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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 amœba to Man
appeared to the philosophers to be obviously a progress--though
whether the amœba 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
Tzŭ 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 Tzŭ 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, however immeasurable it is,
belongs with practice and not with theory. Such theoretic importance
as it may possess is only in relation to human nature, not in relation
to the world at large. The scientific philosophy, therefore, which
aims only at understanding the world and not directly at any other
improvement of human life, cannot take account of ethical notions
without being turned aside from that submission to fact which is the
essence of the scientific temper.


II

If the notion of the universe and the notion of good and evil are
extruded from scientific philosophy, it may be asked what specific
problems remain for the philosopher as opposed to the man of science?
It would be difficult to give a precise answer to this question, but
certain characteristics may be noted as distinguishing the province of
philosophy from that of the special sciences.

In the first place a philosophical proposition must be general. It
must not deal specially with things on the surface of the earth, or
with the solar system, or with any other portion of space and time. It
is this need of generality which has led to the belief that philosophy
deals with the universe as a whole. I do not believe that this belief
is justified, but I do believe that a philosophical proposition must
be applicable to everything that exists or may exist. It might be
supposed that this admission would be scarcely distinguishable from
the view which I wish to reject. This, however, would be an error, and
an important one. The traditional view would make the universe itself
the subject of various predicates which could not be applied to any
particular thing in the universe, and the ascription of such peculiar
predicates to the universe would be the special business of
philosophy. I maintain, on the contrary, that there are no
propositions of which the "universe" is the subject; in other words,
that there is no such thing as the "universe." What I do maintain is
that there are general propositions which may be asserted of each
individual thing, such as the propositions of logic. This does not
involve that all the things there are form a whole which could be
regarded as another thing and be made the subject of predicates. It
involves only the assertion that there are properties which belong to
each separate thing, not that there are properties belonging to the
whole of things collectively. The philosophy which I wish to advocate
may be called logical atomism or absolute pluralism, because, while
maintaining that there are many things, it denies that there is a
whole composed of those things. We shall see, therefore, that
philosophical propositions, instead of being concerned with the whole
of things collectively, are concerned with all things distributively;
and not only must they be concerned with all things, but they must be
concerned with such properties of all things as do not depend upon the
accidental nature of the things that there happen to be, but are true
of any possible world, independently of such facts as can only be
discovered by our senses.

This brings us to a second characteristic of philosophical
propositions, namely, that they must be _a priori_. A philosophical
proposition must be such as can be neither proved nor disproved by
empirical evidence. Too often we find in philosophical books arguments
based upon the course of history, or the convolutions of the brain, or
the eyes of shell-fish. Special and accidental facts of this kind are
irrelevant to philosophy, which must make only such assertions as
would be equally true however the actual world were constituted.

We may sum up these two characteristics of philosophical propositions
by saying that _philosophy is the science of the possible_. But this
statement unexplained is liable to be misleading, since it may be
thought that the possible is something other than the general, whereas
in fact the two are indistinguishable.

Philosophy, if what has been said is correct, becomes indistinguishable
from logic as that word has now come to be used. The study of logic
consists, broadly speaking, of two not very sharply distinguished
portions. On the one hand it is concerned with those general statements
which can be made concerning everything without mentioning any one
thing or predicate or relation, such for example as "if _x_ is a member
of the class α and every member of α is a member of β, then _x_ is a
member of the class β, whatever _x_, α, and β may be." On the other
hand, it is concerned with the analysis and enumeration of logical
_forms_, i.e. with the kinds of propositions that may occur, with the
various types of facts, and with the classification of the constituents
of facts. In this way logic provides an inventory of possibilities, a
repertory of abstractly tenable hypotheses.

It might be thought that such a study would be too vague and too
general to be of any very great importance, and that, if its problems
became at any point sufficiently definite, they would be merged in the
problems of some special science. It appears, however, that this is
not the case. In some problems, for example, the analysis of space and
time, the nature of perception, or the theory of judgment, the
discovery of the logical form of the facts involved is the hardest
part of the work and the part whose performance has been most lacking
hitherto. It is chiefly for want of the right logical hypothesis that
such problems have hitherto been treated in such an unsatisfactory
manner, and have given rise to those contradictions or antinomies in
which the enemies of reason among philosophers have at all times
delighted.

By concentrating attention upon the investigation of logical forms, it
becomes possible at last for philosophy to deal with its problems
piecemeal, and to obtain, as the sciences do, such partial and
probably not wholly correct results as subsequent investigation can
utilise even while it supplements and improves them. Most
philosophies hitherto have been constructed all in one block, in such
a way that, if they were not wholly correct, they were wholly
incorrect, and could not be used as a basis for further
investigations. It is chiefly owing to this fact that philosophy,
unlike science, has hitherto been unprogressive, because each original
philosopher has had to begin the work again from the beginning,
without being able to accept anything definite from the work of his
predecessors. A scientific philosophy such as I wish to recommend will
be piecemeal and tentative like other sciences; above all, it will be
able to invent hypotheses which, even if they are not wholly true,
will yet remain fruitful after the necessary corrections have been
made. This possibility of successive approximations to the truth is,
more than anything else, the source of the triumphs of science, and to
transfer this possibility to philosophy is to ensure a progress in
method whose importance it would be almost impossible to exaggerate.

The essence of philosophy as thus conceived is analysis, not
synthesis. To build up systems of the world, like Heine's German
professor who knit together fragments of life and made an intelligible
system out of them, is not, I believe, any more feasible than the
discovery of the philosopher's stone. What is feasible is the
understanding of general forms, and the division of traditional
problems into a number of separate and less baffling questions.
"Divide and conquer" is the maxim of success here as elsewhere.

Let us illustrate these somewhat general maxims by examining their
application to the philosophy of space, for it is only in application
that the meaning or importance of a method can be understood. Suppose
we are confronted with the problem of space as presented in Kant's
Transcendental Æsthetic, and suppose we wish to discover what are the
elements of the problem and what hope there is of obtaining a solution
of them. It will soon appear that three entirely distinct problems,
belonging to different studies, and requiring different methods for
their solution, have been confusedly combined in the supposed single
problem with which Kant is concerned. There is a problem of logic, a
problem of physics, and a problem of theory of knowledge. Of these
three, the problem of logic can be solved exactly and perfectly; the
problem of physics can probably be solved with as great a degree of
certainty and as great an approach to exactness as can be hoped in an
empirical region; the problem of theory of knowledge, however, remains
very obscure and very difficult to deal with. Let us see how these
three problems arise.

(1) The logical problem has arisen through the suggestions of
non-Euclidean geometry. Given a body of geometrical propositions, it
is not difficult to find a minimum statement of the axioms from which
this body of propositions can be deduced. It is also not difficult, by
dropping or altering some of these axioms, to obtain a more general or
a different geometry, having, from the point of view of pure
mathematics, the same logical coherence and the same title to respect
as the more familiar Euclidean geometry. The Euclidean geometry itself
is true perhaps of actual space (though this is doubtful), but
certainly of an infinite number of purely arithmetical systems, each
of which, from the point of view of abstract logic, has an equal and
indefeasible right to be called a Euclidean space. Thus space as an
object of logical or mathematical study loses its uniqueness; not only
are there many kinds of spaces, but there are an infinity of examples
of each kind, though it is difficult to find any kind of which the
space of physics may be an example, and it is impossible to find any
kind of which the space of physics is certainly an example. As an
illustration of one possible logical system of geometry we may
consider all relations of three terms which are analogous in certain
formal respects to the relation "between" as it appears to be in
actual space. A space is then defined by means of one such three-term
relation. The points of the space are all the terms which have this
relation to something or other, and their order in the space in
question is determined by this relation. The points of one space are
necessarily also points of other spaces, since there are necessarily
other three-term relations having those same points for their field.
The space in fact is not determined by the class of its points, but by
the ordering three-term relation. When enough abstract logical
properties of such relations have been enumerated to determine the
resulting kind of geometry, say, for example, Euclidean geometry, it
becomes unnecessary for the pure geometer in his abstract capacity to
distinguish between the various relations which have all these
properties. He considers the whole class of such relations, not any
single one among them. Thus in studying a given kind of geometry the
pure mathematician is studying a certain class of relations defined by
means of certain abstract logical properties which take the place of
what used to be called axioms. The nature of geometrical _reasoning_
therefore is purely deductive and purely logical; if any special
epistemological peculiarities are to be found in geometry, it must not
be in the reasoning, but in our knowledge concerning the axioms in
some given space.

(2) The physical problem of space is both more interesting and more
difficult than the logical problem. The physical problem may be
stated as follows: to find in the physical world, or to construct from
physical materials, a space of one of the kinds enumerated by the
logical treatment of geometry. This problem derives its difficulty
from the attempt to accommodate to the roughness and vagueness of the
real world some system possessing the logical clearness and exactitude
of pure mathematics. That this can be done with a certain degree of
approximation is fairly evident If I see three people _A_, _B_, and
_C_ sitting in a row, I become aware of the fact which may be
expressed by saying that _B_ is between _A_ and _C_ rather than that
_A_ is between _B_ and _C_, or _C_ is between _A_ and _B_. This
relation of "between" which is thus perceived to hold has some of the
abstract logical properties of those three-term relations which, we
saw, give rise to a geometry, but its properties fail to be exact, and
are not, as empirically given, amenable to the kind of treatment at
which geometry aims. In abstract geometry we deal with points,
straight lines, and planes; but the three people _A_, _B_, and _C_
whom I see sitting in a row are not exactly points, nor is the row
exactly a straight line. Nevertheless physics, which formally assumes
a space containing points, straight lines, and planes, is found
empirically to give results applicable to the sensible world. It must
therefore be possible to find an interpretation of the points,
straight lines, and planes of physics in terms of physical data, or at
any rate in terms of data together with such hypothetical additions as
seem least open to question. Since all data suffer from a lack of
mathematical precision through being of a certain size and somewhat
vague in outline, it is plain that if such a notion as that of a point
is to find any application to empirical material, the point must be
neither a datum nor a hypothetical addition to data, but a
_construction_ by means of data with their hypothetical additions. It
is obvious that any hypothetical filling out of data is less dubious
and unsatisfactory when the additions are closely analogous to data
than when they are of a radically different sort. To assume, for
example, that objects which we see continue, after we have turned away
our eyes, to be more or less analogous to what they were while we were
looking, is a less violent assumption than to assume that such objects
are composed of an infinite number of mathematical points. Hence in
the physical study of the geometry of physical space, points must not
be assumed _ab initio_ as they are in the logical treatment of
geometry, but must be constructed as systems composed of data and
hypothetical analogues of data. We are thus led naturally to define a
physical point as a certain class of those objects which are the
ultimate constituents of the physical world. It will be the class of
all those objects which, as one would naturally say, _contain_ the
point. To secure a definition giving this result, without previously
assuming that physical objects are composed of points, is an agreeable
problem in mathematical logic. The solution of this problem and the
perception of its importance are due to my friend Dr. Whitehead. The
oddity of regarding a point as a class of physical entities wears off
with familiarity, and ought in any case not to be felt by those who
maintain, as practically every one does, that points are mathematical
fictions. The word "fiction" is used glibly in such connexions by many
men who seem not to feel the necessity of explaining how it can come
about that a fiction can be so useful in the study of the actual world
as the points of mathematical physics have been found to be. By our
definition, which regards a point as a class of physical objects, it
is explained both how the use of points can lead to important
physical results, and how we can nevertheless avoid the assumption
that points are themselves entities in the physical world.

Many of the mathematically convenient properties of abstract logical
spaces cannot be either known to belong or known not to belong to the
space of physics. Such are all the properties connected with continuity.
For to know that actual space has these properties would require an
infinite exactness of sense-perception. If actual space is continuous,
there are nevertheless many possible non-continuous spaces which will be
empirically indistinguishable from it; and, conversely, actual space may
be non-continuous and yet empirically indistinguishable from a possible
continuous space. Continuity, therefore, though obtainable in the _a
priori_ region of arithmetic, is not with certainty obtainable in the
space or time of the physical world: whether these are continuous or not
would seem to be a question not only unanswered but for ever
unanswerable. From the point of view of philosophy, however, the
discovery that a question is unanswerable is as complete an answer as
any that could possibly be obtained. And from the point of view of
physics, where no empirical means of distinction can be found, there can
be no empirical objection to the mathematically simplest assumption,
which is that of continuity.

The subject of the physical theory of space is a very large one,
hitherto little explored. It is associated with a similar theory of
time, and both have been forced upon the attention of philosophically
minded physicists by the discussions which have raged concerning the
theory of relativity.

(3) The problem with which Kant is concerned in the Transcendental
Æsthetic is primarily the epistemological problem: "How do we come to
have knowledge of geometry _a priori_?" By the distinction between the
logical and physical problems of geometry, the bearing and scope of
this question are greatly altered. Our knowledge of pure geometry is
_a priori_ but is wholly logical. Our knowledge of physical geometry
is synthetic, but is not _a priori_. Our knowledge of pure geometry is
hypothetical, and does not enable us to assert, for example, that the
axiom of parallels is true in the physical world. Our knowledge of
physical geometry, while it does enable us to assert that this axiom
is approximately verified, does not, owing to the inevitable
inexactitude of observation, enable us to assert that it is verified
_exactly_. Thus, with the separation which we have made between pure
geometry and the geometry of physics, the Kantian problem collapses.
To the question, "How is synthetic _a priori_ knowledge possible?" we
can now reply, at any rate so far as geometry is concerned, "It is not
possible," if "synthetic" means "not deducible from logic alone." Our
knowledge of geometry, like the rest of our knowledge, is derived
partly from logic, partly from sense, and the peculiar position which
in Kant's day geometry appeared to occupy is seen now to be a
delusion. There are still some philosophers, it is true, who maintain
that our knowledge that the axiom of parallels, for example, is true
of actual space, is not to be accounted for empirically, but is as
Kant maintained derived from an _a priori_ intuition. This position is
not logically refutable, but I think it loses all plausibility as soon
as we realise how complicated and derivative is the notion of physical
space. As we have seen, the application of geometry to the physical
world in no way demands that there should really be points and
straight lines among physical entities. The principle of economy,
therefore, demands that we should abstain from assuming the existence
of points and straight lines. As soon, however, as we accept the view
that points and straight lines are complicated constructions by means
of classes of physical entities, the hypothesis that we have an _a
priori_ intuition enabling us to know what happens to straight lines
when they are produced indefinitely becomes extremely strained and
harsh; nor do I think that such an hypothesis would ever have arisen
in the mind of a philosopher who had grasped the nature of physical
space. Kant, under the influence of Newton, adopted, though with some
vacillation, the hypothesis of absolute space, and this hypothesis,
though logically unobjectionable, is removed by Occam's razor, since
absolute space is an unnecessary entity in the explanation of the
physical world. Although, therefore, we cannot refute the Kantian
theory of an _a priori_ intuition, we can remove its grounds one by
one through an analysis of the problem. Thus, here as in many other
philosophical questions, the analytic method, while not capable of
arriving at a demonstrative result, is nevertheless capable of showing
that all the positive grounds in favour of a certain theory are
fallacious and that a less unnatural theory is capable of accounting
for the facts.

Another question by which the capacity of the analytic method can be
shown is the question of realism. Both those who advocate and those
who combat realism seem to me to be far from clear as to the nature of
the problem which they are discussing. If we ask: "Are our objects of
perception _real_ and are they _independent_ of the percipient?" it
must be supposed that we attach some meaning to the words "real" and
"independent," and yet, if either side in the controversy of realism
is asked to define these two words, their answer is pretty sure to
embody confusions such as logical analysis will reveal.

Let us begin with the word "real." There certainly are objects of
perception, and therefore, if the question whether these objects are
real is to be a substantial question, there must be in the world two
sorts of objects, namely, the real and the unreal, and yet the unreal
is supposed to be essentially what there is not. The question what
properties must belong to an object in order to make it real is one to
which an adequate answer is seldom if ever forthcoming. There is of
course the Hegelian answer, that the real is the self-consistent and
that nothing is self-consistent except the Whole; but this answer,
true or false, is not relevant in our present discussion, which moves
on a lower plane and is concerned with the status of objects of
perception among other objects of equal fragmentariness. Objects of
perception are contrasted, in the discussions concerning realism,
rather with psychical states on the one hand and matter on the other
hand than with the all-inclusive whole of things. The question we have
therefore to consider is the question as to what can be meant by
assigning "reality" to some but not all of the entities that make up
the world. Two elements, I think, make up what is felt rather than
thought when the word "reality" is used in this sense. A thing is real
if it persists at times when it is not perceived; or again, a thing is
real when it is correlated with other things in a way which experience
has led us to expect. It will be seen that reality in either of these
senses is by no means necessary to a thing, and that in fact there
might be a whole world in which nothing was real in either of these
senses. It might turn out that the objects of perception failed of
reality in one or both of these respects, without its being in any way
deducible that they are not parts of the external world with which
physics deals. Similar remarks will apply to the word "independent."
Most of the associations of this word are bound up with ideas as to
causation which it is not now possible to maintain. _A_ is independent
of _B_ when _B_ is not an indispensable part of the _cause_ of _A_.
But when it is recognised that causation is nothing more than
correlation, and that there are correlations of simultaneity as well
as of succession, it becomes evident that there is no uniqueness in a
series of casual antecedents of a given event, but that, at any point
where there is a correlation of simultaneity, we can pass from one
line of antecedents to another in order to obtain a new series of
causal antecedents. It will be necessary to specify the causal law
according to which the antecedents are to be considered. I received a
letter the other day from a correspondent who had been puzzled by
various philosophical questions. After enumerating them he says:
"These questions led me from Bonn to Strassburg, where I found
Professor Simmel." Now, it would be absurd to deny that these
questions caused his body to move from Bonn to Strassburg, and yet it
must be supposed that a set of purely mechanical antecedents could
also be found which would account for this transfer of matter from one
place to another. Owing to this plurality of causal series antecedent
to a given event, the notion of _the_ cause becomes indefinite, and
the question of independence becomes correspondingly ambiguous. Thus,
instead of asking simply whether _A_ is independent of _B_, we ought
to ask whether there is a series determined by such and such causal
laws leading from _B_ to _A_. This point is important in connexion
with the particular question of objects of perception. It may be that
no objects quite like those which we perceive ever exist unperceived;
in this case there will be a causal law according to which objects of
perception are not independent of being perceived. But even if this be
the case, it may nevertheless also happen that there are purely
physical causal laws determining the occurrence of objects which are
perceived by means of other objects which perhaps are not perceived.
In that case, in regard to such causal laws objects of perception will
be independent of being perceived. Thus the question whether objects
of perception are independent of being perceived is, as it stands,
indeterminate, and the answer will be yes or no according to the
method adopted of making it determinate. I believe that this confusion
has borne a very large part in prolonging the controversies on this
subject, which might well have seemed capable of remaining for ever
undecided. The view which I should wish to advocate is that objects of
perception do not persist unchanged at times when they are not
perceived, although probably objects more or less resembling them do
exist at such times; that objects of perception are part, and the only
empirically knowable part, of the actual subject-matter of physics,
and are themselves properly to be called physical; that purely
physical laws exist determining the character and duration of objects
of perception without any reference to the fact that they are
perceived; and that in the establishment of such laws the propositions
of physics do not presuppose any propositions of psychology or even
the existence of mind. I do not know whether realists would recognise
such a view as realism. All that I should claim for it is, that it
avoids difficulties which seem to me to beset both realism and
idealism as hitherto advocated, and that it avoids the appeal which
they have made to ideas which logical analysis shows to be ambiguous.
A further defence and elaboration of the positions which I advocate,
but for which time is lacking now, will be found indicated in my book
on _Our Knowledge of the External World_.[22]

The adoption of scientific method in philosophy, if I am not mistaken,
compels us to abandon the hope of solving many of the more ambitious
and humanly interesting problems of traditional philosophy. Some of
these it relegates, though with little expectation of a successful
solution, to special sciences, others it shows to be such as our
capacities are essentially incapable of solving. But there remain a
large number of the recognised problems of philosophy in regard to
which the method advocated gives all those advantages of division into
distinct questions, of tentative, partial, and progressive advance,
and of appeal to principles with which, independently of temperament,
all competent students must agree. The failure of philosophy hitherto
has been due in the main to haste and ambition: patience and modesty,
here as in other sciences, will open the road to solid and durable
progress.

FOOTNOTES:

[19] Bosanquet, _Logic_, ii, p. 211.

[20] _Some Problems of Philosophy_, p 124.

[21] _First Principles_ (1862), Part II, beginning of chap. viii.

[22] Open Court Company, 1914.




VII

THE ULTIMATE CONSTITUENTS OF MATTER[23]


I wish to discuss in this article no less a question than the ancient
metaphysical query, "What is matter?" The question, "What is matter?"
in so far as it concerns philosophy, is, I think, already capable of
an answer which in principle will be as complete as an answer can hope
to be; that is to say, we can separate the problem into an essentially
soluble and an essentially insoluble portion, and we can now see how
to solve the essentially soluble portion, at least as regards its main
outlines. It is these outlines which I wish to suggest in the present
article. My main position, which is realistic, is, I hope and believe,
not remote from that of Professor Alexander, by whose writings on this
subject I have profited greatly.[24] It is also in close accord with
that of Dr. Nunn.[25]

Common sense is accustomed to the division of the world into mind and
matter. It is supposed by all who have never studied philosophy that
the distinction between mind and matter is perfectly clear and easy,
that the two do not at any point overlap, and that only a fool or a
philosopher could be in doubt as to whether any given entity is mental
or material. This simple faith survives in Descartes and in a
somewhat modified form in Spinoza, but with Leibniz it begins to
disappear, and from his day to our own almost every philosopher of
note has criticised and rejected the dualism of common sense. It is my
intention in this article to defend this dualism; but before defending
it we must spend a few moments on the reasons which have prompted its
rejection.

Our knowledge of the material world is obtained by means of the
senses, of sight and touch and so on. At first it is supposed that
things are just as they seem, but two opposite sophistications soon
destroy this naïve belief. On the one hand the physicists cut up
matter into molecules, atoms, corpuscles, and as many more such
subdivisions as their future needs may make them postulate, and the
units at which they arrive are uncommonly different from the visible,
tangible objects of daily life. A unit of matter tends more and more
to be something like an electromagnetic field filling all space,
though having its greatest intensity in a small region. Matter
consisting of such elements is as remote from daily life as any
metaphysical theory. It differs from the theories of metaphysicians
only in the fact that its practical efficacy proves that it contains
some measure of truth and induces business men to invest money on the
strength of it; but, in spite of its connection with the money market,
it remains a metaphysical theory none the less.

The second kind of sophistication to which the world of common sense
has been subjected is derived from the psychologists and
physiologists. The physiologists point out that what we see depends
upon the eye, that what we hear depends upon the ear, and that all our
senses are liable to be affected by anything which affects the brain,
like alcohol or hasheesh. Psychologists point out how much of what we
think we see is supplied by association or unconscious inference, how
much is mental interpretation, and how doubtful is the residuum which
can be regarded as crude datum. From these facts it is argued by the
psychologists that the notion of a datum passively received by the
mind is a delusion, and it is argued by the physiologists that even if
a pure datum of sense could be obtained by the analysis of experience,
still this datum could not belong, as common sense supposes, to the
outer world, since its whole nature is conditioned by our nerves and
sense organs, changing as they change in ways which it is thought
impossible to connect with any change in the matter supposed to be
perceived. This physiologist's argument is exposed to the rejoinder,
more specious than solid, that our knowledge of the existence of the
sense organs and nerves is obtained by that very process which the
physiologist has been engaged in discrediting, since the existence of
the nerves and sense organs is only known through the evidence of the
senses themselves. This argument may prove that some reinterpretation
of the results of physiology is necessary before they can acquire
metaphysical validity. But it does not upset the physiological
argument in so far as this constitutes merely a _reductio ad absurdum_
of naïve realism.

These various lines of argument prove, I think, that some part of the
beliefs of common sense must be abandoned. They prove that, if we take
these beliefs as a whole, we are forced into conclusions which are in
part self-contradictory; but such arguments cannot of themselves
decide what portion of our common-sense beliefs is in need of
correction. Common sense believes that what we see is physical,
outside the mind, and continuing to exist if we shut our eyes or turn
them in another direction. I believe that common sense is right in
regarding what we see as physical and (in one of several possible
senses) outside the mind, but is probably wrong in supposing that it
continues to exist when we are no longer looking at it. It seems to me
that the whole discussion of matter has been obscured by two errors
which support each other. The first of these is the error that what we
see, or perceive through any of our other senses, is subjective: the
second is the belief that what is physical must be persistent.
Whatever physics may regard as the ultimate constituents of matter, it
always supposes these constituents to be indestructible. Since the
immediate data of sense are not indestructible but in a state of
perpetual flux, it is argued that these data themselves cannot be
among the ultimate constituents of matter. I believe this to be a
sheer mistake. The persistent particles of mathematical physics I
regard as logical constructions, symbolic fictions enabling us to
express compendiously very complicated assemblages of facts; and, on
the other hand, I believe that the actual data in sensation, the
immediate objects of sight or touch or hearing, are extra-mental,
purely physical, and among the ultimate constituents of matter.

My meaning in regard to the impermanence of physical entities may
perhaps be made clearer by the use of Bergson's favourite illustration
of the cinematograph. When I first read Bergson's statement that the
mathematician conceives the world after the analogy of a
cinematograph, I had never seen a cinematograph, and my first visit to
one was determined by the desire to verify Bergson's statement, which
I found to be completely true, at least so far as I am concerned.
When, in a picture palace, we see a man rolling down hill, or running
away from the police, or falling into a river, or doing any of those
other things to which men in such places are addicted, we know that
there is not really only one man moving, but a succession of films,
each with a different momentary man. The illusion of persistence
arises only through the approach to continuity in the series of
momentary men. Now what I wish to suggest is that in this respect the
cinema is a better metaphysician than common sense, physics, or
philosophy. The real man too, I believe, however the police may swear
to his identity, is really a series of momentary men, each different
one from the other, and bound together, not by a numerical identity,
but by continuity and certain intrinsic causal laws. And what applies
to men applies equally to tables and chairs, the sun, moon and stars.
Each of these is to be regarded, not as one single persistent entity,
but as a series of entities succeeding each other in time, each
lasting for a very brief period, though probably not for a mere
mathematical instant. In saying this I am only urging the same kind of
division in time as we are accustomed to acknowledge in the case of
space. A body which fills a cubic foot will be admitted to consist of
many smaller bodies, each occupying only a very tiny volume; similarly
a thing which persists for an hour is to be regarded as composed of
many things of less duration. A true theory of matter requires a
division of things into time-corpuscles as well as into
space-corpuscles.

The world may be conceived as consisting of a multitude of entities
arranged in a certain pattern. The entities which are arranged I shall
call "particulars." The arrangement or pattern results from relations
among particulars. Classes or series of particulars, collected
together on account of some property which makes it convenient to be
able to speak of them as wholes, are what I call logical constructions
or symbolic fictions. The particulars are to be conceived, not on the
analogy of bricks in a building, but rather on the analogy of notes
in a symphony. The ultimate constituents of a symphony (apart from
relations) are the notes, each of which lasts only for a very short
time. We may collect together all the notes played by one instrument:
these may be regarded as the analogues of the successive particulars
which common sense would regard as successive states of one "thing."
But the "thing" ought to be regarded as no more "real" or
"substantial" than, for example, the rôle of the trombone. As soon as
"things" are conceived in this manner it will be found that the
difficulties in the way of regarding immediate objects of sense as
physical have largely disappeared.

When people ask, "Is the object of sense mental or physical?" they
seldom have any clear idea either what is meant by "mental" or
"physical," or what criteria are to be applied for deciding whether a
given entity belongs to one class or the other. I do not know how to
give a sharp definition of the word "mental," but something may be
done by enumerating occurrences which are indubitably mental:
believing, doubting, wishing, willing, being pleased or pained, are
certainly mental occurrences; so are what we may call experiences,
seeing, hearing, smelling, perceiving generally. But it does not
follow from this that what is seen, what is heard, what is smelt, what
is perceived, must be mental. When I see a flash of lightning, my
seeing of it is mental, but what I see, although it is not quite the
same as what anybody else sees at the same moment, and although it
seems very unlike what the physicist would describe as a flash of
lightning, is not mental. I maintain, in fact, that if the physicist
could describe truly and fully all that occurs in the physical world
when there is a flash of lightning, it would contain as a constituent
what I see, and also what is seen by anybody else who would commonly
be said to see the same flash. What I mean may perhaps be made plainer
by saying that if my body could remain in exactly the same state in
which it is, although my mind had ceased to exist, precisely that
object which I now see when I see the flash would exist, although of
course I should not see it, since my seeing is mental. The principal
reasons which have led people to reject this view have, I think, been
two: first, that they did not adequately distinguish between my seeing
and what I see; secondly, that the causal dependence of what I see
upon my body has made people suppose that what I see cannot be
"outside" me. The first of these reasons need not detain us, since the
confusion only needs to be pointed out in order to be obviated; but
the second requires some discussion, since it can only be answered by
removing current misconceptions, on the one hand as to the nature of
space, and on the other, as to the meaning of causal dependence.

When people ask whether colours, for example, or other secondary
qualities are inside or outside the mind, they seem to suppose that
their meaning must be clear, and that it ought to be possible to say
yes or no without any further discussion of the terms involved. In
fact, however, such terms as "inside" or "outside" are very ambiguous.
What is meant by asking whether this or that is "in" the mind? The
mind is not like a bag or a pie; it does not occupy a certain region
in space, or, if (in a sense) it does, what is in that region is
presumably part of the brain, which would not be said to be in the
mind. When people say that sensible qualities are in the mind, they do
not mean "spatially contained in" in the sense in which the blackbirds
were in the pie. We might regard the mind as an assemblage of
particulars, namely, what would be called "states of mind," which
would belong together in virtue of some specific common quality. The
common quality of all states of mind would be the quality designated
by the word "mental"; and besides this we should have to suppose that
each separate person's states of mind have some common characteristic
distinguishing them from the states of mind of other people. Ignoring
this latter point, let us ask ourselves whether the quality designated
by the word "mental" does, as a matter of observation, actually belong
to objects of sense, such as colours or noises. I think any candid
person must reply that, however difficult it may be to know what we
mean by "mental," it is not difficult to see that colours and noises
are not mental in the sense of having that intrinsic peculiarity which
belongs to beliefs and wishes and volitions, but not to the physical
world. Berkeley advances on this subject a plausible argument[26]
which seems to me to rest upon an ambiguity in the word "pain." He
argues that the realist supposes the heat which he feels in
approaching a fire to be something outside his mind, but that as he
approaches nearer and nearer to the fire the sensation of heat passes
imperceptibly into pain, and that no one could regard pain as
something outside the mind. In reply to this argument, it should be
observed in the first place that the heat of which we are immediately
aware is not in the fire but in our own body. It is only by inference
that the fire is judged to be the cause of the heat which we feel in
our body. In the second place (and this is the more important point),
when we speak of pain we may mean one of two things: we may mean the
object of the sensation or other experience which has the quality of
being painful, or we may mean the quality of painfulness itself. When
a man says he has a pain in his great toe, what he means is that he
has a sensation associated with his great toe and having the quality
of painfulness. The sensation itself, like every sensation, consists
in experiencing a sensible object, and the experiencing has that
quality of painfulness which only mental occurrences can have, but
which may belong to thoughts or desires, as well as to sensations. But
in common language we speak of the sensible object experienced in a
painful sensation as a pain, and it is this way of speaking which
causes the confusion upon which the plausibility of Berkeley's
argument depends. It would be absurd to attribute the quality of
painfulness to anything non-mental, and hence it comes to be thought
that what we call a pain in the toe must be mental. In fact, however,
it is not the sensible object in such a case which is painful, but the
sensation, that is to say, the experience of the sensible object. As
the heat which we experience from the fire grows greater, the
experience passes gradually from being pleasant to being painful, but
neither the pleasure nor the pain is a quality of the object
experienced as opposed to the experience, and it is therefore a
fallacy to argue that this object must be mental on the ground that
painfulness can only be attributed to what is mental.

If, then, when we say that something is in the mind we mean that it
has a certain recognisable intrinsic characteristic such as belongs to
thoughts and desires, it must be maintained on grounds of immediate
inspection that objects of sense are not in any mind.

A different meaning of "in the mind" is, however, to be inferred from
the arguments advanced by those who regard sensible objects as being
in the mind. The arguments used are, in the main, such as would prove
the causal dependence of objects of sense upon the percipient. Now
the notion of causal dependence is very obscure and difficult, much
more so in fact than is generally realised by philosophers. I shall
return to this point in a moment. For the present, however, accepting
the notion of causal dependence without criticism, I wish to urge that
the dependence in question is rather upon our bodies than upon our
minds. The visual appearance of an object is altered if we shut one
eye, or squint, or look previously at something dazzling; but all
these are bodily acts, and the alterations which they effect are to be
explained by physiology and optics, not by psychology.[27] They are in
fact of exactly the same kind as the alterations effected by
spectacles or a microscope. They belong therefore to the theory of the
physical world, and can have no bearing upon the question whether what
we see is causally dependent upon the mind. What they do tend to
prove, and what I for my part have no wish to deny, is that what we
see is causally dependent upon our body and is not, as crude common
sense would suppose, something which would exist equally if our eyes
and nerves and brain were absent, any more than the visual appearance
presented by an object seen through a microscope would remain if the
microscope were removed. So long as it is supposed that the physical
world is composed of stable and more or less permanent constituents,
the fact that what we see is changed by changes in our body appears to
afford reason for regarding what we see as not an ultimate constituent
of matter. But if it is recognised that the ultimate constituents of
matter are as circumscribed in duration as in spatial extent, the
whole of this difficulty vanishes.

There remains, however, another difficulty, connected with space. When
we look at the sun we wish to know something about the sun itself,
which is ninety-three million miles away; but what we see is dependent
upon our eyes, and it is difficult to suppose that our eyes can affect
what happens at a distance of ninety-three million miles. Physics
tells us that certain electromagnetic waves start from the sun, and
reach our eyes after about eight minutes. They there produce
disturbances in the rods and cones, thence in the optic nerve, thence
in the brain. At the end of this purely physical series, by some odd
miracle, comes the experience which we call "seeing the sun," and it
is such experiences which form the whole and sole reason for our
belief in the optic nerve, the rods and cones, the ninety-three
million miles, the electromagnetic waves, and the sun itself. It is
this curious oppositeness of direction between the order of causation
as affirmed by physics, and the order of evidence as revealed by
theory of knowledge, that causes the most serious perplexities in
regard to the nature of physical reality. Anything that invalidates
our seeing, as a source of knowledge concerning physical reality,
invalidates also the whole of physics and physiology. And yet,
starting from a common-sense acceptance of our seeing, physics has
been led step by step to the construction of the causal chain in which
our seeing is the last link, and the immediate object which we see
cannot be regarded as that initial cause which we believe to be
ninety-three million miles away, and which we are inclined to regard
as the "real" sun.

I have stated this difficulty as forcibly as I can, because I believe
that it can only be answered by a radical analysis and reconstruction
of all the conceptions upon whose employment it depends.

Space, time, matter and cause, are the chief of these conceptions. Let
us begin with the conception of cause.

Causal dependence, as I observed a moment ago, is a conception which
it is very dangerous to accept at its face value. There exists a
notion that in regard to any event there is something which may be
called _the_ cause of that event--some one definite occurrence,
without which the event would have been impossible and with which it
becomes necessary. An event is supposed to be dependent upon its cause
in some way which in it is not dependent upon other things. Thus men
will urge that the mind is dependent upon the brain, or, with equal
plausibility, that the brain is dependent upon the mind. It seems not
improbable that if we had sufficient knowledge we could infer the
state of a man's mind from the state of his brain, or the state of his
brain from the state of his mind. So long as the usual conception of
causal dependence is retained, this state of affairs can be used by
the materialist to urge that the state of our brain causes our
thoughts, and by the idealist to urge that our thoughts cause the
state of our brain. Either contention is equally valid or equally
invalid. The fact seems to be that there are many correlations of the
sort which may be called causal, and that, for example, either a
physical or a mental event can be predicted, theoretically, either
from a sufficient number of physical antecedents or from a sufficient
number of mental antecedents. To speak of _the_ cause of an event is
therefore misleading. Any set of antecedents from which the event can
theoretically be inferred by means of correlations might be called a
cause of the event. But to speak of _the_ cause is to imply a
uniqueness which does not exist.

The relevance of this to the experience which we call "seeing the sun"
is obvious. The fact that there exists a chain of antecedents which
makes our seeing dependent upon the eyes and nerves and brain does not
even tend to show that there is not another chain of antecedents in
which the eyes and nerves and brain as physical things are ignored. If
we are to escape from the dilemma which seemed to arise out of the
physiological causation of what we see when we say we see the sun, we
must find, at least in theory, a way of stating causal laws for the
physical world, in which the units are not material things, such as
the eyes and nerves and brain, but momentary particulars of the same
sort as our momentary visual object when we look at the sun. The sun
itself and the eyes and nerves and brain must be regarded as
assemblages of momentary particulars. Instead of supposing, as we
naturally do when we start from an uncritical acceptance of the
apparent dicta of physics, that _matter_ is what is "really real" in
the physical world, and that the immediate objects of sense are mere
phantasms, we must regard matter as a logical construction, of which
the constituents will be just such evanescent particulars as may, when
an observer happens to be present, become data of sense to that
observer. What physics regards as the sun of eight minutes ago will be
a whole assemblage of particulars, existing at different times,
spreading out from a centre with the velocity of light, and containing
among their number all those visual data which are seen by people who
are now looking at the sun. Thus the sun of eight minutes ago is a
class of particulars, and what I see when I now look at the sun is one
member of this class. The various particulars constituting this class
will be correlated with each other by a certain continuity and certain
intrinsic laws of variation as we pass outwards from the centre,
together with certain modifications correlated extrinsically with
other particulars which are not members of this class. It is these
extrinsic modifications which represent the sort of facts that, in our
former account, appeared as the influence of the eyes and nerves in
modifying the appearance of the sun.[28]

The _prima facie_ difficulties in the way of this view are chiefly
derived from an unduly conventional theory of space. It might seem at
first sight as if we had packed the world much fuller than it could
possibly hold. At every place between us and the sun, we said, there
is to be a particular which is to be a member of the sun as it was a
few minutes ago. There will also, of course, have to be a particular
which is a member of any planet or fixed star that may happen to be
visible from that place. At the place where I am, there will be
particulars which will be members severally of all the "things" I am
now said to be perceiving. Thus throughout the world, everywhere,
there will be an enormous number of particulars coexisting in the same
place. But these troubles result from contenting ourselves too readily
with the merely three-dimensional space to which schoolmasters have
accustomed us. The space of the real world is a space of six
dimensions, and as soon as we realise this we see that there is plenty
of room for all the particulars for which we want to find positions.
In order to realise this we have only to return for a moment from the
polished space of physics to the rough and untidy space of our
immediate sensible experience. The space of one man's sensible objects
is a three-dimensional space. It does not appear probable that two men
ever both perceive at the same time any one sensible object; when they
are said to see the same thing or hear the same noise, there will
always be some difference, however slight, between the actual shapes
seen or the actual sounds heard. If this is so, and if, as is
generally assumed, position in space is purely relative, it follows
that the space of one man's objects and the space of another man's
objects have no place in common, that they are in fact different
spaces, and not merely different parts of one space. I mean by this
that such immediate spatial relations as are perceived to hold
between the different parts of the sensible space perceived by one
man, do not hold between parts of sensible spaces perceived by
different men. There are therefore a multitude of three-dimensional
spaces in the world: there are all those perceived by observers, and
presumably also those which are not perceived, merely because no
observer is suitably situated for perceiving them.

But although these spaces do not have to one another the same kind of
spatial relations as obtain between the parts of one of them, it is
nevertheless possible to arrange these spaces themselves in a
three-dimensional order. This is done by means of the correlated
particulars which we regard as members (or aspects) of one physical
thing. When a number of people are said to see the same object, those
who would be said to be near to the object see a particular occupying
a larger part of their field of vision than is occupied by the
corresponding particular seen by people who would be said to be
farther from the thing. By means of such considerations it is
possible, in ways which need not now be further specified, to arrange
all the different spaces in a three-dimensional series. Since each of
the spaces is itself three-dimensional, the whole world of particulars
is thus arranged in a six-dimensional space, that is to say, six
co-ordinates will be required to assign completely the position of any
given particular, namely, three to assign its position in its own
space and three more to assign the position of its space among the
other spaces.

There are two ways of classifying particulars: we may take together
all those that belong to a given "perspective," or all those that are,
as common sense would say, different "aspects" of the same "thing."
For example, if I am (as is said) seeing the sun, what I see belongs
to two assemblages: (1) the assemblage of all my present objects of
sense, which is what I call a "perspective"; (2) the assemblage of
all the different particulars which would be called aspects of the sun
of eight minutes ago--this assemblage is what I define as _being_ the
sun of eight minutes ago. Thus "perspectives" and "things" are merely
two different ways of classifying particulars. It is to be observed
that there is no _a priori_ necessity for particulars to be
susceptible of this double classification. There may be what might be
called "wild" particulars, not having the usual relations by which the
classification is effected; perhaps dreams and hallucinations are
composed of particulars which are "wild" in this sense.

The exact definition of what is meant by a perspective is not quite
easy. So long as we confine ourselves to visible objects or to objects
of touch we might define the perspective of a given particular as "all
particulars which have a simple (direct) spatial relation to the given
particular." Between two patches of colour which I see now, there is a
direct spatial relation which I equally see. But between patches of
colour seen by different men there is only an indirect constructed
spatial relation by means of the placing of "things" in physical space
(which is the same as the space composed of perspectives). Those
particulars which have direct spatial relations to a given particular
will belong to the same perspective. But if, for example, the sounds
which I hear are to belong to the same perspective with the patches of
colour which I see, there must be particulars which have no direct
spatial relation and yet belong to the same perspective. We cannot
define a perspective as all the data of one percipient at one time,
because we wish to allow the possibility of perspectives which are not
perceived by any one. There will be need, therefore, in defining a
perspective, of some principle derived neither from psychology nor
from space.

Such a principle may be obtained from the consideration of _time_.
The one all-embracing time, like the one all-embracing space, is a
construction; there is no _direct_ time-relation between particulars
belonging to my perspective and particulars belonging to another
man's. On the other hand, any two particulars of which I am aware are
either simultaneous or successive, and their simultaneity or
successiveness is sometimes itself a datum to me. We may therefore
define the perspective to which a given particular belongs as "all
particulars simultaneous with the given particular," where
"simultaneous" is to be understood as a direct simple relation, not
the derivative constructed relation of physics. It may be observed
that the introduction of "local time" suggested by the principle of
relativity has effected, for purely scientific reasons, much the same
multiplication of times as we have just been advocating.

The sum-total of all the particulars that are (directly) either
simultaneous with or before or after a given particular may be defined
as the "biography" to which that particular belongs. It will be
observed that, just as a perspective need not be actually perceived by
any one, so a biography need not be actually lived by any one. Those
biographies that are lived by no one are called "official."

The definition of a "thing" is effected by means of continuity and of
correlations which have a certain differential independence of other
"things." That is to say, given a particular in one perspective, there
will usually in a neighbouring perspective be a very similar
particular, differing from the given particular, to the first order of
small quantities, according to a law involving only the difference of
position of the two perspectives in perspective space, and not any of
the other "things" in the universe. It is this continuity and
differential independence in the law of change as we pass from one
perspective to another that defines the class of particulars which is
to be called "one thing."

Broadly speaking, we may say that the physicist finds it convenient to
classify particulars into "things," while the psychologist finds it
convenient to classify them into "perspectives" and "biographies,"
since one perspective _may_ constitute the momentary data of one
percipient, and one biography _may_ constitute the whole of the data
of one percipient throughout his life.

We may now sum up our discussion. Our object has been to discover as
far as possible the nature of the ultimate constituents of the
physical world. When I speak of the "physical world," I mean, to begin
with, the world dealt with by physics. It is obvious that physics is
an empirical science, giving us a certain amount of knowledge and
based upon evidence obtained through the senses. But partly through
the development of physics itself, partly through arguments derived
from physiology, psychology or metaphysics, it has come to be thought
that the immediate data of sense could not themselves form part of the
ultimate constituents of the physical world, but were in some sense
"mental," "in the mind," or "subjective." The grounds for this view,
in so far as they depend upon physics, can only be adequately dealt
with by rather elaborate constructions depending upon symbolic logic,
showing that out of such materials as are provided by the senses it is
possible to construct classes and series having the properties which
physics assigns to matter. Since this argument is difficult and
technical, I have not embarked upon it in this article. But in so far
as the view that sense-data are "mental" rests upon physiology,
psychology, or metaphysics, I have tried to show that it rests upon
confusions and prejudices--prejudices in favour of permanence in the
ultimate constituents of matter, and confusions derived from unduly
simple notions as to space, from the causal correlation of sense-data
with sense-organs, and from failure to distinguish between sense-data
and sensations. If what we have said on these subjects is valid, the
existence of sense-data is logically independent of the existence of
mind, and is causally dependent upon the _body_ of the percipient
rather than upon his mind. The causal dependence upon the body of the
percipient, we found, is a more complicated matter than it appears to
be, and, like all causal dependence, is apt to give rise to erroneous
beliefs through misconceptions as to the nature of causal correlation.
If we have been right in our contentions, sense-data are merely those
among the ultimate constituents of the physical world, of which we
happen to be immediately aware; they themselves are purely physical,
and all that is mental in connection with them is our awareness of
them, which is irrelevant to their nature and to their place in
physics.

Unduly simple notions as to space have been a great stumbling-block to
realists. When two men look at the same table, it is supposed that
what the one sees and what the other sees are in the same place. Since
the shape and colour are not quite the same for the two men, this
raises a difficulty, hastily solved, or rather covered up, by
declaring what each sees to be purely "subjective"--though it would
puzzle those who use this glib word to say what they mean by it. The
truth seems to be that space--and time also--is much more complicated
than it would appear to be from the finished structure of physics, and
that the one all-embracing three-dimensional space is a logical
construction, obtained by means of correlations from a crude space of
six dimensions. The particulars occupying this six-dimensional space,
classified in one way, form "things," from which with certain further
manipulations we can obtain what physics can regard as matter;
classified in another way, they form "perspectives" and "biographies,"
which may, if a suitable percipient happens to exist, form
respectively the sense-data of a momentary or of a total experience.
It is only when physical "things" have been dissected into series of
classes of particulars, as we have done, that the conflict between the
point of view of physics and the point of view of psychology can be
overcome. This conflict, if what has been said is not mistaken, flows
from different methods of classification, and vanishes as soon as its
source is discovered.

In favour of the theory which I have briefly outlined, I do not claim
that it is _certainly_ true. Apart from the likelihood of mistakes,
much of it is avowedly hypothetical. What I do claim for the theory is
that it _may_ be true, and that this is more than can be said for any
other theory except the closely analogous theory of Leibniz. The
difficulties besetting realism, the confusions obstructing any
philosophical account of physics, the dilemma resulting from
discrediting sense-data, which yet remain the sole source of our
knowledge of the outer world--all these are avoided by the theory
which I advocate. This does not prove the theory to be true, since
probably many other theories might be invented which would have the
same merits. But it does prove that the theory has a better chance of
being true than any of its present competitors, and it suggests that
what can be known with certainty is likely to be discoverable by
taking our theory as a starting-point, and gradually freeing it from
all such assumptions as seem irrelevant, unnecessary, or unfounded. On
these grounds, I recommend it to attention as a hypothesis and a basis
for further work, though not as itself a finished or adequate solution
of the problem with which it deals.

FOOTNOTES:

[23] An address delivered to the Philosophical Society of Manchester
in February, 1915. Reprinted from _The Monist_, July, 1915.

[24] Cf. especially Samuel Alexander, "The Basis of Realism," _British
Academy_, Vol. VI.

[25] "Are Secondary Qualities Independent of Perception?" _Proc.
Arist. Soc._, 1909-10, pp. 191-218.

[26] First dialogue between Hylas and Philonous, _Works_ (Fraser's
edition 1901). I. p. 384.

[27] This point has been well urged by the American realists.

[28] Cf. T.P. Nunn, "Are Secondary Qualities Independent of
Perception?" _Proc. Arist. Soc._, 1909-1910.




VIII

THE RELATION OF SENSE-DATA TO PHYSICS


I. THE PROBLEM STATED

Physics is said to be an empirical science, based upon observation and
experiment.

It is supposed to be verifiable, i.e. capable of calculating
beforehand results subsequently confirmed by observation and
experiment.

What can we learn by observation and experiment?

Nothing, so far as physics is concerned, except immediate data of
sense: certain patches of colour, sounds, tastes, smells, etc., with
certain spatio-temporal relations.

The supposed contents of the physical world are _prima facie_ very
different from these: molecules have no colour, atoms make no noise,
electrons have no taste, and corpuscles do not even smell.

If such objects are to be verified, it must be solely through their
relation to sense-data: they must have some kind of correlation with
sense-data, and must be verifiable through their correlation _alone_.

But how is the correlation itself ascertained? A correlation can only
be ascertained empirically by the correlated objects being constantly
_found_ together. But in our case, only one term of the correlation,
namely, the sensible term, is ever _found_: the other term seems
essentially incapable of being found. Therefore, it would seem, the
correlation with objects of sense, by which physics was to be
verified, is itself utterly and for ever unverifiable.

There are two ways of avoiding this result.

(1) We may say that we know some principle _a priori_, without the
need of empirical verification, e.g. that our sense-data have _causes_
other than themselves, and that something can be known about these
causes by inference from their effects. This way has been often
adopted by philosophers. It may be necessary to adopt this way to some
extent, but in so far as it is adopted physics ceases to be empirical
or based upon experiment and observation alone. Therefore this way is
to be avoided as much as possible.

(2) We may succeed in actually defining the objects of physics as
functions of sense-data. Just in so far as physics leads to
expectations, this _must_ be possible, since we can only _expect_ what
can be experienced. And in so far as the physical state of affairs is
inferred from sense-data, it must be capable of expression as a
function of sense-data. The problem of accomplishing this expression
leads to much interesting logico-mathematical work.

In physics as commonly set forth, sense-data appear as functions of
physical objects: when such-and-such waves impinge upon the eye, we
see such-and-such colours, and so on. But the waves are in fact
inferred from the colours, not vice versa. Physics cannot be regarded
as validly based upon empirical data until the waves have been
expressed as functions of the colours and other sense-data.

Thus if physics is to be verifiable we are faced with the following
problem: Physics exhibits sense-data as functions of physical objects,
but verification is only possible if physical objects can be exhibited
as functions of sense-data. We have therefore to solve the equations
giving sense-data in terms of physical objects, so as to make them
instead give physical objects in terms of sense-data.


II. CHARACTERISTICS OF SENSE-DATA

When I speak of a "sense-datum," I do not mean the whole of what is
given in sense at one time. I mean rather such a part of the whole as
might be singled out by attention: particular patches of colour,
particular noises, and so on. There is some difficulty in deciding
what is to be considered _one_ sense-datum: often attention causes
divisions to appear where, so far as can be discovered, there were no
divisions before. An observed complex fact, such as that this patch of
red is to the left of that patch of blue, is also to be regarded as a
datum from our present point of view: epistemologically, it does not
differ greatly from a simple sense-datum as regards its function in
giving knowledge. Its _logical_ structure is very different, however,
from that of sense: _sense_ gives acquaintance with particulars, and
is thus a two-term relation in which the object can be _named_ but not
_asserted_, and is inherently incapable of truth or falsehood, whereas
the observation of a complex fact, which may be suitably called
perception, is not a two-term relation, but involves the propositional
form on the object-side, and gives knowledge of a truth, not mere
acquaintance with a particular. This logical difference, important as
it is, is not very relevant to our present problem; and it will be
convenient to regard data of perception as included among sense-data
for the purposes of this paper. It is to be observed that the
particulars which are constituents of a datum of perception are always
sense-data in the strict sense.

Concerning sense-data, we know that they are there while they are
data, and this is the epistemological basis of all our knowledge of
external particulars. (The meaning of the word "external" of course
raises problems which will concern us later.) We do not know, except
by means of more or less precarious inferences, whether the objects
which are at one time sense-data continue to exist at times when they
are not data. Sense-data at the times when they are data are all that
we directly and primitively know of the external world; hence in
epistemology the fact that they are _data_ is all-important. But the
fact that they are all that we directly know gives, of course, no
presumption that they are all that there is. If we could construct an
impersonal metaphysic, independent of the accidents of our knowledge
and ignorance, the privileged position of the actual data would
probably disappear, and they would probably appear as a rather
haphazard selection from a mass of objects more or less like them. In
saying this, I assume only that it is probable that there are
particulars with which we are not acquainted. Thus the special
importance of sense-data is in relation to epistemology, not to
metaphysics. In this respect, physics is to be reckoned as
metaphysics: it is impersonal, and nominally pays no special attention
to sense-data. It is only when we ask how physics can be _known_ that
the importance of sense-data re-emerges.


III. SENSIBILIA

I shall give the name _sensibilia_ to those objects which have the
same metaphysical and physical status as sense-data, without
necessarily being data to any mind. Thus the relation of a _sensibile_
to a sense-datum is like that of a man to a husband: a man becomes a
husband by entering into the relation of marriage, and similarly a
_sensibile_ becomes a sense-datum by entering into the relation of
acquaintance. It is important to have both terms; for we wish to
discuss whether an object which is at one time a sense-datum can still
exist at a time when it is not a sense-datum. We cannot ask "Can
sense-data exist without being given?" for that is like asking "Can
husbands exist without being married?" We must ask "Can _sensibilia_
exist without being given?" and also "Can a particular _sensibile_ be
at one time a sense-datum, and at another not?" Unless we have the
word _sensibile_ as well as the word "sense-datum," such questions are
apt to entangle us in trivial logical puzzles.

It will be seen that all sense-data are _sensibilia_. It is a
metaphysical question whether all _sensibilia_ are sense-data, and an
epistemological question whether there exist means of inferring
_sensibilia_ which are not data from those that are.

A few preliminary remarks, to be amplified as we proceed, will serve
to elucidate the use which I propose to make of _sensibilia_.

I regard sense-data as not mental, and as being, in fact, part of the
actual subject-matter of physics. There are arguments, shortly to be
examined, for their subjectivity, but these arguments seem to me only
to prove _physiological_ subjectivity, i.e. causal dependence on the
sense-organs, nerves, and brain. The appearance which a thing presents
to us is causally dependent upon these, in exactly the same way as it
is dependent upon intervening fog or smoke or coloured glass. Both
dependences are contained in the statement that the appearance which a
piece of matter presents when viewed from a given place is a function
not only of the piece of matter, but also of the intervening medium.
(The terms used in this statement--"matter," "view from a given
place," "appearance," "intervening medium"--will all be defined in the
course of the present paper.) We have not the means of ascertaining
how things appear from places not surrounded by brain and nerves and
sense-organs, because we cannot leave the body; but continuity makes
it not unreasonable to suppose that they present _some_ appearance at
such places. Any such appearance would be included among _sensibilia_.
If--_per impossibile_--there were a complete human body with no mind
inside it, all those _sensibilia_ would exist, in relation to that
body, which would be sense-data if there were a mind in the body. What
the mind adds to _sensibilia_, in fact, is _merely_ awareness:
everything else is physical or physiological.


IV. SENSE-DATA ARE PHYSICAL

Before discussing this question it will be well to define the sense in
which the terms "mental" and "physical" are to be used. The word
"physical," in all preliminary discussions, is to be understood as
meaning "what is dealt with by physics." Physics, it is plain, tells
us something about some of the constituents of the actual world; what
these constituents are may be doubtful, but it is they that are to be
called physical, whatever their nature may prove to be.

The definition of the term "mental" is more difficult, and can only be
satisfactorily given after many difficult controversies have been
discussed and decided. For present purposes therefore I must content
myself with assuming a dogmatic answer to these controversies. I shall
call a particular "mental" when it is aware of something, and I shall
call a fact "mental" when it contains a mental particular as a
constituent.

It will be seen that the mental and the physical are not necessarily
mutually exclusive, although I know of no reason to suppose that they
overlap.

The doubt as to the correctness of our definition of the "mental" is
of little importance in our present discussion. For what I am
concerned to maintain is that sense-data are physical, and this being
granted it is a matter of indifference in our present inquiry whether
or not they are also mental. Although I do not hold, with Mach and
James and the "new realists," that the difference between the mental
and the physical is _merely_ one of arrangement, yet what I have to
say in the present paper is compatible with their doctrine and might
have been reached from their standpoint.

In discussions on sense-data, two questions are commonly confused,
namely:

(1) Do sensible objects persist when we are not sensible of them? in
other words, do _sensibilia_ which are data at a certain time
sometimes continue to exist at times when they are not data? And (2)
are sense-data mental or physical?

I propose to assert that sense-data are physical, while yet
maintaining that they probably never persist unchanged after ceasing
to be data. The view that they do not persist is often thought, quite
erroneously in my opinion, to imply that they are mental; and this
has, I believe, been a potent source of confusion in regard to our
present problem. If there were, as some have held, a _logical
impossibility_ in sense-data persisting after ceasing to be data, that
certainly would tend to show that they were mental; but if, as I
contend, their non-persistence is merely a probable inference from
empirically ascertained causal laws, then it carries no such
implication with it, and we are quite free to treat them as part of
the subject-matter of physics.

Logically a sense-datum is an object, a particular of which the
subject is aware. It does not contain the subject as a part, as for
example beliefs and volitions do. The existence of the sense-datum is
therefore not logically dependent upon that of the subject; for the
only way, so far as I know, in which the existence of _A_ can be
_logically_ dependent upon the existence of _B_ is when _B_ is part of
_A_. There is therefore no _a priori_ reason why a particular which is
a sense-datum should not persist after it has ceased to be a datum,
nor why other similar particulars should not exist without ever being
data. The view that sense-data are mental is derived, no doubt, in
part from their physiological subjectivity, but in part also from a
failure to distinguish between sense-data and "sensations." By a
sensation I mean the fact consisting in the subject's awareness of the
sense-datum. Thus a sensation is a complex of which the subject is a
constituent and which therefore is mental. The sense-datum, on the
other hand, stands over against the subject as that external object of
which in sensation the subject is aware. It is true that the
sense-datum is in many cases in the subject's body, but the subject's
body is as distinct from the subject as tables and chairs are, and is
in fact merely a part of the material world. So soon, therefore, as
sense-data are clearly distinguished from sensations, and as their
subjectivity is recognised to be physiological not psychical, the
chief obstacles in the way of regarding them as physical are removed.


V. "SENSIBILIA" AND "THINGS"

But if "sensibilia" are to be recognised as the ultimate constituents
of the physical world, a long and difficult journey is to be performed
before we can arrive either at the "thing" of common sense or at the
"matter" of physics. The supposed impossibility of combining the
different sense-data which are regarded as appearances of the same
"thing" to different people has made it seem as though these
"sensibilia" must be regarded as mere subjective phantasms. A given
table will present to one man a rectangular appearance, while to
another it appears to have two acute angles and two obtuse angles; to
one man it appears brown, while to another, towards whom it reflects
the light, it appears white and shiny. It is said, not wholly without
plausibility, that these different shapes and different colours cannot
co-exist simultaneously in the same place, and cannot therefore both
be constituents of the physical world. This argument I must confess
appeared to me until recently to be irrefutable. The contrary opinion
has, however, been ably maintained by Dr. T.P. Nunn in an article
entitled: "Are Secondary Qualities Independent of Perception?"[29] The
supposed impossibility derives its apparent force from the phrase:
"_in the same place_," and it is precisely in this phrase that its
weakness lies. The conception of space is too often treated in
philosophy--even by those who on reflection would not defend such
treatment--as though it were as given, simple, and unambiguous as
Kant, in his psychological innocence, supposed. It is the unperceived
ambiguity of the word "place" which, as we shall shortly see, has
caused the difficulties to realists and given an undeserved advantage
to their opponents. Two "places" of different kinds are involved in
every sense-datum, namely the place _at_ which it appears and the
place _from_ which it appears. These belong to different spaces,
although, as we shall see, it is possible, with certain limitations,
to establish a correlation between them. What we call the different
appearances of the same thing to different observers are each in a
space private to the observer concerned. No place in the private world
of one observer is identical with a place in the private world of
another observer. There is therefore no question of combining the
different appearances in the one place; and the fact that they cannot
all exist in one place affords accordingly no ground whatever for
questioning their physical reality. The "thing" of common sense may in
fact be identified with the whole class of its appearances--where,
however, we must include among appearances not only those which are
actual sense-data, but also those "sensibilia," if any, which, on
grounds of continuity and resemblance, are to be regarded as belonging
to the same system of appearances, although there happen to be no
observers to whom they are data.

An example may make this clearer. Suppose there are a number of people
in a room, all seeing, as they say, the same tables and chairs, walls
and pictures. No two of these people have exactly the same sense-data,
yet there is sufficient similarity among their data to enable them to
group together certain of these data as appearances of one "thing" to
the several spectators, and others as appearances of another "thing."
Besides the appearances which a given thing in the room presents to
the actual spectators, there are, we may suppose, other appearances
which it would present to other possible spectators. If a man were to
sit down between two others, the appearance which the room would
present to him would be intermediate between the appearances which it
presents to the two others: and although this appearance would not
exist as it is without the sense organs, nerves and brain, of the
newly arrived spectator, still it is not unnatural to suppose that,
from the position which he now occupies, _some_ appearance of the
room existed before his arrival. This supposition, however, need
merely be noticed and not insisted upon.

Since the "thing" cannot, without indefensible partiality, be
identified with any single one of its appearances, it came to be
thought of as something distinct from all of them and underlying them.
But by the principle of Occam's razor, if the class of appearances
will fulfil the purposes for the sake of which the thing was invented
by the prehistoric metaphysicians to whom common sense is due, economy
demands that we should identify the thing with the class of its
appearances. It is not necessary to _deny_ a substance or substratum
underlying these appearances; it is merely expedient to abstain from
asserting this unnecessary entity. Our procedure here is precisely
analogous to that which has swept away from the philosophy of
mathematics the useless menagerie of metaphysical monsters with which
it used to be infested.


VI. CONSTRUCTIONS VERSUS INFERENCES

Before proceeding to analyse and explain the ambiguities of the word
"place," a few general remarks on method are desirable. The supreme
maxim in scientific philosophising is this:

  _Wherever possible, logical constructions are to be substituted
  for inferred entities._

Some examples of the substitution of construction for inference in the
realm of mathematical philosophy may serve to elucidate the uses of
this maxim. Take first the case of irrationals. In old days,
irrationals were inferred as the supposed limits of series of
rationals which had no rational limit; but the objection to this
procedure was that it left the existence of irrationals merely
optative, and for this reason the stricter methods of the present day
no longer tolerate such a definition. We now define an irrational
number as a certain class of ratios, thus constructing it logically by
means of ratios, instead of arriving at it by a doubtful inference
from them. Take again the case of cardinal numbers. Two equally
numerous collections appear to have something in common: this
something is supposed to be their cardinal number. But so long as the
cardinal number is inferred from the collections, not constructed in
terms of them, its existence must remain in doubt, unless in virtue of
a metaphysical postulate _ad hoc_. By defining the cardinal number of
a given collection as the class of all equally numerous collections,
we avoid the necessity of this metaphysical postulate, and thereby
remove a needless element of doubt from the philosophy of arithmetic.
A similar method, as I have shown elsewhere, can be applied to classes
themselves, which need not be supposed to have any metaphysical
reality, but can be regarded as symbolically constructed fictions.

The method by which the construction proceeds is closely analogous in
these and all similar cases. Given a set of propositions nominally
dealing with the supposed inferred entities, we observe the properties
which are required of the supposed entities in order to make these
propositions true. By dint of a little logical ingenuity, we then
construct some logical function of less hypothetical entities which
has the requisite properties. This constructed function we substitute
for the supposed inferred entities, and thereby obtain a new and less
doubtful interpretation of the body of propositions in question. This
method, so fruitful in the philosophy of mathematics, will be found
equally applicable in the philosophy of physics, where, I do not
doubt, it would have been applied long ago but for the fact that all
who have studied this subject hitherto have been completely ignorant
of mathematical logic. I myself cannot claim originality in the
application of this method to physics, since I owe the suggestion and
the stimulus for its application entirely to my friend and
collaborator Dr. Whitehead, who is engaged in applying it to the more
mathematical portions of the region intermediate between sense-data
and the points, instants and particles of physics.

A complete application of the method which substitutes constructions
for inferences would exhibit matter wholly in terms of sense-data, and
even, we may add, of the sense-data of a single person, since the
sense-data of others cannot be known without some element of
inference. This, however, must remain for the present an ideal, to be
approached as nearly as possible, but to be reached, if at all, only
after a long preliminary labour of which as yet we can only see the
very beginning. The inferences which are unavoidable can, however, be
subjected to certain guiding principles. In the first place they
should always be made perfectly explicit, and should be formulated in
the most general manner possible. In the second place the inferred
entities should, whenever this can be done, be similar to those whose
existence is given, rather than, like the Kantian _Ding an sich_,
something wholly remote from the data which nominally support the
inference. The inferred entities which I shall allow myself are of two
kinds: (_a_) the sense-data of other people, in favour of which there
is the evidence of testimony, resting ultimately upon the analogical
argument in favour of minds other than my own; (_b_) the "sensibilia"
which would appear from places where there happen to be no minds, and
which I suppose to be real although they are no one's data. Of these
two classes of inferred entities, the first will probably be allowed
to pass unchallenged. It would give me the greatest satisfaction to be
able to dispense with it, and thus establish physics upon a
solipsistic basis; but those--and I fear they are the majority--in
whom the human affections are stronger than the desire for logical
economy, will, no doubt, not share my desire to render solipsism
scientifically satisfactory. The second class of inferred entities
raises much more serious questions. It may be thought monstrous to
maintain that a thing can present any appearance at all in a place
where no sense organs and nervous structure exist through which it
could appear. I do not myself feel the monstrosity; nevertheless I
should regard these supposed appearances only in the light of a
hypothetical scaffolding, to be used while the edifice of physics is
being raised, though possibly capable of being removed as soon as the
edifice is completed. These "sensibilia" which are not data to anyone
are therefore to be taken rather as an illustrative hypothesis and as
an aid in preliminary statement than as a dogmatic part of the
philosophy of physics in its final form.


VII. PRIVATE SPACE AND THE SPACE OF PERSPECTIVES

We have now to explain the ambiguity in the word "place," and how it
comes that two places of different sorts are associated with every
sense-datum, namely the place _at_ which it is and the place _from_
which it is perceived. The theory to be advocated is closely analogous
to Leibniz's monadology, from which it differs chiefly in being less
smooth and tidy.

The first fact to notice is that, so far as can be discovered, no
sensibile is ever a datum to two people at once. The things seen by
two different people are often closely similar, so similar that the
same _words_ can be used to denote them, without which communication
with others concerning sensible objects would be impossible. But, in
spite of this similarity, it would seem that some difference always
arises from difference in the point of view. Thus each person, so far
as his sense-data are concerned, lives in a private world. This
private world contains its own space, or rather spaces, for it would
seem that only experience teaches us to correlate the space of sight
with the space of touch and with the various other spaces of other
senses. This multiplicity of private spaces, however, though
interesting to the psychologist, is of no great importance in regard
to our present problem, since a merely solipsistic experience enables
us to correlate them into the one private space which embraces all our
own sense-data. The place _at_ which a sense-datum is, is a place in
private space. This place therefore is different from any place in the
private space of another percipient. For if we assume, as logical
economy demands, that all position is relative, a place is only
definable by the things in or around it, and therefore the same place
cannot occur in two private worlds which have no common constituent.
The question, therefore, of combining what we call different
appearances of the same thing in the same place does not arise, and
the fact that a given object appears to different spectators to have
different shapes and colours affords no argument against the physical
reality of all these shapes and colours.

In addition to the private spaces belonging to the private worlds of
different percipients, there is, however, another space, in which one
whole private world counts as a point, or at least as a spatial unit.
This might be described as the space of points of view, since each
private world may be regarded as the appearance which the universe
presents from a certain point of view. I prefer, however, to speak of
it as the space of _perspectives_, in order to obviate the suggestion
that a private world is only real when someone views it. And for the
same reason, when I wish to speak of a private world without assuming
a percipient, I shall call it a "perspective."

We have now to explain how the different perspectives are ordered in
one space. This is effected by means of the correlated "sensibilia"
which are regarded as the appearances, in different perspectives, of
one and the same thing. By moving, and by testimony, we discover that
two different perspectives, though they cannot both contain the same
"sensibilia," may nevertheless contain very similar ones; and the
spatial order of a certain group of "sensibilia" in a private space of
one perspective is found to be identical with, or very similar to, the
spatial order of the correlated "sensibilia" in the private space of
another perspective. In this way one "sensibile" in one perspective is
correlated with one "sensibile" in another. Such correlated
"sensibilia" will be called "appearances of one thing." In Leibniz's
monadology, since each monad mirrored the whole universe, there was in
each perspective a "sensibile" which was an appearance of each thing.
In our system of perspectives, we make no such assumption of
completeness. A given thing will have appearances in some
perspectives, but presumably not in certain others. The "thing" being
defined as the class of its appearances, if κ is the class of
perspectives in which a certain thing θ appears, then θ is a member of
the multiplicative class of κ, κ being a class of mutually exclusive
classes of "sensibilia." And similarly a perspective is a member of
the multiplicative class of the things which appear in it.

The arrangement of perspectives in a space is effected by means of the
differences between the appearances of a given thing in the various
perspectives. Suppose, say, that a certain penny appears in a number
of different perspectives; in some it looks larger and in some
smaller, in some it looks circular, in others it presents the
appearance of an ellipse of varying eccentricity. We may collect
together all those perspectives in which the appearance of the penny
is circular. These we will place on one straight line, ordering them
in a series by the variations in the apparent size of the penny. Those
perspectives in which the penny appears as a straight line of a
certain thickness will similarly be placed upon a plane (though in
this case there will be many different perspectives in which the penny
is of the same size; when one arrangement is completed these will form
a circle concentric with the penny), and ordered as before by the
apparent size of the penny. By such means, all those perspectives in
which the penny presents a visual appearance can be arranged in a
three-dimensional spatial order. Experience shows that the same
spatial order of perspectives would have resulted if, instead of the
penny, we had chosen any other thing which appeared in all the
perspectives in question, or any other method of utilising the
differences between the appearances of the same things in different
perspectives. It is this empirical fact which has made it possible to
construct the one all-embracing space of physics.

The space whose construction has just been explained, and whose
elements are whole perspectives, will be called "perspective-space."


VIII. THE PLACING OF "THINGS" AND "SENSIBILIA" IN PERSPECTIVE SPACE

The world which we have so far constructed is a world of six
dimensions, since it is a three-dimensional series of perspectives,
each of which is itself three-dimensional. We have now to explain the
correlation between the perspective space and the various private
spaces contained within the various perspectives severally. It is by
means of this correlation that the one three-dimensional space of
physics is constructed; and it is because of the unconscious
performance of this correlation that the distinction between
perspective space and the percipient's private space has been blurred,
with disastrous results for the philosophy of physics. Let us revert
to our penny: the perspectives in which the penny appears larger are
regarded as being nearer to the penny than those in which it appears
smaller, but as far as experience goes the apparent size of the penny
will not grow beyond a certain limit, namely, that where (as we say)
the penny is so near the eye that if it were any nearer it could not
be seen. By touch we may prolong the series until the penny touches
the eye, but no further. If we have been travelling along a line of
perspectives in the previously defined sense, we may, however, by
imagining the penny removed, prolong the line of perspectives by
means, say, of another penny; and the same may be done with any other
line of perspectives defined by means of the penny. All these lines
meet in a certain place, that is, in a certain perspective. This
perspective will be defined as "the place where the penny is."

It is now evident in what sense two places in constructed physical
space are associated with a given "sensibile." There is first the
place which is the perspective of which the "sensibile" is a member.
This is the place _from_ which the "sensibile" appears. Secondly there
is the place where the thing is of which the "sensibile" is a member,
in other words an appearance; this is the place _at_ which the
"sensibile" appears. The "sensibile" which is a member of one
perspective is correlated with another perspective, namely, that which
is the place where the thing is of which the "sensibile" is an
appearance. To the psychologist the "place from which" is the more
interesting, and the "sensibile" accordingly appears to him subjective
and where the percipient is. To the physicist the "place at which" is
the more interesting, and the "sensibile" accordingly appears to him
physical and external. The causes, limits and partial justification of
each of these two apparently incompatible views are evident from the
above duplicity of places associated with a given "sensibile."

We have seen that we can assign to a physical thing a place in the
perspective space. In this way different parts of our body acquire
positions in perspective space, and therefore there is a meaning
(whether true or false need not much concern us) in saying that the
perspective to which our sense-data belong is inside our head. Since
our mind is correlated with the perspective to which our sense-data
belong, we may regard this perspective as being the position of our
mind in perspective space. If, therefore, this perspective is, in the
above defined sense, inside our head, there is a good meaning for the
statement that the mind is in the head. We can now say of the various
appearances of a given thing that some of them are nearer to the thing
than others; those are nearer which belong to perspectives that are
nearer to "the place where the thing is." We can thus find a meaning,
true or false, for the statement that more is to be learnt about a
thing by examining it close to than by viewing it from a distance. We
can also find a meaning for the phrase "the things which intervene
between the subject and a thing of which an appearance is a datum to
him." One reason often alleged for the subjectivity of sense-data is
that the appearance of a thing may change when we find it hard to
suppose that the thing itself has changed--for example, when the
change is due to our shutting our eyes, or to our screwing them up so
as to make the thing look double. If the thing is defined as the class
of its appearances (which is the definition adopted above), there is
of course necessarily _some_ change in the thing whenever any one of
its appearances changes. Nevertheless there is a very important
distinction between two different ways in which the appearances may
change. If after looking at a thing I shut my eyes, the appearance of
my eyes changes in every perspective in which there is such an
appearance, whereas most of the appearances of the thing will remain
unchanged. We may say, as a matter of definition, that a thing changes
when, however near to the thing an appearance of it may be, there are
changes in appearances as near as, or still nearer to, the thing. On
the other hand we shall say that the change is in some other thing if
all appearances of the thing which are at not more than a certain
distance from the thing remain unchanged, while only comparatively
distant appearances of the thing are altered. From this consideration
we are naturally led to the consideration of _matter_, which must be
our next topic.


IX. THE DEFINITION OF MATTER

We defined the "physical thing" as the class of its appearances, but
this can hardly be taken as a definition of matter. We want to be able
to express the fact that the appearance of a thing in a given
perspective is causally affected by the matter between the thing and
the perspective. We have found a meaning for "between a thing and a
perspective." But we want matter to be something other than the whole
class of appearances of a thing, in order to state the influence of
matter on appearances.

We commonly assume that the information we get about a thing is more
accurate when the thing is nearer. Far off, we see it is a man; then
we see it is Jones; then we see he is smiling. Complete accuracy would
only be attainable as a limit: if the appearances of Jones as we
approach him tend towards a limit, that limit may be taken to be what
Jones really is. It is obvious that from the point of view of physics
the appearances of a thing close to "count" more than the appearances
far off. We may therefore set up the following tentative definition:

The _matter_ of a given thing is the limit of its appearances as their
distance from the thing diminishes.

It seems probable that there is something in this definition, but it
is not quite satisfactory, because empirically there is no such limit
to be obtained from sense-data. The definition will have to be eked
out by constructions and definitions. But probably it suggests the
right direction in which to look.

We are now in a position to understand in outline the reverse journey
from matter to sense-data which is performed by physics. The
appearance of a thing in a given perspective is a function of the
matter composing the thing and of the intervening matter. The
appearance of a thing is altered by intervening smoke or mist, by blue
spectacles or by alterations in the sense-organs or nerves of the
percipient (which also must be reckoned as part of the intervening
medium). The nearer we approach to the thing, the less its appearance
is affected by the intervening matter. As we travel further and
further from the thing, its appearances diverge more and more from
their initial character; and the causal laws of their divergence are
to be stated in terms of the matter which lies between them and the
thing. Since the appearances at very small distances are less affected
by causes other than the thing itself, we come to think that the limit
towards which these appearances tend as the distance diminishes is
what the thing "really is," as opposed to what it merely seems to be.
This, together with its necessity for the statement of causal laws,
seems to be the source of the entirely erroneous feeling that matter
is more "real" than sense-data.

Consider for example the infinite divisibility of matter. In looking
at a given thing and approaching it, one sense-datum will become
several, and each of these will again divide. Thus _one_ appearance
may represent _many_ things, and to this process there seems no end.
Hence in the limit, when we approach indefinitely near to the thing
there will be an indefinite number of units of matter corresponding to
what, at a finite distance, is only one appearance. This is how
infinite divisibility arises.

The whole causal efficacy of a thing resides in its matter. This is in
some sense an empirical fact, but it would be hard to state it
precisely, because "causal efficacy" is difficult to define.

What can be known empirically about the matter of a thing is only
approximate, because we cannot get to know the appearances of the
thing from very small distances, and cannot accurately infer the limit
of these appearances. But it _is_ inferred _approximately_ by means of
the appearances we can observe. It then turns out that these
appearances can be exhibited by physics as a function of the matter
in our immediate neighbourhood; e.g. the visual appearance of a
distant object is a function of the light-waves that reach the eyes.
This leads to confusions of thought, but offers no real difficulty.

One appearance, of a visible object for example, is not sufficient to
determine its other simultaneous appearances, although it goes a
certain distance towards determining them. The determination of the
hidden structure of a thing, so far as it is possible at all, can only
be effected by means of elaborate dynamical inferences.


X. TIME[30]

It seems that the one all-embracing time is a construction, like the
one all-embracing space. Physics itself has become conscious of this
fact through the discussions connected with relativity.

Between two perspectives which both belong to one person's experience,
there will be a direct time-relation of before and after. This
suggests a way of dividing history in the same sort of way as it is
divided by different experiences, but without introducing experience
or anything mental: we may define a "biography" as everything that is
(directly) earlier or later than, or simultaneous with, a given
"sensibile." This will give a series of perspectives, which _might_
all form parts of one person's experience, though it is not necessary
that all or any of them should actually do so. By this means, the
history of the world is divided into a number of mutually exclusive
biographies.

We have now to correlate the times in the different biographies. The
natural thing would be to say that the appearances of a given
(momentary) thing in two different perspectives belonging to different
biographies are to be taken as simultaneous; but this is not
convenient. Suppose _A_ shouts to _B_, and _B_ replies as soon as he
hears _A's_ shout. Then between _A's_ hearing of his own shout and his
hearing of _B's_ there is an interval; thus if we made _A's_ and _B's_
hearing of the same shout exactly simultaneous with each other, we
should have events exactly simultaneous with a given event but not
with each other. To obviate this, we assume a "velocity of sound."
That is, we assume that the time when _B_ hears _A's_ shout is
half-way between the time when _A_ hears his own shout and the time
when he hears _B's_. In this way the correlation is effected.

What has been said about sound applies of course equally to light. The
general principle is that the appearances, in different perspectives,
which are to be grouped together as constituting what a certain thing
is at a certain moment, are not to be all regarded as being at that
moment. On the contrary they spread outward from the thing with
various velocities according to the nature of the appearances. Since
no _direct_ means exist of correlating the time in one biography with
the time in another, this temporal grouping of the appearances
belonging to a given thing at a given moment is in part conventional.
Its motive is partly to secure the verification of such maxims as that
events which are exactly simultaneous with the same event are exactly
simultaneous with one another, partly to secure convenience in the
formulation of causal laws.


XI. THE PERSISTENCE OF THINGS AND MATTER

Apart from any of the fluctuating hypotheses of physics, three main
problems arise in connecting the world of physics with the world of
sense, namely:

    1. the construction of a single space;
    2. the construction of a single time;
    3. the construction of permanent things or matter.

We have already considered the first and second of these problems; it
remains to consider the third.

We have seen how correlated appearances in different perspectives are
combined to form one "thing" at one moment in the all-embracing time
of physics. We have now to consider how appearances at different times
are combined as belonging to one "thing," and how we arrive at the
persistent "matter" of physics. The assumption of permanent substance,
which technically underlies the procedure of physics, cannot of course
be regarded as metaphysically legitimate: just as the one thing
simultaneously seen by many people is a construction, so the one thing
seen at different times by the same or different people must be a
construction, being in fact nothing but a certain grouping of certain
"sensibilia."

We have seen that the momentary state of a "thing" is an assemblage of
"sensibilia," in different perspectives, not all simultaneous in the
one constructed time, but spreading out from "the place where the
thing is" with velocities depending upon the nature of the
"sensibilia." The time _at_ which the "thing" is in this state is the
lower limit of the times at which these appearances occur. We have now
to consider what leads us to speak of another set of appearances as
belonging to the same "thing" at a different time.

For this purpose, we may, at least to begin with, confine ourselves
within a single biography. If we can always say when two "sensibilia"
in a given biography are appearances of one thing, then, since we have
seen how to connect "sensibilia" in different biographies as
appearances of the same momentary state of a thing, we shall have all
that is necessary for the complete construction of the history of a
thing.

It is to be observed, to begin with, that the identity of a thing for
common sense is not always correlated with the identity of matter for
physics. A human body is one persisting thing for common sense, but
for physics its matter is constantly changing. We may say, broadly,
that the common-sense conception is based upon continuity in
appearances at the ordinary distances of sense-data, while the
physical conception is based upon the continuity of appearances at
very small distances from the thing. It is probable that the
common-sense conception is not capable of complete precision. Let us
therefore concentrate our attention upon the conception of the
persistence of matter in physics.

The first characteristic of two appearances of the same piece of
matter at different times is _continuity_. The two appearances must be
connected by a series of intermediaries, which, if time and space form
compact series, must themselves form a compact series. The colour of
the leaves is different in autumn from what it is in summer; but we
believe that the change occurs gradually, and that, if the colours are
different at two given times, there are intermediate times at which
the colours are intermediate between those at the given times.

But there are two considerations that are important as regards
continuity.

First, it is largely hypothetical. We do not observe any one thing
continuously, and it is merely a hypothesis to assume that, while we
are not observing it, it passes through conditions intermediate
between those in which it is perceived. During uninterrupted
observation, it is true, continuity is nearly verified; but even here,
when motions are very rapid, as in the case of explosions, the
continuity is not actually capable of direct verification. Thus we can
only say that the sense-data are found to _permit_ a hypothetical
complement of "sensibilia" such as will preserve continuity, and that
therefore there _may_ be such a complement. Since, however, we have
already made such use of hypothetical "sensibilia," we will let this
point pass, and admit such "sensibilia" as are required to preserve
continuity.

Secondly, continuity is not a sufficient criterion of material
identity. It is true that in many cases, such as rocks, mountains,
tables, chairs, etc., where the appearances change slowly, continuity
is sufficient, but in other cases, such as the parts of an
approximately homogeneous fluid, it fails us utterly. We can travel by
sensibly continuous gradations from any one drop of the sea at any one
time to any other drop at any other time. We infer the motions of
sea-water from the effects of the current, but they cannot be inferred
from direct sensible observation together with the assumption of
continuity.

The characteristic required in addition to continuity is conformity
with the laws of dynamics. Starting from what common sense regards as
persistent things, and making only such modifications as from time to
time seem reasonable, we arrive at assemblages of "sensibilia" which
are found to obey certain simple laws, namely those of dynamics. By
regarding "sensibilia" at different times as belonging to the same
piece of matter, we are able to define _motion_, which presupposes the
assumption or construction of something persisting throughout the
time of the motion. The motions which are regarded as occurring,
during a period in which all the "sensibilia" and the times of their
appearance are given, will be different according to the manner in
which we combine "sensibilia" at different times as belonging to the
same piece of matter. Thus even when the whole history of the world is
given in every particular, the question what motions take place is
still to a certain extent arbitrary even after the assumption of
continuity. Experience shows that it is possible to determine motions
in such a way as to satisfy the laws of dynamics, and that this
determination, roughly and on the whole, is fairly in agreement with
the common-sense opinions about persistent things. This determination,
therefore, is adopted, and leads to a criterion by which we can
determine, sometimes practically, sometimes only theoretically,
whether two appearances at different times are to be regarded as
belonging to the same piece of matter. The persistence of all matter
throughout all time can, I imagine, be secured by definition.

To recommend this conclusion, we must consider what it is that is
proved by the empirical success of physics. What is proved is that its
hypotheses, though unverifiable where they go beyond sense-data, are
at no point in contradiction with sense-data, but, on the contrary,
are ideally such as to render all sense-data calculable when a
sufficient collection of "sensibilia" is given. Now physics has found
it empirically possible to collect sense-data into series, each series
being regarded as belonging to one "thing," and behaving, with regard
to the laws of physics, in a way in which series not belonging to one
thing would in general not behave. If it is to be unambiguous whether
two appearances belong to the same thing or not, there must be only
one way of grouping appearances so that the resulting things obey the
laws of physics. It would be very difficult to prove that this is the
case, but for our present purposes we may let this point pass, and
assume that there is only one way. Thus we may lay down the following
definition: _Physical things are those series of appearances whose
matter obeys the laws of physics_. That such series exist is an
empirical fact, which constitutes the verifiability of physics.


XII. ILLUSIONS, HALLUCINATIONS, AND DREAMS

It remains to ask how, in our system, we are to find a place for
sense-data which apparently fail to have the usual connection with the
world of physics. Such sense-data are of various kinds, requiring
somewhat different treatment. But all are of the sort that would be
called "unreal," and therefore, before embarking upon the discussion,
certain logical remarks must be made upon the conceptions of reality
and unreality.

Mr. A. Wolf[31] says:

  "The conception of mind as a system of transparent activities is,
  I think, also untenable because of its failure to account for the
  very possibility of dreams and hallucinations. It seems impossible
  to realise how a bare, transparent activity can be directed to
  what is not there, to apprehend what is not given."

This statement is one which, probably, most people would endorse. But
it is open to two objections. First it is difficult to see how an
activity, however un-"transparent," can be directed towards a nothing:
a term of a relation cannot be a mere nonentity. Secondly, no reason
is given, and I am convinced that none can be given, for the assertion
that dream-objects are not "there" and not "given." Let us take the
second point first.

(1) The belief that dream-objects are not given comes, I think, from
failure to distinguish, as regards waking life, between the
sense-datum and the corresponding "thing." In dreams, there is no such
corresponding "thing" as the dreamer supposes; if, therefore, the
"thing" were given in waking life, as e.g. Meinong maintains,[32] then
there would be a difference in respect of givenness between dreams and
waking life. But if, as we have maintained, what is given is never the
thing, but merely one of the "sensibilia" which compose the thing,
then what we apprehend in a dream is just as much given as what we
apprehend in waking life.

Exactly the same argument applies as to the dream-objects being
"there." They have their position in the private space of the
perspective of the dreamer; where they fail is in their correlation
with other private spaces and therefore with perspective space. But in
the only sense in which "there" can be a datum, they are "there" just
as truly as any of the sense-data of waking life.

(2) The conception of "illusion" or "unreality," and the correlative
conception of "reality," are generally used in a way which embodies
profound logical confusions. Words that go in pairs, such as "real"
and "unreal," "existent" and "non-existent," "valid" and "invalid,"
etc., are all derived from the one fundamental pair, "true" and
"false." Now "true" and "false" are applicable only--except in
derivative significations--to _propositions_. Thus wherever the above
pairs can be significantly applied, we must be dealing either with
propositions or with such incomplete phrases as only acquire meaning
when put into a context which, with them, forms a proposition. Thus
such pairs of words can be applied to _descriptions_,[33] but not to
proper names: in other words, they have no application whatever to
data, but only to entities or non-entities described in terms of data.

Let us illustrate by the terms "existence" and "non-existence." Given
any datum _x_, it is meaningless either to assert or to deny that _x_
"exists." We might be tempted to say: "Of course _x_ exists, for
otherwise it could not be a datum." But such a statement is really
meaningless, although it is significant and true to say "My present
sense-datum exists," and it may also be true that "_x_ is my present
sense-datum." The inference from these two propositions to "_x_
exists" is one which seems irresistible to people unaccustomed to
logic; yet the apparent proposition inferred is not merely false, but
strictly meaningless. To say "My present sense-datum exists" is to say
(roughly): "There is an object of which 'my present sense-datum' is a
description." But we cannot say: "There is an object of which '_x_' is
a description," because '_x_' is (in the case we are supposing) a
name, not a description. Dr. Whitehead and I have explained this point
fully elsewhere (_loc. cit._) with the help of symbols, without which
it is hard to understand; I shall not therefore here repeat the
demonstration of the above propositions, but shall proceed with their
application to our present problem.

The fact that "existence" is only applicable to descriptions is
concealed by the use of what are grammatically proper names in a way
which really transforms them into descriptions. It is, for example, a
legitimate question whether Homer existed; but here "Homer" means
"the author of the Homeric poems," and is a description. Similarly we
may ask whether God exists; but then "God" means "the Supreme Being"
or "the _ens realissimum_" or whatever other description we may
prefer. If "God" were a proper name, God would have to be a datum; and
then no question could arise as to His existence. The distinction
between existence and other predicates, which Kant obscurely felt, is
brought to light by the theory of descriptions, and is seen to remove
"existence" altogether from the fundamental notions of metaphysics.

What has been said about "existence" applies equally to "reality,"
which may, in fact, be taken as synonymous with "existence."
Concerning the immediate objects in illusions, hallucinations, and
dreams, it is meaningless to ask whether they "exist" or are "real."
There they are, and that ends the matter. But we may legitimately
inquire as to the existence or reality of "things" or other
"sensibilia" inferred from such objects. It is the unreality of these
"things" and other "sensibilia," together with a failure to notice
that they are not data, which has led to the view that the objects of
dreams are unreal.

We may now apply these considerations in detail to the stock arguments
against realism, though what is to be said will be mainly a repetition
of what others have said before.

(1) We have first the variety of normal appearances, supposed to be
incompatible. This is the case of the different shapes and colours
which a given thing presents to different spectators. Locke's water
which seems both hot and cold belongs to this class of cases. Our
system of different perspectives fully accounts for these cases, and
shows that they afford no argument against realism.

(2) We have cases where the correlation between different senses is
unusual. The bent stick in water belongs here. People say it looks
bent but is straight: this only means that it is straight to the
touch, though bent to sight. There is no "illusion," but only a false
inference, if we think that the stick would feel bent to the touch.
The stick would look just as bent in a photograph, and, as Mr.
Gladstone used to say, "the photograph cannot lie."[34] The case of
seeing double also belongs here, though in this case the cause of the
unusual correlation is physiological, and would therefore not operate
in a photograph. It is a mistake to ask whether the "thing" is
duplicated when we see it double. The "thing" is a whole system of
"sensibilia," and it is only those visual "sensibilia" which are data
to the percipient that are duplicated. The phenomenon has a purely
physiological explanation; indeed, in view of our having two eyes, it
is in less need of explanation than the single visual sense-datum
which we normally obtain from the things on which we focus.

(3) We come now to cases like dreams, which may, at the moment of
dreaming, contain nothing to arouse suspicion, but are condemned on the
ground of their supposed incompatibility with earlier and later data. Of
course it often happens that dream-objects fail to behave in the
accustomed manner: heavy objects fly, solid objects melt, babies turn
into pigs or undergo even greater changes. But none of these unusual
occurrences _need_ happen in a dream, and it is not on account of such
occurrences that dream-objects are called "unreal." It is their lack of
continuity with the dreamer's past and future that makes him, when he
wakes, condemn them; and it is their lack of correlation with other
private worlds that makes others condemn them. Omitting the latter
ground, our reason for condemning them is that the "things" which we
infer from them cannot be combined according to the laws of physics with
the "things" inferred from waking sense-data. This might be used to
condemn the "things" inferred from the data of dreams. Dream-data are no
doubt appearances of "things," but not of such "things" as the dreamer
supposes. I have no wish to combat psychological theories of dreams,
such as those of the psycho-analysts. But there certainly are cases
where (whatever psychological causes may contribute) the presence of
physical causes also is very evident. For instance, a door banging may
produce a dream of a naval engagement, with images of battleships and
sea and smoke. The whole dream will be an appearance of the door
banging, but owing to the peculiar condition of the body (especially the
brain) during sleep, this appearance is not that expected to be produced
by a door banging, and thus the dreamer is led to entertain false
beliefs. But his sense-data are still physical, and are such as a
completed physics would include and calculate.

(4) The last class of illusions are those which cannot be discovered
within one person's experience, except through the discovery of
discrepancies with the experiences of others. Dreams might conceivably
belong to this class, if they were jointed sufficiently neatly into
waking life; but the chief instances are recurrent sensory
hallucinations of the kind that lead to insanity. What makes the
patient, in such cases, become what others call insane is the fact
that, within his own experience, there is nothing to show that the
hallucinatory sense-data do not have the usual kind of connection with
"sensibilia" in other perspectives. Of course he may learn this
through testimony, but he probably finds it simpler to suppose that
the testimony is untrue and that he is being wilfully deceived. There
is, so far as I can see, no theoretical criterion by which the patient
can decide, in such a case, between the two equally satisfactory
hypotheses of his madness and of his friends' mendacity.

From the above instances it would appear that abnormal sense-data, of
the kind which we regard as deceptive, have intrinsically just the
same status as any others, but differ as regards their correlations or
causal connections with other "sensibilia" and with "things." Since
the usual correlations and connections become part of our unreflective
expectations, and even seem, except to the psychologist, to form part
of our data, it comes to be thought, mistakenly, that in such cases
the data are unreal, whereas they are merely the causes of false
inferences. The fact that correlations and connections of unusual
kinds occur adds to the difficulty of inferring things from sense and
of expressing physics in terms of sense-data. But the unusualness
would seem to be always physically or physiologically explicable, and
therefore raises only a complication, not a philosophical objection.

I conclude, therefore, that no valid objection exists to the view
which regards sense-data as part of the actual substance of the
physical world, and that, on the other hand, this view is the only one
which accounts for the empirical verifiability of physics. In the
present paper, I have given only a rough preliminary sketch. In
particular, the part played by _time_ in the construction of the
physical world is, I think, more fundamental than would appear from
the above account. I should hope that, with further elaboration, the
part played by unperceived "sensibilia" could be indefinitely
diminished, probably by invoking the history of a "thing" to eke out
the inferences derivable from its momentary appearance.

FOOTNOTES:

[29] _Proc. Arist. Soc._, 1909-1910, pp. 191-218.

[30] On this subject, compare _A Theory of Time and Space_, by Mr.
A.A. Robb (Camb. Univ. Press), which first suggested to me the views
advocated here, though I have, for present purposes, omitted what is
most interesting and novel in his theory. Mr. Robb has given a sketch
of his theory in a pamphlet with the same title (Heffer and Sons,
Cambridge, 1913).

[31] "Natural Realism and Present Tendencies in Philosophy," _Proc.
Arist. Soc._, 1908-1909, p. 165.

[32] _Die Erfahrungsgrundlagen unseres Wissens_, p. 28.

[33] Cf. _Principia Mathematica_, Vol. I, * 14, and Introduction,
Chap. III. For the definition of _existence_, cf. * 14. 02.

[34] Cf. Edwin B. Holt, _The Place of Illusory Experience in a
Realistic World._ "The New Realism," p. 303, both on this point and as
regards _seeing double_.




IX

ON THE NOTION OF CAUSE


In the following paper I wish, first, to maintain that the word
"cause" is so inextricably bound up with misleading associations as to
make its complete extrusion from the philosophical vocabulary
desirable; secondly, to inquire what principle, if any, is employed in
science in place of the supposed "law of causality" which philosophers
imagine to be employed; thirdly, to exhibit certain confusions,
especially in regard to teleology and determinism, which appear to me
to be connected with erroneous notions as to causality.

All philosophers, of every school, imagine that causation is one of
the fundamental axioms or postulates of science, yet, oddly enough, in
advanced sciences such as gravitational astronomy, the word "cause"
never occurs. Dr. James Ward, in his _Naturalism and Agnosticism_,
makes this a ground of complaint against physics: the business of
those who wish to ascertain the ultimate truth about the world, he
apparently thinks, should be the discovery of causes, yet physics
never even seeks them. To me it seems that philosophy ought not to
assume such legislative functions, and that the reason why physics has
ceased to look for causes is that, in fact, there are no such things.
The law of causality, I believe, like much that passes muster among
philosophers, is a relic of a bygone age, surviving, like the
monarchy, only because it is erroneously supposed to do no harm. In
order to find out what philosophers commonly understand by "cause," I
consulted Baldwin's _Dictionary_, and was rewarded beyond my
expectations, for I found the following three mutually incompatible
definitions:--

  "CAUSALITY. (1) The necessary connection of events in the
     time-series....

  "CAUSE (notion of). Whatever may be included in the thought or
     perception of a process as taking place in consequence of
     another process....

  "CAUSE AND EFFECT. (1) Cause and effect ... are correlative terms
     denoting any two distinguishable things, phases, or aspects of
     reality, which are so related to each other that whenever the
     first ceases to exist the second comes into existence
     immediately after, and whenever the second comes into existence
     the first has ceased to exist immediately before."

Let us consider these three definitions in turn. The first, obviously,
is unintelligible without a definition of "necessary." Under this
head, Baldwin's _Dictionary_ gives the following:--

  "NECESSARY. That is necessary which not only is true, but would
     be true under all circumstances. Something more than brute
     compulsion is, therefore, involved in the conception; there is
     a general law under which the thing takes place."

The notion of cause is so intimately connected with that of necessity
that it will be no digression to linger over the above definition,
with a view to discovering, if possible, _some_ meaning of which it is
capable; for, as it stands, it is very far from having any definite
signification.

The first point to notice is that, if any meaning is to be given to
the phrase "would be true under all circumstances," the subject of it
must be a propositional function, not a proposition.[35] A
proposition is simply true or false, and that ends the matter: there
can be no question of "circumstances." "Charles I's head was cut off"
is just as true in summer as in winter, on Sundays as on Mondays. Thus
when it is worth saying that something "would be true under all
circumstances," the something in question must be a propositional
function, i.e. an expression containing a variable, and becoming a
proposition when a value is assigned to the variable; the varying
"circumstances" alluded to are then the different values of which the
variable is capable. Thus if "necessary" means "what is true under all
circumstances," then "if _x_ is a man, _x_ is mortal" is necessary,
because it is true for any possible value of _x_. Thus we should be
led to the following definition:--

  "NECESSARY is a predicate of a propositional function, meaning
     that it is true for all possible values of its argument or
     arguments."

Unfortunately, however, the definition in Baldwin's _Dictionary_ says
that what is necessary is not only "true under all circumstances" but
is also "true." Now these two are incompatible. Only propositions can
be "true," and only propositional functions can be "true under all
circumstances." Hence the definition as it stands is nonsense. What is
meant seems to be this: "A proposition is necessary when it is a value
of a propositional function which is true under all circumstances,
i.e. for all values of its argument or arguments." But if we adopt
this definition, the same proposition will be necessary or contingent
according as we choose one or other of its terms as the argument to
our propositional function. For example, "if Socrates is a man,
Socrates is mortal," is necessary if Socrates is chosen as argument,
but not if _man_ or _mortal_ is chosen. Again, "if Socrates is a man,
Plato is mortal," will be necessary if either Socrates or _man_ is
chosen as argument, but not if Plato or _mortal_ is chosen. However,
this difficulty can be overcome by specifying the constituent which is
to be regarded as argument, and we thus arrive at the following
definition:

"A proposition is _necessary_ with respect to a given constituent if
it remains true when that constituent is altered in any way compatible
with the proposition remaining significant."

We may now apply this definition to the definition of causality quoted
above. It is obvious that the argument must be the time at which the
earlier event occurs. Thus an instance of causality will be such as:
"If the event [Math: e_{1}] occurs at the time [Math: t_{1}], it will
be followed by the event [Math: e_{2}]." This proposition is intended
to be necessary with respect to [Math: t_{1}], i.e. to remain true
however [Math: t_{1}] may be varied. Causality, as a universal law,
will then be the following: "Given any event [Math: t_{1}], there is
an event [Math: e_{2}] such that, whenever [Math: t_{1}] occurs,
[Math: e_{2}] occurs later." But before this can be considered
precise, we must specify how much later [Math: e_{2}] is to occur.
Thus the principle becomes:--

"Given any event [Math: e_{1}], there is an event [Math: e_{2}] and a
time-interval τ such that, whenever [Math: e_{1}] occurs, [Math:
e_{2}] follows after an interval τ."

I am not concerned as yet to consider whether this law is true or
false. For the present, I am merely concerned to discover what the law
of causality is supposed to be. I pass, therefore, to the other
definitions quoted above.

The second definition need not detain us long, for two reasons. First,
because it is psychological: not the "thought or perception" of a
process, but the process itself, must be what concerns us in
considering causality. Secondly, because it is circular: in speaking
of a process as "taking place in consequence of" another process, it
introduces the very notion of cause which was to be defined.

The third definition is by far the most precise; indeed as regards
clearness it leaves nothing to be desired. But a great difficulty is
caused by the temporal contiguity of cause and effect which the
definition asserts. No two instants are contiguous, since the
time-series is compact; hence either the cause or the effect or both
must, if the definition is correct, endure for a finite time; indeed,
by the wording of the definition it is plain that both are assumed to
endure for a finite time. But then we are faced with a dilemma: if the
cause is a process involving change within itself, we shall require
(if causality is universal) causal relations between its earlier and
later parts; moreover, it would seem that only the later parts can be
relevant to the effect, since the earlier parts are not contiguous to
the effect, and therefore (by the definition) cannot influence the
effect. Thus we shall be led to diminish the duration of the cause
without limit, and however much we may diminish it, there will still
remain an earlier part which might be altered without altering the
effect, so that the true cause, as defined, will not have been
reached, for it will be observed that the definition excludes
plurality of causes. If, on the other hand, the cause is purely
static, involving no change within itself, then, in the first place,
no such cause is to be found in nature, and in the second place, it
seems strange--too strange to be accepted, in spite of bare logical
possibility--that the cause, after existing placidly for some time,
should suddenly explode into the effect, when it might just as well
have done so at any earlier time, or have gone on unchanged without
producing its effect. This dilemma, therefore, is fatal to the view
that cause and effect can be contiguous in time; if there are causes
and effects, they must be separated by a finite time-interval τ, as
was assumed in the above interpretation of the first definition.

What is essentially the same statement of the law of causality as the
one elicited above from the first of Baldwin's definitions is given by
other philosophers. Thus John Stuart Mill says:--

"The Law of Causation, the recognition of which is the main pillar of
inductive science, is but the familiar truth, that invariability of
succession is found by observation to obtain between every fact in
nature and some other fact which has preceded it."[36]

And Bergson, who has rightly perceived that the law as stated by
philosophers is worthless, nevertheless continues to suppose that it
is used in science. Thus he says:--

"Now, it is argued, this law [the law of causality] means that every
phenomenon is determined by its conditions, or, in other words, that
the same causes produce the same effects."[37]

And again:--

"We perceive physical phenomena, and these phenomena obey laws. This
means: (1) That phenomena _a_, _b_, _c_, _d_, previously perceived,
can occur again in the same shape; (2) that a certain phenomenon P,
which appeared after the conditions _a_, _b_, _c_, _d_, and after
these conditions only, will not fail to recur as soon as the same
conditions are again present."[38]

A great part of Bergson's attack on science rests on the assumption
that it employs this principle. In fact, it employs no such principle,
but philosophers--even Bergson--are too apt to take their views on
science from each other, not from science. As to what the principle
is, there is a fair consensus among philosophers of different schools.
There are, however, a number of difficulties which at once arise. I
omit the question of plurality of causes for the present, since other
graver questions have to be considered. Two of these, which are forced
on our attention by the above statement of the law, are the
following:--

  (1) What is meant by an "event"?

  (2) How long may the time-interval be between cause and effect?

(1) An "event," in the statement of the law, is obviously intended to
be something that is likely to recur since otherwise the law becomes
trivial. It follows that an "event" is not a particular, but some
universal of which there may be many instances. It follows also that
an "event" must be something short of the whole state of the universe,
since it is highly improbable that this will recur. What is meant by
an "event" is something like striking a match, or dropping a penny
into the slot of an automatic machine. If such an event is to recur,
it must not be defined too narrowly: we must not state with what
degree of force the match is to be struck, nor what is to be the
temperature of the penny. For if such considerations were relevant,
our "event" would occur at most once, and the law would cease to give
information. An "event," then, is a universal defined sufficiently
widely to admit of many particular occurrences in time being instances
of it.

(2) The next question concerns the time-interval. Philosophers, no
doubt, think of cause and effect as contiguous in time, but this, for
reasons already given, is impossible. Hence, since there are no
infinitesimal time-intervals, there must be some finite lapse of time
τ between cause and effect. This, however, at once raises insuperable
difficulties. However short we make the interval τ, something may
happen during this interval which prevents the expected result. I put
my penny in the slot, but before I can draw out my ticket there is an
earthquake which upsets the machine and my calculations. In order to
be sure of the expected effect, we must know that there is nothing in
the environment to interfere with it. But this means that the supposed
cause is not, by itself, adequate to insure the effect. And as soon as
we include the environment, the probability of repetition is
diminished, until at last, when the whole environment is included, the
probability of repetition becomes almost _nil_.

In spite of these difficulties, it must, of course, be admitted that
many fairly dependable regularities of sequence occur in daily life.
It is these regularities that have suggested the supposed law of
causality; where they are found to fail, it is thought that a better
formulation could have been found which would have never failed. I am
far from denying that there may be such sequences which in fact never
do fail. It may be that there will never be an exception to the rule
that when a stone of more than a certain mass, moving with more than a
certain velocity, comes in contact with a pane of glass of less than
a certain thickness, the glass breaks. I also do not deny that the
observation of such regularities, even when they are not without
exceptions, is useful in the infancy of a science: the observation
that unsupported bodies in air usually fall was a stage on the way to
the law of gravitation. What I deny is that science assumes the
existence of invariable uniformities of sequence of this kind, or that
it aims at discovering them. All such uniformities, as we saw, depend
upon a certain vagueness in the definition of the "events." That
bodies fall is a vague qualitative statement; science wishes to know
how fast they fall. This depends upon the shape of the bodies and the
density of the air. It is true that there is more nearly uniformity
when they fall in a vacuum; so far as Galileo could observe, the
uniformity is then complete. But later it appeared that even there the
latitude made a difference, and the altitude. Theoretically, the
position of the sun and moon must make a difference. In short, every
advance in a science takes us farther away from the crude uniformities
which are first observed, into greater differentiation of antecedent
and consequent, and into a continually wider circle of antecedents
recognised as relevant.

The principle "same cause, same effect," which philosophers imagine to
be vital to science, is therefore utterly otiose. As soon as the
antecedents have been given sufficiently fully to enable the
consequent to be calculated with some exactitude, the antecedents have
become so complicated that it is very unlikely they will ever recur.
Hence, if this were the principle involved, science would remain
utterly sterile.

The importance of these considerations lies partly in the fact that
they lead to a more correct account of scientific procedure, partly in
the fact that they remove the analogy with human volition which makes
the conception of cause such a fruitful source of fallacies. The
latter point will become clearer by the help of some illustrations.
For this purpose I shall consider a few maxims which have played a
great part in the history of philosophy.

(1) "Cause and effect must more or less resemble each other." This
principle was prominent in the philosophy of occasionalism, and is
still by no means extinct. It is still often thought, for example,
that mind could not have grown up in a universe which previously
contained nothing mental, and one ground for this belief is that
matter is too dissimilar from mind to have been able to cause it. Or,
more particularly, what are termed the nobler parts of our nature are
supposed to be inexplicable, unless the universe always contained
something at least equally noble which could cause them. All such
views seem to depend upon assuming some unduly simplified law of
causality; for, in any legitimate sense of "cause" and "effect,"
science seems to show that they are usually very widely dissimilar,
the "cause" being, in fact, two states of the whole universe, and the
"effect" some particular event.

(2) "Cause is analogous to volition, since there must be an
intelligible _nexus_ between cause and effect." This maxim is, I
think, often unconsciously in the imaginations of philosophers who
would reject it when explicitly stated. It is probably operative in
the view we have just been considering, that mind could not have
resulted from a purely material world. I do not profess to know what
is meant by "intelligible"; it seems to mean "familiar to
imagination." Nothing is less "intelligible," in any other sense, than
the connection between an act of will and its fulfilment. But
obviously the sort of nexus desired between cause and effect is such
as could only hold between the "events" which the supposed law of
causality contemplates; the laws which replace causality in such a
science as physics leave no room for any two events between which a
nexus could be sought.

(3) "The cause _compels_ the effect in some sense in which the effect
does not compel the cause." This belief seems largely operative in the
dislike of determinism; but, as a matter of fact, it is connected with
our second maxim, and falls as soon as that is abandoned. We may
define "compulsion" as follows: "Any set of circumstances is said to
compel A when A desires to do something which the circumstances
prevent, or to abstain from something which the circumstances cause."
This presupposes that some meaning has been found for the word
"cause"--a point to which I shall return later. What I want to make
clear at present is that compulsion is a very complex notion,
involving thwarted desire. So long as a person does what he wishes to
do, there is no compulsion, however much his wishes may be calculable
by the help of earlier events. And where desire does not come in,
there can be no question of compulsion. Hence it is, in general,
misleading to regard the cause as compelling the effect.

A vaguer form of the same maxim substitutes the word "determine" for
the word "compel"; we are told that the cause _determines_ the effect
in a sense in which the effect does not _determine_ the cause. It is
not quite clear what is meant by "determining"; the only precise
sense, so far as I know, is that of a function or one-many relation.
If we admit plurality of causes, but not of effects, that is, if we
suppose that, given the cause, the effect must be such and such, but,
given the effect, the cause may have been one of many alternatives,
then we may say that the cause determines the effect, but not the
effect the cause. Plurality of causes, however, results only from
conceiving the effect vaguely and narrowly and the cause precisely and
widely. Many antecedents may "cause" a man's death, because his death
is vague and narrow. But if we adopt the opposite course, taking as
the "cause" the drinking of a dose of arsenic, and as the "effect" the
whole state of the world five minutes later, we shall have plurality
of effects instead of plurality of causes. Thus the supposed lack of
symmetry between "cause" and "effect" is illusory.

(4) "A cause cannot operate when it has ceased to exist, because what
has ceased to exist is nothing." This is a common maxim, and a still
more common unexpressed prejudice. It has, I fancy, a good deal to do
with the attractiveness of Bergson's "_durée_": since the past has
effects now, it must still exist in some sense. The mistake in this
maxim consists in the supposition that causes "operate" at all. A
volition "operates" when what it wills takes place; but nothing can
operate except a volition. The belief that causes "operate" results
from assimilating them, consciously or unconsciously, to volitions. We
have already seen that, if there are causes at all, they must be
separated by a finite interval of time from their effects, and thus
cause their effects after they have ceased to exist.

It may be objected to the above definition of a volition "operating"
that it only operates when it "causes" what it wills, not when it
merely happens to be followed by what it wills. This certainly
represents the usual view of what is meant by a volition "operating,"
but as it involves the very view of causation which we are engaged in
combating, it is not open to us as a definition. We may say that a
volition "operates" when there is some law in virtue of which a
similar volition in rather similar circumstances will usually be
followed by what it wills. But this is a vague conception, and
introduces ideas which we have not yet considered. What is chiefly
important to notice is that the usual notion of "operating" is not
open to us if we reject, as I contend that we should, the usual notion
of causation.

(5) "A cause cannot operate except where it is." This maxim is very
widespread; it was urged against Newton, and has remained a source of
prejudice against "action at a distance." In philosophy it has led to
a denial of transient action, and thence to monism or Leibnizian
monadism. Like the analogous maxim concerning temporal contiguity, it
rests upon the assumption that causes "operate," i.e. that they are in
some obscure way analogous to volitions. And, as in the case of
temporal contiguity, the inferences drawn from this maxim are wholly
groundless.

I return now to the question, What law or laws can be found to take
the place of the supposed law of causality?

First, without passing beyond such uniformities of sequence as are
contemplated by the traditional law, we may admit that, if any such
sequence has been observed in a great many cases, and has never been
found to fail, there is an inductive probability that it will be found
to hold in future cases. If stones have hitherto been found to break
windows, it is probable that they will continue to do so. This, of
course, assumes the inductive principle, of which the truth may
reasonably be questioned; but as this principle is not our present
concern, I shall in this discussion treat it as indubitable. We may
then say, in the case of any such frequently observed sequence, that
the earlier event is the _cause_ and the later event the _effect_.

Several considerations, however, make such special sequences very
different from the traditional relation of cause and effect. In the
first place, the sequence, in any hitherto unobserved instance, is no
more than probable, whereas the relation of cause and effect was
supposed to be necessary. I do not mean by this merely that we are not
sure of having discovered a true case of cause and effect; I mean
that, even when we have a case of cause and effect in our present
sense, all that is meant is that on grounds of observation, it is
probable that when one occurs the other will also occur. Thus in our
present sense, A may be the cause of B even if there actually are
cases where B does not follow A. Striking a match will be the cause of
its igniting, in spite of the fact that some matches are damp and fail
to ignite.

In the second place, it will not be assumed that _every_ event has
some antecedent which is its cause in this sense; we shall only
believe in causal sequences where we find them, without any
presumption that they always are to be found.

In the third place, _any_ case of sufficiently frequent sequence will
be causal in our present sense; for example, we shall not refuse to
say that night is the cause of day. Our repugnance to saying this
arises from the ease with which we can imagine the sequence to fail,
but owing to the fact that cause and effect must be separated by a
finite interval of time, _any_ such sequence _might_ fail through the
interposition of other circumstances in the interval. Mill, discussing
this instance of night and day, says:--

"It is necessary to our using the word cause, that we should believe
not only that the antecedent always _has_ been followed by the
consequent, but that as long as the present constitution of things
endures, it always _will_ be so."[39]

In this sense, we shall have to give up the hope of finding causal
laws such as Mill contemplated; any causal sequence which we have
observed may at any moment be falsified without a falsification of any
laws of the kind that the more advanced sciences aim at establishing.

In the fourth place, such laws of probable sequence, though useful in
daily life and in the infancy of a science, tend to be displaced by
quite different laws as soon as a science is successful. The law of
gravitation will illustrate what occurs in any advanced science. In
the motions of mutually gravitating bodies, there is nothing that can
be called a cause, and nothing that can be called an effect; there is
merely a formula. Certain differential equations can be found, which
hold at every instant for every particle of the system, and which,
given the configuration and velocities at one instant, or the
configurations at two instants, render the configuration at any other
earlier or later instant theoretically calculable. That is to say, the
configuration at any instant is a function of that instant and the
configurations at two given instants. This statement holds throughout
physics, and not only in the special case of gravitation. But there is
nothing that could be properly called "cause" and nothing that could
be properly called "effect" in such a system.

No doubt the reason why the old "law of causality" has so long
continued to pervade the books of philosophers is simply that the idea
of a function is unfamiliar to most of them, and therefore they seek
an unduly simplified statement. There is no question of repetitions of
the "same" cause producing the "same" effect; it is not in any
sameness of causes and effects that the constancy of scientific law
consists, but in sameness of relations. And even "sameness of
relations" is too simple a phrase; "sameness of differential
equations" is the only correct phrase. It is impossible to state this
accurately in non-mathematical language; the nearest approach would be
as follows: "There is a constant relation between the state of the
universe at any instant and the rate of change in the rate at which
any part of the universe is changing at that instant, and this
relation is many-one, i.e. such that the rate of change in the rate of
change is determinate when the state of the universe is given." If the
"law of causality" is to be something actually discoverable in the
practice of science, the above proposition has a better right to the
name than any "law of causality" to be found in the books of
philosophers.

In regard to the above principle, several observations must be made--

(1) No one can pretend that the above principle is _a priori_ or
self-evident or a "necessity of thought." Nor is it, in any sense, a
premiss of science: it is an empirical generalisation from a number of
laws which are themselves empirical generalisations.

(2) The law makes no difference between past and future: the future
"determines" the past in exactly the same sense in which the past
"determines" the future. The word "determine," here, has a purely
logical significance: a certain number of variables "determine"
another variable if that other variable is a function of them.

(3) The law will not be empirically verifiable unless the course of
events within some sufficiently small volume will be approximately
the same in any two states of the universe which only differ in regard
to what is at a considerable distance from the small volume in
question. For example, motions of planets in the solar system must be
approximately the same however the fixed stars may be distributed,
provided that all the fixed stars are very much farther from the sun
than the planets are. If gravitation varied directly as the distance,
so that the most remote stars made the most difference to the motions
of the planets, the world might be just as regular and just as much
subject to mathematical laws as it is at present, but we could never
discover the fact.

(4) Although the old "law of causality" is not assumed by science,
something which we may call the "uniformity of nature" is assumed, or
rather is accepted on inductive grounds. The uniformity of nature does
not assert the trivial principle "same cause, same effect," but the
principle of the permanence of laws. That is to say, when a law
exhibiting, e.g. an acceleration as a function of the configuration
has been found to hold throughout the observable past, it is expected
that it will continue to hold in the future, or that, if it does not
itself hold, there is some other law, agreeing with the supposed law
as regards the past, which will hold for the future. The ground of
this principle is simply the inductive ground that it has been found
to be true in very many instances; hence the principle cannot be
considered certain, but only probable to a degree which cannot be
accurately estimated.

The uniformity of nature, in the above sense, although it is assumed
in the practice of science, must not, in its generality, be regarded
as a kind of major premiss, without which all scientific reasoning
would be in error. The assumption that _all_ laws of nature are
permanent has, of course, less probability than the assumption that
this or that particular law is permanent; and the assumption that a
particular law is permanent for all time has less probability than the
assumption that it will be valid up to such and such a date. Science,
in any given case, will assume what the case requires, but no more. In
constructing the _Nautical Almanac_ for 1915 it will assume that the
law of gravitation will remain true up to the end of that year; but it
will make no assumption as to 1916 until it comes to the next volume
of the almanac. This procedure is, of course, dictated by the fact
that the uniformity of nature is not known _a priori_, but is an
empirical generalisation, like "all men are mortal." In all such
cases, it is better to argue immediately from the given particular
instances to the new instance, than to argue by way of a major
premiss; the conclusion is only probable in either case, but acquires
a higher probability by the former method than by the latter.

In all science we have to distinguish two sorts of laws: first, those
that are empirically verifiable but probably only approximate;
secondly, those that are not verifiable, but may be exact. The law of
gravitation, for example, in its applications to the solar system, is
only empirically verifiable when it is assumed that matter outside the
solar system may be ignored for such purposes; we believe this to be
only approximately true, but we cannot empirically verify the law of
universal gravitation which we believe to be exact. This point is very
important in connection with what we may call "relatively isolated
systems." These may be defined as follows:--

A system relatively isolated during a given period is one which,
within some assignable margin of error, will behave in the same way
throughout that period, however the rest of the universe may be
constituted.

A system may be called "practically isolated" during a given period
if, although there _might_ be states of the rest of the universe which
would produce more than the assigned margin of error, there is reason
to believe that such states do not in fact occur.

Strictly speaking, we ought to specify the respect in which the system
is relatively isolated. For example, the earth is relatively isolated
as regards falling bodies, but not as regards tides; it is
_practically_ isolated as regards economic phenomena, although, if
Jevons' sunspot theory of commercial crises had been true, it would
not have been even practically isolated in this respect.

It will be observed that we cannot prove in advance that a system is
isolated. This will be inferred from the observed fact that
approximate uniformities can be stated for this system alone. If the
complete laws for the whole universe were known, the isolation of a
system could be deduced from them; assuming, for example, the law of
universal gravitation, the practical isolation of the solar system in
this respect can be deduced by the help of the fact that there is very
little matter in its neighbourhood. But it should be observed that
isolated systems are only important as providing a possibility of
_discovering_ scientific laws; they have no theoretical importance in
the finished structure of a science.

The case where one event A is said to "cause" another event B, which
philosophers take as fundamental, is really only the most simplified
instance of a practically isolated system. It may happen that, as a
result of general scientific laws, whenever A occurs throughout a
certain period, it is followed by B; in that case, A and B form a
system which is practically isolated throughout that period. It is,
however, to be regarded as a piece of good fortune if this occurs; it
will always be due to special circumstances, and would not have been
true if the rest of the universe had been different though subject to
the same laws.

The essential function which causality has been supposed to perform is
the possibility of inferring the future from the past, or, more
generally, events at any time from events at certain assigned times.
Any system in which such inference is possible may be called a
"deterministic" system. We may define a deterministic system as
follows:--

  A system is said to be "deterministic" when, given certain data,
  [Math: e_{1}, e_{2}, ..., e_{n}, at times t_{1}, t_{2}, ...,
  t_{n}] respectively, concerning this system, if [Math: E_{t}] is
  the state of the system at any time _t_, there is a functional
  relation of the form

[Math: E_{t} = f (e_{1}, t_{1}, e_{2}, t_{2}, ..., e_{n}, t_{n}, t)]. (A)

  The system will be "deterministic throughout a given period" if
  _t_, in the above formula, may be any time within that period,
  though outside that period the formula may be no longer true. If
  the universe, as a whole, is such a system, determinism is true of
  the universe; if not, not. A system which is part of a
  deterministic system I shall call "determined"; one which is not
  part of any such system I shall call "capricious."

The events [Math: e_{1}, e_{2}, ..., e_{n}] I shall call "determinants"
of the system. It is to be observed that a system which has one set of
determinants will in general have many. In the case of the motions of
the planets, for example, the configurations of the solar system at any
two given times will be determinants.

We may take another illustration from the hypothesis of
psycho-physical parallelism. Let us assume, for the purposes of this
illustration, that to a given state of brain a given state of mind
always corresponds, and vice versa, i.e. that there is a one-one
relation between them, so that each is a function of the other. We may
also assume, what is practically certain, that to a given state of a
certain brain a given state of the whole material universe
corresponds, since it is highly improbable that a given brain is ever
twice in exactly the same state. Hence there will be a one-one
relation between the state of a given person's mind and the state of
the whole material universe. It follows that, if _n_ states of the
material universe are determinants of the material universe, then _n_
states of a given man's mind are determinants of the whole material
and mental universe--assuming, that is to say, that psycho-physical
parallelism is true.

The above illustration is important in connection with a certain
confusion which seems to have beset those who have philosophised on
the relation of mind and matter. It is often thought that, if the
state of the mind is determinate when the state of the brain is given,
and if the material world forms a deterministic system, then mind is
"subject" to matter in some sense in which matter is not "subject" to
mind. But if the state of the brain is also determinate when the state
of the mind is given, it must be exactly as true to regard matter as
subject to mind as it would be to regard mind as subject to matter. We
could, theoretically, work out the history of mind without ever
mentioning matter, and then, at the end, deduce that matter must
meanwhile have gone through the corresponding history. It is true that
if the relation of brain to mind were many-one, not one-one, there
would be a one-sided dependence of mind on brain, while conversely, if
the relation were one-many, as Bergson supposes, there would be a
one-aided dependence of brain on mind. But the dependence involved is,
in any case, only logical; it does not mean that we shall be
compelled to do things we desire not to do, which is what people
instinctively imagine it to mean.

As another illustration we may take the case of mechanism and
teleology. A system may be defined as "mechanical" when it has a set
of determinants that are purely material, such as the positions of
certain pieces of matter at certain times. It is an open question
whether the world of mind and matter, as we know it, is a mechanical
system or not; let us suppose, for the sake of argument, that it is a
mechanical system. This supposition--so I contend--throws no light
whatever on the question whether the universe is or is not a
"teleological" system. It is difficult to define accurately what is
meant by a "teleological" system, but the argument is not much
affected by the particular definition we adopt. Broadly, a
teleological system is one in which purposes are realised, i.e. in
which certain desires--those that are deeper or nobler or more
fundamental or more universal or what not--are followed by their
realisation. Now the fact--if it be a fact--that the universe is
mechanical has no bearing whatever on the question whether it is
teleological in the above sense. There might be a mechanical system in
which all wishes were realised, and there might be one in which all
wishes were thwarted. The question whether, or how far, our actual
world is teleological, cannot, therefore, be settled by proving that
it is mechanical, and the desire that it should be teleological is no
ground for wishing it to be not mechanical.

There is, in all these questions, a very great difficulty in avoiding
confusion between what we can infer and what is in fact determined.
Let us consider, for a moment, the various senses in which the future
may be "determined." There is one sense--and a very important one--in
which it is determined quite independently of scientific laws, namely,
the sense that it will be what it will be. We all regard the past as
determined simply by the fact that it has happened; but for the
accident that memory works backward and not forward, we should regard
the future as equally determined by the fact that it will happen.
"But," we are told, "you cannot alter the past, while you can to some
extent alter the future." This view seems to me to rest upon just
those errors in regard to causation which it has been my object to
remove. You cannot make the past other than it was--true, but this is
a mere application of the law of contradiction. If you already know
what the past was, obviously it is useless to wish it different. But
also you cannot make the future other than it will be; this again is
an application of the law of contradiction. And if you happen to know
the future--e.g. in the case of a forthcoming eclipse--it is just as
useless to wish it different as to wish the past different. "But," it
will be rejoined, "our wishes can _cause_ the future, sometimes, to be
different from what it would be if they did not exist, and they can
have no such effect upon the past." This, again, is a mere tautology.
An effect being _defined_ as something subsequent to its cause,
obviously we can have no _effect_ upon the past. But that does not
mean that the past would not have been different if our present wishes
had been different. Obviously, our present wishes are conditioned by
the past, and therefore could not have been different unless the past
had been different; therefore, if our present wishes were different,
the past would be different. Of course, the past cannot be different
from what it was, but no more can our present wishes be different from
what they are; this again is merely the law of contradiction. The
facts seem to be merely (1) that wishing generally depends upon
ignorance, and is therefore commoner in regard to the future than in
regard to the past; (2) that where a wish concerns the future, it and
its realisation very often form a "practically independent system,"
i.e. many wishes regarding the future are realised. But there seems no
doubt that the main difference in our feelings arises from the
accidental fact that the past but not the future can be known by
memory.

Although the sense of "determined" in which the future is determined
by the mere fact that it will be what it will be is sufficient (at
least so it seems to me) to refute some opponents of determinism,
notably M. Bergson and the pragmatists, yet it is not what most people
have in mind when they speak of the future as determined. What they
have in mind is a formula by means of which the future can be
exhibited, and at least theoretically calculated, as a function of the
past. But at this point we meet with a great difficulty, which besets
what has been said above about deterministic systems, as well as what
is said by others.

If formulæ of any degree of complexity, however great, are admitted,
it would seem that any system, whose state at a given moment is a
function of certain measurable quantities, must be a deterministic
system. Let us consider, in illustration, a single material particle,
whose co-ordinates at time _t_ are [Math: x_{t}, y_{t}, z_{t}]. Then,
however, the particle moves, there must be, theoretically, functions
[Math: f_{1}, f_{2}, f_{3}], such that

[Math: x_{t} = f_{t}(t), y_{t} = f_{2}(t), z_{t} = f_{3}(t).]

It follows that, theoretically, the whole state of the material
universe at time _t_ must be capable of being exhibited as a function
of _t_. Hence our universe will be deterministic in the sense defined
above. But if this be true, no information is conveyed about the
universe in stating that it is deterministic. It is true that the
formulæ involved may be of strictly infinite complexity, and therefore
not practically capable of being written down or apprehended. But
except from the point of view of our knowledge, this might seem to be
a detail: in itself, if the above considerations are sound, the
material universe _must_ be deterministic, _must_ be subject to laws.

This, however, is plainly not what was intended. The difference
between this view and the view intended may be seen as follows. Given
some formula which fits the facts hitherto--say the law of
gravitation--there will be an infinite number of other formulæ, not
empirically distinguishable from it in the past, but diverging from it
more and more in the future. Hence, even assuming that there are
persistent laws, we shall have no reason for assuming that the law of
the inverse square will hold in future; it may be some other hitherto
indistinguishable law that will hold. We cannot say that _every_ law
which has held hitherto must hold in the future, because past facts
which obey one law will also obey others, hitherto indistinguishable
but diverging in future. Hence there must, at every moment, be laws
hitherto unbroken which are now broken for the first time. What
science does, in fact, is to select the _simplest_ formula that will
fit the facts. But this, quite obviously, is merely a methodological
precept, not a law of Nature. If the simplest formula ceases, after a
time, to be applicable, the simplest formula that remains applicable
is selected, and science has no sense that an axiom has been
falsified. We are thus left with the brute fact that, in many
departments of science, quite simple laws have hitherto been found to
hold. This fact cannot be regarded as having any _a priori_ ground,
nor can it be used to support inductively the opinion that the same
laws will continue; for at every moment laws hitherto true are being
falsified, though in the advanced sciences these laws are less simple
than those that have remained true. Moreover it would be fallacious to
argue inductively from the state of the advanced sciences to the
future state of the others, for it may well be that the advanced
sciences are advanced simply because, hitherto, their subject-matter
has obeyed simple and easily ascertainable laws, while the
subject-matter of other sciences has not done so.

The difficulty we have been considering seems to be met partly, if not
wholly, by the principle that the _time_ must not enter explicitly
into our formulæ. All mechanical laws exhibit acceleration as a
function of configuration, not of configuration and time jointly; and
this principle of the irrelevance of the time may be extended to all
scientific laws. In fact we might interpret the "uniformity of nature"
as meaning just this, that no scientific law involves the time as an
argument, unless, of course, it is given in an integrated form, in
which case _lapse_ of time, though not absolute time, may appear in
our formulæ. Whether this consideration suffices to overcome our
difficulty completely, I do not know; but in any case it does much to
diminish it.

It will serve to illustrate what has been said if we apply it to the
question of free will.

(1) Determinism in regard to the will is the doctrine that our
volitions belong to some deterministic system, i.e. are "determined"
in the sense defined above. Whether this doctrine is true or false, is
a mere question of fact; no _a priori_ considerations (if our previous
discussions have been correct) can exist on either side. On the one
hand, there is no _a priori_ category of causality, but merely certain
observed uniformities. As a matter of fact, there are observed
uniformities in regard to volitions; thus there is some empirical
evidence that volitions are determined. But it would be very rash to
maintain that the evidence is overwhelming, and it is quite possible
that some volitions, as well as some other things, are not determined,
except in the sense in which we found that everything must be
determined.

(2) But, on the other hand, the subjective sense of freedom, sometimes
alleged against determinism, has no bearing on the question whatever.
The view that it has a bearing rests upon the belief that causes
compel their effects, or that nature enforces obedience to its laws as
governments do. These are mere anthropomorphic superstitions, due to
assimilation of causes with volitions and of natural laws with human
edicts. We feel that our will is not compelled, but that only means
that it is not other than we choose it to be. It is one of the
demerits of the traditional theory of causality that it has created an
artificial opposition between determinism and the freedom of which we
are introspectively conscious.

(3) Besides the general question whether volitions are determined,
there is the further question whether they are _mechanically_
determined, i.e. whether they are part of what was above defined as a
mechanical system. This is the question whether they form part of a
system with purely material determinants, i.e. whether there are laws
which, given certain material data, make all volitions functions of
those data. Here again, there is empirical evidence up to a point, but
it is not conclusive in regard to all volitions. It is important to
observe, however that even if volitions are part of a mechanical
system, this by no means implies any supremacy of matter over mind. It
may well be that the same system which is susceptible of material
determinants is also susceptible of mental determinants; thus a
mechanical system may be determined by sets of volitions, as well as
by sets of material facts. It would seem, therefore, that the reasons
which make people dislike the view that volitions are mechanically
determined are fallacious.

(4) The notion of _necessity_, which is often associated with
determinism, is a confused notion not legitimately deducible from
determinism. Three meanings are commonly confounded when necessity is
spoken of:--

(α) An _action_ is necessary when it will be performed however much
the agent may wish to do otherwise. Determinism does not imply that
actions are necessary in this sense.

(β) A _propositional function_ is necessary when all its values are
true. This sense is not relevant to our present discussion.

(γ) A _proposition_ is necessary with respect to a given constituent
when it is the value, with that constituent as argument, of a
necessary propositional function, in other words, when it remains true
however that constituent may be varied. In this sense, in a
deterministic system, the connection of a volition with its
determinants is necessary, if the time at which the determinants occur
be taken as the constituent to be varied, the time-interval between
the determinants and the volition being kept constant. But this sense
of necessity is purely logical, and has no emotional importance.

We may now sum up our discussion of causality. We found first that the
law of causality, as usually stated by philosophers, is false, and is
not employed in science. We then considered the nature of scientific
laws, and found that, instead of stating that one event A is always
followed by another event B, they stated functional relations between
certain events at certain times, which we called determinants, and
other events at earlier or later times or at the same time. We were
unable to find any _a priori_ category involved: the existence of
scientific laws appeared as a purely empirical fact, not necessarily
universal, except in a trivial and scientifically useless form. We
found that a system with one set of determinants may very likely have
other sets of a quite different kind, that, for example, a
mechanically determined system may also be teleologically or
volitionally determined. Finally we considered the problem of free
will: here we found that the reasons for supposing volitions to be
determined are strong but not conclusive, and we decided that even if
volitions are mechanically determined, that is no reason for denying
freedom in the sense revealed by introspection, or for supposing that
mechanical events are not determined by volitions. The problem of free
will _versus_ determinism is therefore, if we were right, mainly
illusory, but in part not yet capable of being decisively solved.

FOOTNOTES:

[35] A propositional function is an expression containing a variable,
or undetermined constituent, and becoming a proposition as soon as a
definite value is assigned to the variable. Examples are: "A is A,"
"_x_ is a number." The variable is called the _argument_ of the
function.

[36] _Logic_, Bk. III, Chap. V, § 2.

[37] _Time and Free Will_, p. 199.

[38] _Time and Free Will._ p. 202.

[39] _Loc. cit._, § 6




X

KNOWLEDGE BY ACQUAINTANCE AND KNOWLEDGE BY DESCRIPTION


The object of the following paper is to consider what it is that we
know in cases where we know propositions about "the so-and-so" without
knowing who or what the so-and-so is. For example, I know that the
candidate who gets most votes will be elected, though I do not know
who is the candidate who will get most votes. The problem I wish to
consider is: What do we know in these cases, where the subject is
merely described? I have considered this problem elsewhere[40] from a
purely logical point of view; but in what follows I wish to consider
the question in relation to theory of knowledge as well as in relation
to logic, and in view of the above-mentioned logical discussions, I
shall in this paper make the logical portion as brief as possible.

In order to make clear the antithesis between "acquaintance" and
"description," I shall first of all try to explain what I mean by
"acquaintance." I say that I am _acquainted_ with an object when I
have a direct cognitive relation to that object, i.e. when I am
directly aware of the object itself. When I speak of a cognitive
relation here, I do not mean the sort of relation which constitutes
judgment, but the sort which constitutes presentation. In fact, I
think the relation of subject and object which I call acquaintance is
simply the converse of the relation of object and subject which
constitutes presentation. That is, to say that S has acquaintance with
O is essentially the same thing as to say that O is presented to S.
But the associations and natural extensions of the word _acquaintance_
are different from those of the word _presentation_. To begin with, as
in most cognitive words, it is natural to say that I am acquainted
with an object even at moments when it is not actually before my mind,
provided it has been before my mind, and will be again whenever
occasion arises. This is the same sense in which I am said to know
that 2+2=4 even when I am thinking of something else. In the second
place, the word _acquaintance_ is designed to emphasise, more than the
word _presentation_, the relational character of the fact with which
we are concerned. There is, to my mind, a danger that, in speaking of
presentation, we may so emphasise the object as to lose sight of the
subject. The result of this is either to lead to the view that there
is no subject, whence we arrive at materialism; or to lead to the view
that what is presented is part of the subject, whence we arrive at
idealism, and should arrive at solipsism but for the most desperate
contortions. Now I wish to preserve the dualism of subject and object
in my terminology, because this dualism seems to me a fundamental fact
concerning cognition. Hence I prefer the word _acquaintance_ because
it emphasises the need of a subject which is acquainted.

When we ask what are the kinds of objects with which we are
acquainted, the first and most obvious example is _sense-data_. When I
see a colour or hear a noise, I have direct acquaintance with the
colour or the noise. The sense-datum with which I am acquainted in
these cases is generally, if not always, complex. This is
particularly obvious in the case of sight. I do not mean, of course,
merely that the supposed physical object is complex, but that the
direct sensible object is complex and contains parts with spatial
relations. Whether it is possible to be aware of a complex without
being aware of its constituents is not an easy question, but on the
whole it would seem that there is no reason why it should not be
possible. This question arises in an acute form in connection with
self-consciousness, which we must now briefly consider.

In introspection, we seem to be immediately aware of varying
complexes, consisting of objects in various cognitive and conative
relations to ourselves. When I see the sun, it often happens that I
am aware of my seeing the sun, in addition to being aware of the sun;
and when I desire food, it often happens that I am aware of my desire
for food. But it is hard to discover any state of mind in which I am
aware of myself alone, as opposed to a complex of which I am a
constituent. The question of the nature of self-consciousness is too
large and too slightly connected with our subject, to be argued at
length here. It is difficult, but probably not impossible, to account
for plain facts if we assume that we do not have acquaintance with
ourselves. It is plain that we are not only _acquainted_ with the
complex "Self-acquainted-with-A," but we also _know_ the proposition
"I am acquainted with A." Now here the complex has been analysed, and
if "I" does not stand for something which is a direct object of
acquaintance, we shall have to suppose that "I" is something known by
description. If we wished to maintain the view that there is no
acquaintance with Self, we might argue as follows: We are acquainted
with _acquaintance_, and we know that it is a relation. Also we are
acquainted with a complex in which we perceive that acquaintance is
the relating relation. Hence we know that this complex must have a
constituent which is that which is acquainted, i.e. must have a
subject-term as well as an object-term. This subject-term we define
as "I." Thus "I" means "the subject-term in awarenesses of which _I_
am aware." But as a definition this cannot be regarded as a happy
effort. It would seem necessary, therefore, either to suppose that I
am acquainted with myself, and that "I," therefore, requires no
definition, being merely the proper name of a certain object, or to
find some other analysis of self-consciousness. Thus self-consciousness
cannot be regarded as throwing light on the question whether we can
know a complex without knowing its constituents. This question,
however, is not important for our present purposes, and I shall
therefore not discuss it further.

The awarenesses we have considered so far have all been awarenesses of
particular existents, and might all in a large sense be called
sense-data. For, from the point of view of theory of knowledge,
introspective knowledge is exactly on a level with knowledge derived
from sight or hearing. But, in addition to awareness of the above kind
of objects, which may be called awareness of _particulars_; we have
also (though not quite in the same sense) what may be called awareness
of _universals_. Awareness of universals is called _conceiving_, and a
universal of which we are aware is called a _concept_. Not only are we
aware of particular yellows, but if we have seen a sufficient number
of yellows and have sufficient intelligence, we are aware of the
universal _yellow_; this universal is the subject in such judgments as
"yellow differs from blue" or "yellow resembles blue less than green
does." And the universal yellow is the predicate in such judgments as
"this is yellow," where "this" is a particular sense-datum. And
universal relations, too, are objects of awarenesses; up and down,
before and after, resemblance, desire, awareness itself, and so on,
would seem to be all of them objects of which we can be aware.

In regard to relations, it might be urged that we are never aware of
the universal relation itself, but only of complexes in which it is a
constituent. For example, it may be said that we do not know directly
such a relation as _before_, though we understand such a proposition
as "this is before that," and may be directly aware of such a complex
as "this being before that." This view, however, is difficult to
reconcile with the fact that we often know propositions in which the
relation is the subject, or in which the relata are not definite given
objects, but "anything." For example, we know that if one thing is
before another, and the other before a third, then the first is before
the third; and here the things concerned are not definite things, but
"anything." It is hard to see how we could know such a fact about
"before" unless we were acquainted with "before," and not merely with
actual particular cases of one given object being before another given
object. And more directly: A judgment such as "this is before that,"
where this judgment is derived from awareness of a complex,
constitutes an analysis, and we should not understand the analysis if
we were not acquainted with the meaning of the terms employed. Thus we
must suppose that we are acquainted with the meaning of "before," and
not merely with instances of it.

There are thus at least two sorts of objects of which we are aware,
namely, particulars and universals. Among particulars I include all
existents, and all complexes of which one or more constituents are
existents, such as this-before-that, this-above-that,
the-yellowness-of-this. Among universals I include all objects of
which no particular is a constituent. Thus the disjunction
"universal-particular" includes all objects. We might also call it the
disjunction "abstract-concrete." It is not quite parallel with the
opposition "concept-percept," because things remembered or imagined
belong with particulars, but can hardly be called percepts. (On the
other hand, universals with which we are acquainted may be identified
with concepts.)

It will be seen that among the objects with which we are acquainted
are not included physical objects (as opposed to sense-data), nor
other people's minds. These things are known to us by what I call
"knowledge by description," which we must now consider.

By a "description" I mean any phrase of the form "a so-and-so" or "the
so-and-so." A phrase of the form "a so-and-so" I shall call an
"ambiguous" description; a phrase of the form "the so-and-so" (in the
singular) I shall call a "definite" description. Thus "a man" is an
ambiguous description, and "the man with the iron mask" is a definite
description. There are various problems connected with ambiguous
descriptions, but I pass them by, since they do not directly concern
the matter I wish to discuss. What I wish to discuss is the nature of
our knowledge concerning objects in cases where we know that there is
an object answering to a definite description, though we are not
_acquainted_ with any such object. This is a matter which is concerned
exclusively with _definite_ descriptions. I shall, therefore, in the
sequel, speak simply of "descriptions" when I mean "definite
descriptions." Thus a description will mean any phrase of the form
"the so-and-so" in the singular.

I shall say that an object is "known by description" when we know that
it is "_the_ so-and-so," i.e. when we know that there is one object,
and no more, having a certain property; and it will generally be
implied that we do not have knowledge of the same object by
acquaintance. We know that the man with the iron mask existed, and
many propositions are known about him; but we do not know who he was.
We know that the candidate who gets most votes will be elected, and in
this case we are very likely also acquainted (in the only sense in
which one can be acquainted with some one else) with the man who is,
in fact, the candidate who will get most votes, but we do not know
which of the candidates he is, i.e. we do not know any proposition of
the form "A is the candidate who will get most votes" where A is one
of the candidates by name. We shall say that we have "_merely_
descriptive knowledge" of the so-and-so when, although we know that
the so-and-so exists, and although we may possibly be acquainted with
the object which is, in fact, the so-and-so, yet we do not know any
proposition "_a_ is the so-and-so," where _a_ is something with which
we are acquainted.

When we say "the so-and-so exists," we mean that there is just one
object which is the so-and-so. The proposition "_a_ is the so-and-so"
means that _a_ has the property so-and-so, and nothing else has. "Sir
Joseph Larmor is the Unionist candidate" means "Sir Joseph Larmor is a
Unionist candidate, and no one else is." "The Unionist candidate
exists" means "some one is a Unionist candidate, and no one else is."
Thus, when we are acquainted with an object which we know to be the
so-and-so, we know that the so-and-so exists but we may know that the
so-and-so exists when we are not acquainted with any object which we
know to be the so-and-so, and even when we are not acquainted with any
object which, in fact, is the so-and-so.

Common words, even proper names, are usually really descriptions. That
is to say, the thought in the mind of a person using a proper name
correctly can generally only be expressed explicitly if we replace the
proper name by a description. Moreover, the description required to
express the thought will vary for different people, or for the same
person at different times. The only thing constant (so long as the
name is rightly used) is the object to which the name applies. But so
long as this remains constant, the particular description involved
usually makes no difference to the truth or falsehood of the
proposition in which the name appears.

Let us take some illustrations. Suppose some statement made about
Bismarck. Assuming that there is such a thing as direct acquaintance
with oneself, Bismarck himself might have used his name directly to
designate the particular person with whom he was acquainted. In this
case, if he made a judgment about himself, he himself might be a
constituent of the judgment. Here the proper name has the direct use
which it always wishes to have, as simply standing for a certain
object, and not for a description of the object. But if a person who
knew Bismarck made a judgment about him, the case is different. What
this person was acquainted with were certain sense-data which he
connected (rightly, we will suppose) with Bismarck's body. His body as
a physical object, and still more his mind, were only known as the
body and the mind connected with these sense-data. That is, they were
known by description. It is, of course, very much a matter of chance
which characteristics of a man's appearance will come into a friend's
mind when he thinks of him; thus the description actually in the
friend's mind is accidental. The essential point is that he knows that
the various descriptions all apply to the same entity, in spite of
not being acquainted with the entity in question.

When we, who did not know Bismarck, make a judgment about him, the
description in our minds will probably be some more or less vague mass
of historical knowledge--far more, in most cases, than is required to
identify him. But, for the sake of illustration, let us assume that we
think of him as "the first Chancellor of the German Empire." Here all
the words are abstract except "German." The word "German" will again
have different meanings for different people. To some it will recall
travels in Germany, to some the look of Germany on the map, and so on.
But if we are to obtain a description which we know to be applicable,
we shall be compelled, at some point, to bring in a reference to a
particular with which we are acquainted. Such reference is involved in
any mention of past, present, and future (as opposed to definite
dates), or of here and there, or of what others have told us. Thus it
would seem that, in some way or other, a description known to be
applicable to a particular must involve some reference to a particular
with which we are acquainted, if our knowledge about the thing
described is not to be merely what follows logically from the
description. For example, "the most long-lived of men" is a
description which must apply to some man, but we can make no judgments
concerning this man which involve knowledge about him beyond what the
description gives. If, however, we say, "the first Chancellor of the
German Empire was an astute diplomatist," we can only be assured of
the truth of our judgment in virtue of something with which we are
acquainted--usually a testimony heard or read. Considered
psychologically, apart from the information we convey to others, apart
from the fact about the actual Bismarck, which gives importance to
our judgment, the thought we really have contains the one or more
particulars involved, and otherwise consists wholly of concepts. All
names of places--London, England, Europe, the earth, the Solar
System--similarly involve, when used, descriptions which start from
some one or more particulars with which we are acquainted. I suspect
that even the Universe, as considered by metaphysics, involves such a
connection with particulars. In logic, on the contrary, where we are
concerned not merely with what does exist, but with whatever might or
could exist or be, no reference to actual particulars is involved.

It would seem that, when we make a statement about something only
known by description, we often _intend_ to make our statement, not in
the form involving the description, but about the actual thing
described. That is to say, when we say anything about Bismarck, we
should like, if we could, to make the judgment which Bismarck alone
can make, namely, the judgment of which he himself is a constituent.
In this we are necessarily defeated, since the actual Bismarck is
unknown to us. But we know that there is an object B called Bismarck,
and that B was an astute diplomatist. We can thus _describe_ the
proposition we should like to affirm, namely, "B was an astute
diplomatist," where B is the object which was Bismarck. What enables
us to communicate in spite of the varying descriptions we employ is
that we know there is a true proposition concerning the actual
Bismarck, and that, however we may vary the description (so long as
the description is correct), the proposition described is still the
same. This proposition, which is described and is known to be true, is
what interests us; but we are not acquainted with the proposition
itself, and do not know _it_, though we know it is true.

It will be seen that there are various stages in the removal from
acquaintance with particulars: there is Bismarck to people who knew
him, Bismarck to those who only know of him through history, the man
with the iron mask, the longest-lived of men. These are progressively
further removed from acquaintance with particulars, and there is a
similar hierarchy in the region of universals. Many universals, like
many particulars, are only known to us by description. But here, as in
the case of particulars, knowledge concerning what is known by
description is ultimately reducible to knowledge concerning what is
known by acquaintance.

The fundamental epistemological principle in the analysis of
propositions containing descriptions is this: _Every proposition which
we can understand must be composed wholly of constituents with which
we are acquainted._ From what has been said already, it will be plain
why I advocate this principle, and how I propose to meet the case of
propositions which at first sight contravene it. Let us begin with the
reasons for supposing the principle true.

The chief reason for supposing the principle true is that it seems
scarcely possible to believe that we can make a judgment or entertain
a supposition without knowing what it is that we are judging or
supposing about. If we make a judgment about (say) Julius Cæsar, it is
plain that the actual person who was Julius Cæsar is not a constituent
of the judgment. But before going further, it may be well to explain
what I mean when I say that this or that is a constituent of a
judgment, or of a proposition which we understand. To begin with
judgments: a judgment, as an occurrence, I take to be a relation of a
mind to several entities, namely, the entities which compose what is
judged. If, e.g. I judge that A loves B, the judgment as an event
consists in the existence, at a certain moment, of a specific
four-term relation, called _judging_, between me and A and love and B.
That is to say, at the time when I judge, there is a certain complex
whose terms are myself and A and love and B, and whose relating
relation is _judging_. My reasons for this view have been set forth
elsewhere,[41] and I shall not repeat them here. Assuming this view of
judgment, the constituents of the judgment are simply the constituents
of the complex which is the judgment. Thus, in the above case, the
constituents are myself and A and love and B and judging. But myself
and judging are constituents shared by all my judgments; thus the
_distinctive_ constituents of the particular judgment in question are
A and love and B. Coming now to what is meant by "understanding a
proposition," I should say that there is another relation possible
between me and A and love and B, which is called my _supposing_ that A
loves B.[42] When we can _suppose_ that A loves B, we "understand the
proposition" _A loves B_. Thus we often understand a proposition in
cases where we have not enough knowledge to make a judgment.
Supposing, like judging, is a many-term relation, of which a mind is
one term. The other terms of the relation are called the constituents
of the proposition supposed. Thus the principle which I enunciated may
be re-stated as follows: _Whenever a relation of supposing or judging
occurs, the terms to which the supposing or judging mind is related by
the relation of supposing or judging must be terms with which the mind
in question is acquainted._ This is merely to say that we cannot make
a judgment or a supposition without knowing what it is that we are
making our judgment or supposition about. It seems to me that the
truth of this principle is evident as soon as the principle is
understood; I shall, therefore, in what follows, assume the principle,
and use it as a guide in analysing judgments that contain
descriptions.

Returning now to Julius Cæsar, I assume that it will be admitted that
he himself is not a constituent of any judgment which I can make. But
at this point it is necessary to examine the view that judgments are
composed of something called "ideas," and that it is the "idea" of
Julius Cæsar that is a constituent of my judgment. I believe the
plausibility of this view rests upon a failure to form a right theory
of descriptions. We may mean by my "idea" of Julius Cæsar the things
that I know about him, e.g. that he conquered Gaul, was assassinated
on the Ides of March, and is a plague to schoolboys. Now I am
admitting, and indeed contending, that in order to discover what is
actually in my mind when I judge about Julius Cæsar, we must
substitute for the proper name a description made up of some of the
things I know about him. (A description which will often serve to
express my thought is "the man whose name was _Julius Cæsar_." For
whatever else I may have forgotten about him, it is plain that when I
mention him I have not forgotten that that was his name.) But although
I think the theory that judgments consist of ideas may have been
suggested in some such way, yet I think the theory itself is
fundamentally mistaken. The view seems to be that there is some
mental existent which may be called the "idea" of something outside
the mind of the person who has the idea, and that, since judgment is a
mental event, its constituents must be constituents of the mind of the
person judging. But in this view ideas become a veil between us and
outside things--we never really, in knowledge, attain to the things we
are supposed to be knowing about, but only to the ideas of those
things. The relation of mind, idea, and object, on this view, is
utterly obscure, and, so far as I can see, nothing discoverable by
inspection warrants the intrusion of the idea between the mind and the
object. I suspect that the view is fostered by the dislike of
relations, and that it is felt the mind could not know objects unless
there were something "in" the mind which could be called the state of
knowing the object. Such a view, however, leads at once to a vicious
endless regress, since the relation of idea to object will have to be
explained by supposing that the idea itself has an idea of the object,
and so on _ad infinitum_. I therefore see no reason to believe that,
when we are acquainted with an object, there is in us something which
can be called the "idea" of the object. On the contrary, I hold that
acquaintance is wholly a relation, not demanding any such constituent
of the mind as is supposed by advocates of "ideas." This is, of
course, a large question, and one which would take us far from our
subject if it were adequately discussed. I therefore content myself
with the above indications, and with the corollary that, in judging,
the actual objects concerning which we judge, rather than any supposed
purely mental entities, are constituents of the complex which is the
judgment.

When, therefore, I say that we must substitute for "Julius Cæsar" some
description of Julius Cæsar, in order to discover the meaning of a
judgment nominally about him, I am not saying that we must substitute
an idea. Suppose our description is "the man whose name was _Julius
Cæsar_." Let our judgment be "Julius Cæsar was assassinated." Then it
becomes "the man whose name was _Julius Cæsar_ was assassinated." Here
_Julius Cæsar_ is a noise or shape with which we are acquainted, and
all the other constituents of the judgment (neglecting the tense in
"was") are _concepts_ with which we are acquainted. Thus our judgment
is wholly reduced to constituents with which we are acquainted, but
Julius Cæsar himself has ceased to be a constituent of our judgment.
This, however, requires a proviso, to be further explained shortly,
namely that "the man whose name was _Julius Cæsar_" must not, as a
whole, be a constituent of our judgment, that is to say, this phrase
must not, as a whole, have a meaning which enters into the judgment.
Any right analysis of the judgment, therefore, must break up this
phrase, and not treat it as a subordinate complex which is part of the
judgment. The judgment "the man whose name was _Julius Cæsar_ was
assassinated" may be interpreted as meaning "one and only one man was
called _Julius Cæsar_, and that one was assassinated." Here it is
plain that there is no constituent corresponding to the phrase "the
man whose name was _Julius Cæsar_." Thus there is no reason to regard
this phrase as expressing a constituent of the judgment, and we have
seen that this phrase must be broken up if we are to be acquainted
with all the constituents of the judgment. This conclusion, which we
have reached from considerations concerned with the theory of
knowledge, is also forced upon us by logical considerations, which
must now be briefly reviewed.

It is common to distinguish two aspects, _meaning_ and _denotation_,
such phrases as "the author of Waverley." The meaning will be a
certain complex, consisting (at least) of authorship and Waverley with
some relation; the denotation will be Scott. Similarly "featherless
bipeds" will have a complex meaning, containing as constituents the
presence of two feet and the absence of feathers, while its denotation
will be the class of men. Thus when we say "Scott is the author of
Waverley" or "men are the same as featherless bipeds," we are
asserting an identity of denotation, and this assertion is worth
making because of the diversity of meaning.[43] I believe that the
duality of meaning and denotation, though capable of a true
interpretation, is misleading if taken as fundamental. The denotation,
I believe, is not a constituent of the proposition, except in the case
of proper names, i.e. of words which do not assign a property to an
object, but merely and solely name it. And I should hold further that,
in this sense, there are only two words which are strictly proper
names of particulars, namely, "I" and "this."[44]

One reason for not believing the denotation to be a constituent of the
proposition is that we may know the proposition even when we are not
acquainted with the denotation. The proposition "the author of
Waverley is a novelist" was known to people who did not know that "the
author of Waverley" denoted Scott. This reason has been already
sufficiently emphasised.

A second reason is that propositions concerning "the so-and-so" are
possible even when "the so-and-so" has no denotation. Take, e.g. "the
golden mountain does not exist" or "the round square is
self-contradictory." If we are to preserve the duality of meaning and
denotation, we have to say, with Meinong, that there are such objects
as the golden mountain and the round square, although these objects do
not have being. We even have to admit that the existent round square
is existent, but does not exist.[45] Meinong does not regard this as a
contradiction, but I fail to see that it is not one. Indeed, it seems
to me evident that the judgment "there is no such object as the round
square" does not presuppose that there is such an object. If this is
admitted, however, we are led to the conclusion that, by parity of
form, no judgment concerning "the so-and-so" actually involves the
so-and-so as a constituent.

Miss Jones[46] contends that there is no difficulty in admitting
contradictory predicates concerning such an object as "the present
King of France," on the ground that this object is in itself
contradictory. Now it might, of course, be argued that this object,
unlike the round square, is not self-contradictory, but merely
non-existent. This, however, would not go to the root of the matter.
The real objection to such an argument is that the law of
contradiction ought not to be stated in the traditional form "A is not
both B and not B," but in the form "no proposition is both true and
false." The traditional form only applies to certain propositions,
namely, to those which attribute a predicate to a subject. When the
law is stated of propositions, instead of being stated concerning
subjects and predicates, it is at once evident that propositions about
the present King of France or the round square can form no exception,
but are just as incapable of being both true and false as other
propositions. Miss Jones[47] argues that "Scott is the author of
Waverley" asserts identity of denotation between _Scott_ and _the
author of Waverley_. But there is some difficulty in choosing among
alternative meanings of this contention. In the first place, it should
be observed that _the author of Waverley_ is not a _mere_ name, like
_Scott_. _Scott_ is merely a noise or shape conventionally used to
designate a certain person; it gives us no information about that
person, and has nothing that can be called meaning as opposed to
denotation. (I neglect the fact, considered above, that even proper
names, as a rule, really stand for descriptions.) But _the author of
Waverley_ is not merely conventionally a name for Scott; the element
of mere convention belongs here to the separate words, _the_ and
_author_ and _of_ and _Waverley_. Given what these words stand for,
_the author of Waverley_ is no longer arbitrary. When it is said that
Scott is the author of Waverley, we are not stating that these are two
_names_ for one man, as we should be if we said "Scott is Sir Walter."
A man's name is what he is called, but however much Scott had been
called the author of Waverley, that would not have made him be the
author; it was necessary for him actually to write Waverley, which was
a fact having nothing to do with names.

If, then, we are asserting identity of denotation, we must not mean by
_denotation_ the mere relation of a name to the thing named. In fact,
it would be nearer to the truth to say that the _meaning_ of "Scott"
is the _denotation_ of "the author of Waverley." The relation of
"Scott" to Scott is that "Scott" means Scott, just as the relation of
"author" to the concept which is so called is that "author" means this
concept. Thus if we distinguish meaning and denotation in "the author
of Waverley," we shall have to say that "Scott" has meaning but not
denotation. Also when we say "Scott is the author of Waverley," the
_meaning_ of "the author of Waverley" is relevant to our assertion.
For if the denotation alone were relevant, any other phrase with the
same denotation would give the same proposition. Thus "Scott is the
author of Marmion" would be the same proposition as "Scott is the
author of Waverley." But this is plainly not the case, since from the
first we learn that Scott wrote Marmion and from the second we learn
that he wrote Waverley, but the first tells us nothing about Waverley
and the second nothing about Marmion. Hence the meaning of "the author
of Waverley," as opposed to the denotation, is certainly relevant to
"Scott is the author of Waverley."

We have thus agreed that "the author of Waverley" is not a mere name,
and that its meaning is relevant in propositions in which it occurs.
Thus if we are to say, as Miss Jones does, that "Scott is the author
of Waverley" asserts an identity of denotation, we must regard the
denotation of "the author of Waverley" as the denotation of what is
_meant_ by "the author of Waverley." Let us call the meaning of "the
author of Waverley" M. Thus M is what "the author of Waverley" means.
Then we are to suppose that "Scott is the author of Waverley" means
"Scott is the denotation of M." But here we are explaining our
proposition by another of the same form, and thus we have made no
progress towards a real explanation. "The denotation of M," like "the
author of Waverley," has both meaning and denotation, on the theory we
are examining. If we call its meaning M', our proposition becomes
"Scott is the denotation of M'." But this leads at once to an endless
regress. Thus the attempt to regard our proposition as asserting
identity of denotation breaks down, and it becomes imperative to find
some other analysis. When this analysis has been completed, we shall
be able to reinterpret the phrase "identity of denotation," which
remains obscure so long as it is taken as fundamental.

The first point to observe is that, in any proposition about "the
author of Waverley," provided Scott is not explicitly mentioned, the
denotation itself, i.e. Scott, does not occur, but only the concept of
denotation, which will be represented by a variable. Suppose we say
"the author of Waverley was the author of Marmion," we are certainly
not saying that both were Scott--we may have forgotten that there was
such a person as Scott. We are saying that there is some man who was
the author of Waverley and the author of Marmion. That is to say,
there is some one who wrote Waverley and Marmion, and no one else
wrote them. Thus the identity is that of a variable, i.e. of an
indefinite subject, "some one." This is why we can understand
propositions about "the author of Waverley," without knowing who he
was. When we say "the author of Waverley was a poet," we mean "one and
only one man wrote Waverley, and he was a poet"; when we say "the
author of Waverley was Scott" we mean "one and only one man wrote
Waverley, and he was Scott." Here the identity is between a variable,
i.e. an indeterminate subject ("he"), and Scott; "the author of
Waverley" has been analysed away, and no longer appears as a
constituent of the proposition.[48]

The reason why it is imperative to analyse away the phrase "the author
of Waverley" may be stated as follows. It is plain that when we say
"the author of Waverley is the author of Marmion," the _is_ expresses
identity. We have seen also that the common _denotation_, namely
Scott, is not a constituent of this proposition, while the _meanings_
(if any) of "the author of Waverley" and "the author of Marmion" are
not identical. We have seen also that, in any sense in which the
meaning of a word is a constituent of a proposition in whose verbal
expression the word occurs, "Scott" means the actual man Scott, in the
same sense (so far as concerns our present discussion) in which
"author" means a certain universal. Thus, if "the author of Waverley"
were a subordinate complex in the above proposition, its _meaning_
would have to be what was said to be identical with the _meaning_ of
"the author of Marmion." This is plainly not the case; and the only
escape is to say that "the author of Waverley" does not, by itself,
have a meaning, though phrases of which it is part do have a meaning.
That is, in a right analysis of the above proposition, "the author of
Waverley" must disappear. This is effected when the above proposition
is analysed as meaning: "Some one wrote Waverley and no one else did,
and that some one also wrote Marmion and no one else did." This may be
more simply expressed by saying that the propositional function "_x_
wrote Waverley and Marmion, and no one else did" is capable of truth,
i.e. some value of _x_ makes it true, but no other value does. Thus
the true subject of our judgment is a propositional function, i.e. a
complex containing an undetermined constituent, and becoming a
proposition as soon as this constituent is determined.

We may now define the denotation of a phrase. If we know that the
proposition "_a_ is the so-and-so" is true, i.e. that _a_ is so-and-so
and nothing else is, we call _a_ the denotation of the phrase "the
so-and-so." A very great many of the propositions we naturally make
about "the so-and-so" will remain true or remain false if we
substitute _a_ for "the so-and-so," where _a_ is the denotation of
"the so-and-so." Such propositions will also remain true or remain
false if we substitute for "the so-and-so" any other phrase having the
same denotation. Hence, as practical men, we become interested in the
denotation more than in the description, since the denotation decides
as to the truth or falsehood of so many statements in which the
description occurs. Moreover, as we saw earlier in considering the
relations of description and acquaintance, we often wish to reach the
denotation, and are only hindered by lack of acquaintance: in such
cases the description is merely the means we employ to get as near as
possible to the denotation. Hence it naturally comes to be supposed
that the denotation is part of the proposition in which the
description occurs. But we have seen, both on logical and on
epistemological grounds, that this is an error. The actual object (if
any) which is the denotation is not (unless it is explicitly
mentioned) a constituent of propositions in which descriptions occur;
and this is the reason why, in order to understand such propositions,
we need acquaintance with the constituents of the description, but do
not need acquaintance with its denotation. The first result of
analysis, when applied to propositions whose grammatical subject is
"the so-and-so," is to substitute a variable as subject; i.e. we
obtain a proposition of the form: "There is _something_ which alone is
so-and-so, and that _something_ is such-and-such." The further
analysis of propositions concerning "the so-and-so" is thus merged in
the problem of the nature of the variable, i.e. of the meanings of
_some_, _any_, and _all_. This is a difficult problem, concerning
which I do not intend to say anything at present.

To sum up our whole discussion. We began by distinguishing two sorts
of knowledge of objects, namely, knowledge by _acquaintance_ and
knowledge by _description_. Of these it is only the former that brings
the object itself before the mind. We have acquaintance with
sense-data, with many universals, and possibly with ourselves, but not
with physical objects or other minds. We have _descriptive_ knowledge
of an object when we know that it is _the_ object having some property
or properties with which we are acquainted; that is to say, when we
know that the property or properties in question belong to one object
and no more, we are said to have knowledge of that one object by
description, whether or not we are acquainted with the object. Our
knowledge of physical objects and of other minds is only knowledge by
description, the descriptions involved being usually such as involve
sense-data. All propositions intelligible to us, whether or not they
primarily concern things only known to us by description, are composed
wholly of constituents with which we are acquainted, for a constituent
with which we are not acquainted is unintelligible to us. A judgment,
we found, is not composed of mental constituents called "ideas," but
consists of an occurrence whose constituents are a mind[49] and
certain objects, particulars or universals. (One at least must be a
universal.) When a judgment is rightly analysed, the objects which are
constituents of it must all be objects with which the mind which is a
constituent of it is acquainted. This conclusion forces us to analyse
descriptive phrases occurring in propositions, and to say that the
objects denoted by such phrases are not constituents of judgments in
which such phrases occur (unless these objects are explicitly
mentioned). This leads us to the view (recommended also on purely
logical grounds) that when we say "the author of Marmion was the
author of Waverley," Scott himself is not a constituent of our
judgment, and that the judgment cannot be explained by saying that it
affirms identity of denotation with diversity of meaning. It also,
plainly, does not assert identity of meaning. Such judgments,
therefore, can only be analysed by breaking up the descriptive
phrases, introducing a variable, and making propositional functions
the ultimate subjects. In fact, "the so-and-so is such-and-such" will
mean that "_x_ is so-and-so and nothing else is, and _x_ is
such-and-such" is capable of truth. The analysis of such judgments
involves many fresh problems, but the discussion of these problems is
not undertaken in the present paper.

FOOTNOTES:

[40] See references later.

[41] _Philosophical Essays_, "The Nature of Truth." I have been
persuaded by Mr. Wittgenstein that this theory is somewhat unduly
simple, but the modification which I believe it to require does not
affect the above argument [1917].

[42] Cf. Meinong, _Ueber Annahmen_, _passim_. I formerly supposed,
contrary to Meinong's view, that the relationship of supposing might
be merely that of presentation. In this view I now think I was
mistaken, and Meinong is right. But my present view depends upon the
theory that both in judgment and in assumption there is no single
Objective, but the several constituents of the judgment or assumption
are in a many-term relation to the mind.

[43] This view has been recently advocated by Miss E.E.C. Jones. "A
New Law of Thought and its Implications," _Mind_, January, 1911.

[44] I should now exclude "I" from proper names in the strict sense,
and retain only "this" [1917].

[45] Meinong, _Ueber Annahmen_, 2nd ed., Leipzig, 1910, p. 141.

[46] _Mind_, July, 1910, p. 380.

[47] _Mind_, July, 1910, p. 379.

[48] The theory which I am advocating is set forth fully, with the
logical grounds in its favour, in _Principia Mathematica_, Vol. I.
Introduction, Chap. III; also, less fully, in _Mind_, October, 1905.

[49] I use this phrase merely to denote the something psychological
which enters into judgment, without intending to prejudge the question
as to what this something is.




INDEX


Achilles and the tortoise, 80 ff, 89 ff

Acquaintance, the relation of, 209 ff

Alexander, 125

American Realists, the, 134

Aristotle, 42, 76, 97


Bacon, 41

Bergson, 14 ff, 22, 105, 128, 185 ff, 203

Berkeley, 97, 132

Blake, 1

Bosanquet, 99

Broad, 89 _n_


Calculus, the, 82

Cantor, Georg, 64, 81 ff, 85, 91

Carlyle, 50, 82

Cause, the conception of, 135 _n_, 180 ff

Christianity and renunciation, 51

Chuang Tzŭ, 106

Construction of permanent things and matter, 169 ff

Constructions, logical, 155 ff


Darwin, 15, 23, 43

Dedekind, 64, 81 ff, 85

Descartes, 97, 126

Descriptions, 175, 214 ff


Education, 37 ff

Euclid, 62, 92, 94

Evolutionism, 23 ff, 28


Fano, 93

Faraday, 34

Free will, 205 ff

Frege, 78 _n_


Galileo, 42

Gladstone, 177

Good and evil, 26 ff


Hegel, 8, 10, 18, 85, 97, 105 ff

Heine, 113

Heraclitus, 1 ff, 10

Hertz, 34

Holt, 177 _n_

Hume, 1, 97


Infinite, the mathematical, 84 ff


James, William, 100

Jones, Miss E.E.C., 224 _n_, 225

Judgment, 219 ff


Kant, 85, 96, 97, 99, 118 ff

Knowledge by acquaintance, 209 ff;
  by description, 214 ff


Laplace, 23

Leibniz, 76, 79, 82 ff, 97, 126, 144, 160

Locke, 97

Logic, the laws of, 68 ff


Macaulay and Taylor's theorem, 95

Malthus, 43

Mathematics, 58 ff;
  and the Metaphysicians, 74 ff;
  and logic, 75 ff;
  and the infinitesimal, 82 ff

Matter, the nature of, 125 ff;
  definition of, 164 ff

Maxwell, 34

Meaning and denotation, 223 ff

Meinong, 174, 220 _n_, 225

Militarism, 50

Mill, 185, 193 ff

Mysticism and logic, 1 ff


Necessity, the notion of, 207 ff

Nietzsche, 22, 50

Nunn, 125, 137 _n_, 153


Parmenides, 7 ff, 18, 21

Particulars, awareness of, 210 ff

Peano, 78 ff, 93 ff

Perspectives, 139 ff;
  the space of, 158 ff

Philosophy and logic, 111

Physics, sense-data and, 145 ff

Pierce, 76 _n_

Plato, 1 ff, 10, 30, 60, 97

Pragmatism, 22, 105


Realism and the analytic method, 120 ff

Reason and intuition, 12 ff

Relatives, the logic of, 76

Robb, 167 _n_


Santayana, 20

Sense-data, 147, 210 ff;
  and physics, 145 ff

Sensibilia, 148 ff

Space, 138 ff;
  private, 158 ff;
  the logical problem, 114 ff;
  the problem in physics, 115 ff;
  the epistemological problem, 118 ff

Systems, deterministic, 199;
  practically isolated, 198;
  relatively isolated, 197;
  mechanical, 201


Time, 10, 21 ff, 141 ff, 167 ff

Tristram Shandy, the paradox of, 90 ff


Unity and Plurality, 18 ff

Universals, awareness of, 212 ff


Ward, 180

Weierstrass, 80, 82, 95

Whitehead, 117, 157, 175

Wolf, 173


Zeno the Eleatic, 64, 80, 84, 89 ff




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