professed mathematicians.
Geometry, like Arithmetic, has been subsumed, in recent times, under
the general study of order. It was formerly supposed that Geometry was
the study of the nature of the space in which we live, and accordingly
it was urged, by those who held that what exists can only be known
empirically, that Geometry should really be regarded as belonging to
applied mathematics. But it has gradually appeared, by the increase of
non-Euclidean systems, that Geometry throws no more light upon the
nature of space than Arithmetic throws upon the population of the
United States. Geometry is a whole collection of deductive sciences
based on a corresponding collection of sets of axioms. One set of
axioms is Euclid's; other equally good sets of axioms lead to other
results. Whether Euclid's axioms are true, is a question as to which
the pure mathematician is indifferent; and, what is more, it is a
question which it is theoretically impossible to answer with certainty
in the affirmative. It might possibly be shown, by very careful
measurements, that Euclid's axioms are false; but no measurements
could ever assure us (owing to the errors of observation) that they
are exactly true. Thus the geometer leaves to the man of science to
decide, as best he may, what axioms are most nearly true in the actual
world. The geometer takes any set of axioms that seem interesting, and
deduces their consequences. What defines Geometry, in this sense, is
that the axioms must give rise to a series of more than one dimension.
And it is thus that Geometry becomes a department in the study of
order.
In Geometry, as in other parts of mathematics, Peano and his disciples
have done work of the very greatest merit as regards principles.
Formerly, it was held by philosophers and mathematicians alike that
the proofs in Geometry depended on the figure; nowadays, this is known
to be false. In the best books there are no figures at all. The
reasoning proceeds by the strict rules of formal logic from a set of
axioms laid down to begin with. If a figure is used, all sorts of
things seem obviously to follow, which no formal reasoning can prove
from the explicit axioms, and which, as a matter of fact, are only
accepted because they are obvious. By banishing the figure, it becomes
possible to discover _all_ the axioms that are needed; and in this way
all sorts of possibilities, which would have otherwise remained
undetected, are brought to light.
One great advance, from the point of view of correctness, has been
made by introducing points as they are required, and not starting, as
was formerly done, by assuming the whole of space. This method is due
partly to Peano, partly to another Italian named Fano. To those
unaccustomed to it, it has an air of somewhat wilful pedantry. In this
way, we begin with the following axioms: (1) There is a class of
entities called _points_. (2) There is at least one point. (3) If _a_
be a point, there is at least one other point besides _a_. Then we
bring in the straight line joining two points, and begin again with
(4), namely, on the straight line joining _a_ and _b_, there is at
least one other point besides _a_ and _b_. (5) There is at least one
point not on the line _ab_. And so we go on, till we have the means of
obtaining as many points as we require. But the word _space_, as Peano
humorously remarks, is one for which Geometry has no use at all.
The rigid methods employed by modern geometers have deposed Euclid
from his pinnacle of correctness. It was thought, until recent times,
that, as Sir Henry Savile remarked in 1621, there were only two
blemishes in Euclid, the theory of parallels and the theory of
proportion. It is now known that these are almost the only points in
which Euclid is free from blemish. Countless errors are involved in
his first eight propositions. That is to say, not only is it doubtful
whether his axioms are true, which is a comparatively trivial matter,
but it is certain that his propositions do not follow from the axioms
which he enunciates. A vastly greater number of axioms, which Euclid
unconsciously employs, are required for the proof of his propositions.
Even in the first proposition of all, where he constructs an
equilateral triangle on a given base, he uses two circles which are
assumed to intersect. But no explicit axiom assures us that they do
so, and in some kinds of spaces they do not always intersect. It is
quite doubtful whether our space belongs to one of these kinds or not.
Thus Euclid fails entirely to prove his point in the very first
proposition. As he is certainly not an easy author, and is terribly
long-winded, he has no longer any but an historical interest. Under
these circumstances, it is nothing less than a scandal that he should
still be taught to boys in England.[17] A book should have either
intelligibility or correctness; to combine the two is impossible, but
to lack both is to be unworthy of such a place as Euclid has occupied
in education.
The most remarkable result of modern methods in mathematics is the
importance of symbolic logic and of rigid formalism. Mathematicians,
under the influence of Weierstrass, have shown in modern times a care
for accuracy, and an aversion to slipshod reasoning, such as had not
been known among them previously since the time of the Greeks. The
great inventions of the seventeenth century--Analytical Geometry and
the Infinitesimal Calculus--were so fruitful in new results that
mathematicians had neither time nor inclination to examine their
foundations. Philosophers, who should have taken up the task, had too
little mathematical ability to invent the new branches of mathematics
which have now been found necessary for any adequate discussion. Thus
mathematicians were only awakened from their "dogmatic slumbers" when
Weierstrass and his followers showed that many of their most cherished
propositions are in general false. Macaulay, contrasting the certainty
of mathematics with the uncertainty of philosophy, asks who ever heard
of a reaction against Taylor's theorem? If he had lived now, he
himself might have heard of such a reaction, for this is precisely one
of the theorems which modern investigations have overthrown. Such rude
shocks to mathematical faith have produced that love of formalism
which appears, to those who are ignorant of its motive, to be mere
outrageous pedantry.
The proof that all pure mathematics, including Geometry, is nothing
but formal logic, is a fatal blow to the Kantian philosophy. Kant,
rightly perceiving that Euclid's propositions could not be deduced
from Euclid's axioms without the help of the figures, invented a
theory of knowledge to account for this fact; and it accounted so
successfully that, when the fact is shown to be a mere defect in
Euclid, and not a result of the nature of geometrical reasoning,
Kant's theory also has to be abandoned. The whole doctrine of _a
priori_ intuitions, by which Kant explained the possibility of pure
mathematics, is wholly inapplicable to mathematics in its present
form. The Aristotelian doctrines of the schoolmen come nearer in
spirit to the doctrines which modern mathematics inspire; but the
schoolmen were hampered by the fact that their formal logic was very
defective, and that the philosophical logic based upon the syllogism
showed a corresponding narrowness. What is now required is to give the
greatest possible development to mathematical logic, to allow to the
full the importance of relations, and then to found upon this secure
basis a new philosophical logic, which may hope to borrow some of the
exactitude and certainty of its mathematical foundation. If this can
be successfully accomplished, there is every reason to hope that the
near future will be as great an epoch in pure philosophy as the
immediate past has been in the principles of mathematics. Great
triumphs inspire great hopes; and pure thought may achieve, within our
generation, such results as will place our time, in this respect, on a
level with the greatest age of Greece.[18]
FOOTNOTES:
[11] This subject is due in the main to Mr. C.S. Peirce.
[12] I ought to have added Frege, but his writings were unknown to me
when this article was written. [Note added in 1917.]
[13] Professor of Mathematics in the University of Berlin. He died in
1897.
[14] [Note added in 1917.] Although some infinite numbers are greater
than some others, it cannot be proved that of any two infinite numbers
one must be the greater.
[15] Cantor was not guilty of a fallacy on this point. His proof that
there is no greatest number is valid. The solution of the puzzle is
complicated and depends upon the theory of types, which is explained
in _Principia Mathematica_, Vol. I (Camb. Univ. Press, 1910). [Note
added in 1917.]
[16] This must not be regarded as a historically correct account of
what Zeno actually had in mind. It is a new argument for his
conclusion, not the argument which influenced him. On this point, see
e.g. C.D. Broad, "Note on Achilles and the Tortoise," _Mind_, N.S.,
Vol. XXII, pp. 318-19. Much valuable work on the interpretation of
Zeno has been done since this article was written. [Note added in
1917.]
[17] Since the above was written, he has ceased to be used as a
textbook. But I fear many of the books now used are so bad that the
change is no great improvement. [Note added in 1917.]
[18] The greatest age of Greece was brought to an end by the
Peloponnesian War. [Note added in 1917.]
VI
ON SCIENTIFIC METHOD IN PHILOSOPHY
When we try to ascertain the motives which have led men to the
investigation of philosophical questions, we find that, broadly
speaking, they can be divided into two groups, often antagonistic, and
leading to very divergent systems. These two groups of motives are, on
the one hand, those derived from religion and ethics, and, on the
other hand, those derived from science. Plato, Spinoza, and Hegel may
be taken as typical of the philosophers whose interests are mainly
religious and ethical, while Leibniz, Locke, and Hume may be taken as
representatives of the scientific wing. In Aristotle, Descartes,
Berkeley, and Kant we find both groups of motives strongly present.
Herbert Spencer, in whose honour we are assembled to-day, would
naturally be classed among scientific philosophers: it was mainly from
science that he drew his data, his formulation of problems, and his
conception of method. But his strong religious sense is obvious in
much of his writing, and his ethical pre-occupations are what make him
value the conception of evolution--that conception in which, as a
whole generation has believed, science and morals are to be united in
fruitful and indissoluble marriage.
It is my belief that the ethical and religious motives in spite of
the splendidly imaginative systems to which they have given rise, have
been on the whole a hindrance to the progress of philosophy, and ought
now to be consciously thrust aside by those who wish to discover
philosophical truth. Science, originally, was entangled in similar
motives, and was thereby hindered in its advances. It is, I maintain,
from science, rather than from ethics and religion, that philosophy
should draw its inspiration.
But there are two different ways in which a philosophy may seek to
base itself upon science. It may emphasise the most general _results_
of science, and seek to give even greater generality and unity to
these results. Or it may study the _methods_ of science, and seek to
apply these methods, with the necessary adaptations, to its own
peculiar province. Much philosophy inspired by science has gone astray
through preoccupation with the _results_ momentarily supposed to have
been achieved. It is not results, but _methods_ that can be
transferred with profit from the sphere of the special sciences to the
sphere of philosophy. What I wish to bring to your notice is the
possibility and importance of applying to philosophical problems
certain broad principles of method which have been found successful in
the study of scientific questions.
The opposition between a philosophy guided by scientific method and a
philosophy dominated by religious and ethical ideas may be illustrated
by two notions which are very prevalent in the works of philosophers,
namely the notion of _the universe_, and the notion of _good and
evil_. A philosopher is expected to tell us something about the nature
of the universe as a whole, and to give grounds for either optimism or
pessimism. Both these expectations seem to me mistaken. I believe the
conception of "the universe" to be, as its etymology indicates, a
mere relic of pre-Copernican astronomy: and I believe the question of
optimism and pessimism to be one which the philosopher will regard as
outside his scope, except, possibly, to the extent of maintaining that
it is insoluble.
In the days before Copernicus, the conception of the "universe" was
defensible on scientific grounds: the diurnal revolution of the
heavenly bodies bound them together as all parts of one system, of
which the earth was the centre. Round this apparent scientific fact,
many human desires rallied: the wish to believe Man important in the
scheme of things, the theoretical desire for a comprehensive
understanding of the Whole, the hope that the course of nature might
be guided by some sympathy with our wishes. In this way, an ethically
inspired system of metaphysics grew up, whose anthropocentrism was
apparently warranted by the geocentrism of astronomy. When Copernicus
swept away the astronomical basis of this system of thought, it had
grown so familiar, and had associated itself so intimately with men's
aspirations, that it survived with scarcely diminished force--survived
even Kant's "Copernican revolution," and is still now the unconscious
premiss of most metaphysical systems.
The oneness of the world is an almost undiscussed postulate of most
metaphysics. "Reality is not merely one and self-consistent, but is a
system of reciprocally determinate parts"[19]--such a statement would
pass almost unnoticed as a mere truism. Yet I believe that it embodies
a failure to effect thoroughly the "Copernican revolution," and that
the apparent oneness of the world is merely the oneness of what is
seen by a single spectator or apprehended by a single mind. The
Critical Philosophy, although it intended to emphasise the subjective
element in many apparent characteristics of the world, yet, by
regarding the world in itself as unknowable, so concentrated attention
upon the subjective representation that its subjectivity was soon
forgotten. Having recognised the categories as the work of the mind,
it was paralysed by its own recognition, and abandoned in despair the
attempt to undo the work of subjective falsification. In part, no
doubt, its despair was well founded, but not, I think, in any absolute
or ultimate sense. Still less was it a ground for rejoicing, or for
supposing that the nescience to which it ought to have given rise
could be legitimately exchanged for a metaphysical dogmatism.
I
As regards our present question, namely, the question of the unity of
the world, the right method, as I think, has been indicated by William
James.[20] "Let us now turn our backs upon ineffable or unintelligible
ways of accounting for the world's oneness, and inquire whether,
instead of being a principle, the 'oneness' affirmed may not merely be
a name like 'substance' descriptive of the fact that certain _specific
and verifiable connections_ are found among the parts of the
experiential flux.... We can easily conceive of things that shall have
no connection whatever with each other. We may assume them to inhabit
different times and spaces, as the dreams of different persons do even
now. They may be so unlike and incommensurable, and so inert towards
one another, as never to jostle or interfere. Even now there may
actually be whole universes so disparate from ours that we who know
ours have no means of perceiving that they exist. We conceive their
diversity, however; and by that fact the whole lot of them form what
is known in logic as 'a universe of discourse.' To form a universe of
discourse argues, as this example shows, no further kind of connexion.
The importance attached by certain monistic writers to the fact that
any chaos may become a universe by merely being named, is to me
incomprehensible." We are thus left with two kinds of unity in the
experienced world; the one what we may call the epistemological unity,
due merely to the fact that my experienced world is what _one_
experience selects from the sum total of existence: the other that
tentative and partial unity exhibited in the prevalence of scientific
laws in those portions of the world which science has hitherto
mastered. Now a generalisation based upon either of these kinds of
unity would be fallacious. That the things which we experience have
the common property of being experienced by us is a truism from which
obviously nothing of importance can be deducible: it is clearly
fallacious to draw from the fact that whatever we experience is
experienced the conclusion that therefore everything must be
experienced. The generalisation of the second kind of unity, namely,
that derived from scientific laws, would be equally fallacious, though
the fallacy is a trifle less elementary. In order to explain it let us
consider for a moment what is called the reign of law. People often
speak as though it were a remarkable fact that the physical world is
subject to invariable laws. In fact, however, it is not easy to see
how such a world could fail to obey general laws. Taking any arbitrary
set of points in space, there is a function of the time corresponding
to these points, i.e. expressing the motion of a particle which
traverses these points: this function may be regarded as a general law
to which the behaviour of such a particle is subject. Taking all such
functions for all the particles in the universe, there will be
theoretically some one formula embracing them all, and this formula
may be regarded as the single and supreme law of the spatio-temporal
world. Thus what is surprising in physics is not the existence of
general laws, but their extreme simplicity. It is not the uniformity
of nature that should surprise us, for, by sufficient analytic
ingenuity, any conceivable course of nature might be shown to exhibit
uniformity. What should surprise us is the fact that the uniformity is
simple enough for us to be able to discover it. But it is just this
characteristic of simplicity in the laws of nature hitherto discovered
which it would be fallacious to generalise, for it is obvious that
simplicity has been a part cause of their discovery, and can,
therefore, give no ground for the supposition that other undiscovered
laws are equally simple.
The fallacies to which these two kinds of unity have given rise
suggest a caution as regards all use in philosophy of general
_results_ that science is supposed to have achieved. In the first
place, in generalising these results beyond past experience, it is
necessary to examine very carefully whether there is not some reason
making it more probable that these results should hold of all that has
been experienced than that they should hold of things universally. The
sum total of what is experienced by mankind is a selection from the
sum total of what exists, and any general character exhibited by this
selection may be due to the manner of selecting rather than to the
general character of that from which experience selects. In the second
place, the most general results of science are the least certain and
the most liable to be upset by subsequent research. In utilizing these
results as the basis of a philosophy, we sacrifice the most valuable
and remarkable characteristic of scientific method, namely, that,
although almost everything in science is found sooner or later to
require some correction, yet this correction is almost always such as
to leave untouched, or only slightly modified, the greater part of the
results which have been deduced from the premiss subsequently
discovered to be faulty. The prudent man of science acquires a certain
instinct as to the kind of uses which may be made of present
scientific beliefs without incurring the danger of complete and utter
refutation from the modifications likely to be introduced by
subsequent discoveries. Unfortunately the use of scientific
generalisations of a sweeping kind as the basis of philosophy is just
that kind of use which an instinct of scientific caution would avoid,
since, as a rule, it would only lead to true results if the
generalisation upon which it is based stood in _no_ need of
correction.
We may illustrate these general considerations by means of two
examples, namely, the conservation of energy and the principle of
evolution.
(1) Let us begin with the conservation of energy, or, as Herbert
Spencer used to call it, the persistence of force. He says:[21]
"Before taking a first step in the rational interpretation of
Evolution, it is needful to recognise, not only the facts that
Matter is indestructible and Motion continuous, but also the fact
that Force persists. An attempt to assign the _causes_ of
Evolution would manifestly be absurd if that agency to which the
metamorphosis in general and in detail is due, could either come
into existence or cease to exist. The succession of phenomena
would in such case be altogether arbitrary, and deductive Science
impossible."
This paragraph illustrates the kind of way in which the philosopher is
tempted to give an air of absoluteness and necessity to empirical
generalisations, of which only the approximate truth in the regions
hitherto investigated can be guaranteed by the unaided methods of
science. It is very often said that the persistence of something or
other is a necessary presupposition of all scientific investigation,
and this presupposition is then thought to be exemplified in some
quantity which physics declares to be constant. There are here, as it
seems to me, three distinct errors. First, the detailed scientific
investigation of nature does not _presuppose_ any such general laws as
its results are found to verify. Apart from particular observations,
science need presuppose nothing except the general principles of
logic, and these principles are not laws of nature, for they are
merely hypothetical, and apply not only to the actual world but to
whatever is _possible_. The second error consists in the
identification of a constant quantity with a persistent entity. Energy
is a certain function of a physical system, but is not a thing or
substance persisting throughout the changes of the system. The same is
true of mass, in spite of the fact that mass has often been defined as
_quantity of matter_. The whole conception of quantity, involving, as
it does, numerical measurement based largely upon conventions, is far
more artificial, far more an embodiment of mathematical convenience,
than is commonly believed by those who philosophise on physics. Thus
even if (which I cannot for a moment admit) the persistence of some
entity were among the necessary postulates of science, it would be a
sheer error to infer from this the constancy of any physical quantity,
or the _a priori_ necessity of any such constancy which may be
empirically discovered. In the third place, it has become more and
more evident with the progress of physics that large generalisations,
such as the conservation of energy or mass, are far from certain and
are very likely only approximate. Mass, which used to be regarded as
the most indubitable of physical quantities, is now generally believed
to vary according to velocity, and to be, in fact, a vector quantity
which at a given moment is different in different directions. The
detailed conclusions deduced from the supposed constancy of mass for
such motions as used to be studied in physics will remain very nearly
exact, and therefore over the field of the older investigations very
little modification of the older results is required. But as soon as
such a principle as the conservation of mass or of energy is erected
into a universal _a priori_ law, the slightest failure in absolute
exactness is fatal, and the whole philosophic structure raised upon
this foundation is necessarily ruined. The prudent philosopher,
therefore, though he may with advantage study the methods of physics,
will be very chary of basing anything upon what happen at the moment
to be the most general results apparently obtained by those methods.
(2) The philosophy of evolution, which was to be our second example,
illustrates the same tendency to hasty generalisation, and also
another sort, namely, the undue preoccupation with ethical notions.
There are two kinds of evolutionist philosophy, of which both Hegel
and Spencer represent the older and less radical kind, while
Pragmatism and Bergson represent the more modern and revolutionary
variety. But both these sorts of evolutionism have in common the
emphasis on _progress_, that is, upon a continual change from the
worse to the better, or from the simpler to the more complex. It
would be unfair to attribute to Hegel any scientific motive or
foundation, but all the other evolutionists, including Hegel's modern
disciples, have derived their impetus very largely from the history of
biological development. To a philosophy which derives a law of
universal progress from this history there are two objections. First,
that this history itself is concerned with a very small selection of
facts confined to an infinitesimal fragment of space and time, and
even on scientific grounds probably not an average sample of events in
the world at large. For we know that decay as well as growth is a
normal occurrence in the world. An extra-terrestrial philosopher, who
had watched a single youth up to the age of twenty-one and had never
come across any other human being, might conclude that it is the
nature of human beings to grow continually taller and wiser in an
indefinite progress towards perfection; and this generalisation would
be just as well founded as the generalisation which evolutionists base
upon the previous history of this planet. Apart, however, from this
scientific objection to evolutionism, there is another, derived from
the undue admixture of ethical notions in the very idea of progress
from which evolutionism derives its charm. Organic life, we are told,
has developed gradually from the protozoon to the philosopher, and
this development, we are assured, is indubitably an advance.
Unfortunately it is the philosopher, not the protozoon, who gives us
this assurance, and we can have no security that the impartial
outsider would agree with the philosopher's self-complacent
assumption. This point has been illustrated by the philosopher Chuang
Tz[)u] in the following instructive anecdote:
"The Grand Augur, in his ceremonial robes, approached the shambles
and thus addressed the pigs: 'How can you object to die? I shall
fatten you for three months. I shall discipline myself for ten
days and fast for three. I shall strew fine grass, and place you
bodily upon a carved sacrificial dish. Does not this satisfy you?'
Then, speaking from the pigs' point of view, he continued: 'It is
better, perhaps, after all, to live on bran and escape the
shambles....'
'But then,' added he, speaking from his own point of view, 'to
enjoy honour when alive one would readily die on a war-shield or
in the headsman's basket.'
So he rejected the pigs' point of view and adopted his own point
of view. In what sense, then, was he different from the pigs?"
I much fear that the evolutionists too often resemble the Grand Augur
and the pigs.
The ethical element which has been prominent in many of the most
famous systems of philosophy is, in my opinion, one of the most
serious obstacles to the victory of scientific method in the
investigation of philosophical questions. Human ethical notions, as
Chuang Tz[)u] perceived, are essentially anthropocentric, and involve,
when used in metaphysics, an attempt, however veiled, to legislate for
the universe on the basis of the present desires of men. In this way
they interfere with that receptivity to fact which is the essence of
the scientific attitude towards the world. To regard ethical notions
as a key to the understanding of the world is essentially
pre-Copernican. It is to make man, with the hopes and ideals which he
happens to have at the present moment, the centre of the universe and
the interpreter of its supposed aims and purposes. Ethical metaphysics
is fundamentally an attempt, however disguised, to give legislative
force to our own wishes. This may, of course, be questioned, but I
think that it is confirmed by a consideration of the way in which
ethical notions arise. Ethics is essentially a product of the
gregarious instinct, that is to say, of the instinct to co-operate
with those who are to form our own group against those who belong to
other groups. Those who belong to our own group are good; those who
belong to hostile groups are wicked. The ends which are pursued by our
own group are desirable ends, the ends pursued by hostile groups are
nefarious. The subjectivity of this situation is not apparent to the
gregarious animal, which feels that the general principles of justice
are on the side of its own herd. When the animal has arrived at the
dignity of the metaphysician, it invents ethics as the embodiment of
its belief in the justice of its own herd. So the Grand Augur invokes
ethics as the justification of Augurs in their conflicts with pigs.
But, it may be said, this view of ethics takes no account of such
truly ethical notions as that of self-sacrifice. This, however, would
be a mistake. The success of gregarious animals in the struggle for
existence depends upon co-operation within the herd, and co-operation
requires sacrifice, to some extent, of what would otherwise be the
interest of the individual. Hence arises a conflict of desires and
instincts, since both self-preservation and the preservation of the
herd are biological ends to the individual. Ethics is in origin the
art of recommending to others the sacrifices required for co-operation
with oneself. Hence, by reflexion, it comes, through the operation of
social justice, to recommend sacrifices by oneself, but all ethics,
however refined, remains more or less subjective. Even vegetarians do
not hesitate, for example, to save the life of a man in a fever,
although in doing so they destroy the lives of many millions of
microbes. The view of the world taken by the philosophy derived from
ethical notions is thus never impartial and therefore never fully
scientific. As compared with science, it fails to achieve the
imaginative liberation from self which is necessary to such
understanding of the world as man can hope to achieve, and the
philosophy which it inspires is always more or less parochial, more or
less infected with the prejudices of a time and a place.
I do not deny the importance or value, within its own sphere, of the
kind of philosophy which is inspired by ethical notions. The ethical
work of Spinoza, for example, appears to me of the very highest
significance, but what is valuable in such work is not any
metaphysical theory as to the nature of the world to which it may give
rise, nor indeed anything which can be proved or disproved by
argument. What is valuable is the indication of some new way of
feeling towards life and the world, some way of feeling by which our
own existence can acquire more of the characteristics which we must
deeply desire. The value of such work, howeve
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