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Title: Mysticism and Logic and Other Essays

Author: Bertrand Russell

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BERTRAND RUSSELL



MYSTICISM ANDLOGIC

AND OTHER ESSAYS

_LONDON_ GEORGE ALLEN & UNWIN LTDRUSKIN HOUSE MUSEUM STREET

MYSTICISM AND LOGICAND 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 toreprint them.

The essay on "Mysticism and Logic" appeared in the _Hibbert Journal_ for July, 1914. "The Place of Science in a Liberal Education" appearedin two numbers of _The New Statesman_, May 24 and 31, 1913. "The FreeMan's Worship" and "The Study of Mathematics" were included in aformer collection (now out of print), _Philosophical Essays_, alsopublished 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 theoreticalEthics, the position advocated in "The Free Man's Worship" is notquite identical with that which I hold now: I feel less convinced than

I did then of the objectivity of good and evil. But the generalattitude towards life which is suggested in that essay still seems tome, in the main, the one which must be adopted in times of stress anddifficulty by those who have no dogmatic religious beliefs, if inwarddefeat 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." Somepoints in this essay require modification in view of later work.



These are indicated in footnotes. Its tone is partly explained by thefact that the editor begged me to make the article "as romantic aspossible."

All the above essays are entirely popular, but those that follow aresomewhat more technical. "On Scientific Method in Philosophy" was theHerbert Spencer lecture at Oxford in 1914, and was published by theClarendon Press, which has kindly allowed me to include it in thiscollection. "The Ultimate Constituents of Matter" was an address tothe Manchester Philosophical Society, early in 1915, and was publishedin the _Monist_ in July of that year. The essay on "The Relation ofSense-data to Physics" was written in January, 1914, and firstappeared in No. 4 of that year's volume of _Scientia_, anInternational Review of Scientific Synthesis, edited by M. EugenioRignano, published monthly by Messrs. Williams and Norgate, London,Nicola Zanichelli, Bologna, and Félix Alcan, Paris. The essay "On theNotion of Cause" was the presidential address to the AristotelianSociety 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 publishedin 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 meansof thought, has been developed, from the first, by the union andconflict of two very different human impulses, the one urging mentowards mysticism, the other urging them towards science. Some menhave achieved greatness through one of these impulses alone, othersthrough the other alone: in Hume, for example, the scientific impulsereigns quite unchecked, while in Blake a strong hostility to scienceco-exists with profound mystic insight. But the greatest men who havebeen philosophers have felt the need both of science and of mysticism:the attempt to harmonise the two was what made their life, and whatalways must, for all its arduous uncertainty, make philosophy, to someminds, a greater thing than either science or religion.

Before attempting an explicit characterisation of the scientific andthe mystical impulses, I will illustrate them by examples from twophilosophers whose greatness lies in the very intimate blending whichthey achieved. The two philosophers I mean are Heraclitus and Plato.

Heraclitus, as every one knows, was a believer in universal flux: timebuilds and destroys all things. From the few fragments that remain, itis not easy to discover how he arrived at his opinions, but there aresome sayings that strongly suggest scientific observation as thesource.

"The things that can be seen, heard, and learned," he says, "are whatI prize the most." This is the language of the empiricist, to whomobservation is the sole guarantee of truth. "The sun is new everyday," is another fragment; and this opinion, in spite of itsparadoxical 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 thenight. Actual observation must also have suggested to him his centraldoctrine, that Fire is the one permanent substance, of which allvisible things are passing phases. In combustion we see things changeutterly, 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 ormen has made; but it was ever, is now, and ever shall be, anever-living Fire, with measures kindling, and measures going out."

"The transformations of Fire are, first of all, sea; and half of thesea is earth, half whirlwind."

This theory, though no longer one which science can accept, isnevertheless scientific in spirit. Science, too, might have inspiredthe famous saying to which Plato alludes: "You cannot step twice intothe same rivers; for fresh waters are ever flowing in upon you." Butwe find also another statement among the extant fragments: "We stepand do not step into the same rivers; we are and are not."

The comparison of this statement, which is mystical, with the onequoted by Plato, which is scientific, shows how intimately the twotendencies are blended in the system of Heraclitus. Mysticism is, in



essence, little more than a certain intensity and depth of feeling inregard to what is believed about the universe; and this kind offeeling leads Heraclitus, on the basis of his science, to strangelypoignant 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 despoticlord of the world, with all the irresponsible frivolity of a child. Itis mysticism, too, which leads Heraclitus to assert the identity ofopposites: "Good and ill are one," he says; and again: "To God allthings are fair and good and right, but men hold some things wrong andsome right."

Much of mysticism underlies the ethics of Heraclitus. It is true thata 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 toget, 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 areenough to show the character of the man: the facts of science, as theyappeared to him, fed the flame in his soul, and in its light he sawinto the depths of the world by the reflection of his own dancingswiftly penetrating fire. In such a nature we see the true union ofthe 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 impulseis distinctly the stronger of the two, and secures ultimate victorywhenever the conflict is sharp. His description of the cave is the

classical statement of belief in a knowledge and reality truer andmore 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 isan identification of the good with the truly real, which becameembodied in the philosophical tradition, and is still largelyoperative in our own day. In thus allowing a legislative function tothe good, Plato produced a divorce between philosophy and science,from which, in my opinion, both have suffered ever since and are stillsuffering. The man of science, whatever his hopes may be, must laythem aside while he studies nature; and the philosopher, if he is toachieve truth, must do the same. Ethical considerations can onlylegitimately appear when the truth has been ascertained: they can andshould appear as determining our feeling towards the truth, and ourmanner of ordering our lives in view of the truth, but not asthemselves dictating what the truth is to be.

There are passages in Plato--among those which illustrate thescientific side of his mind--where he seems clearly aware of this. Themost noteworthy is the one in which Socrates, as a young man, isexplaining the theory of ideas to Parmenides.

After Socrates has explained that there is an idea of the good, butnot of such things as hair and mud and dirt, Parmenides advises him"not to despise even the meanest things," and this advice shows thegenuine scientific temper. It is with this impartial temper that themystic's apparent insight into a higher reality and a hidden good hasto be combined if philosophy is to realise its greatest possibilities.And it is failure in this respect that has made so much of idealisticphilosophy thin, lifeless, and insubstantial. It is only in marriagewith the world that our ideals can bear fruit: divorced from it, theyremain barren. But marriage with the world is not to be achieved by anideal which shrinks from fact, or demands in advance that the worldshall conform to its desires.

Parmenides himself is the source of a peculiarly interesting strainof mysticism which pervades Plato's thought--the mysticism which maybe called "logical" because it is embodied in theories on logic. Thisform of mysticism, which appears, so far as the West is concerned, tohave originated with Parmenides, dominates the reasonings of all thegreat mystical metaphysicians from his day to that of Hegel and hismodern disciples. Reality, he says, is uncreated, indestructible,unchanging, indivisible; it is "immovable in the bonds of mightychains, without beginning and without end; since coming into being andpassing away have been driven afar, and true belief has cast themaway." The fundamental principle of his inquiry is stated in asentence 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 samething that can be thought and that can be." And again: "It needs mustbe that what can be thought and spoken of is; for it is possible forit to be, and it is not possible for what is nothing to be." Theimpossibility of change follows from this principle; for what is pastcan be spoken of, and therefore, by the principle, still is.

Mystical philosophy, in all ages and in all parts of the world, ischaracterised by certain beliefs which are illustrated by thedoctrines we have been considering.



There is, first, the belief in insight as against discursive analyticknowledge: the belief in a way of wisdom, sudden, penetrating,coercive, which is contrasted with the slow and fallible study ofoutward appearance by a science relying wholly upon the senses. Allwho are capable of absorption in an inward passion must haveexperienced at times the strange feeling of unreality in commonobjects, the loss of contact with daily things, in which the solidityof the outer world is lost, and the soul seems, in utter loneliness,to bring forth, out of its own depths, the mad dance of fantasticphantoms which have hitherto appeared as independently real andliving. This is the negative side of the mystic's initiation: thedoubt concerning common knowledge, preparing the way for the receptionof what seems a higher wisdom. Many men to whom this negativeexperience is familiar do not pass beyond it, but for the mystic it ismerely the gateway to an ampler world.

The mystic insight begins with the sense of a mystery unveiled, of ahidden wisdom now suddenly become certain beyond the possibility of adoubt. The sense of certainty and revelation comes earlier than anydefinite belief. The definite beliefs at which mystics arrive are theresult of reflection upon the inarticulate experience gained in themoment of insight. Often, beliefs which have no real connection withthis 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 temporarycharacter, which no doubt become amalgamated with what was essentiallymystical in virtue of their subjective certainty. We may ignore suchinessential accretions, and confine ourselves to the beliefs which allmystics share.

The first and most direct outcome of the moment of illumination isbelief in the possibility of a way of knowledge which may be calledrevelation or insight or intuition, as contrasted with sense, reason,and analysis, which are regarded as blind guides leading to the morassof illusion. Closely connected with this belief is the conception of aReality behind the world of appearance and utterly different from it.

This Reality is regarded with an admiration often amounting toworship; it is felt to be always and everywhere close at hand, thinlyveiled by the shows of sense, ready, for the receptive mind, to shinein its glory even through the apparent folly and wickedness of Man.The poet, the artist, and the lover are seekers after that glory: thehaunting beauty that they pursue is the faint reflection of its sun.But the mystic lives in the full light of the vision: what othersdimly seek he knows, with a knowledge beside which all other knowledgeis ignorance.

The second characteristic of mysticism is its belief in unity, and itsrefusal to admit opposition or division anywhere. We found Heraclitussaying "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 thesimultaneous assertion of contradictory propositions, such as: "Westep and do not step into the same rivers; we are and are not." Theassertion of Parmenides, that reality is one and indivisible, comesfrom the same impulse towards unity. In Plato, this impulse is lessprominent, being held in check by his theory of ideas; but itreappears, so far as his logic permits, in the doctrine of the primacyof 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 allis one, the distinction of past and future must be illusory. We haveseen this doctrine prominent in Parmenides; and among moderns it isfundamental in the systems of Spinoza and Hegel.

The last of the doctrines of mysticism which we have to consider isits belief that all evil is mere appearance, an illusion produced bythe divisions and oppositions of the analytic intellect. Mysticismdoes not maintain that such things as cruelty, for example, are good,but it denies that they are real: they belong to that lower world ofphantoms from which we are to be liberated by the insight of thevision. Sometimes--for example in Hegel, and at least verbally inSpinoza--not only evil, but good also, is regarded as illusory, thoughnevertheless the emotional attitude towards what is held to be Realityis such as would naturally be associated with the belief that Realityis good. What is, in all cases, ethically characteristic of mysticismis absence of indignation or protest, acceptance with joy, disbeliefin the ultimate truth of the division into two hostile camps, the goodand the bad. This attitude is a direct outcome of the nature of themystical experience: with its sense of unity is associated a feelingof infinite peace. Indeed it may be suspected that the feeling ofpeace produces, as feelings do in dreams, the whole system ofassociated beliefs which make up the body of mystic doctrine. But thisis 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 ofmysticism, 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 seemsto me mistaken, I yet believe that, by sufficient restraint, there isan 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 isthe truth, mysticism is to be commended as an attitude towards life,not as a creed about the world. The meta-physical creed, I shallmaintain, is a mistaken outcome of the emotion, although this emotion,as colouring and informing all other thoughts and feelings, is theinspirer of whatever is best in Man. Even the cautious and patientinvestigation of truth by science, which seems the very antithesis ofthe 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. Ihave no wish to deny it, nor even to declare that the insight whichreveals it is not a genuine insight. What I do wish to maintain--andit is here that the scientific attitude becomes imperative--is thatinsight, untested and unsupported, is an insufficient guarantee of



truth, in spite of the fact that much of the most important truth isfirst suggested by its means. It is common to speak of an oppositionbetween instinct and reason; in the eighteenth century, the oppositionwas drawn in favour of reason, but under the influence of Rousseau andthe romantic movement instinct was given the preference, first bythose who rebelled against artificial forms of government and thought,and then, as the purely rationalistic defence of traditional theologybecame increasingly difficult, by all who felt in science a menace tocreeds which they associated with a spiritual outlook on life and theworld. Bergson, under the name of "intuition," has raised instinct tothe position of sole arbiter of metaphysical truth. But in fact theopposition of instinct and reason is mainly illusory. Instinct,intuition, or insight is what first leads to the beliefs whichsubsequent reason confirms or confutes; but the confirmation, where itis possible, consists, in the last analysis, of agreement with otherbeliefs no less instinctive. Reason is a harmonising, controllingforce rather than a creative one. Even in the most purely logicalrealm, it is insight that first arrives at what is new.

Where instinct and reason do sometimes conflict is in regard to singlebeliefs, held instinctively, and held with such determination that nodegree of inconsistency with other beliefs leads to their abandonment.Instinct, like all human faculties, is liable to error. Those in whomreason is weak are often unwilling to admit this as regards

themselves, though all admit it in regard to others. Where instinct isleast liable to error is in practical matters as to which rightjudgment is a help to survival: friendship and hostility in others,for instance, are often felt with extraordinary discrimination throughvery careful disguises. But even in such matters a wrong impressionmay be given by reserve or flattery; and in matters less directlypractical, such as philosophy deals with, very strong instinctivebeliefs are sometimes wholly mistaken, as we may come to know throughtheir perceived inconsistency with other equally strong beliefs. It issuch considerations that necessitate the harmonising mediation ofreason, which tests our beliefs by their mutual compatibility, andexamines, in doubtful cases, the possible sources of error on the oneside 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 ofinstinct to the exclusion of other more commonplace but not lesstrustworthy 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 toBergson's advocacy of "intuition" as against "intellect." There are,he says, "two profoundly different ways of knowing a thing. The firstimplies that we move round the object: the second that we enter intoit. The first depends on the point of view at which we are placed andon the symbols by which we express ourselves. The second neitherdepends on a point of view nor relies on any symbol. The first kind ofknowledge may be said to stop at the _relative_; the second, in those

cases where it is possible, to attain the _absolute_."[4] The secondof these, which is intuition, is, he says, "the kind of _intellectualsympathy_ by which one places oneself within an object in order tocoincide with what is unique in it and therefore inexpressible" (p.6). In illustration, he mentions self-knowledge: "there is onereality, at least, which we all seize from within, by intuition andnot by simple analysis. It is our own personality in its flowingthrough time--our self which endures" (p. 8). The rest of Bergson'sphilosophy consists in reporting, through the imperfect medium ofwords, the knowledge gained by intuition, and the consequent complete



condemnation of all the pretended knowledge derived from science andcommon sense.

This procedure, since it takes sides in a conflict of instinctivebeliefs, stands in need of justification by proving the greatertrustworthiness of the beliefs on one side than of those on the other.Bergson attempts this justification in two ways, first by explainingthat intellect is a purely practical faculty to secure biologicalsuccess, secondly by mentioning remarkable feats of instinct inanimals and by pointing out characteristics of the world which, thoughintuition can apprehend them, are baffling to intellect as heinterprets it.

Of Bergson's theory that intellect is a purely practical faculty,developed in the struggle for survival, and not a source of truebeliefs, we may say, first, that it is only through intellect that weknow of the struggle for survival and of the biological ancestry ofman: if the intellect is misleading, the whole of this merely inferredhistory is presumably untrue. If, on the other hand, we agree with himin thinking that evolution took place as Darwin believed, then it isnot only intellect, but all our faculties, that have been developedunder the stress of practical utility. Intuition is seen at its bestwhere it is directly useful, for example in regard to other people'scharacters and dispositions. Bergson apparently holds that capacity

for this kind of knowledge is less explicable by the struggle forexistence than, for example, capacity for pure mathematics. Yet thesavage deceived by false friendship is likely to pay for his mistakewith his life; whereas even in the most civilised societies men arenot put to death for mathematical incompetence. All the most strikingof his instances of intuition in animals have a very direct survivalvalue. The fact is, of course, that both intuition and intellect havebeen developed because they are useful, and that, speaking broadly,they are useful when they give truth and become harmful when they givefalsehood. Intellect, in civilised man, like artistic capacity, hasoccasionally been developed beyond the point where it is useful to theindividual; intuition, on the other hand, seems on the whole todiminish 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. Butthose who see in these facts a recommendation of intuition ought toreturn to running wild in the woods, dyeing themselves with woad andliving on hips and haws.

Let us next examine whether intuition possesses any such infallibilityas Bergson claims for it. The best instance of it, according to him,is our acquaintance with ourselves; yet self-knowledge is proverbiallyrare and difficult. Most men, for example, have in their naturemeannesses, vanities, and envies of which they are quite unconscious,though even their best friends can perceive them without anydifficulty. It is true that intuition has a convincingness which is

lacking to intellect: while it is present, it is almost impossible todoubt its truth. But if it should appear, on examination, to be atleast as fallible as intellect, its greater subjective certaintybecomes a demerit, making it only the more irresistibly deceptive.Apart from self-knowledge, one of the most notable examples ofintuition is the knowledge people believe themselves to possess ofthose with whom they are in love: the wall between differentpersonalities seems to become transparent, and people think they seeinto another soul as into their own. Yet deception in such cases isconstantly practised with success; and even where there is no



intentional deception, experience gradually proves, as a rule, thatthe supposed insight was illusory, and that the slower more gropingmethods of the intellect are in the long run more reliable.

Bergson maintains that intellect can only deal with things in so faras they resemble what has been experienced in the past, whileintuition has the power of apprehending the uniqueness and noveltythat always belong to each fresh moment. That there is somethingunique and new at every moment, is certainly true; it is also truethat 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, anddoes not require, so far as I can see, any special faculty ofintuition for its apprehension. It is neither intellect nor intuition,but sensation, that supplies new data; but when the data are new inany remarkable manner, intellect is much more capable of dealing withthem than intuition would be. The hen with a brood of ducklings nodoubt has intuition which seems to place her inside them, and notmerely to know them analytically; but when the ducklings take to thewater, the whole apparent intuition is seen to be illusory, and thehen is left helpless on the shore. Intuition, in fact, is an aspectand development of instinct, and, like all instinct, is admirable inthose customary surroundings which have moulded the habits of theanimal in question, but totally incompetent as soon as the

surroundings are changed in a way which demands some non-habitual modeof action.

The theoretical understanding of the world, which is the aim ofphilosophy, is not a matter of great practical importance to animals,or to savages, or even to most civilised men. It is hardly to besupposed, therefore, that the rapid, rough and ready methods ofinstinct or intuition will find in this field a favourable ground fortheir application. It is the older kinds of activity, which bring outour kinship with remote generations of animal and semi-humanancestors, that show intuition at its best. In such matters asself-preservation and love, intuition will act sometimes (though notalways) with a swiftness and precision which are astonishing to the

critical intellect. But philosophy is not one of the pursuits whichillustrate our affinity with the past: it is a highly refined, highlycivilised pursuit, demanding, for its success, a certain liberationfrom the life of instinct, and even, at times, a certain aloofnessfrom all mundane hopes and fears. It is not in philosophy, therefore,that we can hope to see intuition at its best. On the contrary, sincethe true objects of philosophy, and the habit of thought demanded fortheir apprehension, are strange, unusual, and remote, it is here, morealmost than anywhere else, that intellect proves superior tointuition, and that quick unanalysed convictions are least deservingof uncritical acceptance.

In advocating the scientific restraint and balance, as against the

self-assertion of a confident reliance upon intuition, we are onlyurging, in the sphere of knowledge, that largeness of contemplation,that impersonal disinterestedness, and that freedom from practicalpreoccupations which have been inculcated by all the great religionsof the world. Thus our conclusion, however it may conflict with theexplicit beliefs of many mystics, is, in essence, not contrary to thespirit which inspires those beliefs, but rather the outcome of thisvery spirit as applied in the realm of thought.



II. UNITY AND PLURALITY

One of the most convincing aspects of the mystic illumination is theapparent revelation of the oneness of all things, giving rise topantheism 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 oneindivisible Whole, and that what seem to be its parts, if consideredas substantial and self-existing, are mere illusion. The conceptionof a Reality quite other than the world of appearance, a reality one,indivisible, and unchanging, was introduced into Western philosophy byParmenides, not, nominally at least, for mystical or religiousreasons, but on the basis of a logical argument as to theimpossibility of not-being, and most subsequent metaphysical systemsare 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. Ishall not here repeat these criticisms, since they are lengthy anddifficult, but shall instead attempt an analysis of the state of mindfrom which mystical logic has arisen.

Belief in a reality quite different from what appears to the sensesarises with irresistible force in certain moods, which are the source

of most mysticism, and of most metaphysics. While such a mood isdominant, the need of logic is not felt, and accordingly the morethoroughgoing mystics do not employ logic, but appeal directly to theimmediate deliverance of their insight. But such fully developedmysticism is rare in the West. When the intensity of emotionalconviction subsides, a man who is in the habit of reasoning willsearch for logical grounds in favour of the belief which he finds inhimself. But since the belief already exists, he will be veryhospitable to any ground that suggests itself. The paradoxesapparently 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 inaccordance with insight. The resulting logic has rendered mostphilosophers incapable of giving any account of the world of science

and daily life. If they had been anxious to give such an account, theywould probably have discovered the errors of their logic; but most ofthem were less anxious to understand the world of science and dailylife than to convict it of unreality in the interests of asuper-sensible "real" world.

It is in this way that logic has been pursued by those of the greatphilosophers who were mystics. But since they usually took for grantedthe supposed insight of the mystic emotion, their logical doctrineswere presented with a certain dryness, and were believed by theirdisciples to be quite independent of the sudden illumination fromwhich they sprang. Nevertheless their origin clung to them, and theyremained--to borrow a useful word from Mr. Santayana--"malicious" in

regard to the world of science and common sense. It is only so that wecan account for the complacency with which philosophers have acceptedthe inconsistency of their doctrines with all the common andscientific facts which seem best established and most worthy ofbelief.

The logic of mysticism shows, as is natural, the defects which areinherent in anything malicious. The impulse to logic, not felt whilethe mystic mood is dominant, reasserts itself as the mood fades, butwith a desire to retain the vanishing insight, or at least to prove



that it _was_ insight, and that what seems to contradict it isillusion. The logic which thus arises is not quite disinterested orcandid, and is inspired by a certain hatred of the daily world towhich it is to be applied. Such an attitude naturally does not tend tothe best results. Everyone knows that to read an author simply inorder to refute him is not the way to understand him; and to read thebook of Nature with a conviction that it is all illusion is just asunlikely to lead to understanding. If our logic is to find the commonworld intelligible, it must not be hostile, but must be inspired by agenuine acceptance such as is not usually to be found amongmetaphysicians.

III. TIME

The unreality of time is a cardinal doctrine of many metaphysicalsystems, often nominally based, as already by Parmenides, upon logicalarguments, but originally derived, at any rate in the founders of newsystems, from the certainty which is born in the moment of mysticinsight. 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 verycommon one: it gave rise to the metaphysical notion of substance, andfinds, even now, a wholly illegitimate satisfaction in such scientificdoctrines 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 theworld of sense is illusory must, I think, be regarded as fallacious.Nevertheless there is some sense--easier to feel than to state--inwhich time is an unimportant and superficial characteristic ofreality. Past and future must be acknowledged to be as real as thepresent, and a certain emancipation from slavery to time is essential

to philosophic thought. The importance of time is rather practicalthan theoretical, rather in relation to our desires than in relationto truth. A truer image of the world, I think, is obtained bypicturing things as entering into the stream of time from an eternalworld outside, than from a view which regards time as the devouringtyrant of all that is. Both in thought and in feeling, even thoughtime be real, to realise the unimportance of time is the gate ofwisdom.

That this is the case may be seen at once by asking ourselves why ourfeelings towards the past are so different from our feelings towardsthe future. The reason for this difference is wholly practical: ourwishes can affect the future but not the past, the future is to some

extent subject to our power, while the past is unalterably fixed. Butevery future will some day be past: if we see the past truly now, itmust, when it was still future, have been just what we now see it tobe, and what is now future must be just what we shall see it to bewhen it has become past. The felt difference of quality between pastand future, therefore, is not an intrinsic difference, but only adifference in relation to us: to impartial contemplation, it ceases toexist. And impartiality of contemplation is, in the intellectualsphere, that very same virtue of disinterestedness which, in thesphere of action, appears as justice and unselfishness. Whoever wishes



to see the world truly, to rise in thought above the tyranny ofpractical desires, must learn to overcome the difference of attitudetowards past and future, and to survey the whole stream of time in onecomprehensive vision.

The kind of way in which, as it seems to me, time ought not to enterinto our theoretic philosophical thought, may be illustrated by thephilosophy which has become associated with the idea of evolution, andwhich is exemplified by Nietzsche, pragmatism, and Bergson. Thisphilosophy, on the basis of the development which has led from thelowest forms of life up to man, sees in _progress_ the fundamental lawof the universe, and thus admits the difference between _earlier_ and

_later_ into the very citadel of its contemplative outlook. With itspast and future history of the world, conjectural as it is, I do notwish to quarrel. But I think that, in the intoxication of a quicksuccess, much that is required for a true understanding of theuniverse has been forgotten. Something of Hellenism, something, too,of Oriental resignation, must be combined with its hurrying Westernself-assertion before it can emerge from the ardour of youth into themature wisdom of manhood. In spite of its appeals to science, the truescientific philosophy, I think, is something more arduous and morealoof, appealing to less mundane hopes, and requiring a severerdiscipline for its successful practice.

Darwin's _Origin of Species_ persuaded the world that the differencebetween different species of animals and plants is not the fixedimmutable difference that it appears to be. The doctrine of naturalkinds, which had rendered classification easy and definite, which wasenshrined in the Aristotelian tradition, and protected by its supposednecessity for orthodox dogma, was suddenly swept away for ever out ofthe biological world. The difference between man and the loweranimals, which to our human conceit appears enormous, was shown to bea gradual achievement, involving intermediate being who could not withcertainty be placed either within or without the human family. The sunand the planets had already been shown by Laplace to be very probablyderived from a primitive more or less undifferentiated nebula. Thusthe old fixed landmarks became wavering and indistinct, and all sharp

outlines were blurred. Things and species lost their boundaries, andnone could say where they began or where they ended.

But if human conceit was staggered for a moment by its kinship withthe ape, it soon found a way to reassert itself, and that way is the"philosophy" of evolution. A process which led from the amúba to Manappeared to the philosophers to be obviously a progress--thoughwhether the amúba would agree with this opinion is not known. Hencethe cycle of changes which science had shown to be the probablehistory of the past was welcomed as revealing a law of developmenttowards good in the universe--an evolution or unfolding of an ideaslowly embodying itself in the actual. But such a view, though itmight satisfy Spencer and those whom we may call Hegelian

evolutionists, could not be accepted as adequate by the morewhole-hearted votaries of change. An ideal to which the worldcontinuously approaches is, to these minds, too dead and static to beinspiring. Not only the aspiration, but the ideal too, must change anddevelop with the course of evolution: there must be no fixed goal, buta continual fashioning of fresh needs by the impulse which is life andwhich alone gives unity to the process.

Life, in this philosophy, is a continuous stream, in which alldivisions are artificial and unreal. Separate things, beginnings and



endings, are mere convenient fictions: there is only smooth unbrokentransition. The beliefs of to-day may count as true to-day, if theycarry us along the stream; but to-morrow they will be false, and mustbe replaced by new beliefs to meet the new situation. All our thinkingconsists of convenient fictions, imaginary congealings of the stream:reality flows on in spite of all our fictions, and though it can belived, it cannot be conceived in thought. Somehow, without explicitstatement, the assurance is slipped in that the future, though wecannot foresee it, will be better than the past or the present: thereader is like the child which expects a sweet because it has beentold to open its mouth and shut its eyes. Logic, mathematics, physicsdisappear in this philosophy, because they are too "static"; what isreal is no impulse and movement towards a goal which, like therainbow, recedes as we advance, and makes every place different whenit reaches it from what it appeared to be at a distance.

I do not propose to enter upon a technical examination of thisphilosophy. I wish only to maintain that the motives and interestswhich inspire it are so exclusively practical, and the problems withwhich it deals are so special, that it can hardly be regarded astouching any of the questions that, to my mind, constitute genuinephilosophy.

The predominant interest of evolutionism is in the question of human

destiny, or at least of the destiny of Life. It is more interested inmorality and happiness than in knowledge for its own sake. It must beadmitted that the same may be said of many other philosophies, andthat a desire for the kind of knowledge which philosophy can give isvery rare. But if philosophy is to attain truth, it is necessary firstand foremost that philosophers should acquire the disinterestedintellectual curiosity which characterises the genuine man of science.Knowledge concerning the future--which is the kind of knowledge thatmust be sought if we are to know about human destiny--is possiblewithin certain narrow limits. It is impossible to say how much thelimits may be enlarged with the progress of science. But what isevident is that any proposition about the future belongs by itssubject-matter to some particular science, and is to be ascertained,

if at all, by the methods of that science. Philosophy is not a shortcut to the same kind of results as those of the other sciences: if itis to be a genuine study, it must have a province of its own, and aimat results which the other sciences can neither prove nor disprove.

Evolutionism, in basing itself upon the notion of _progress_, which ischange from the worse to the better, allows the notion of time, as itseems to me, to become its tyrant rather than its servant, and therebyloses that impartiality of contemplation which is the source of allthat is best in philosophic thought and feeling. Metaphysicians, as wesaw, have frequently denied altogether the reality of time. I do notwish to do this; I wish only to preserve the mental outlook whichinspired the denial, the attitude which, in thought, regards the past

as having the same reality as the present and the same importance asthe future. "In so far," says Spinoza,[6] "as the mind conceives athing according to the dictate of reason, it will be equally affectedwhether the idea is that of a future, past, or present thing." It isthis "conceiving according to the dictate of reason" that I findlacking in the philosophy which is based on evolution.

IV. GOOD AND EVIL



Mysticism maintains that all evil is illusory, and sometimes maintainsthe same view as regards good, but more often holds that all Realityis good. Both views are to be found in Heraclitus: "Good and ill areone," he says, but again, "To God all things are fair and good andright, but men hold some things wrong and some right." A similartwofold position is to be found in Spinoza, but he uses the word"perfection" when he means to speak of the good that is not merelyhuman. "By reality and perfection I mean the same thing," he says;[7]but elsewhere we find the definition: "By _good_ I shall mean thatwhich we certainly know to be useful to us."[8] Thus perfectionbelongs to Reality in its own nature, but goodness is relative toourselves and our needs, and disappears in an impartial survey. Somesuch distinction, I think, is necessary in order to understand theethical outlook of mysticism: there is a lower mundane kind of goodand evil, which divides the world of appearance into what seem to beconflicting parts; but there is also a higher, mystical kind of good,which belongs to Reality and is not opposed by any correlative kind ofevil.

It is difficult to give a logically tenable account of this positionwithout recognising that good and evil are subjective, that what isgood is merely that towards which we have one kind of feeling, andwhat is evil is merely that towards which we have another kind offeeling. 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 adistinction of good and evil, or at least of better and worse. Butthis distinction, like everything pertaining to action, belongs towhat mysticism regards as the world of illusion, if only because it isessentially concerned with time. In our contemplative life, whereaction is not called for, it is possible to be impartial, and toovercome the ethical dualism which action requires. So long as weremain _merely_ impartial, we may be content to say that both the goodand the evil of action are illusions. But if, as we must do if we havethe 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 thatthis higher good belongs to the whole world as it is in reality. Inthis way the twofold attitude and the apparent vacillation ofmysticism are explained and justified.

The possibility of this universal love and joy in all that exists isof supreme importance for the conduct and happiness of life, and givesinestimable value to the mystic emotion, apart from any creeds whichmay 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 itdoes not reveal anything about the non-human, or about the nature ofthe universe in general. Good and bad, and even the higher good thatmysticism finds everywhere, are the reflections of our own emotions onother things, not part of the substance of things as they are inthemselves. And therefore an impartial contemplation, freed from allpre-occupation with Self, will not judge things good or bad, althoughit is very easily combined with that feeling of universal love whichleads the mystic to say that the whole world is good.



The philosophy of evolution, through the notion of progress, is boundup with the ethical dualism of the worse and the better, and is thusshut out, not only from the kind of survey which discards good andevil altogether from its view, but also from the mystical belief inthe goodness of everything. In this way the distinction of good andevil, like time, becomes a tyrant in this philosophy, and introducesinto thought the restless selectiveness of action. Good and evil, liketime, are, it would seem, not general or fundamental in the world ofthought, but late and highly specialised members of the intellectualhierarchy.

Although, as we saw, mysticism can be interpreted so as to agree withthe view that good and evil are not intellectually fundamental, itmust be admitted that here we are no longer in verbal agreement withmost of the great philosophers and religious teachers of the past. Ibelieve, however, that the elimination of ethical considerations fromphilosophy is both scientifically necessary and--though this may seema paradox--an ethical advance. Both these contentions must be brieflydefended.

The hope of satisfaction to our more human desires--the hope ofdemonstrating that the world has this or that desirable ethicalcharacteristic--is not one which, so far as I can see, a scientificphilosophy can do anything whatever to satisfy. The difference between

a good world and a bad one is a difference in the particularcharacteristics of the particular things that exist in these worlds:it is not a sufficiently abstract difference to come within theprovince of philosophy. Love and hate, for example, are ethicalopposites, but to philosophy they are closely analogous attitudestowards objects. The general form and structure of those attitudestowards objects which constitute mental phenomena is a problem forphilosophy, but the difference between love and hate is not adifference of form or structure, and therefore belongs rather to thespecial science of psychology than to philosophy. Thus the ethicalinterests which have often inspired philosophers must remain in thebackground: 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 remindourselves that a similar change has been found necessary in all theother sciences. The physicist or chemist is not now required to provethe ethical importance of his ions or atoms; the biologist is notexpected to prove the utility of the plants or animals which hedissects. In pre-scientific ages this was not the case. Astronomy, forexample, was studied because men believed in astrology: it was thoughtthat the movements of the planets had the most direct and importantbearing upon the lives of human beings. Presumably, when this beliefdecayed and the disinterested study of astronomy began, many who hadfound astrology absorbingly interesting decided that astronomy had too

little human interest to be worthy of study. Physics, as it appears inPlato's Timæus for example, is full of ethical notions: it is anessential part of its purpose to show that the earth is worthy ofadmiration. The modern physicist, on the contrary, though he has nowish to deny that the earth is admirable, is not concerned, asphysicist, with its ethical attributes: he is merely concerned to findout facts, not to consider whether they are good or bad. Inpsychology, the scientific attitude is even more recent and moredifficult than in the physical sciences: it is natural to considerthat human nature is either good or bad, and to suppose that the



difference between good and bad, so all-important in practice, must beimportant in theory also. It is only during the last century that anethically neutral psychology has grown up; and here too, ethicalneutrality has been essential to scientific success.

In philosophy, hitherto, ethical neutrality has been seldom sought andhardly ever achieved. Men have remembered their wishes, and havejudged philosophies in relation to their wishes. Driven from theparticular sciences, the belief that the notions of good and evil mustafford a key to the understanding of the world has sought a refuge inphilosophy. But even from this last refuge, if philosophy is not toremain a set of pleasing dreams, this belief must be driven forth. Itis a commonplace that happiness is not best achieved by those whoseek 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 onlyto know the facts are more likely to achieve good than those who viewthe world through the distorting medium of their own desires.

We are thus brought back to our seeming paradox, that a philosophywhich does not seek to impose upon the world its own conceptions ofgood and evil is not only more likely to achieve truth, but is alsothe outcome of a higher ethical standpoint than one which, likeevolutionism and most traditional systems, is perpetually appraisingthe universe and seeking to find in it an embodiment of present

ideals. In religion, and in every deeply serious view of the world andof human destiny, there is an element of submission, a realisation ofthe limits of human power, which is somewhat lacking in the modernworld, with its quick material successes and its insolent belief inthe boundless possibilities of progress. "He that loveth his lifeshall lose it"; and there is danger lest, through a too confident loveof life, life itself should lose much of what gives it its highestworth. The submission which religion inculcates in action isessentially the same in spirit as that which science teaches inthought; and the ethical neutrality by which its victories have beenachieved 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 attitudetowards the world. Insistence on belief in an external realisation ofthe good is a form of self-assertion, which, while it cannot securethe external good which it desires, can seriously impair the inwardgood which lies within our power, and destroy that reverence towardsfact which constitutes both what is valuable in humility and what isfruitful in the scientific temper.

Human beings cannot, of course, wholly transcend human nature;something subjective, if only the interest that determines thedirection of our attention, must remain in all our thought. Butscientific philosophy comes nearer to objectivity than any other humanpursuit, 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; butexperience has shown that friendliness and hostility are not theconceptions by which the world is to be understood. Scientificphilosophy thus represents, though as yet only in a nascent condition,a higher form of thought than any pre-scientific belief orimagination, and, like every approach to self-transcendence, it bringswith it a rich reward in increase of scope and breadth andcomprehension. Evolutionism, in spite of its appeals to particularscientific facts, fails to be a truly scientific philosophy because of



its slavery to time, its ethical preoccupations, and its predominantinterest in our mundane concerns and destiny. A truly scientificphilosophy will be more humble, more piecemeal, more arduous, offeringless glitter of outward mirage to flatter fallacious hopes, but moreindifferent to fate, and more capable of accepting the world withoutthe tyrannous imposition of our human and temporary demands.

FOOTNOTES:

[1] All the above quotations are from Burnet's _Early GreekPhilosophy_, (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, havebeen printed in a course of Lowell lectures _On our knowledge of theexternal world_, published by the Open Court Publishing Company. But Ihave left them here, as this is the context for which they wereoriginally 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 avarying selection of sensational triumphs, such as wireless telegraphyand aeroplanes, radio-activity and the marvels of modern alchemy. It isnot of this aspect of science that I wish to speak. Science, in thisaspect, consists of detached up-to-date fragments, interesting onlyuntil they are replaced by something newer and more up-to-date,displaying nothing of the systems of patiently constructed knowledgeout of which, almost as a casual incident, have come the practicallyuseful results which interest the man in the street. The increasedcommand over the forces of nature which is derived from science is

undoubtedly an amply sufficient reason for encouraging scientificresearch, but this reason has been so often urged and is so easilyappreciated that other reasons, to my mind quite as important, are aptto be overlooked. It is with these other reasons, especially with theintrinsic value of a scientific habit of mind in forming our outlookon the world, that I shall be concerned in what follows.

The instance of wireless telegraphy will serve to illustrate thedifference between the two points of view. Almost all the seriousintellectual labour required for the possibility of this invention is



due to three men--Faraday, Maxwell, and Hertz. In alternating layersof experiment and theory these three men built up the modern theory ofelectromagnetism, and demonstrated the identity of light withelectromagnetic waves. The system which they discovered is one ofprofound intellectual interest, bringing together and unifying anendless variety of apparently detached phenomena, and displaying acumulative mental power which cannot but afford delight to everygenerous spirit. The mechanical details which remained to be adjustedin order to utilise their discoveries for a practical system oftelegraphy demanded, no doubt, very considerable ingenuity, but hadnot that broad sweep and that universality which could give themintrinsic interest as an object of disinterested contemplation.

From the point of view of training the mind, of giving thatwell-informed, impersonal outlook which constitutes culture in thegood sense of this much-misused word, it seems to be generally heldindisputable that a literary education is superior to one based onscience. Even the warmest advocates of science are apt to rest theirclaims on the contention that culture ought to be sacrificed toutility. Those men of science who respect culture, when they associatewith men learned in the classics, are apt to admit, not merelypolitely, but sincerely, a certain inferiority on their side,compensated doubtless by the services which science renders tohumanity, but none the less real. And so long as this attitude exists

among men of science, it tends to verify itself: the intrinsicallyvaluable aspects of science tend to be sacrificed to the merelyuseful, and little attempt is made to preserve that leisurely,systematic survey by which the finer quality of mind is formed andnourished.

But even if there be, in present fact, any such inferiority as issupposed in the educational value of science, this is, I believe, notthe fault of science itself, but the fault of the spirit in whichscience is taught. If its full possibilities were realised by thosewho teach it, I believe that its capacity of producing those habits ofmind which constitute the highest mental excellence would be at leastas great as that of literature, and more particularly of Greek and

Latin literature. In saying this I have no wish whatever to disparagea classical education. I have not myself enjoyed its benefits, and myknowledge of Greek and Latin authors is derived almost wholly fromtranslations. But I am firmly persuaded that the Greeks fully deserveall the admiration that is bestowed upon them, and that it is a verygreat and serious loss to be unacquainted with their writings. It isnot by attacking them, but by drawing attention to neglectedexcellences in science, that I wish to conduct my argument.

One defect, however, does seem inherent in a purely classicaleducation--namely, a too exclusive emphasis on the past. By the studyof what is absolutely ended and can never be renewed, a habit ofcriticism towards the present and the future is engendered. The

qualities in which the present excels are qualities to which the studyof the past does not direct attention, and to which, therefore, thestudent of Greek civilisation may easily become blind. In what is newand growing there is apt to be something crude, insolent, even alittle vulgar, which is shocking to the man of sensitive taste;quivering from the rough contact, he retires to the trim gardens of apolished past, forgetting that they were reclaimed from the wildernessby men as rough and earth-soiled as those from whom he shrinks in hisown day. The habit of being unable to recognise merit until it isdead 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 througheveryday surroundings to the essential splendour of contemporarythings, 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 morepugnacious atmosphere of the West, where the champions of past andfuture fight a never-ending battle, instead of combining to seek outthe merits of both.

This consideration, which militates not only against the exclusivestudy of the classics, but against every form of culture which hasbecome static, traditional, and academic, leads inevitably to thefundamental question: What is the true end of education? But beforeattempting to answer this question it will be well to define the sensein which we are to use the word "education." For this purpose I shalldistinguish the sense in which I mean to use it from two others, bothperfectly legitimate, the one broader and the other narrower than thesense in which I mean to use the word.

In the broader sense, education will include not only what we learnthrough instruction, but all that we learn through personalexperience--the formation of character through the education of life.Of this aspect of education, vitally important as it is, I will saynothing, since its consideration would introduce topics quite foreignto the question with which we are concerned.

In the narrower sense, education may be confined to instruction, theimparting of definite information on various subjects, because suchinformation, in and for itself, is useful in daily life. Elementaryeducation--reading, writing, and arithmetic--is almost wholly of thiskind. 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 _theformation, by means of instruction, of certain mental habits and acertain outlook on life and the world_. It remains to ask ourselves,what mental habits, and what sort of outlook, can be hoped for as theresult of instruction? When we have answered this question we canattempt to decide what science has to contribute to the formation ofthe habits and outlook which we desire.

Our whole life is built about a certain number--not a very smallnumber--of primary instincts and impulses. Only what is in some wayconnected with these instincts and impulses appears to us desirable orimportant; there is no faculty, whether "reason" or "virtue" or

whatever it may be called, that can take our active life and our hopesand fears outside the region controlled by these first movers of alldesire. Each of them is like a queen-bee, aided by a hive of workersgathering honey; but when the queen is gone the workers languish anddie, and the cells remain empty of their expected sweetness. So witheach primary impulse in civilised man: it is surrounded and protectedby a busy swarm of attendant derivative desires, which store up in itsservice whatever honey the surrounding world affords. But if thequeen-impulse dies, the death-dealing influence, though retarded alittle by habit, spreads slowly through all the subsidiary impulses,



and a whole tract of life becomes inexplicably colourless. What wasformerly full of zest, and so obviously worth doing that it raised noquestions, has now grown dreary and purposeless: with a sense ofdisillusion we inquire the meaning of life, and decide, perhaps, thatall is vanity. The search for an outside meaning that can _compel_ aninner response must always be disappointed: all "meaning" must be atbottom related to our primary desires, and when they are extinct nomiracle can restore to the world the value which they reflected uponit.

The purpose of education, therefore, cannot be to create any primaryimpulse which is lacking in the uneducated; the purpose can only be toenlarge the scope of those that human nature provides, by increasingthe number and variety of attendant thoughts, and by showing where themost permanent satisfaction is to be found. Under the impulse of aCalvinistic horror of the "natural man," this obvious truth has beentoo often misconceived in the training of the young; "nature" has beenfalsely regarded as excluding all that is best in what is natural, andthe endeavour to teach virtue has led to the production of stunted andcontorted hypocrites instead of full-grown human beings. From suchmistakes in education a better psychology or a kinder heart isbeginning to preserve the present generation; we need, therefore,waste no more words on the theory that the purpose of education is tothwart or eradicate nature.

But although nature must supply the initial force of desire, nature isnot, in the civilised man, the spasmodic, fragmentary, and yet violentset of impulses that it is in the savage. Each impulse has itsconstitutional ministry of thought and knowledge and reflection,through which possible conflicts of impulses are foreseen, andtemporary impulses are controlled by the unifying impulse which may becalled wisdom. In this way education destroys the crudity ofinstinct, and increases through knowledge the wealth and variety ofthe individual's contacts with the outside world, making him no longeran isolated fighting unit, but a citizen of the universe, embracingdistant countries, remote regions of space, and vast stretches of pastand future within the circle of his interests. It is this simultaneous

softening in the insistence of desire and enlargement of its scopethat is the chief moral end of education.

Closely connected with this moral end is the more purely intellectualaim of education, the endeavour to make us see and imagine the worldin an objective manner, as far as possible as it is in itself, and notmerely through the distorting medium of personal desire. The completeattainment 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 andour outlook on the world, is to be judged successful in proportion asits 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 thenature of the non-human world as it is in itself apart from ourdesires and interests. If this standard is admitted, we can return tothe consideration of science, inquiring how far science contributes tosuch an aim, and whether it is in any respect superior to its rivalsin educational practice.

II



Two opposite and at first sight conflicting merits belong to scienceas against literature and art. The one, which is not inherentlynecessary, but is certainly true at the present day, is hopefulnessas to the future of human achievement, and in particular as to theuseful work that may be accomplished by any intelligent student. Thismerit and the cheerful outlook which it engenders prevent what mightotherwise be the depressing effect of another aspect of science, to mymind also a merit, and perhaps its greatest merit--I mean theirrelevance of human passions and of the whole subjective apparatuswhere scientific truth is concerned. Each of these reasons forpreferring the study of science requires some amplification. Let usbegin with the first.

In the study of literature or art our attention is perpetually rivetedupon the past: the men of Greece or of the Renaissance did better thanany men do now; the triumphs of former ages, so far from facilitatingfresh triumphs in our own age, actually increase the difficulty offresh triumphs by rendering originality harder of attainment; not onlyis artistic achievement not cumulative, but it seems even to dependupon a certain freshness and _naïveté_ of impulse and vision whichcivilisation tends to destroy. Hence comes, to those who have beennourished on the literary and artistic productions of former ages, acertain peevishness and undue fastidiousness towards the present, fromwhich there seems no escape except into the deliberate vandalism which

ignores tradition and in the search after originality achieves onlythe eccentric. But in such vandalism there is none of the simplicityand spontaneity out of which great art springs: theory is still thecanker in its core, and insincerity destroys the advantages of amerely pretended ignorance.

The despair thus arising from an education which suggests nopre-eminent mental activity except that of artistic creation is whollyabsent from an education which gives the knowledge of scientificmethod. The discovery of scientific method, except in puremathematics, is a thing of yesterday; speaking broadly, we may saythat 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 whichthey are no longer, as in art, dependent for progress upon theappearance of continually greater genius, for in science thesuccessors stand upon the shoulders of their predecessors; where oneman of supreme genius has invented a method, a thousand lesser men canapply it. No transcendent ability is required in order to make usefuldiscoveries 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 bedone without genius; in science even a very moderate capacity cancontribute 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 canapply the method with fresh vigour, unimpaired by the previous labourof perfecting it; but the mental calibre of the thought required fortheir work, however brilliant, is not so great as that required by thefirst inventor of the method. There are in science immense numbers ofdifferent methods, appropriate to different classes of problems; butover and above them all, there is something not easily definable,which may be called _the_ method of science. It was formerly customaryto identify this with the inductive method, and to associate it withthe name of Bacon. But the true inductive method was not discovered by



Bacon, and the true method of science is something which includesdeduction as much as induction, logic and mathematics as much asbotany and geology. I shall not attempt the difficult task of statingwhat the scientific method is, but I will try to indicate the temperof mind out of which the scientific method grows, which is the secondof the two merits that were mentioned above as belonging to ascientific education.

The kernel of the scientific outlook is a thing so simple, so obvious,so seemingly trivial, that the mention of it may almost excitederision. The kernel of the scientific outlook is the refusal toregard our own desires, tastes, and interests as affording a key tothe understanding of the world. Stated thus baldly, this may seem nomore than a trite truism. But to remember it consistently in mattersarousing our passionate partisanship is by no means easy, especiallywhere the available evidence is uncertain and inconclusive. A fewillustrations will make this clear.

Aristotle, I understand, considered that the stars must move incircles because the circle is the most perfect curve. In the absenceof evidence to the contrary, he allowed himself to decide a questionof fact by an appeal to æsthetico-moral considerations. In such a caseit is at once obvious to us that this appeal was unjustifiable. Weknow 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 inaccurate ellipses, or in any other kind of simply describable curve.This may be painful to a certain hankering after simplicity of patternin the universe, but we know that in astronomy such feelings areirrelevant. Easy as this knowledge seems now, we owe it to the courageand insight of the first inventors of scientific method, and moreespecially of Galileo.

We may take as another illustration Malthus's doctrine of population.This illustration is all the better for the fact that his actualdoctrine is now known to be largely erroneous. It is not hisconclusions that are valuable, but the temper and method of hisinquiry. As everyone knows, it was to him that Darwin owed an

essential part of his theory of natural selection, and this was onlypossible because Malthus's outlook was truly scientific. His greatmerit lies in considering man not as the object of praise or blame,but as a part of nature, a thing with a certain characteristicbehaviour from which certain consequences must follow. If thebehaviour is not quite what Malthus supposed, if the consequences arenot quite what he inferred, that may falsify his conclusions, but doesnot impair the value of his method. The objections which were madewhen his doctrine was new--that it was horrible and depressing, thatpeople ought not to act as he said they did, and so on--were all suchas implied an unscientific attitude of mind; as against all of them,his calm determination to treat man as a natural phenomenon marks animportant advance over the reformers of the eighteenth century and the

Revolution.

Under the influence of Darwinism the scientific attitude towards manhas now become fairly common, and is to some people quite natural,though to most it is still a difficult and artificial intellectualcontortion. There is however, one study which is as yet almost whollyuntouched by the scientific spirit--I mean the study of philosophy.Philosophers and the public imagine that the scientific spirit mustpervade pages that bristle with allusions to ions, germ-plasms, andthe eyes of shell-fish. But as the devil can quote Scripture, so the



philosopher can quote science. The scientific spirit is not an affairof quotation, of externally acquired information, any more thanmanners are an affair of the etiquette-book. The scientific attitudeof mind involves a sweeping away of all other desires in the interestsof the desire to know--it involves suppression of hopes and fears,loves and hates, and the whole subjective emotional life, until webecome subdued to the material, able to see it frankly, withoutpreconceptions, without bias, without any wish except to see it as itis, and without any belief that what it is must be determined by somerelation, positive or negative, to what we should like it to be, or towhat 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 almostall attempts to conceive the universe as a whole. Mind, or some aspectof it--thought or will or sentience--has been regarded as the patternafter which the universe is to be conceived, for no better reason, atbottom, than that such a universe would not seem strange, and wouldgive us the cosy feeling that every place is like home. To conceivethe universe as essentially progressive or essentially deteriorating,for example, is to give to our hopes and fears a cosmic importancewhich _may_, of course, be justified, but which we have as yet noreason to suppose justified. Until we have learnt to think of it inethically neutral terms, we have not arrived at a scientific attitude

in philosophy; and until we have arrived at such an attitude, it ishardly to be hoped that philosophy will achieve any solid results.

I have spoken so far largely of the negative aspect of the scientificspirit, but it is from the positive aspect that its value is derived.The instinct of constructiveness, which is one of the chief incentivesto artistic creation, can find in scientific systems a satisfactionmore massive than any epic poem. Disinterested curiosity, which is thesource of almost all intellectual effort, finds with astonisheddelight that science can unveil secrets which might well have seemedfor ever undiscoverable. The desire for a larger life and widerinterests, for an escape from private circumstances, and even from thewhole recurring human cycle of birth and death, is fulfilled by the

impersonal cosmic outlook of science as by nothing else. To all thesemust be added, as contributing to the happiness of the man of science,the admiration of splendid achievement, and the consciousness ofinestimable utility to the human race. A life devoted to science istherefore a happy life, and its happiness is derived from the verybest sources that are open to dwellers on this troubled and passionateplanet.

III

A FREE MAN'S WORSHIP[9]

To Dr. Faustus in his study Mephistopheles told the history of theCreation, saying:

"The endless praises of the choirs of angels had begun to growwearisome; for, after all, did he not deserve their praise? Had he notgiven them endless joy? Would it not be more amusing to obtainundeserved praise, to be worshipped by beings whom he tortured? He



smiled inwardly, and resolved that the great drama should beperformed.

"For countless ages the hot nebula whirled aimlessly through space. Atlength it began to take shape, the central mass threw off planets, theplanets cooled, boiling seas and burning mountains heaved and tossed,from black masses of cloud hot sheets of rain deluged the barely solidcrust. And now the first germ of life grew in the depths of the ocean,and developed rapidly in the fructifying warmth into vast foresttrees, huge ferns springing from the damp mould, sea monstersbreeding, fighting, devouring, and passing away. And from themonsters, as the play unfolded itself, Man was born, with the power ofthought, the knowledge of good and evil, and the cruel thirst forworship. 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 oflife before Death's inexorable decree. And Man said: 'There is ahidden purpose, could we but fathom it, and the purpose is good; forwe must reverence something, and in the visible world there is nothingworthy of reverence.' And Man stood aside from the struggle, resolvingthat God intended harmony to come out of chaos by human efforts. Andwhen he followed the instincts which God had transmitted to him fromhis ancestry of beasts of prey, he called it Sin, and asked God toforgive him. But he doubted whether he could be justly forgiven, untilhe 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, thatthereby the future might be better. And he gave God thanks for thestrength that enabled him to forgo even the joys that were possible.And God smiled; and when he saw that Man had become perfect inrenunciation and worship, he sent another sun through the sky, whichcrashed into Man's sun; and all returned again to nebula.

"'Yes,' he murmured, 'it was a good play; I will have it performedagain.'"

Such, in outline, but even more purposeless, more void of meaning, isthe world which Science presents for our belief. Amid such a world, ifanywhere, our ideals henceforward must find a home. That Man is the

product of causes which had no prevision of the end they wereachieving; that his origin, his growth, his hopes and fears, his lovesand his beliefs, are but the outcome of accidental collocations ofatoms; that no fire, no heroism, no intensity of thought and feeling,can preserve an individual life beyond the grave; that all the laboursof the ages, all the devotion, all the inspiration, all the noondaybrightness of human genius, are destined to extinction in the vastdeath of the solar system, and that the whole temple of Man'sachievement must inevitably be buried beneath the débris of a universein ruins--all these things, if not quite beyond dispute, are yet sonearly certain, that no philosophy which rejects them can hope tostand. Only within the scaffolding of these truths, only on the firmfoundation of unyielding despair, can the soul's habitation henceforth

be safely built.

How, in such an alien and inhuman world, can so powerless a creatureas Man preserve his aspirations untarnished? A strange mystery it isthat Nature, omnipotent but blind, in the revolutions of her secularhurryings through the abysses of space, has brought forth at last achild, subject still to her power, but gifted with sight, withknowledge of good and evil, with the capacity of judging all the worksof his unthinking Mother. In spite of Death, the mark and seal of theparental control, Man is yet free, during his brief years, to examine,



to criticise, to know, and in imagination to create. To him alone, inthe world with which he is acquainted, this freedom belongs; and inthis lies his superiority to the resistless forces that control hisoutward life.

The savage, like ourselves, feels the oppression of his impotencebefore the powers of Nature; but having in himself nothing that herespects more than Power, he is willing to prostrate himself beforehis 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 placatingthe jealous gods: surely, the trembling believer thinks, when what ismost precious has been freely given, their lust for blood must beappeased, and more will not be required. The religion of Moloch--assuch creeds may be generically called--is in essence the cringingsubmission of the slave, who dare not, even in his heart, allow thethought that his master deserves no adulation. Since the independenceof ideals is not yet acknowledged, Power may be freely worshipped, andreceive an unlimited respect, despite its wanton infliction of pain.

But gradually, as morality grows bolder, the claim of the ideal worldbegins to be felt; and worship, if it is not to cease, must be givento gods of another kind than those created by the savage. Some, thoughthey feel the demands of the ideal, will still consciously reject

them, still urging that naked Power is worthy of worship. Such is theattitude inculcated in God's answer to Job out of the whirlwind: thedivine power and knowledge are paraded, but of the divine goodnessthere is no hint. Such also is the attitude of those who, in our ownday, base their morality upon the struggle for survival, maintainingthat the survivors are necessarily the fittest. But others, notcontent with an answer so repugnant to the moral sense, will adopt theposition which we have become accustomed to regard as speciallyreligious, maintaining that, in some hidden manner, the world of factis really harmonious with the world of ideals. Thus Man creates God,all-powerful and all-good, the mystic unity of what is and what shouldbe.

But the world of fact, after all, is not good; and, in submitting ourjudgment to it, there is an element of slavishness from which ourthoughts must be purged. For in all things it is well to exalt thedignity of Man, by freeing him as far as possible from the tyranny ofnon-human Power. When we have realised that Power is largely bad, thatman, with his knowledge of good and evil, is but a helpless atom in aworld which has no such knowledge, the choice is again presented tous: Shall we worship Force, or shall we worship Goodness? Shall ourGod exist and be evil, or shall he be recognised as the creation ofour own conscience?

The answer to this question is very momentous, and affects profoundlyour whole morality. The worship of Force, to which Carlyle and

Nietzsche and the creed of Militarism have accustomed us, is theresult of failure to maintain our own ideals against a hostileuniverse: it is itself a prostrate submission to evil, a sacrifice ofour best to Moloch. If strength indeed is to be respected, let usrespect rather the strength of those who refuse that false"recognition of facts" which fails to recognise that facts are oftenbad. Let us admit that, in the world we know, there are many thingsthat would be better otherwise, and that the ideals to which we do andmust adhere are not realised in the realm of matter. Let us preserveour respect for truth, for beauty, for the ideal of perfection which



life does not permit us to attain, though none of these things meetwith the approval of the unconscious universe. If Power is bad, as itseems to be, let us reject it from our hearts. In this lies Man's truefreedom: in determination to worship only the God created by our ownlove of the good, to respect only the heaven which inspires theinsight of our best moments. In action, in desire, we must submitperpetually to the tyranny of outside forces; but in thought, inaspiration, we are free, free from our fellow-men, free from the pettyplanet on which our bodies impotently crawl, free even, while we live,from the tyranny of death. Let us learn, then, that energy of faithwhich enables us to live constantly in the vision of the good; and letus descend, in action, into the world of fact, with that vision alwaysbefore us.

When first the opposition of fact and ideal grows fully visible, aspirit of fiery revolt, of fierce hatred of the gods, seems necessaryto the assertion of freedom. To defy with Promethean constancy ahostile universe, to keep its evil always in view, always activelyhated, to refuse no pain that the malice of Power can invent, appearsto be the duty of all who will not bow before the inevitable. Butindignation is still a bondage, for it compels our thoughts to beoccupied with an evil world; and in the fierceness of desire fromwhich rebellion springs there is a kind of self-assertion which it isnecessary for the wise to overcome. Indignation is a submission of our

thoughts, but not of our desires; the Stoic freedom in which wisdomconsists is found in the submission of our desires, but not of ourthoughts. From the submission of our desires springs the virtue ofresignation; from the freedom of our thoughts springs the whole worldof art and philosophy, and the vision of beauty by which, at last, wehalf reconquer the reluctant world. But the vision of beauty ispossible only to unfettered contemplation, to thoughts not weighted bythe load of eager wishes; and thus Freedom comes only to those who nolonger ask of life that it shall yield them any of those personalgoods that are subject to the mutations of Time.

Although the necessity of renunciation is evidence of the existence ofevil, yet Christianity, in preaching it, has shown a wisdom exceeding

that of the Promethean philosophy of rebellion. It must be admittedthat, of the things we desire, some, though they prove impossible, areyet real goods; others, however, as ardently longed for, do not formpart of a fully purified ideal. The belief that what must be renouncedis bad, though sometimes false, is far less often false than untamedpassion supposes; and the creed of religion, by providing a reasonfor proving that it is never false, has been the means of purifyingour 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. Toevery man comes, sooner or later, the great renunciation. For theyoung, there is nothing unattainable; a good thing desired with the

whole force of a passionate will, and yet impossible, is to them notcredible. Yet, by death, by illness, by poverty, or by the voice ofduty, we must learn, each one of us, that the world was not made forus, and that, however beautiful may be the things we crave, Fate maynevertheless forbid them. It is the part of courage, when misfortunecomes, to bear without repining the ruin of our hopes, to turn awayour thoughts from vain regrets. This degree of submission to Power isnot 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 ownideals. Haunting foreshadowings of the temple appear in the realm ofimagination, in music, in architecture, in the untroubled kingdom ofreason, and in the golden sunset magic of lyrics, where beauty shinesand glows, remote from the touch of sorrow, remote from the fear ofchange, remote from the failures and disenchantments of the world offact. In the contemplation of these things the vision of heaven willshape itself in our hearts, giving at once a touchstone to judge theworld about us, and an inspiration by which to fashion to our needswhatever is not incapable of serving as a stone in the sacred temple.

Except for those rare spirits that are born without sin, there is acavern of darkness to be traversed before that temple can be entered.The gate of the cavern is despair, and its floor is paved with thegravestones of abandoned hopes. There Self must die; there theeagerness, the greed of untamed desire must be slain, for only so canthe soul be freed from the empire of Fate. But out of the cavern theGate of Renunciation leads again to the daylight of wisdom, by whoseradiance a new insight, a new joy, a new tenderness, shine forth togladden the pilgrim's heart.

When, without the bitterness of impotent rebellion, we have learntboth to resign ourselves to the outward rule of Fate and to recognisethat the non-human world is unworthy of our worship, it becomes

possible at last so to transform and refashion the unconsciousuniverse, so to transmute it in the crucible of imagination, that anew image of shining gold replaces the old idol of clay. In all themultiform facts of the world--in the visual shapes of trees andmountains and clouds, in the events of the life of man, even in thevery omnipotence of Death--the insight of creative idealism can findthe reflection of a beauty which its own thoughts first made. In thisway mind asserts its subtle mastery over the thoughtless forces ofNature. The more evil the material with which it deals, the morethwarting to untrained desire, the greater is its achievement ininducing the reluctant rock to yield up its hidden treasures, theprouder its victory in compelling the opposing forces to swell thepageant of its triumph. Of all the arts, Tragedy is the proudest, the

most triumphant; for it builds its shining citadel in the very centreof the enemy's country, on the very summit of his highest mountain;from its impregnable watchtowers, his camps and arsenals, his columnsand forts, are all revealed; within its walls the free life continues,while the legions of Death and Pain and Despair, and all the servilecaptains of tyrant Fate, afford the burghers of that dauntless citynew spectacles of beauty. Happy those sacred ramparts, thrice happythe dwellers on that all-seeing eminence. Honour to those bravewarriors who, through countless ages of warfare, have preserved for usthe priceless heritage of liberty, and have kept undefiled bysacrilegious 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, andin the irrevocableness of a vanished past, there is a sacredness, anoverpowering awe, a feeling of the vastness, the depth, theinexhaustible mystery of existence, in which, as by some strangemarriage of pain, the sufferer is bound to the world by bonds ofsorrow. In these moments of insight, we lose all eagerness oftemporary desire, all struggling and striving for petty ends, all carefor the little trivial things that, to a superficial view, make up thecommon life of day by day; we see, surrounding the narrow raft



illumined by the flickering light of human comradeship, the dark oceanon whose rolling waves we toss for a brief hour; from the great nightwithout, a chill blast breaks in upon our refuge; all the lonelinessof humanity amid hostile forces is concentrated upon the individualsoul, which must struggle alone, with what of courage it can command,against the whole weight of a universe that cares nothing for itshopes and fears. Victory, in this struggle with the powers ofdarkness, is the true baptism into the glorious company of heroes, thetrue initiation into the overmastering beauty of human existence. Fromthat 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 forceswhose puppets we seem to be--Death and change, the irrevocableness ofthe past, and the powerlessness of man before the blind hurry of theuniverse from vanity to vanity--to feel these things and know them isto conquer them.

This is the reason why the Past has such magical power. The beauty ofits motionless and silent pictures is like the enchanted purity oflate autumn, when the leaves, though one breath would make them fall,still glow against the sky in golden glory. The Past does not changeor strive; like Duncan, after life's fitful fever it sleeps well; whatwas 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 comparisonwith the forces of Nature. The slave is doomed to worship Time andFate and Death, because they are greater than anything he finds inhimself, and because all his thoughts are of things which they devour.But, great as they are, to think of them greatly, to feel theirpassionless splendour, is greater still. And such thought makes usfree men; we no longer bow before the inevitable in Orientalsubjection, but we absorb it, and make it a part of ourselves. Toabandon the struggle for private happiness, to expel all eagerness oftemporary desire, to burn with passion for eternal things--this is

emancipation, and this is the free man's worship. And this liberationis effected by a contemplation of Fate; for Fate itself is subdued bythe mind which leaves nothing to be purged by the purifying fire ofTime.

United with his fellow-men by the strongest of all ties, the tie of acommon 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 along march through the night, surrounded by invisible foes, torturedby weariness and pain, towards a goal that few can hope to reach, andwhere none may tarry long. One by one, as they march, our comradesvanish from our sight, seized by the silent orders of omnipotentDeath. 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 ontheir path, to lighten their sorrows by the balm of sympathy, to givethem the pure joy of a never-tiring affection, to strengthen failingcourage, to instil faith in hours of despair. Let us not weigh ingrudging scales their merits and demerits, but let us think only oftheir need--of the sorrows, the difficulties, perhaps the blindnesses,that make the misery of their lives; let us remember that they arefellow-sufferers in the same darkness, actors in the same tragedy withourselves. And so, when their day is over, when their good and theirevil 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 wasthe cause; but wherever a spark of the divine fire kindled in theirhearts, we were ready with encouragement, with sympathy, with bravewords 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 ofdestruction, omnipotent matter rolls on its relentless way; for Man,condemned to-day to lose his dearest, to-morrow himself to passthrough the gate of darkness, it remains only to cherish, ere yet theblow falls, the lofty thoughts that ennoble his little day; disdainingthe coward terrors of the slave of Fate, to worship at the shrine thathis own hands have built; undismayed by the empire of chance, topreserve a mind free from the wanton tyranny that rules his outwardlife; proudly defiant of the irresistible forces that tolerate, for amoment, his knowledge and his condemnation, to sustain alone, a wearybut unyielding Atlas, the world that his own ideals have fashioneddespite 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 thequestion should be asked from time to time, What is its purpose andideal? In what way does it contribute to the beauty of humanexistence? As respects those pursuits which contribute only remotely,by providing the mechanism of life, it is well to be reminded that notthe mere fact of living is to be desired, but the art of living in the

contemplation of great things. Still more in regard to thoseavocations which have no end outside themselves, which are to bejustified, if at all, as actually adding to the sum of the world'spermanent possessions, it is necessary to keep alive a knowledge oftheir aims, a clear prefiguring vision of the temple in which creativeimagination is to be embodied.

The fulfilment of this need, in what concerns the studies forming thematerial upon which custom has decided to train the youthful mind, isindeed sadly remote--so remote as to make the mere statement of such aclaim appear preposterous. Great men, fully alive to the beauty of thecontemplations to whose service their lives are devoted, desiring thatothers may share in their joys, persuade mankind to impart to the

successive generations the mechanical knowledge without which it isimpossible to cross the threshold. Dry pedants possess themselves ofthe privilege of instilling this knowledge: they forget that it is toserve but as a key to open the doors of the temple; though they spendtheir lives on the steps leading up to those sacred doors, they turntheir backs upon the temple so resolutely that its very existence isforgotten, and the eager youth, who would press forward to beinitiated to its domes and arches, is bidden to turn back and countthe steps.



Mathematics, perhaps more even than the study of Greece and Rome, hassuffered from this oblivion of its due place in civilisation. Althoughtradition has decreed that the great bulk of educated men shall knowat least the elements of the subject, the reasons for which thetradition arose are forgotten, buried beneath a great rubbish-heap ofpedantries and trivialities. To those who inquire as to the purpose ofmathematics, the usual answer will be that it facilitates the makingof machines, the travelling from place to place, and the victory overforeign nations, whether in war or commerce. If it be objected thatthese ends--all of which are of doubtful value--are not furthered bythe merely elementary study imposed upon those who do not becomeexpert mathematicians, the reply, it is true, will probably be thatmathematics trains the reasoning faculties. Yet the very men who makethis reply are, for the most part, unwilling to abandon the teachingof definite fallacies, known to be such, and instinctively rejected bythe unsophisticated mind of every intelligent learner. And thereasoning faculty itself is generally conceived, by those who urge itscultivation, as merely a means for the avoidance of pitfalls and ahelp in the discovery of rules for the guidance of practical life. Allthese are undeniably important achievements to the credit ofmathematics; yet it is none of these that entitles mathematics to aplace in every liberal education. Plato, we know, regarded thecontemplation of mathematical truths as worthy of the Deity; andPlato realised, more perhaps than any other single man, what those

elements are in human life which merit a place in heaven. There is inmathematics, he says, "something which is _necessary_ and cannot beset aside ... and, if I mistake not, of divine necessity; for as tothe 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 someuse or knowledge of which a man cannot become a God to the world, nora spirit, nor yet a hero, nor able earnestly to think and care forman" (_Laws_, p. 818).[10] Such was Plato's judgment of mathematics;but the mathematicians do not read Plato, while those who read himknow no mathematics, and regard his opinion upon this question asmerely a curious aberration.

Mathematics, rightly viewed, possesses not only truth, but supremebeauty--a beauty cold and austere, like that of sculpture, withoutappeal to any part of our weaker nature, without the gorgeoustrappings of painting or music, yet sublimely pure, and capable of astern perfection such as only the greatest art can show. The truespirit of delight, the exaltation, the sense of being more than man,which is the touchstone of the highest excellence, is to be found inmathematics as surely as in poetry. What is best in mathematicsdeserves not merely to be learnt as a task, but to be assimilated as apart of daily thought, and brought again and again before the mindwith ever-renewed encouragement. Real life is, to most men, a longsecond-best, a perpetual compromise between the ideal and the

possible; but the world of pure reason knows no compromise, nopractical limitations, no barrier to the creative activity embodyingin splendid edifices the passionate aspiration after the perfect fromwhich all great work springs. Remote from human passions, remote evenfrom the pitiful facts of nature, the generations have graduallycreated an ordered cosmos, where pure thought can dwell as in itsnatural home, and where one, at least, of our nobler impulses canescape 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 toirrepressible instincts, which were better than avowed beliefs, hasbeen moulded by an unconscious taste; but much also has been spoilt byfalse notions of what was fitting. The characteristic excellence ofmathematics is only to be found where the reasoning is rigidlylogical: the rules of logic are to mathematics what those of structureare to architecture. In the most beautiful work, a chain of argumentis presented in which every link is important on its own account, inwhich there is an air of ease and lucidity throughout, and thepremises achieve more than would have been thought possible, by meanswhich appear natural and inevitable. Literature embodies what isgeneral in particular circumstances whose universal significanceshines through their individual dress; but mathematics endeavours topresent whatever is most general in its purity, without any irrelevanttrappings.

How should the teaching of mathematics be conducted so as tocommunicate to the learner as much as possible of this high ideal?Here experience must, in a great measure, be our guide; but somemaxims may result from our consideration of the ultimate purpose to beachieved.

One of the chief ends served by mathematics, when rightly taught, isto awaken the learner's belief in reason, his confidence in the truth

of what has been demonstrated, and in the value of demonstration. Thispurpose is not served by existing instruction; but it is easy to seeways in which it might be served. At present, in what concernsarithmetic, the boy or girl is given a set of rules, which presentthemselves as neither true nor false, but as merely the will of theteacher, the way in which, for some unfathomable reason, the teacherprefers to have the game played. To some degree, in a study of suchdefinite practical utility, this is no doubt unavoidable; but as soonas possible, the reasons of rules should be set forth by whatevermeans most readily appeal to the childish mind. In geometry, insteadof the tedious apparatus of fallacious proofs for obvious truismswhich constitutes the beginning of Euclid, the learner should beallowed at first to assume the truth of everything obvious, and should

be instructed in the demonstrations of theorems which are at oncestartling and easily verifiable by actual drawing, such as those inwhich it is shown that three or more lines meet in a point. In thisway belief is generated; it is seen that reasoning may lead tostartling conclusions, which nevertheless the facts will verify; andthus the instinctive distrust of whatever is abstract or rational isgradually overcome. Where theorems are difficult, they should be firsttaught as exercises in geometrical drawing, until the figure hasbecome thoroughly familiar; it will then be an agreeable advance to betaught the logical connections of the various lines or circles thatoccur. It is desirable also that the figure illustrating a theoremshould be drawn in all possible cases and shapes, that so the abstractrelations with which geometry is concerned may of themselves emerge

as the residue of similarity amid such great apparent diversity. Inthis way the abstract demonstrations should form but a small part ofthe instruction, and should be given when, by familiarity withconcrete illustrations, they have come to be felt as the naturalembodiment of visible fact. In this early stage proofs should not begiven with pedantic fullness; definitely fallacious methods, such asthat of superposition, should be rigidly excluded from the first, butwhere, without such methods, the proof would be very difficult, theresult should be rendered acceptable by arguments and illustrationswhich are explicitly contrasted with demonstrations.



In the beginning of algebra, even the most intelligent child finds, asa rule, very great difficulty. The use of letters is a mystery, whichseems to have no purpose except mystification. It is almostimpossible, at first, not to think that every letter stands for someparticular number, if only the teacher would reveal _what_ number itstands for. The fact is, that in algebra the mind is first taught toconsider general truths, truths which are not asserted to hold only ofthis or that particular thing, but of any one of a whole group ofthings. It is in the power of understanding and discovering suchtruths that the mastery of the intellect over the whole world ofthings actual and possible resides; and ability to deal with thegeneral as such is one of the gifts that a mathematical educationshould bestow. But how little, as a rule, is the teacher of algebraable to explain the chasm which divides it from arithmetic, and howlittle is the learner assisted in his groping efforts atcomprehension! Usually the method that has been adopted in arithmeticis continued: rules are set forth, with no adequate explanation oftheir grounds; the pupil learns to use the rules blindly, andpresently, when he is able to obtain the answer that the teacherdesires, he feels that he has mastered the difficulties of thesubject. But of inner comprehension of the processes employed he hasprobably acquired almost nothing.

When algebra has been learnt, all goes smoothly until we reach thosestudies in which the notion of infinity is employed--the infinitesimalcalculus and the whole of higher mathematics. The solution of thedifficulties which formerly surrounded the mathematical infinite isprobably the greatest achievement of which our own age has to boast.Since the beginnings of Greek thought these difficulties have beenknown; in every age the finest intellects have vainly endeavoured toanswer the apparently unanswerable questions that had been asked byZeno the Eleatic. At last Georg Cantor has found the answer, and hasconquered for the intellect a new and vast province which had beengiven over to Chaos and old Night. It was assumed as self-evident,until Cantor and Dedekind established the opposite, that if, from anycollection of things, some were taken away, the number of things left

must always be less than the original number of things. Thisassumption, as a matter of fact, holds only of finite collections; andthe rejection of it, where the infinite is concerned, has been shownto remove all the difficulties that had hitherto baffled human reasonin this matter, and to render possible the creation of an exactscience of the infinite. This stupendous fact ought to produce arevolution in the higher teaching of mathematics; it has itself addedimmeasurably to the educational value of the subject, and it has atlast given the means of treating with logical precision many studieswhich, until lately, were wrapped in fallacy and obscurity. By thosewho were educated on the old lines, the new work is considered to beappallingly difficult, abstruse, and obscure; and it must be confessedthat 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 andinquiring minds, has facilitated the mastery of higher mathematics;for hitherto, it has been necessary to learn, by a long process ofsophistication, to give assent to arguments which, on firstacquaintance, were rightly judged to be confused and erroneous. So farfrom producing a fearless belief in reason, a bold rejection ofwhatever failed to fulfil the strictest requirements of logic, amathematical training, during the past two centuries, encouraged thebelief that many things, which a rigid inquiry would reject as



fallacious, must yet be accepted because they work in what themathematician calls "practice." By this means, a timid, compromisingspirit, or else a sacerdotal belief in mysteries not intelligible tothe profane, has been bred where reason alone should have ruled. Allthis it is now time to sweep away; let those who wish to penetrateinto the arcana of mathematics be taught at once the true theory inall its logical purity, and in the concatenation established by thevery essence of the entities concerned.

If we are considering mathematics as an end in itself, and not as atechnical training for engineers, it is very desirable to preserve thepurity and strictness of its reasoning. Accordingly those who haveattained a sufficient familiarity with its easier portions should beled backward from propositions to which they have assented asself-evident to more and more fundamental principles from which whathad previously appeared as premises can be deduced. They should betaught--what the theory of infinity very aptly illustrates--that manypropositions seem self-evident to the untrained mind which,nevertheless, a nearer scrutiny shows to be false. By this means theywill be led to a sceptical inquiry into first principles, anexamination of the foundations upon which the whole edifice ofreasoning is built, or, to take perhaps a more fitting metaphor, thegreat 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 alsohow it grows out of the central principles of logic. Questions of thisnature can now be answered with a precision and certainty which wereformerly quite impossible; and in the chains of reasoning that theanswer requires the unity of all mathematical studies at last unfoldsitself.

In the great majority of mathematical text-books there is a total lackof unity in method and of systematic development of a central theme.Propositions of very diverse kinds are proved by whatever means arethought most easily intelligible, and much space is devoted to merecuriosities which in no way contribute to the main argument. But inthe greatest works, unity and inevitability are felt as in the

unfolding of a drama; in the premisses a subject is proposed forconsideration, and in every subsequent step some definite advance ismade towards mastery of its nature. The love of system, ofinterconnection, which is perhaps the inmost essence of theintellectual impulse, can find free play in mathematics as nowhereelse. The learner who feels this impulse must not be repelled by anarray of meaningless examples or distracted by amusing oddities, butmust be encouraged to dwell upon central principles, to becomefamiliar with the structure of the various subjects which are putbefore him, to travel easily over the steps of the more importantdeductions. In this way a good tone of mind is cultivated, andselective attention is taught to dwell by preference upon what isweighty and essential.

When the separate studies into which mathematics is divided have eachbeen viewed as a logical whole, as a natural growth from thepropositions which constitute their principles, the learner will beable to understand the fundamental science which unifies andsystematises the whole of deductive reasoning. This is symboliclogic--a study which, though it owes its inception to Aristotle, isyet, in its wider developments, a product, almost wholly, of thenineteenth century, and is indeed, in the present day, still growingwith great rapidity. The true method of discovery in symbolic logic,



another sense we and the whole universe of existing things areindependent of mathematics. The apprehension of this purely idealcharacter is indispensable, if we are to understand rightly the placeof mathematics as one among the arts. It was formerly supposed thatpure reason could decide, in some respects, as to the nature of theactual world: geometry, at least, was thought to deal with the spacein which we live. But we now know that pure mathematics can neverpronounce upon questions of actual existence: the world of reason, ina sense, controls the world of fact, but it is not at any pointcreative of fact, and in the application of its results to the worldin time and space, its certainty and precision are lost amongapproximations and working hypotheses. The objects considered bymathematicians have, in the past, been mainly of a kind suggested byphenomena; but from such restrictions the abstract imagination shouldbe wholly free. A reciprocal liberty must thus be accorded: reasoncannot dictate to the world of facts, but the facts cannot restrictreason's privilege of dealing with whatever objects its love of beautymay cause to seem worthy of consideration. Here, as elsewhere, webuild 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 adiscovery.

It is very desirable, in instruction, not merely to persuade thestudent of the accuracy of important theorems, but to persuade him in

the way which itself has, of all possible ways, the most beauty. Thetrue interest of a demonstration is not, as traditional modes ofexposition suggest, concentrated wholly in the result; where this doesoccur, it must be viewed as a defect, to be remedied, if possible, byso generalising the steps of the proof that each becomes important inand for itself. An argument which serves only to prove a conclusion islike 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 conquestof new realms, is responsible for the undue emphasis upon resultswhich prevails in mathematical instruction. The better way is topropose some theme for consideration--in geometry, a figure havingimportant properties; in analysis, a function of which the study is

illuminating, and so on. Whenever proofs depend upon some only of themarks by which we define the object to be studied, these marks shouldbe 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 theessential principles in virtue of which the thesis is true. It is amerit in Euclid that he advances as far as he is able to go withoutemploying the axiom of parallels--not, as is often said, because thisaxiom is inherently objectionable, but because, in mathematics, everynew axiom diminishes the generality of the resulting theorems, and thegreatest 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 uponphilosophy has, in the past, been most notable, but most varied; inthe seventeenth century, idealism and rationalism, in the eighteenth,materialism and sensationalism, seemed equally its offspring. Of theeffect which it is likely to have in the future it would be very rashto say much; but in one respect a good result appears probable.Against that kind of scepticism which abandons the pursuit of idealsbecause the road is arduous and the goal not certainly attainable,mathematics, within its own sphere, is a complete answer. Too often itis 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, byhis own peculiarities, his own taste and bias; that there is noexternal kingdom of truth to which, by patience and discipline, we mayat last obtain admittance, but only truth for me, for you, for everyseparate person. By this habit of mind one of the chief ends of humaneffort is denied, and the supreme virtue of candour, of fearlessacknowledgment of what is, disappears from our moral vision. Of suchscepticism mathematics is a perpetual reproof; for its edifice oftruths stands unshakable and inexpungable to all the weapons ofdoubting cynicism.

The effects of mathematics upon practical life, though they should notbe regarded as the motive of our studies, may be used to answer adoubt to which the solitary student must always be liable. In a worldso full of evil and suffering, retirement into the cloister ofcontemplation, to the enjoyment of delights which, however noble, mustalways be for the few only, cannot but appear as a somewhat selfishrefusal to share the burden imposed upon others by accidents in whichjustice plays no part. Have any of us the right, we ask, to withdrawfrom present evils, to leave our fellow-men unaided, while we live alife which, though arduous and austere, is yet plainly good in its ownnature? When these questions arise, the true answer is, no doubt, thatsome must keep alive the sacred fire, some must preserve, in everygeneration, the haunting vision which shadows forth the goal of so

much striving. But when, as must sometimes occur, this answer seemstoo cold, when we are almost maddened by the spectacle of sorrows towhich we bring no help, then we may reflect that indirectly themathematician often does more for human happiness than any of his morepractically active contemporaries. The history of science abundantlyproves that a body of abstract propositions--even if, as in the caseof conic sections, it remains two thousand years without effect upondaily life--may yet, at any moment, be used to cause a revolution inthe habitual thoughts and occupations of every citizen. The use ofsteam and electricity--to take striking instances--is renderedpossible only by mathematics. In the results of abstract thought theworld possesses a capital of which the employment in enriching thecommon round has no hitherto discoverable limits. Nor does experience

give any means of deciding what parts of mathematics will be founduseful. Utility, therefore, can be only a consolation in moments ofdiscouragement, not a guide in directing our studies.

For the health of the moral life, for ennobling the tone of an age ora nation, the austerer virtues have a strange power, exceeding thepower of those not informed and purified by thought. Of these austerervirtues the love of truth is the chief, and in mathematics, more thanelsewhere, the love of truth may find encouragement for waning faith.Every great study is not only an end in itself, but also a means ofcreating and sustaining a lofty habit of mind; and this purpose shouldbe kept always in view throughout the teaching and learning ofmathematics.

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 ofsteam and evolution, might have derived a more legitimate title tofame from the discovery of pure mathematics. This science, like mostothers, was baptised long before it was born; and thus we find writersbefore the nineteenth century alluding to what they called puremathematics. But if they had been asked what this subject was, theywould 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, ourancestors were completely in the dark.

Pure mathematics was discovered by Boole, in a work which he calledthe _Laws of Thought_ (1854). This work abounds in asseverations thatit is not mathematical, the fact being that Boole was too modest tosuppose his book the first ever written on mathematics. He was alsomistaken 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 wascurious that no one should ever have thought in such a way before. Hisbook was in fact concerned with formal logic, and this is the samething as mathematics.

Pure mathematics consists entirely of assertions to the effect that,if such and such a proposition is true of _anything_, then such andsuch another proposition is true of that thing. It is essential not todiscuss whether the first proposition is really true, and not tomention what the anything is, of which it is supposed to be true. Boththese points would belong to applied mathematics. We start, in puremathematics, from certain rules of inference, by which we can inferthat _if_ one proposition is true, then so is some other proposition.These rules of inference constitute the major part of the principlesof formal logic. We then take any hypothesis that seems amusing, anddeduce its consequences. _If_ our hypothesis is about _anything_, andnot about some one or more particular things, then our deductions

constitute mathematics. Thus mathematics may be defined as the subjectin which we never know what we are talking about, nor whether what weare saying is true. People who have been puzzled by the beginnings ofmathematics will, I hope, find comfort in this definition, and willprobably agree that it is accurate.

As one of the chief triumphs of modern mathematics consists in havingdiscovered what mathematics really is, a few more words on thissubject may not be amiss. It is common to start any branch ofmathematics--for instance, Geometry--with a certain number ofprimitive ideas, supposed incapable of definition, and a certainnumber of primitive propositions or axioms, supposed incapable ofproof. Now the fact is that, though there are indefinables and

indemonstrables in every branch of applied mathematics, there are nonein pure mathematics except such as belong to general logic. Logic,broadly speaking, is distinguished by the fact that its propositionscan be put into a form in which they apply to anything whatever. Allpure mathematics--Arithmetic, Analysis, and Geometry--is built up bycombinations of the primitive ideas of logic, and its propositions arededuced from the general axioms of logic, such as the syllogism andthe other rules of inference. And this is no longer a dream or anaspiration. On the contrary, over the greater and more difficult partof the domain of mathematics, it has been already accomplished; in the



few remaining cases, there is no special difficulty, and it is nowbeing rapidly achieved. Philosophers have disputed for ages whethersuch deduction was possible; mathematicians have sat down and made thededuction. For the philosophers there is now nothing left but gracefulacknowledgments.

The subject of formal logic, which has thus at last shown itself to beidentical with mathematics, was, as every one knows, invented byAristotle, and formed the chief study (other than theology) of theMiddle Ages. But Aristotle never got beyond the syllogism, which is avery small part of the subject, and the schoolmen never got beyondAristotle. If any proof were required of our superiority to themediæval doctors, it might be found in this. Throughout the MiddleAges, almost all the best intellects devoted themselves to formallogic, whereas in the nineteenth century only an infinitesimalproportion of the world's thought went into this subject.Nevertheless, in each decade since 1850 more has been done to advancethe subject than in the whole period from Aristotle to Leibniz. Peoplehave discovered how to make reasoning symbolic, as it is in Algebra,so that deductions are effected by mathematical rules. They havediscovered many rules besides the syllogism, and a new branch oflogic, called the Logic of Relatives,[11] has been invented to dealwith topics that wholly surpassed the powers of the old logic, thoughthey form the chief contents of mathematics.

It is not easy for the lay mind to realise the importance of symbolismin discussing the foundations of mathematics, and the explanation mayperhaps seem strangely paradoxical. The fact is that symbolism isuseful because it makes things difficult. (This is not true of theadvanced parts of mathematics, but only of the beginnings.) What wewish to know is, what can be deduced from what. Now, in thebeginnings, everything is self-evident; and it is very hard to seewhether one self-evident proposition follows from another or not.Obviousness is always the enemy to correctness. Hence we invent somenew and difficult symbolism, in which nothing seems obvious. Then weset up certain rules for operating on the symbols, and the whole thingbecomes mechanical. In this way we find out what must be taken as

premiss and what can be demonstrated or defined. For instance, thewhole of Arithmetic and Algebra has been shown to require threeindefinable notions and five indemonstrable propositions. But withouta symbolism it would have been very hard to find this out. It is soobvious that two and two are four, that we can hardly make ourselvessufficiently sceptical to doubt whether it can be proved. And the sameholds in other cases where self-evident things are to be proved.

But the proof of self-evident propositions may seem, to theuninitiated, a somewhat frivolous occupation. To this we might replythat it is often by no means self-evident that one obvious propositionfollows from another obvious proposition; so that we are reallydiscovering new truths when we prove what is evident by a method which

is not evident. But a more interesting retort is, that since peoplehave tried to prove obvious propositions, they have found that many ofthem are false. Self-evidence is often a mere will-o'-the-wisp, whichis 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 apart, or that a number is increased by adding one to it. But thesepropositions are now known to be usually false. Most numbers areinfinite, and if a number is infinite you may add ones to it as longas you like without disturbing it in the least. One of the merits of aproof 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 ourown day, is an Italian, Professor Peano, of the University ofTurin.[12] He has reduced the greater part of mathematics (and he orhis followers will, in time, have reduced the whole) to strictsymbolic form, in which there are no words at all. In the ordinarymathematical books, there are no doubt fewer words than most readerswould wish. Still, little phrases occur, such as _therefore, let usassume, consider_, or _hence it follows_. All these, however, are aconcession, and are swept away by Professor Peano. For instance, if wewish to learn the whole of Arithmetic, Algebra, the Calculus, andindeed 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 theseideas mean, it is necessary to know if you wish to become anarithmetician. But after symbols have been invented for these threeideas, not another word is required in the whole development. Allfuture 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 thatwhat 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 puremathematics (including Geometry) are compounded. Professor Peano, whois assisted by a very able school of young Italian disciples, hasshown how this may be done; and although the method which he hasinvented is capable of being carried a good deal further than he hascarried it, the honour of the pioneer must belong to him.

Two hundred years ago, Leibniz foresaw the science which Peano hasperfected, and endeavoured to create it. He was prevented fromsucceeding by respect for the authority of Aristotle, whom he couldnot believe guilty of definite, formal fallacies; but the subjectwhich he desired to create now exists, in spite of the patronisingcontempt with which his schemes have been treated by all superior

persons. From this "Universal Characteristic," as he called it, hehoped for a solution of all problems, and an end to all disputes. "Ifcontroversies were to arise," he says, "there would be no more need ofdisputation between two philosophers than between two accountants. Forit would suffice to take their pens in their hands, to sit down totheir desks, and to say to each other (with a friend as witness, ifthey liked), 'Let us calculate.'" This optimism has now appeared to besomewhat excessive; there still are problems whose solution isdoubtful, and disputes which calculation cannot decide. But over anenormous field of what was formerly controversial, Leibniz's dream hasbecome sober fact. In the whole philosophy of mathematics, which usedto be at least as full of doubt as any other part of philosophy, orderand certainty have replaced the confusion and hesitation which

formerly reigned. Philosophers, of course, have not yet discoveredthis fact, and continue to write on such subjects in the old way. Butmathematicians, at least in Italy, have now the power of treating theprinciples of mathematics in an exact and masterly manner, by means ofwhich the certainty of mathematics extends also to mathematicalphilosophy. Hence many of the topics which used to be placed among thegreat mysteries--for example, the natures of infinity, of continuity,of space, time and motion--are now no longer in any degree open todoubt or discussion. Those who wish to know the nature of these thingsneed only read the works of such men as Peano or Georg Cantor; they



will there find exact and indubitable expositions of all these quondammysteries.

In this capricious world, nothing is more capricious than posthumousfame. One of the most notable examples of posterity's lack of judgmentis the Eleatic Zeno. This man, who may be regarded as the founder ofthe philosophy of infinity, appears in Plato's Parmenides in theprivileged position of instructor to Socrates. He invented fourarguments, all immeasurably subtle and profound, to prove that motionis impossible, that Achilles can never overtake the tortoise, and thatan arrow in flight is really at rest. After being refuted byAristotle, and by every subsequent philosopher from that day to ourown, these arguments were reinstated, and made the basis of amathematical renaissance, by a German professor, who probably neverdreamed of any connection between himself and Zeno. Weierstrass,[13]by strictly banishing from mathematics the use of infinitesimals, hasat last shown that we live in an unchanging world, and that the arrowin its flight is truly at rest. Zeno's only error lay in inferring (ifhe did infer) that, because there is no such thing as a state ofchange, therefore the world is in the same state at any one time as atany other. This is a consequence which by no means follows; and inthis respect, the German mathematician is more constructive than theingenious Greek. Weierstrass has been able, by embodying his views inmathematics, where familiarity with truth eliminates the vulgar

prejudices of common sense, to invest Zeno's paradoxes with therespectable air of platitudes; and if the result is less delightful tothe lover of reason than Zeno's bold defiance, it is at any rate morecalculated to appease the mass of academic mankind.

Zeno was concerned, as a matter of fact, with three problems, eachpresented by motion, but each more abstract than motion, and capableof a purely arithmetical treatment. These are the problems of theinfinitesimal, the infinite, and continuity. To state clearly thedifficulties involved, was to accomplish perhaps the hardest part ofthe philosopher's task. This was done by Zeno. From him to our ownday, the finest intellects of each generation in turn attacked theproblems, but achieved, broadly speaking, nothing. In our own time,

however, three men--Weierstrass, Dedekind, and Cantor--have not merelyadvanced the three problems, but have completely solved them. Thesolutions, for those acquainted with mathematics, are so clear as toleave no longer the slightest doubt or difficulty. This achievement isprobably the greatest of which our age has to boast; and I know of noage (except perhaps the golden age of Greece) which has a moreconvincing proof to offer of the transcendent genius of its great men.Of the three problems, that of the infinitesimal was solved byWeierstrass; the solution of the other two was begun by Dedekind, anddefinitively accomplished by Cantor.

The infinitesimal played formerly a great part in mathematics. It wasintroduced by the Greeks, who regarded a circle as differing

infinitesimally from a polygon with a very large number of very smallequal sides. It gradually grew in importance, until, when Leibnizinvented the Infinitesimal Calculus, it seemed to become thefundamental notion of all higher mathematics. Carlyle tells, in his

_Frederick the Great_, how Leibniz used to discourse to Queen SophiaCharlotte of Prussia concerning the infinitely little, and how shewould reply that on that subject she needed no instruction--thebehaviour of courtiers had made her thoroughly familiar with it. Butphilosophers and mathematicians--who for the most part had lessacquaintance with courts--continued to discuss this topic, though



without making any advance. The Calculus required continuity, andcontinuity was supposed to require the infinitely little; but nobodycould discover what the infinitely little might be. It was plainly notquite zero, because a sufficiently large number of infinitesimals,added together, were seen to make up a finite whole. But nobody couldpoint out any fraction which was not zero, and yet not finite. Thusthere was a deadlock. But at last Weierstrass discovered that theinfinitesimal was not needed at all, and that everything could beaccomplished without it. Thus there was no longer any need to supposethat there was such a thing. Nowadays, therefore, mathematicians aremore dignified than Leibniz: instead of talking about the infinitelysmall, they talk about the infinitely great--a subject which, howeverappropriate to monarchs, seems, unfortunately, to interest them evenless than the infinitely little interested the monarchs to whomLeibniz discoursed.

The banishment of the infinitesimal has all sorts of odd consequences,to which one has to become gradually accustomed. For example, there isno such thing as the next moment. The interval between one moment andthe next would have to be infinitesimal, since, if we take two momentswith a finite interval between them, there are always other moments inthe interval. Thus if there are to be no infinitesimals, no twomoments are quite consecutive, but there are always other momentsbetween any two. Hence there must be an infinite number of moments

between any two; because if there were a finite number one would benearest the first of the two moments, and therefore next to it. Thismight be thought to be a difficulty; but, as a matter of fact, it ishere that the philosophy of the infinite comes in, and makes allstraight.

The same sort of thing happens in space. If any piece of matter be cutin two, and then each part be halved, and so on, the bits will becomesmaller and smaller, and can theoretically be made as small as weplease. However small they may be, they can still be cut up and madesmaller still. But they will always have _some_ finite size, howeversmall they may be. We never reach the infinitesimal in this way, andno finite number of divisions will bring us to points. Nevertheless

there _are_ points, only these are not to be reached by successivedivisions. Here again, the philosophy of the infinite shows us howthis is possible, and why points are not infinitesimal lengths.

As regards motion and change, we get similarly curious results. Peopleused to think that when a thing changes, it must be in a state ofchange, and that when a thing moves, it is in a state of motion. Thisis now known to be a mistake. When a body moves, all that can be saidis that it is in one place at one time and in another at another. Wemust not say that it will be in a neighbouring place at the nextinstant, since there is no next instant. Philosophers often tell usthat when a body is in motion, it changes its position within theinstant. 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 notof the kind to which philosophers are accustomed to give weight, andthey have continued down to our own day to repeat the same phraseswhich roused the Eleatic's destructive ardour. It was only recentlythat it became possible to explain motion in detail in accordance withZeno's platitude, and in opposition to the philosopher's paradox. Wemay now at last indulge the comfortable belief that a body in motionis just as truly where it is as a body at rest. Motion consists merelyin the fact that bodies are sometimes in one place and sometimes inanother, and that they are at intermediate places at intermediate



times. Only those who have waded through the quagmire of philosophicspeculation on this subject can realise what a liberation from antiqueprejudices is involved in this simple and straightforward commonplace.

The philosophy of the infinitesimal, as we have just seen, is mainlynegative. People used to believe in it, and now they have found outtheir mistake. The philosophy of the infinite, on the other hand, iswholly positive. It was formerly supposed that infinite numbers, andthe mathematical infinite generally, were self-contradictory. But asit was obvious that there were infinities--for example, the number ofnumbers--the contradictions of infinity seemed unavoidable, andphilosophy seemed to have wandered into a "cul-de-sac." Thisdifficulty led to Kant's antinomies, and hence, more or lessindirectly, to much of Hegel's dialectic method. Almost all currentphilosophy is upset by the fact (of which very few philosophers are asyet aware) that all the ancient and respectable contradictions in thenotion of the infinite have been once for all disposed of. The methodby which this has been done is most interesting and instructive. Inthe first place, though people had talked glibly about infinity eversince the beginnings of Greek thought, nobody had ever thought ofasking, What is infinity? If any philosopher had been asked for adefinition of infinity, he might have produced some unintelligiblerigmarole, but he would certainly not have been able to give adefinition that had any meaning at all. Twenty years ago, roughly

speaking, Dedekind and Cantor asked this question, and, what is moreremarkable, they answered it. They found, that is to say, a perfectlyprecise definition of an infinite number or an infinite collection ofthings. This was the first and perhaps the greatest step. It thenremained to examine the supposed contradictions in this notion. HereCantor proceeded in the only proper way. He took pairs ofcontradictory propositions, in which both sides of the contradictionwould be usually regarded as demonstrable, and he strictly examinedthe supposed proofs. He found that all proofs adverse to infinityinvolved a certain principle, at first sight obviously true, butdestructive, in its consequences, of almost all mathematics. Theproofs favourable to infinity, on the other hand, involved noprinciple 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 apart. This maxim is true of finite numbers. For example, Englishmenare only some among Europeans, and there are fewer Englishmen thanEuropeans. But when we come to infinite numbers, this is no longertrue. This breakdown of the maxim gives us the precise definition ofinfinity. A collection of terms is infinite when it contains as partsother collections which have just as many terms as it has. If you cantake away some of the terms of a collection, without diminishing thenumber of terms, then there are an infinite number of terms in the

collection. For example, there are just as many even numbers as thereare numbers altogether, since every number can be doubled. This may beseen by putting odd and even numbers together in one row, and evennumbers 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 therow above, because there is one below for each one above. This



property, which was formerly thought to be a contradiction, is nowtransformed into a harmless definition of infinity, and shows, in theabove case, that the number of finite numbers is infinite.

But the uninitiated may wonder how it is possible to deal with anumber which cannot be counted. It is impossible to count up _all_ thenumbers, one by one, because, however many we may count, there arealways more to follow. The fact is that counting is a very vulgar andelementary way of finding out how many terms there are in acollection. And in any case, counting gives us what mathematicianscall the _ordinal_ number of our terms; that is to say, it arrangesour terms in an order or series, and its result tells us what type ofseries results from this arrangement. In other words, it is impossibleto count things without counting some first and others afterwards, sothat counting always has to do with order. Now when there are only afinite number of terms, we can count them in any order we like; butwhen there are an infinite number, what corresponds to counting willgive us quite different results according to the way in which we carryout the operation. Thus the ordinal number, which results from what,in a general sense may be called counting, depends not only upon howmany 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 orderand counting them, but by a different method, which tells us, to beginwith, whether two collections have the same number of terms, or, ifnot, which is the greater.[14] It does not tell us, in the way inwhich counting does, _what_ number of terms a collection has; but ifwe define a number as the number of terms in such and such acollection, then this method enables us to discover whether some othercollection that may be mentioned has more or fewer terms. Anillustration will show how this is done. If there existed some countryin which, for one reason or another, it was impossible to take acensus, but in which it was known that every man had a wife and everywoman a husband, then (provided polygamy was not a nationalinstitution) we should know, without counting, that there were exactly

as many men as there were women in that country, neither more norless. This method can be applied generally. If there is some relationwhich, like marriage, connects the things in one collection each withone of the things in another collection, and vice versa, then the twocollections have the same number of terms. This was the way in whichwe found that there are as many even numbers as there are numbers.Every number can be doubled, and every even number can be halved, andeach process gives just one number corresponding to the one that isdoubled or halved. And in this way we can find any number ofcollections each of which has just as many terms as there are finitenumbers. 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 theprocess, then our collection must have just as many terms as there are

finite numbers. This is the general method by which the numbers ofinfinite collections are defined.

But it must not be supposed that all infinite numbers are equal. Onthe contrary, there are infinitely more infinite numbers than finiteones. There are more ways of arranging the finite numbers in differenttypes of series than there are finite numbers. There are probably morepoints in space and more moments in time than there are finitenumbers. 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 arewhole numbers or fractions. There are probably exactly as many pointsin space as there are irrational numbers, and exactly as many pointson a line a millionth of an inch long as in the whole of infinitespace. There is a greatest of all infinite numbers, which is thenumber of things altogether, of every sort and kind. It is obviousthat there cannot be a greater number than this, because, ifeverything has been taken, there is nothing left to add. Cantor has aproof 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 subtlefallacy, which I hope to explain in some future work.[15]

We can now understand why Zeno believed that Achilles cannot overtakethe tortoise and why as a matter of fact he can overtake it. We shallsee 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 roadat the same time, the tortoise (as is only fair) being allowed ahandicap. Let Achilles go twice as fast as the tortoise, or ten timesor a hundred times as fast. Then he will never reach the tortoise. Forat every moment the tortoise is somewhere and Achilles is somewhere;and neither is ever twice in the same place while the race is goingon. 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 anyother moment. But if Achilles were to catch up with the tortoise, theplaces where the tortoise would have been would be only part of theplaces where Achilles would have been. Here, we must suppose, Zenoappealed 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 inmore 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 hecan never catch the tortoise. This argument is strictly correct, if weallow the axiom that the whole has more terms than the part. As theconclusion is absurd, the axiom must be rejected, and then all goeswell. But there is no good word to be said for the philosophers of thepast two thousand years and more, who have all allowed the axiom and

denied the conclusion.

The retention of this axiom leads to absolute contradictions, whileits rejection leads only to oddities. Some of these oddities, it mustbe confessed, are very odd. One of them, which I call the paradox ofTristram Shandy, is the converse of the Achilles, and shows that thetortoise, if you give him time, will go just as far as Achilles.Tristram Shandy, as we know, employed two years in chronicling thefirst two days of his life, and lamented that, at this rate, materialwould accumulate faster than he could deal with it, so that, as yearswent 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 ofhis task, then, even if his life had continued as event fully as it

began, no part of his biography would have remained unwritten. Forconsider: the hundredth day will be described in the hundredth year,the thousandth in the thousandth year, and so on. Whatever day we maychoose as so far on that he cannot hope to reach it, that day will bedescribed in the corresponding year. Thus any day that may bementioned will be written up sooner or later, and therefore no part ofthe biography will remain permanently unwritten. This paradoxical butperfectly true proposition depends upon the fact that the number ofdays in all time is no greater than the number of years.



Thus on the subject of infinity it is impossible to avoid conclusionswhich at first sight appear paradoxical, and this is the reason why somany philosophers have supposed that there were inherentcontradictions in the infinite. But a little practice enables one tograsp the true principles of Cantor's doctrine, and to acquire new andbetter instincts as to the true and the false. The oddities thenbecome no odder than the people at the antipodes, who used to bethought impossible because they would find it so inconvenient to standon their heads.

The solution of the problems concerning infinity has enabled Cantor tosolve also the problems of continuity. Of this, as of infinity, he hasgiven a perfectly precise definition, and has shown that there are nocontradictions in the notion so defined. But this subject is sotechnical that it is impossible to give any account of it here.

The notion of continuity depends upon that of _order_, sincecontinuity is merely a particular type of order. Mathematics has, inmodern times, brought order into greater and greater prominence. Informer days, it was supposed (and philosophers are still apt tosuppose) that quantity was the fundamental notion of mathematics. Butnowadays, quantity is banished altogether, except from one littlecorner of Geometry, while order more and more reigns supreme. Theinvestigation of different kinds of series and their relations is now

a very large part of mathematics, and it has been found that thisinvestigation can be conducted without any reference to quantity, and,for the most part, without any reference to number. All types ofseries are capable of formal definition, and their properties can bededuced from the principles of symbolic logic by means of the Algebraof Relatives. The notion of a limit, which is fundamental in thegreater part of higher mathematics, used to be defined by means ofquantity, as a term to which the terms of some series approximate asnearly as we please. But nowadays the limit is defined quitedifferently, and the series which it limits may not approximate to itat all. This improvement also is due to Cantor, and it is one whichhas 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 infinitedistance from it. The study of different types of series is a generalsubject of which the study of ordinal numbers (mentioned above) is aspecial and very interesting branch. But the unavoidabletechnicalities of this subject render it impossible to explain to anybut professed mathematicians.

Geometry, like Arithmetic, has been subsumed, in recent times, underthe general study of order. It was formerly supposed that Geometry wasthe study of the nature of the space in which we live, and accordinglyit was urged, by those who held that what exists can only be knownempirically, that Geometry should really be regarded as belonging toapplied mathematics. But it has gradually appeared, by the increase of

non-Euclidean systems, that Geometry throws no more light upon thenature of space than Arithmetic throws upon the population of theUnited States. Geometry is a whole collection of deductive sciencesbased on a corresponding collection of sets of axioms. One set ofaxioms is Euclid's; other equally good sets of axioms lead to otherresults. Whether Euclid's axioms are true, is a question as to whichthe pure mathematician is indifferent; and, what is more, it is aquestion which it is theoretically impossible to answer with certaintyin the affirmative. It might possibly be shown, by very carefulmeasurements, that Euclid's axioms are false; but no measurements



could ever assure us (owing to the errors of observation) that theyare exactly true. Thus the geometer leaves to the man of science todecide, as best he may, what axioms are most nearly true in the actualworld. The geometer takes any set of axioms that seem interesting, anddeduces their consequences. What defines Geometry, in this sense, isthat 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 oforder.

In Geometry, as in other parts of mathematics, Peano and his discipleshave done work of the very greatest merit as regards principles.Formerly, it was held by philosophers and mathematicians alike thatthe proofs in Geometry depended on the figure; nowadays, this is knownto be false. In the best books there are no figures at all. Thereasoning proceeds by the strict rules of formal logic from a set ofaxioms laid down to begin with. If a figure is used, all sorts ofthings seem obviously to follow, which no formal reasoning can provefrom the explicit axioms, and which, as a matter of fact, are onlyaccepted because they are obvious. By banishing the figure, it becomespossible to discover _all_ the axioms that are needed; and in this wayall sorts of possibilities, which would have otherwise remainedundetected, 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, aswas formerly done, by assuming the whole of space. This method is duepartly to Peano, partly to another Italian named Fano. To thoseunaccustomed to it, it has an air of somewhat wilful pedantry. In thisway, we begin with the following axioms: (1) There is a class ofentities 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 webring in the straight line joining two points, and begin again with(4), namely, on the straight line joining _a_ and _b_, there is atleast one other point besides _a_ and _b_. (5) There is at least onepoint not on the line _ab_. And so we go on, till we have the means ofobtaining as many points as we require. But the word _space_, as Peanohumorously remarks, is one for which Geometry has no use at all.

The rigid methods employed by modern geometers have deposed Euclidfrom his pinnacle of correctness. It was thought, until recent times,that, as Sir Henry Savile remarked in 1621, there were only twoblemishes in Euclid, the theory of parallels and the theory ofproportion. It is now known that these are almost the only points inwhich Euclid is free from blemish. Countless errors are involved inhis first eight propositions. That is to say, not only is it doubtfulwhether his axioms are true, which is a comparatively trivial matter,but it is certain that his propositions do not follow from the axiomswhich he enunciates. A vastly greater number of axioms, which Euclidunconsciously 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 areassumed to intersect. But no explicit axiom assures us that they doso, and in some kinds of spaces they do not always intersect. It isquite doubtful whether our space belongs to one of these kinds or not.Thus Euclid fails entirely to prove his point in the very firstproposition. As he is certainly not an easy author, and is terriblylong-winded, he has no longer any but an historical interest. Underthese circumstances, it is nothing less than a scandal that he shouldstill be taught to boys in England.[17] A book should have eitherintelligibility or correctness; to combine the two is impossible, but



to lack both is to be unworthy of such a place as Euclid has occupiedin education.

The most remarkable result of modern methods in mathematics is theimportance of symbolic logic and of rigid formalism. Mathematicians,under the influence of Weierstrass, have shown in modern times a carefor accuracy, and an aversion to slipshod reasoning, such as had notbeen known among them previously since the time of the Greeks. Thegreat inventions of the seventeenth century--Analytical Geometry andthe Infinitesimal Calculus--were so fruitful in new results thatmathematicians had neither time nor inclination to examine theirfoundations. Philosophers, who should have taken up the task, had toolittle mathematical ability to invent the new branches of mathematicswhich have now been found necessary for any adequate discussion. Thusmathematicians were only awakened from their "dogmatic slumbers" whenWeierstrass and his followers showed that many of their most cherishedpropositions are in general false. Macaulay, contrasting the certaintyof mathematics with the uncertainty of philosophy, asks who ever heardof a reaction against Taylor's theorem? If he had lived now, hehimself might have heard of such a reaction, for this is precisely oneof the theorems which modern investigations have overthrown. Such rudeshocks to mathematical faith have produced that love of formalismwhich appears, to those who are ignorant of its motive, to be mereoutrageous pedantry.

The proof that all pure mathematics, including Geometry, is nothingbut formal logic, is a fatal blow to the Kantian philosophy. Kant,rightly perceiving that Euclid's propositions could not be deducedfrom Euclid's axioms without the help of the figures, invented atheory of knowledge to account for this fact; and it accounted sosuccessfully that, when the fact is shown to be a mere defect inEuclid, and not a result of the nature of geometrical reasoning,Kant's theory also has to be abandoned. The whole doctrine of _apriori_ intuitions, by which Kant explained the possibility of puremathematics, is wholly inapplicable to mathematics in its presentform. The Aristotelian doctrines of the schoolmen come nearer inspirit to the doctrines which modern mathematics inspire; but the

schoolmen were hampered by the fact that their formal logic was verydefective, and that the philosophical logic based upon the syllogismshowed a corresponding narrowness. What is now required is to give thegreatest possible development to mathematical logic, to allow to thefull the importance of relations, and then to found upon this securebasis a new philosophical logic, which may hope to borrow some of theexactitude and certainty of its mathematical foundation. If this canbe successfully accomplished, there is every reason to hope that thenear future will be as great an epoch in pure philosophy as theimmediate past has been in the principles of mathematics. Greattriumphs inspire great hopes; and pure thought may achieve, within ourgeneration, such results as will place our time, in this respect, on alevel 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 mewhen this article was written. [Note added in 1917.]

[13] Professor of Mathematics in the University of Berlin. He died in1897.



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

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

[16] This must not be regarded as a historically correct account ofwhat Zeno actually had in mind. It is a new argument for hisconclusion, not the argument which influenced him. On this point, seee.g. C.D. Broad, "Note on Achilles and the Tortoise," _Mind_, N.S.,Vol. XXII, pp. 318-19. Much valuable work on the interpretation ofZeno has been done since this article was written. [Note added in1917.]

[17] Since the above was written, he has ceased to be used as atextbook. But I fear many of the books now used are so bad that thechange 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 theinvestigation of philosophical questions, we find that, broadlyspeaking, they can be divided into two groups, often antagonistic, and

leading to very divergent systems. These two groups of motives are, onthe one hand, those derived from religion and ethics, and, on theother hand, those derived from science. Plato, Spinoza, and Hegel maybe taken as typical of the philosophers whose interests are mainlyreligious and ethical, while Leibniz, Locke, and Hume may be taken asrepresentatives 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, wouldnaturally be classed among scientific philosophers: it was mainly fromscience that he drew his data, his formulation of problems, and hisconception of method. But his strong religious sense is obvious inmuch of his writing, and his ethical pre-occupations are what make him

value the conception of evolution--that conception in which, as awhole generation has believed, science and morals are to be united infruitful and indissoluble marriage.

It is my belief that the ethical and religious motives in spite ofthe splendidly imaginative systems to which they have given rise, havebeen on the whole a hindrance to the progress of philosophy, and oughtnow to be consciously thrust aside by those who wish to discoverphilosophical truth. Science, originally, was entangled in similarmotives, and was thereby hindered in its advances. It is, I maintain,



from science, rather than from ethics and religion, that philosophyshould draw its inspiration.

But there are two different ways in which a philosophy may seek tobase itself upon science. It may emphasise the most general _results_ of science, and seek to give even greater generality and unity tothese results. Or it may study the _methods_ of science, and seek toapply these methods, with the necessary adaptations, to its ownpeculiar province. Much philosophy inspired by science has gone astraythrough preoccupation with the _results_ momentarily supposed to havebeen achieved. It is not results, but _methods_ that can betransferred with profit from the sphere of the special sciences to thesphere of philosophy. What I wish to bring to your notice is thepossibility and importance of applying to philosophical problemscertain broad principles of method which have been found successful inthe study of scientific questions.

The opposition between a philosophy guided by scientific method and aphilosophy dominated by religious and ethical ideas may be illustratedby two notions which are very prevalent in the works of philosophers,namely the notion of _the universe_, and the notion of _good andevil_. A philosopher is expected to tell us something about the natureof the universe as a whole, and to give grounds for either optimism orpessimism. Both these expectations seem to me mistaken. I believe the

conception of "the universe" to be, as its etymology indicates, amere relic of pre-Copernican astronomy: and I believe the question ofoptimism and pessimism to be one which the philosopher will regard asoutside his scope, except, possibly, to the extent of maintaining thatit is insoluble.

In the days before Copernicus, the conception of the "universe" wasdefensible on scientific grounds: the diurnal revolution of theheavenly bodies bound them together as all parts of one system, ofwhich the earth was the centre. Round this apparent scientific fact,many human desires rallied: the wish to believe Man important in thescheme of things, the theoretical desire for a comprehensiveunderstanding of the Whole, the hope that the course of nature might

be guided by some sympathy with our wishes. In this way, an ethicallyinspired system of metaphysics grew up, whose anthropocentrism wasapparently warranted by the geocentrism of astronomy. When Copernicusswept away the astronomical basis of this system of thought, it hadgrown so familiar, and had associated itself so intimately with men'saspirations, that it survived with scarcely diminished force--survivedeven Kant's "Copernican revolution," and is still now the unconsciouspremiss of most metaphysical systems.

The oneness of the world is an almost undiscussed postulate of mostmetaphysics. "Reality is not merely one and self-consistent, but is asystem of reciprocally determinate parts"[19]--such a statement wouldpass almost unnoticed as a mere truism. Yet I believe that it embodies

a failure to effect thoroughly the "Copernican revolution," and thatthe apparent oneness of the world is merely the oneness of what isseen by a single spectator or apprehended by a single mind. TheCritical Philosophy, although it intended to emphasise the subjectiveelement in many apparent characteristics of the world, yet, byregarding the world in itself as unknowable, so concentrated attentionupon the subjective representation that its subjectivity was soonforgotten. Having recognised the categories as the work of the mind,it was paralysed by its own recognition, and abandoned in despair theattempt to undo the work of subjective falsification. In part, no



doubt, its despair was well founded, but not, I think, in any absoluteor ultimate sense. Still less was it a ground for rejoicing, or forsupposing that the nescience to which it ought to have given risecould be legitimately exchanged for a metaphysical dogmatism.

I

As regards our present question, namely, the question of the unity ofthe world, the right method, as I think, has been indicated by WilliamJames.[20] "Let us now turn our backs upon ineffable or unintelligibleways of accounting for the world's oneness, and inquire whether,instead of being a principle, the 'oneness' affirmed may not merely bea name like 'substance' descriptive of the fact that certain _specificand verifiable connections_ are found among the parts of theexperiential flux.... We can easily conceive of things that shall haveno connection whatever with each other. We may assume them to inhabitdifferent times and spaces, as the dreams of different persons do evennow. They may be so unlike and incommensurable, and so inert towardsone another, as never to jostle or interfere. Even now there mayactually be whole universes so disparate from ours that we who knowours have no means of perceiving that they exist. We conceive theirdiversity, however; and by that fact the whole lot of them form whatis 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 thatany chaos may become a universe by merely being named, is to meincomprehensible." We are thus left with two kinds of unity in theexperienced 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 thattentative and partial unity exhibited in the prevalence of scientificlaws in those portions of the world which science has hithertomastered. Now a generalisation based upon either of these kinds ofunity would be fallacious. That the things which we experience havethe common property of being experienced by us is a truism from whichobviously nothing of importance can be deducible: it is clearly

fallacious to draw from the fact that whatever we experience isexperienced the conclusion that therefore everything must beexperienced. The generalisation of the second kind of unity, namely,that derived from scientific laws, would be equally fallacious, thoughthe fallacy is a trifle less elementary. In order to explain it let usconsider for a moment what is called the reign of law. People oftenspeak as though it were a remarkable fact that the physical world issubject to invariable laws. In fact, however, it is not easy to seehow such a world could fail to obey general laws. Taking any arbitraryset of points in space, there is a function of the time correspondingto these points, i.e. expressing the motion of a particle whichtraverses these points: this function may be regarded as a general lawto which the behaviour of such a particle is subject. Taking all such

functions for all the particles in the universe, there will betheoretically some one formula embracing them all, and this formulamay be regarded as the single and supreme law of the spatio-temporalworld. Thus what is surprising in physics is not the existence ofgeneral laws, but their extreme simplicity. It is not the uniformityof nature that should surprise us, for, by sufficient analyticingenuity, any conceivable course of nature might be shown to exhibituniformity. What should surprise us is the fact that the uniformity issimple enough for us to be able to discover it. But it is just thischaracteristic of simplicity in the laws of nature hitherto discovered



which it would be fallacious to generalise, for it is obvious thatsimplicity has been a part cause of their discovery, and can,therefore, give no ground for the supposition that other undiscoveredlaws are equally simple.

The fallacies to which these two kinds of unity have given risesuggest a caution as regards all use in philosophy of general

_results_ that science is supposed to have achieved. In the firstplace, in generalising these results beyond past experience, it isnecessary to examine very carefully whether there is not some reasonmaking it more probable that these results should hold of all that hasbeen experienced than that they should hold of things universally. Thesum total of what is experienced by mankind is a selection from thesum total of what exists, and any general character exhibited by thisselection may be due to the manner of selecting rather than to thegeneral character of that from which experience selects. In the secondplace, the most general results of science are the least certain andthe most liable to be upset by subsequent research. In utilizing theseresults as the basis of a philosophy, we sacrifice the most valuableand remarkable characteristic of scientific method, namely, that,although almost everything in science is found sooner or later torequire some correction, yet this correction is almost always such asto leave untouched, or only slightly modified, the greater part of theresults which have been deduced from the premiss subsequently

discovered to be faulty. The prudent man of science acquires a certaininstinct as to the kind of uses which may be made of presentscientific beliefs without incurring the danger of complete and utterrefutation from the modifications likely to be introduced bysubsequent discoveries. Unfortunately the use of scientificgeneralisations of a sweeping kind as the basis of philosophy is justthat kind of use which an instinct of scientific caution would avoid,since, as a rule, it would only lead to true results if thegeneralisation upon which it is based stood in _no_ need ofcorrection.

We may illustrate these general considerations by means of twoexamples, namely, the conservation of energy and the principle of

evolution.

(1) Let us begin with the conservation of energy, or, as HerbertSpencer 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 istempted to give an air of absoluteness and necessity to empiricalgeneralisations, of which only the approximate truth in the regionshitherto investigated can be guaranteed by the unaided methods ofscience. It is very often said that the persistence of something orother is a necessary presupposition of all scientific investigation,and this presupposition is then thought to be exemplified in somequantity which physics declares to be constant. There are here, as it



seems to me, three distinct errors. First, the detailed scientificinvestigation of nature does not _presuppose_ any such general laws asits results are found to verify. Apart from particular observations,science need presuppose nothing except the general principles oflogic, and these principles are not laws of nature, for they aremerely hypothetical, and apply not only to the actual world but towhatever is _possible_. The second error consists in theidentification of a constant quantity with a persistent entity. Energyis a certain function of a physical system, but is not a thing orsubstance persisting throughout the changes of the system. The same istrue of mass, in spite of the fact that mass has often been defined as

_quantity of matter_. The whole conception of quantity, involving, asit does, numerical measurement based largely upon conventions, is farmore artificial, far more an embodiment of mathematical convenience,than is commonly believed by those who philosophise on physics. Thuseven if (which I cannot for a moment admit) the persistence of someentity were among the necessary postulates of science, it would be asheer error to infer from this the constancy of any physical quantity,or the _a priori_ necessity of any such constancy which may beempirically discovered. In the third place, it has become more andmore evident with the progress of physics that large generalisations,such as the conservation of energy or mass, are far from certain andare very likely only approximate. Mass, which used to be regarded asthe most indubitable of physical quantities, is now generally believed

to vary according to velocity, and to be, in fact, a vector quantitywhich at a given moment is different in different directions. Thedetailed conclusions deduced from the supposed constancy of mass forsuch motions as used to be studied in physics will remain very nearlyexact, and therefore over the field of the older investigations verylittle modification of the older results is required. But as soon assuch a principle as the conservation of mass or of energy is erectedinto a universal _a priori_ law, the slightest failure in absoluteexactness is fatal, and the whole philosophic structure raised uponthis 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 momentto 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 alsoanother sort, namely, the undue preoccupation with ethical notions.There are two kinds of evolutionist philosophy, of which both Hegeland Spencer represent the older and less radical kind, whilePragmatism and Bergson represent the more modern and revolutionaryvariety. But both these sorts of evolutionism have in common theemphasis on _progress_, that is, upon a continual change from theworse to the better, or from the simpler to the more complex. Itwould be unfair to attribute to Hegel any scientific motive orfoundation, but all the other evolutionists, including Hegel's moderndisciples, have derived their impetus very largely from the history of

biological development. To a philosophy which derives a law ofuniversal progress from this history there are two objections. First,that this history itself is concerned with a very small selection offacts confined to an infinitesimal fragment of space and time, andeven on scientific grounds probably not an average sample of events inthe world at large. For we know that decay as well as growth is anormal occurrence in the world. An extra-terrestrial philosopher, whohad watched a single youth up to the age of twenty-one and had nevercome across any other human being, might conclude that it is thenature of human beings to grow continually taller and wiser in an



indefinite progress towards perfection; and this generalisation wouldbe just as well founded as the generalisation which evolutionists baseupon the previous history of this planet. Apart, however, from thisscientific objection to evolutionism, there is another, derived fromthe undue admixture of ethical notions in the very idea of progressfrom which evolutionism derives its charm. Organic life, we are told,has developed gradually from the protozoon to the philosopher, andthis development, we are assured, is indubitably an advance.Unfortunately it is the philosopher, not the protozoon, who gives usthis assurance, and we can have no security that the impartialoutsider would agree with the philosopher's self-complacentassumption. This point has been illustrated by the philosopher ChuangTzŭ 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 Augurand the pigs.

The ethical element which has been prominent in many of the mostfamous systems of philosophy is, in my opinion, one of the mostserious obstacles to the victory of scientific method in the

investigation of philosophical questions. Human ethical notions, asChuang Tzŭ perceived, are essentially anthropocentric, and involve,when used in metaphysics, an attempt, however veiled, to legislate forthe universe on the basis of the present desires of men. In this waythey interfere with that receptivity to fact which is the essence ofthe scientific attitude towards the world. To regard ethical notionsas a key to the understanding of the world is essentiallypre-Copernican. It is to make man, with the hopes and ideals which hehappens to have at the present moment, the centre of the universe andthe interpreter of its supposed aims and purposes. Ethical metaphysicsis fundamentally an attempt, however disguised, to give legislativeforce to our own wishes. This may, of course, be questioned, but Ithink that it is confirmed by a consideration of the way in which

ethical notions arise. Ethics is essentially a product of thegregarious instinct, that is to say, of the instinct to co-operatewith those who are to form our own group against those who belong toother groups. Those who belong to our own group are good; those whobelong to hostile groups are wicked. The ends which are pursued by ourown group are desirable ends, the ends pursued by hostile groups arenefarious. The subjectivity of this situation is not apparent to thegregarious animal, which feels that the general principles of justiceare on the side of its own herd. When the animal has arrived at thedignity of the metaphysician, it invents ethics as the embodiment of



its belief in the justice of its own herd. So the Grand Augur invokesethics as the justification of Augurs in their conflicts with pigs.But, it may be said, this view of ethics takes no account of suchtruly ethical notions as that of self-sacrifice. This, however, wouldbe a mistake. The success of gregarious animals in the struggle forexistence depends upon co-operation within the herd, and co-operationrequires sacrifice, to some extent, of what would otherwise be theinterest of the individual. Hence arises a conflict of desires andinstincts, since both self-preservation and the preservation of theherd are biological ends to the individual. Ethics is in origin theart of recommending to others the sacrifices required for co-operationwith oneself. Hence, by reflexion, it comes, through the operation ofsocial justice, to recommend sacrifices by oneself, but all ethics,however refined, remains more or less subjective. Even vegetarians donot hesitate, for example, to save the life of a man in a fever,although in doing so they destroy the lives of many millions ofmicrobes. The view of the world taken by the philosophy derived fromethical notions is thus never impartial and therefore never fullyscientific. As compared with science, it fails to achieve theimaginative liberation from self which is necessary to suchunderstanding of the world as man can hope to achieve, and thephilosophy which it inspires is always more or less parochial, more orless infected with the prejudices of a time and a place.

I do not deny the importance or value, within its own sphere, of thekind of philosophy which is inspired by ethical notions. The ethicalwork of Spinoza, for example, appears to me of the very highestsignificance, but what is valuable in such work is not anymetaphysical theory as to the nature of the world to which it may giverise, nor indeed anything which can be proved or disproved byargument. What is valuable is the indication of some new way offeeling towards life and the world, some way of feeling by which ourown existence can acquire more of the characteristics which we mustdeeply desire. The value of such work, however immeasurable it is,belongs with practice and not with theory. Such theoretic importanceas it may possess is only in relation to human nature, not in relationto the world at large. The scientific philosophy, therefore, which

aims only at understanding the world and not directly at any otherimprovement of human life, cannot take account of ethical notionswithout being turned aside from that submission to fact which is theessence of the scientific temper.

II

If the notion of the universe and the notion of good and evil areextruded from scientific philosophy, it may be asked what specificproblems remain for the philosopher as opposed to the man of science?It would be difficult to give a precise answer to this question, butcertain 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. Itmust not deal specially with things on the surface of the earth, orwith the solar system, or with any other portion of space and time. Itis this need of generality which has led to the belief that philosophydeals with the universe as a whole. I do not believe that this beliefis justified, but I do believe that a philosophical proposition mustbe applicable to everything that exists or may exist. It might besupposed that this admission would be scarcely distinguishable from



the view which I wish to reject. This, however, would be an error, andan important one. The traditional view would make the universe itselfthe subject of various predicates which could not be applied to anyparticular thing in the universe, and the ascription of such peculiarpredicates to the universe would be the special business ofphilosophy. I maintain, on the contrary, that there are nopropositions of which the "universe" is the subject; in other words,that there is no such thing as the "universe." What I do maintain isthat there are general propositions which may be asserted of eachindividual thing, such as the propositions of logic. This does notinvolve that all the things there are form a whole which could beregarded as another thing and be made the subject of predicates. Itinvolves only the assertion that there are properties which belong toeach separate thing, not that there are properties belonging to thewhole of things collectively. The philosophy which I wish to advocatemay be called logical atomism or absolute pluralism, because, whilemaintaining that there are many things, it denies that there is awhole composed of those things. We shall see, therefore, thatphilosophical propositions, instead of being concerned with the wholeof things collectively, are concerned with all things distributively;and not only must they be concerned with all things, but they must beconcerned with such properties of all things as do not depend upon theaccidental nature of the things that there happen to be, but are trueof any possible world, independently of such facts as can only be

discovered by our senses.

This brings us to a second characteristic of philosophicalpropositions, namely, that they must be _a priori_. A philosophicalproposition must be such as can be neither proved nor disproved byempirical evidence. Too often we find in philosophical books argumentsbased upon the course of history, or the convolutions of the brain, orthe eyes of shell-fish. Special and accidental facts of this kind areirrelevant to philosophy, which must make only such assertions aswould be equally true however the actual world were constituted.

We may sum up these two characteristics of philosophical propositionsby saying that _philosophy is the science of the possible_. But this

statement unexplained is liable to be misleading, since it may bethought that the possible is something other than the general, whereasin fact the two are indistinguishable.

Philosophy, if what has been said is correct, becomes indistinguishablefrom logic as that word has now come to be used. The study of logicconsists, broadly speaking, of two not very sharply distinguishedportions. On the one hand it is concerned with those general statementswhich can be made concerning everything without mentioning any onething or predicate or relation, such for example as "if _x_ is a memberof the class a

nd every member of a is

member of b, then _x_ is amem er of the class b, whatever _x_, a,

nd b may e." On the otherhand, it is concerned with the analysis and enumeration of logical

_forms_, i.e. with the kinds of propositions that may occur, with thevarious types of facts, and with the classification of the constituentsof facts. In this way logic provides an inventory of possi ilities, arepertory of a stractly tena le hypotheses.

It might e thought that such a study would e too vague and toogeneral to e of any very great importance, and that, if its pro lems ecame at any point sufficiently definite, they would e merged in thepro lems of some special science. It appears, however, that this isnot the case. In some pro lems, for example, the analysis of space and



time, the nature of perception, or the theory of judgment, thediscovery of the logical form of the facts involved is the hardestpart of the work and the part whose performance has een most lackinghitherto. It is chiefly for want of the right logical hypothesis thatsuch pro lems have hitherto een treated in such an unsatisfactorymanner, and have given rise to those contradictions or antinomies inwhich the enemies of reason among philosophers have at all timesdelighted.

By concentrating attention upon the investigation of logical forms, it ecomes possi le at last for philosophy to deal with its pro lemspiecemeal, and to o tain, as the sciences do, such partial andpro a ly not wholly correct results as su sequent investigation canutilise even while it supplements and improves them. Mostphilosophies hitherto have een constructed all in one lock, in sucha way that, if they were not wholly correct, they were whollyincorrect, and could not e used as a asis for furtherinvestigations. It is chiefly owing to this fact that philosophy,unlike science, has hitherto een unprogressive, ecause each originalphilosopher has had to egin the work again from the eginning,without eing a le to accept anything definite from the work of hispredecessors. A scientific philosophy such as I wish to recommend will e piecemeal and tentative like other sciences; a ove all, it will ea le to invent hypotheses which, even if they are not wholly true,

will yet remain fruitful after the necessary corrections have

eenmade. This possi ility of successive approximations to the truth is,more than anything else, the source of the triumphs of science, and totransfer this possi ility to philosophy is to ensure a progress inmethod whose importance it would e almost impossi le to exaggerate.

The essence of philosophy as thus conceived is analysis, notsynthesis. To uild up systems of the world, like Heine's Germanprofessor who knit together fragments of life and made an intelligi lesystem out of them, is not, I elieve, any more feasi le than thediscovery of the philosopher's stone. What is feasi le is theunderstanding of general forms, and the division of traditionalpro lems into a num er of separate and less affling questions.

"Divide and conquer" is the maxim of success here as elsewhere.

Let us illustrate these somewhat general maxims y examining theirapplication to the philosophy of space, for it is only in applicationthat the meaning or importance of a method can e understood. Supposewe are confronted with the pro lem of space as presented in Kant'sTranscendental Æsthetic, and suppose we wish to discover what are theelements of the pro lem and what hope there is of o taining a solutionof them. It will soon appear that three entirely distinct pro lems, elonging to different studies, and requiring different methods fortheir solution, have een confusedly com ined in the supposed singlepro lem with which Kant is concerned. There is a pro lem of logic, apro lem of physics, and a pro lem of theory of knowledge. Of these

three, the pro

lem of logic can

e solved exactly and perfectly; thepro lem of physics can pro a ly e solved with as great a degree ofcertainty and as great an approach to exactness as can e hoped in anempirical region; the pro lem of theory of knowledge, however, remainsvery o scure and very difficult to deal with. Let us see how thesethree pro lems arise.

(1) The logical pro lem has arisen through the suggestions ofnon-Euclidean geometry. Given a ody of geometrical propositions, itis not difficult to find a minimum statement of the axioms from which



this ody of propositions can e deduced. It is also not difficult, ydropping or altering some of these axioms, to o tain a more general ora different geometry, having, from the point of view of puremathematics, the same logical coherence and the same title to respectas the more familiar Euclidean geometry. The Euclidean geometry itselfis true perhaps of actual space (though this is dou tful), utcertainly of an infinite num er of purely arithmetical systems, eachof which, from the point of view of a stract logic, has an equal andindefeasi le right to e called a Euclidean space. Thus space as ano ject of logical or mathematical study loses its uniqueness; not onlyare there many kinds of spaces, ut there are an infinity of examplesof each kind, though it is difficult to find any kind of which thespace of physics may e an example, and it is impossi le to find anykind of which the space of physics is certainly an example. As anillustration of one possi le logical system of geometry we mayconsider all relations of three terms which are analogous in certainformal respects to the relation " etween" as it appears to e inactual space. A space is then defined y means of one such three-termrelation. The points of the space are all the terms which have thisrelation to something or other, and their order in the space inquestion is determined y this relation. The points of one space arenecessarily also points of other spaces, since there are necessarilyother three-term relations having those same points for their field.The space in fact is not determined y the class of its points, ut y

the ordering three-term relation. When enough a

stract logicalproperties of such relations have een enumerated to determine theresulting kind of geometry, say, for example, Euclidean geometry, it ecomes unnecessary for the pure geometer in his a stract capacity todistinguish etween the various relations which have all theseproperties. He considers the whole class of such relations, not anysingle one among them. Thus in studying a given kind of geometry thepure mathematician is studying a certain class of relations defined ymeans of certain a stract logical properties which take the place ofwhat used to e called axioms. The nature of geometrical _reasoning_ therefore is purely deductive and purely logical; if any specialepistemological peculiarities are to e found in geometry, it must not e in the reasoning, ut in our knowledge concerning the axioms in

some given space.

(2) The physical pro lem of space is oth more interesting and moredifficult than the logical pro lem. The physical pro lem may estated as follows: to find in the physical world, or to construct fromphysical materials, a space of one of the kinds enumerated y thelogical treatment of geometry. This pro lem derives its difficultyfrom the attempt to accommodate to the roughness and vagueness of thereal world some system possessing the logical clearness and exactitudeof pure mathematics. That this can e done with a certain degree ofapproximation is fairly evident If I see three people _A_, _B_, and _C_ sitting in a row, I ecome aware of the fact which may eexpressed y saying that _B_ is etween _A_ and _C_ rather than that

_A_ is

etween _B_ and _C_, or _C_ is

etween _A_ and _B_. Thisrelation of " etween" which is thus perceived to hold has some of thea stract logical properties of those three-term relations which, wesaw, give rise to a geometry, ut its properties fail to e exact, andare not, as empirically given, amena le to the kind of treatment atwhich geometry aims. In a stract geometry we deal with points,straight lines, and planes; ut the three people _A_, _B_, and _C_ whom I see sitting in a row are not exactly points, nor is the rowexactly a straight line. Nevertheless physics, which formally assumesa space containing points, straight lines, and planes, is found



empirically to give results applica le to the sensi le world. It musttherefore e possi le to find an interpretation of the points,straight lines, and planes of physics in terms of physical data, or atany rate in terms of data together with such hypothetical additions asseem least open to question. Since all data suffer from a lack ofmathematical precision through eing of a certain size and somewhatvague in outline, it is plain that if such a notion as that of a pointis to find any application to empirical material, the point must eneither a datum nor a hypothetical addition to data, ut a _construction_ y means of data with their hypothetical additions. Itis o vious that any hypothetical filling out of data is less du iousand unsatisfactory when the additions are closely analogous to datathan when they are of a radically different sort. To assume, forexample, that o jects which we see continue, after we have turned awayour eyes, to e more or less analogous to what they were while we werelooking, is a less violent assumption than to assume that such o jectsare composed of an infinite num er of mathematical points. Hence inthe physical study of the geometry of physical space, points must not e assumed _a initio_ as they are in the logical treatment ofgeometry, ut must e constructed as systems composed of data andhypothetical analogues of data. We are thus led naturally to define aphysical point as a certain class of those o jects which are theultimate constituents of the physical world. It will e the class ofall those o jects which, as one would naturally say, _contain_ the

point. To secure a definition giving this result, without previouslyassuming that physical o jects are composed of points, is an agreea lepro lem in mathematical logic. The solution of this pro lem and theperception of its importance are due to my friend Dr. Whitehead. Theoddity of regarding a point as a class of physical entities wears offwith familiarity, and ought in any case not to e felt y those whomaintain, as practically every one does, that points are mathematicalfictions. The word "fiction" is used gli ly in such connexions y manymen who seem not to feel the necessity of explaining how it can comea out that a fiction can e so useful in the study of the actual worldas the points of mathematical physics have een found to e. By ourdefinition, which regards a point as a class of physical o jects, itis explained oth how the use of points can lead to important

physical results, and how we can nevertheless avoid the assumptionthat points are themselves entities in the physical world.

Many of the mathematically convenient properties of a stract logicalspaces cannot e either known to elong or known not to elong to thespace of physics. Such are all the properties connected with continuity.For to know that actual space has these properties would require aninfinite exactness of sense-perception. If actual space is continuous,there are nevertheless many possi le non-continuous spaces which will eempirically indistinguisha le from it; and, conversely, actual space may e non-continuous and yet empirically indistinguisha le from a possi lecontinuous space. Continuity, therefore, though o taina le in the _apriori_ region of arithmetic, is not with certainty o taina le in the

space or time of the physical world: whether these are continuous or notwould seem to e a question not only unanswered ut for everunanswera le. From the point of view of philosophy, however, thediscovery that a question is unanswera le is as complete an answer asany that could possi ly e o tained. And from the point of view ofphysics, where no empirical means of distinction can e found, there can e no empirical o jection to the mathematically simplest assumption,which is that of continuity.

The su ject of the physical theory of space is a very large one,



hitherto little explored. It is associated with a similar theory oftime, and oth have een forced upon the attention of philosophicallyminded physicists y the discussions which have raged concerning thetheory of relativity.

(3) The pro lem with which Kant is concerned in the TranscendentalÆsthetic is primarily the epistemological pro lem: "How do we come tohave knowledge of geometry _a priori_?" By the distinction etween thelogical and physical pro lems of geometry, the earing and scope ofthis question are greatly altered. Our knowledge of pure geometry is _a priori_ ut is wholly logical. Our knowledge of physical geometryis synthetic, ut is not _a priori_. Our knowledge of pure geometry ishypothetical, and does not ena le us to assert, for example, that theaxiom of parallels is true in the physical world. Our knowledge ofphysical geometry, while it does ena le us to assert that this axiomis approximately verified, does not, owing to the inevita leinexactitude of o servation, ena le us to assert that it is verified _exactly_. Thus, with the separation which we have made etween puregeometry and the geometry of physics, the Kantian pro lem collapses.To the question, "How is synthetic _a priori_ knowledge possi le?" wecan now reply, at any rate so far as geometry is concerned, "It is notpossi le," if "synthetic" means "not deduci le from logic alone." Ourknowledge of geometry, like the rest of our knowledge, is derivedpartly from logic, partly from sense, and the peculiar position which

in Kant's day geometry appeared to occupy is seen now to

e adelusion. There are still some philosophers, it is true, who maintainthat our knowledge that the axiom of parallels, for example, is trueof actual space, is not to e accounted for empirically, ut is asKant maintained derived from an _a priori_ intuition. This position isnot logically refuta le, ut I think it loses all plausi ility as soonas we realise how complicated and derivative is the notion of physicalspace. As we have seen, the application of geometry to the physicalworld in no way demands that there should really e points andstraight lines among physical entities. The principle of economy,therefore, demands that we should a stain from assuming the existenceof points and straight lines. As soon, however, as we accept the viewthat points and straight lines are complicated constructions y means

of classes of physical entities, the hypothesis that we have an _apriori_ intuition ena ling us to know what happens to straight lineswhen they are produced indefinitely ecomes extremely strained andharsh; nor do I think that such an hypothesis would ever have arisenin the mind of a philosopher who had grasped the nature of physicalspace. Kant, under the influence of Newton, adopted, though with somevacillation, the hypothesis of a solute space, and this hypothesis,though logically uno jectiona le, is removed y Occam's razor, sincea solute space is an unnecessary entity in the explanation of thephysical world. Although, therefore, we cannot refute the Kantiantheory of an _a priori_ intuition, we can remove its grounds one yone through an analysis of the pro lem. Thus, here as in many otherphilosophical questions, the analytic method, while not capa le of

arriving at a demonstrative result, is nevertheless capa

le of showingthat all the positive grounds in favour of a certain theory arefallacious and that a less unnatural theory is capa le of accountingfor the facts.

Another question y which the capacity of the analytic method can eshown is the question of realism. Both those who advocate and thosewho com at realism seem to me to e far from clear as to the nature ofthe pro lem which they are discussing. If we ask: "Are our o jects ofperception _real_ and are they _independent_ of the percipient?" it



must e supposed that we attach some meaning to the words "real" and"independent," and yet, if either side in the controversy of realismis asked to define these two words, their answer is pretty sure toem ody confusions such as logical analysis will reveal.

Let us egin with the word "real." There certainly are o jects ofperception, and therefore, if the question whether these o jects arereal is to e a su stantial question, there must e in the world twosorts of o jects, namely, the real and the unreal, and yet the unrealis supposed to e essentially what there is not. The question whatproperties must elong to an o ject in order to make it real is one towhich an adequate answer is seldom if ever forthcoming. There is ofcourse the Hegelian answer, that the real is the self-consistent andthat nothing is self-consistent except the Whole; ut this answer,true or false, is not relevant in our present discussion, which moveson a lower plane and is concerned with the status of o jects ofperception among other o jects of equal fragmentariness. O jects ofperception are contrasted, in the discussions concerning realism,rather with psychical states on the one hand and matter on the otherhand than with the all-inclusive whole of things. The question we havetherefore to consider is the question as to what can e meant yassigning "reality" to some ut not all of the entities that make upthe world. Two elements, I think, make up what is felt rather thanthought 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 isreal when it is correlated with other things in a way which experiencehas led us to expect. It will e seen that reality in either of thesesenses is y no means necessary to a thing, and that in fact theremight e a whole world in which nothing was real in either of thesesenses. It might turn out that the o jects of perception failed ofreality in one or oth of these respects, without its eing in any waydeduci le that they are not parts of the external world with whichphysics deals. Similar remarks will apply to the word "independent."Most of the associations of this word are ound up with ideas as tocausation which it is not now possi le to maintain. _A_ is independentof _B_ when _B_ is not an indispensa le 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 wellas of succession, it ecomes evident that there is no uniqueness in aseries of casual antecedents of a given event, ut that, at any pointwhere there is a correlation of simultaneity, we can pass from oneline of antecedents to another in order to o tain a new series ofcausal antecedents. It will e necessary to specify the causal lawaccording to which the antecedents are to e considered. I received aletter the other day from a correspondent who had een puzzled yvarious philosophical questions. After enumerating them he says:"These questions led me from Bonn to Strass urg, where I foundProfessor Simmel." Now, it would e a surd to deny that thesequestions caused his ody to move from Bonn to Strass urg, and yet itmust e supposed that a set of purely mechanical antecedents could

also

e found which would account for this transfer of matter from oneplace to another. Owing to this plurality of causal series antecedentto a given event, the notion of _the_ cause ecomes indefinite, andthe question of independence ecomes correspondingly am iguous. Thus,instead of asking simply whether _A_ is independent of _B_, we oughtto ask whether there is a series determined y such and such causallaws leading from _B_ to _A_. This point is important in connexionwith the particular question of o jects of perception. It may e thatno o jects quite like those which we perceive ever exist unperceived;in this case there will e a causal law according to which o jects of



perception are not independent of eing perceived. But even if this ethe case, it may nevertheless also happen that there are purelyphysical causal laws determining the occurrence of o jects which areperceived y means of other o jects which perhaps are not perceived.In that case, in regard to such causal laws o jects of perception will e independent of eing perceived. Thus the question whether o jectsof perception are independent of eing perceived is, as it stands,indeterminate, and the answer will e yes or no according to themethod adopted of making it determinate. I elieve that this confusionhas orne a very large part in prolonging the controversies on thissu ject, which might well have seemed capa le of remaining for everundecided. The view which I should wish to advocate is that o jects ofperception do not persist unchanged at times when they are notperceived, although pro a ly o jects more or less resem ling them doexist at such times; that o jects of perception are part, and the onlyempirically knowa le part, of the actual su ject-matter of physics,and are themselves properly to e called physical; that purelyphysical laws exist determining the character and duration of o jectsof perception without any reference to the fact that they areperceived; and that in the esta lishment of such laws the propositionsof physics do not presuppose any propositions of psychology or eventhe existence of mind. I do not know whether realists would recognisesuch a view as realism. All that I should claim for it is, that itavoids difficulties which seem to me to eset oth realism and

idealism as hitherto advocated, and that it avoids the appeal whichthey have made to ideas which logical analysis shows to e am iguous.A further defence and ela oration of the positions which I advocate, ut for which time is lacking now, will e found indicated in my ookon _Our Knowledge of the External World_.[22]

The adoption of scientific method in philosophy, if I am not mistaken,compels us to a andon the hope of solving many of the more am itiousand humanly interesting pro lems of traditional philosophy. Some ofthese it relegates, though with little expectation of a successfulsolution, to special sciences, others it shows to e such as ourcapacities are essentially incapa le of solving. But there remain alarge num er of the recognised pro lems of philosophy in regard to

which the method advocated gives all those advantages of division intodistinct 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 hithertohas een due in the main to haste and am ition: patience and modesty,here as in other sciences, will open the road to solid and dura leprogress.

FOOTNOTES:

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

[20] _Some Pro lems of Philosophy_, p 124.

[21] _First Principles_ (1862), Part II, eginning 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 ancientmetaphysical query, "What is matter?" The question, "What is matter?"in so far as it concerns philosophy, is, I think, already capa le ofan answer which in principle will e as complete as an answer can hopeto e; that is to say, we can separate the pro lem into an essentiallysolu le and an essentially insolu le portion, and we can now see howto solve the essentially solu le portion, at least as regards its mainoutlines. It is these outlines which I wish to suggest in the presentarticle. My main position, which is realistic, is, I hope and elieve,not remote from that of Professor Alexander, y whose writings on thissu ject I have profited greatly.[24] It is also in close accord withthat of Dr. Nunn.[25]

Common sense is accustomed to the division of the world into mind andmatter. It is supposed y all who have never studied philosophy thatthe distinction etween mind and matter is perfectly clear and easy,that the two do not at any point overlap, and that only a fool or aphilosopher could e in dou t as to whether any given entity is mentalor material. This simple faith survives in Descartes and in asomewhat modified form in Spinoza, ut with Lei niz it egins todisappear, and from his day to our own almost every philosopher of

note has criticised and rejected the dualism of common sense. It is myintention in this article to defend this dualism; ut efore defendingit we must spend a few moments on the reasons which have prompted itsrejection.

Our knowledge of the material world is o tained y means of thesenses, of sight and touch and so on. At first it is supposed thatthings are just as they seem, ut two opposite sophistications soondestroy this naïve elief. On the one hand the physicists cut upmatter into molecules, atoms, corpuscles, and as many more suchsu divisions as their future needs may make them postulate, and theunits at which they arrive are uncommonly different from the visi le,tangi le o jects of daily life. A unit of matter tends more and more

to

e something like an electromagnetic field filling all space,though having its greatest intensity in a small region. Matterconsisting of such elements is as remote from daily life as anymetaphysical theory. It differs from the theories of metaphysiciansonly in the fact that its practical efficacy proves that it containssome measure of truth and induces usiness men to invest money on thestrength of it; ut, 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 sensehas een su jected is derived from the psychologists andphysiologists. The physiologists point out that what we see dependsupon the eye, that what we hear depends upon the ear, and that all our

senses are lia

le to

e affected

y anything which affects the

rain,like alcohol or hasheesh. Psychologists point out how much of what wethink we see is supplied y association or unconscious inference, howmuch is mental interpretation, and how dou tful is the residuum whichcan e regarded as crude datum. From these facts it is argued y thepsychologists that the notion of a datum passively received y themind is a delusion, and it is argued y the physiologists that even ifa pure datum of sense could e o tained y the analysis of experience,still this datum could not elong, as common sense supposes, to theouter world, since its whole nature is conditioned y our nerves and



sense organs, changing as they change in ways which it is thoughtimpossi le to connect with any change in the matter supposed to eperceived. This physiologist's argument is exposed to the rejoinder,more specious than solid, that our knowledge of the existence of thesense organs and nerves is o tained y that very process which thephysiologist has een engaged in discrediting, since the existence ofthe nerves and sense organs is only known through the evidence of thesenses themselves. This argument may prove that some reinterpretationof the results of physiology is necessary efore they can acquiremetaphysical validity. But it does not upset the physiologicalargument in so far as this constitutes merely a _reductio ad a surdum_ of naïve realism.

These various lines of argument prove, I think, that some part of the eliefs of common sense must e a andoned. They prove that, if we takethese eliefs as a whole, we are forced into conclusions which are inpart self-contradictory; ut such arguments cannot of themselvesdecide what portion of our common-sense eliefs is in need ofcorrection. Common sense elieves that what we see is physical,outside the mind, and continuing to exist if we shut our eyes or turnthem in another direction. I elieve that common sense is right inregarding what we see as physical and (in one of several possi lesenses) outside the mind, ut is pro a ly wrong in supposing that itcontinues to exist when we are no longer looking at it. It seems to me

that the whole discussion of matter has

een o

scured

y two errorswhich support each other. The first of these is the error that what wesee, or perceive through any of our other senses, is su jective: thesecond is the elief that what is physical must e persistent.Whatever physics may regard as the ultimate constituents of matter, italways supposes these constituents to e indestructi le. Since theimmediate data of sense are not indestructi le ut in a state ofperpetual flux, it is argued that these data themselves cannot eamong the ultimate constituents of matter. I elieve this to e asheer mistake. The persistent particles of mathematical physics Iregard as logical constructions, sym olic fictions ena ling us toexpress compendiously very complicated assem lages of facts; and, onthe other hand, I elieve that the actual data in sensation, the

immediate o

jects 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 mayperhaps e made clearer y the use of Bergson's favourite illustrationof the cinematograph. When I first read Bergson's statement that themathematician conceives the world after the analogy of acinematograph, I had never seen a cinematograph, and my first visit toone was determined y the desire to verify Bergson's statement, whichI found to e completely true, at least so far as I am concerned.When, in a picture palace, we see a man rolling down hill, or runningaway from the police, or falling into a river, or doing any of thoseother things to which men in such places are addicted, we know that

there is not really only one man moving,

ut a succession of films,each with a different momentary man. The illusion of persistencearises only through the approach to continuity in the series ofmomentary men. Now what I wish to suggest is that in this respect thecinema is a etter metaphysician than common sense, physics, orphilosophy. The real man too, I elieve, however the police may swearto his identity, is really a series of momentary men, each differentone from the other, and ound together, not y a numerical identity, ut y continuity and certain intrinsic causal laws. And what appliesto men applies equally to ta les and chairs, the sun, moon and stars.



Each of these is to e regarded, not as one single persistent entity, ut as a series of entities succeeding each other in time, eachlasting for a very rief period, though pro a ly not for a meremathematical instant. In saying this I am only urging the same kind ofdivision in time as we are accustomed to acknowledge in the case ofspace. A ody which fills a cu ic foot will e admitted to consist ofmany smaller odies, each occupying only a very tiny volume; similarlya thing which persists for an hour is to e regarded as composed ofmany things of less duration. A true theory of matter requires adivision of things into time-corpuscles as well as intospace-corpuscles.

The world may e conceived as consisting of a multitude of entitiesarranged in a certain pattern. The entities which are arranged I shallcall "particulars." The arrangement or pattern results from relationsamong particulars. Classes or series of particulars, collectedtogether on account of some property which makes it convenient to ea le to speak of them as wholes, are what I call logical constructionsor sym olic fictions. The particulars are to e conceived, not on theanalogy of ricks in a uilding, ut rather on the analogy of notesin a symphony. The ultimate constituents of a symphony (apart fromrelations) are the notes, each of which lasts only for a very shorttime. We may collect together all the notes played y one instrument:these may e regarded as the analogues of the successive particulars

which common sense would regard as successive states of one "thing."But the "thing" ought to e regarded as no more "real" or"su stantial" than, for example, the rôle of the trom one. As soon as"things" are conceived in this manner it will e found that thedifficulties in the way of regarding immediate o jects of sense asphysical have largely disappeared.

When people ask, "Is the o ject of sense mental or physical?" theyseldom have any clear idea either what is meant y "mental" or"physical," or what criteria are to e applied for deciding whether agiven entity elongs to one class or the other. I do not know how togive a sharp definition of the word "mental," ut something may edone y enumerating occurrences which are indu ita ly mental:

elieving, dou

ting, wishing, willing,

eing pleased or pained, arecertainly mental occurrences; so are what we may call experiences,seeing, hearing, smelling, perceiving generally. But it does notfollow from this that what is seen, what is heard, what is smelt, whatis perceived, must e mental. When I see a flash of lightning, myseeing of it is mental, ut what I see, although it is not quite thesame as what any ody else sees at the same moment, and although itseems very unlike what the physicist would descri e as a flash oflightning, is not mental. I maintain, in fact, that if the physicistcould descri e truly and fully all that occurs in the physical worldwhen there is a flash of lightning, it would contain as a constituentwhat I see, and also what is seen y any ody else who would commonly e said to see the same flash. What I mean may perhaps e made plainer

y saying that if my

ody could remain in exactly the same state inwhich it is, although my mind had ceased to exist, precisely thato ject which I now see when I see the flash would exist, although ofcourse I should not see it, since my seeing is mental. The principalreasons which have led people to reject this view have, I think, eentwo: first, that they did not adequately distinguish etween my seeingand what I see; secondly, that the causal dependence of what I seeupon my ody has made people suppose that what I see cannot e"outside" me. The first of these reasons need not detain us, since theconfusion only needs to e pointed out in order to e o viated; ut



the second requires some discussion, since it can only e answered yremoving current misconceptions, on the one hand as to the nature ofspace, and on the other, as to the meaning of causal dependence.

When people ask whether colours, for example, or other secondaryqualities are inside or outside the mind, they seem to suppose thattheir meaning must e clear, and that it ought to e possi le to sayyes or no without any further discussion of the terms involved. Infact, however, such terms as "inside" or "outside" are very am iguous.What is meant y asking whether this or that is "in" the mind? Themind is not like a ag or a pie; it does not occupy a certain regionin space, or, if (in a sense) it does, what is in that region ispresuma ly part of the rain, which would not e said to e in themind. When people say that sensi le qualities are in the mind, they donot mean "spatially contained in" in the sense in which the lack irdswere in the pie. We might regard the mind as an assem lage ofparticulars, namely, what would e called "states of mind," whichwould elong together in virtue of some specific common quality. Thecommon quality of all states of mind would e the quality designated y the word "mental"; and esides this we should have to suppose thateach separate person's states of mind have some common characteristicdistinguishing them from the states of mind of other people. Ignoringthis latter point, let us ask ourselves whether the quality designated y the word "mental" does, as a matter of o servation, actually elong

to o

jects of sense, such as colours or noises. I think any candidperson must reply that, however difficult it may e to know what wemean y "mental," it is not difficult to see that colours and noisesare not mental in the sense of having that intrinsic peculiarity which elongs to eliefs and wishes and volitions, ut not to the physicalworld. Berkeley advances on this su ject a plausi le argument[26]which seems to me to rest upon an am iguity in the word "pain." Heargues that the realist supposes the heat which he feels inapproaching a fire to e something outside his mind, ut that as heapproaches nearer and nearer to the fire the sensation of heat passesimpercepti ly into pain, and that no one could regard pain assomething outside the mind. In reply to this argument, it should eo served in the first place that the heat of which we are immediately

aware is not in the fire

ut in our own

ody. It is only

y inferencethat the fire is judged to e the cause of the heat which we feel inour ody. 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 theo ject of the sensation or other experience which has the quality of eing painful, or we may mean the quality of painfulness itself. Whena man says he has a pain in his great toe, what he means is that hehas a sensation associated with his great toe and having the qualityof painfulness. The sensation itself, like every sensation, consistsin experiencing a sensi le o ject, and the experiencing has thatquality of painfulness which only mental occurrences can have, utwhich may elong to thoughts or desires, as well as to sensations. Butin common language we speak of the sensi le o ject experienced in a

painful sensation as a pain, and it is this way of speaking whichcauses the confusion upon which the plausi ility of Berkeley'sargument depends. It would e a surd to attri ute the quality ofpainfulness to anything non-mental, and hence it comes to e thoughtthat what we call a pain in the toe must e mental. In fact, however,it is not the sensi le o ject in such a case which is painful, ut thesensation, that is to say, the experience of the sensi le o ject. Asthe heat which we experience from the fire grows greater, theexperience passes gradually from eing pleasant to eing painful, utneither the pleasure nor the pain is a quality of the o ject



experienced as opposed to the experience, and it is therefore afallacy to argue that this o ject must e mental on the ground thatpainfulness can only e attri uted to what is mental.

If, then, when we say that something is in the mind we mean that ithas a certain recognisa le intrinsic characteristic such as elongs tothoughts and desires, it must e maintained on grounds of immediateinspection that o jects of sense are not in any mind.

A different meaning of "in the mind" is, however, to e inferred fromthe arguments advanced y those who regard sensi le o jects as eingin the mind. The arguments used are, in the main, such as would provethe causal dependence of o jects of sense upon the percipient. Nowthe notion of causal dependence is very o scure and difficult, muchmore so in fact than is generally realised y philosophers. I shallreturn to this point in a moment. For the present, however, acceptingthe notion of causal dependence without criticism, I wish to urge thatthe dependence in question is rather upon our odies than upon ourminds. The visual appearance of an o ject is altered if we shut oneeye, or squint, or look previously at something dazzling; ut allthese are odily acts, and the alterations which they effect are to eexplained y physiology and optics, not y psychology.[27] They are infact of exactly the same kind as the alterations effected yspectacles or a microscope. They elong therefore to the theory of the

physical world, and can have no

earing upon the question whether whatwe see is causally dependent upon the mind. What they do tend toprove, and what I for my part have no wish to deny, is that what wesee is causally dependent upon our ody and is not, as crude commonsense would suppose, something which would exist equally if our eyesand nerves and rain were a sent, any more than the visual appearancepresented y an o ject seen through a microscope would remain if themicroscope were removed. So long as it is supposed that the physicalworld is composed of sta le and more or less permanent constituents,the fact that what we see is changed y changes in our ody appears toafford reason for regarding what we see as not an ultimate constituentof matter. But if it is recognised that the ultimate constituents ofmatter are as circumscri ed in duration as in spatial extent, the

whole of this difficulty vanishes.

There remains, however, another difficulty, connected with space. Whenwe look at the sun we wish to know something a out the sun itself,which is ninety-three million miles away; ut what we see is dependentupon our eyes, and it is difficult to suppose that our eyes can affectwhat happens at a distance of ninety-three million miles. Physicstells us that certain electromagnetic waves start from the sun, andreach our eyes after a out eight minutes. They there producedistur ances in the rods and cones, thence in the optic nerve, thencein the rain. At the end of this purely physical series, y some oddmiracle, comes the experience which we call "seeing the sun," and itis such experiences which form the whole and sole reason for our

elief in the optic nerve, the rods and cones, the ninety-threemillion miles, the electromagnetic waves, and the sun itself. It isthis curious oppositeness of direction etween the order of causationas affirmed y physics, and the order of evidence as revealed ytheory of knowledge, that causes the most serious perplexities inregard to the nature of physical reality. Anything that invalidatesour 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 een led step y step to the construction of the causal chain in which



our seeing is the last link, and the immediate o ject which we seecannot e regarded as that initial cause which we elieve to eninety-three million miles away, and which we are inclined to regardas the "real" sun.

I have stated this difficulty as forci ly as I can, ecause I elievethat it can only e answered y a radical analysis and reconstructionof all the conceptions upon whose employment it depends.

Space, time, matter and cause, are the chief of these conceptions. Letus egin with the conception of cause.

Causal dependence, as I o served a moment ago, is a conception whichit is very dangerous to accept at its face value. There exists anotion that in regard to any event there is something which may ecalled _the_ cause of that event--some one definite occurrence,without which the event would have een impossi le and with which it ecomes necessary. An event is supposed to e dependent upon its causein some way which in it is not dependent upon other things. Thus menwill urge that the mind is dependent upon the rain, or, with equalplausi ility, that the rain is dependent upon the mind. It seems notimpro a le that if we had sufficient knowledge we could infer thestate of a man's mind from the state of his rain, or the state of his rain from the state of his mind. So long as the usual conception of

causal dependence is retained, this state of affairs can

e used

ythe materialist to urge that the state of our rain causes ourthoughts, and y the idealist to urge that our thoughts cause thestate of our rain. Either contention is equally valid or equallyinvalid. The fact seems to e that there are many correlations of thesort which may e called causal, and that, for example, either aphysical or a mental event can e predicted, theoretically, eitherfrom a sufficient num er of physical antecedents or from a sufficientnum er of mental antecedents. To speak of _the_ cause of an event istherefore misleading. Any set of antecedents from which the event cantheoretically e inferred y means of correlations might e called acause of the event. But to speak of _the_ cause is to imply auniqueness which does not exist.

The relevance of this to the experience which we call "seeing the sun"is o vious. The fact that there exists a chain of antecedents whichmakes our seeing dependent upon the eyes and nerves and rain does noteven tend to show that there is not another chain of antecedents inwhich the eyes and nerves and rain as physical things are ignored. Ifwe are to escape from the dilemma which seemed to arise out of thephysiological causation of what we see when we say we see the sun, wemust find, at least in theory, a way of stating causal laws for thephysical world, in which the units are not material things, such asthe eyes and nerves and rain, ut momentary particulars of the samesort as our momentary visual o ject when we look at the sun. The sunitself and the eyes and nerves and rain must e regarded as

assem

lages of momentary particulars. Instead of supposing, as wenaturally do when we start from an uncritical acceptance of theapparent dicta of physics, that _matter_ is what is "really real" inthe physical world, and that the immediate o jects of sense are merephantasms, we must regard matter as a logical construction, of whichthe constituents will e just such evanescent particulars as may, whenan o server happens to e present, ecome data of sense to thato server. What physics regards as the sun of eight minutes ago will ea whole assem lage of particulars, existing at different times,spreading out from a centre with the velocity of light, and containing



among their num er all those visual data which are seen y people whoare now looking at the sun. Thus the sun of eight minutes ago is aclass of particulars, and what I see when I now look at the sun is onemem er of this class. The various particulars constituting this classwill e correlated with each other y a certain continuity and certainintrinsic laws of variation as we pass outwards from the centre,together with certain modifications correlated extrinsically withother particulars which are not mem ers of this class. It is theseextrinsic modifications which represent the sort of facts that, in ourformer account, appeared as the influence of the eyes and nerves inmodifying the appearance of the sun.[28]

The _prima facie_ difficulties in the way of this view are chieflyderived from an unduly conventional theory of space. It might seem atfirst sight as if we had packed the world much fuller than it couldpossi ly hold. At every place etween us and the sun, we said, thereis to e a particular which is to e a mem er of the sun as it was afew minutes ago. There will also, of course, have to e a particularwhich is a mem er of any planet or fixed star that may happen to evisi le from that place. At the place where I am, there will eparticulars which will e mem ers severally of all the "things" I amnow said to e perceiving. Thus throughout the world, everywhere,there will e an enormous num er of particulars coexisting in the sameplace. But these trou les result from contenting ourselves too readily

with the merely three-dimensional space to which schoolmasters haveaccustomed us. The space of the real world is a space of sixdimensions, and as soon as we realise this we see that there is plentyof 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 thepolished space of physics to the rough and untidy space of ourimmediate sensi le experience. The space of one man's sensi le o jectsis a three-dimensional space. It does not appear pro a le that two menever oth perceive at the same time any one sensi le o ject; when theyare said to see the same thing or hear the same noise, there willalways e some difference, however slight, etween the actual shapesseen or the actual sounds heard. If this is so, and if, as isgenerally assumed, position in space is purely relative, it follows

that the space of one man's o

jects and the space of another man'so jects have no place in common, that they are in fact differentspaces, and not merely different parts of one space. I mean y thisthat such immediate spatial relations as are perceived to hold etween the different parts of the sensi le space perceived y oneman, do not hold etween parts of sensi le spaces perceived ydifferent men. There are therefore a multitude of three-dimensionalspaces in the world: there are all those perceived y o servers, andpresuma ly also those which are not perceived, merely ecause noo server is suita ly situated for perceiving them.

But although these spaces do not have to one another the same kind ofspatial relations as o tain etween the parts of one of them, it is

nevertheless possi

le to arrange these spaces themselves in athree-dimensional order. This is done y means of the correlatedparticulars which we regard as mem ers (or aspects) of one physicalthing. When a num er of people are said to see the same o ject, thosewho would e said to e near to the o ject see a particular occupyinga larger part of their field of vision than is occupied y thecorresponding particular seen y people who would e said to efarther from the thing. By means of such considerations it ispossi le, in ways which need not now e further specified, to arrangeall the different spaces in a three-dimensional series. Since each of



the spaces is itself three-dimensional, the whole world of particularsis thus arranged in a six-dimensional space, that is to say, sixco-ordinates will e required to assign completely the position of anygiven particular, namely, three to assign its position in its ownspace and three more to assign the position of its space among theother spaces.

There are two ways of classifying particulars: we may take togetherall those that elong 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 elongsto two assem lages: (1) the assem lage of all my present o jects ofsense, which is what I call a "perspective"; (2) the assem lage ofall the different particulars which would e called aspects of the sunof eight minutes ago--this assem lage is what I define as _ eing_ thesun of eight minutes ago. Thus "perspectives" and "things" are merelytwo different ways of classifying particulars. It is to e o servedthat there is no _a priori_ necessity for particulars to esuscepti le of this dou le classification. There may e what might ecalled "wild" particulars, not having the usual relations y which theclassification is effected; perhaps dreams and hallucinations arecomposed of particulars which are "wild" in this sense.

The exact definition of what is meant y a perspective is not quite

easy. So long as we confine ourselves to visi

le o

jects or to o

jectsof touch we might define the perspective of a given particular as "allparticulars which have a simple (direct) spatial relation to the givenparticular." Between two patches of colour which I see now, there is adirect spatial relation which I equally see. But etween patches ofcolour seen y different men there is only an indirect constructedspatial relation y means of the placing of "things" in physical space(which is the same as the space composed of perspectives). Thoseparticulars which have direct spatial relations to a given particularwill elong to the same perspective. But if, for example, the soundswhich I hear are to elong to the same perspective with the patches ofcolour which I see, there must e particulars which have no directspatial relation and yet elong to the same perspective. We cannot

define a perspective as all the data of one percipient at one time, ecause we wish to allow the possi ility of perspectives which are notperceived y any one. There will e need, therefore, in defining aperspective, of some principle derived neither from psychology norfrom space.

Such a principle may e o tained from the consideration of _time_.The one all-em racing time, like the one all-em racing space, is aconstruction; there is no _direct_ time-relation etween particulars elonging to my perspective and particulars elonging to anotherman's. On the other hand, any two particulars of which I am aware areeither simultaneous or successive, and their simultaneity orsuccessiveness is sometimes itself a datum to me. We may therefore

define the perspective to which a given particular

elongs as "allparticulars simultaneous with the given particular," where"simultaneous" is to e understood as a direct simple relation, notthe derivative constructed relation of physics. It may e o servedthat the introduction of "local time" suggested y the principle ofrelativity has effected, for purely scientific reasons, much the samemultiplication of times as we have just een advocating.

The sum-total of all the particulars that are (directly) eithersimultaneous with or efore or after a given particular may e defined



as the " iography" to which that particular elongs. It will eo served that, just as a perspective need not e actually perceived yany one, so a iography need not e actually lived y any one. Those iographies that are lived y no one are called "official."

The definition of a "thing" is effected y means of continuity and ofcorrelations which have a certain differential independence of other"things." That is to say, given a particular in one perspective, therewill usually in a neigh ouring perspective e a very similarparticular, differing from the given particular, to the first order ofsmall quantities, according to a law involving only the difference ofposition of the two perspectives in perspective space, and not any ofthe other "things" in the universe. It is this continuity anddifferential independence in the law of change as we pass from oneperspective to another that defines the class of particulars which isto e called "one thing."

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

We may now sum up our discussion. Our o

ject has

een to discover asfar as possi le the nature of the ultimate constituents of thephysical world. When I speak of the "physical world," I mean, to eginwith, the world dealt with y physics. It is o vious that physics isan empirical science, giving us a certain amount of knowledge and ased upon evidence o tained through the senses. But partly throughthe development of physics itself, partly through arguments derivedfrom physiology, psychology or metaphysics, it has come to e thoughtthat the immediate data of sense could not themselves form part of theultimate constituents of the physical world, ut were in some sense"mental," "in the mind," or "su jective." The grounds for this view,in so far as they depend upon physics, can only e adequately dealtwith y rather ela orate constructions depending upon sym olic logic,

showing that out of such materials as are provided

y the senses it ispossi le to construct classes and series having the properties whichphysics assigns to matter. Since this argument is difficult andtechnical, I have not em arked upon it in this article. But in so faras the view that sense-data are "mental" rests upon physiology,psychology, or metaphysics, I have tried to show that it rests uponconfusions and prejudices--prejudices in favour of permanence in theultimate constituents of matter, and confusions derived from undulysimple notions as to space, from the causal correlation of sense-datawith sense-organs, and from failure to distinguish etween sense-dataand sensations. If what we have said on these su jects is valid, theexistence of sense-data is logically independent of the existence ofmind, and is causally dependent upon the _ ody_ of the percipient

rather than upon his mind. The causal dependence upon the

ody of thepercipient, we found, is a more complicated matter than it appears to e, and, like all causal dependence, is apt to give rise to erroneous eliefs through misconceptions as to the nature of causal correlation.If we have een right in our contentions, sense-data are merely thoseamong the ultimate constituents of the physical world, of which wehappen to e immediately aware; they themselves are purely physical,and all that is mental in connection with them is our awareness ofthem, which is irrelevant to their nature and to their place inphysics.



Unduly simple notions as to space have een a great stum ling- lock torealists. When two men look at the same ta le, it is supposed thatwhat the one sees and what the other sees are in the same place. Sincethe shape and colour are not quite the same for the two men, thisraises a difficulty, hastily solved, or rather covered up, ydeclaring what each sees to e purely "su jective"--though it wouldpuzzle those who use this gli word to say what they mean y it. Thetruth seems to e that space--and time also--is much more complicatedthan it would appear to e from the finished structure of physics, andthat the one all-em racing three-dimensional space is a logicalconstruction, o tained y means of correlations from a crude space ofsix dimensions. The particulars occupying this six-dimensional space,classified in one way, form "things," from which with certain furthermanipulations we can o tain what physics can regard as matter;classified in another way, they form "perspectives" and " iographies,"which may, if a suita le percipient happens to exist, formrespectively the sense-data of a momentary or of a total experience.It is only when physical "things" have een dissected into series ofclasses of particulars, as we have done, that the conflict etween thepoint of view of physics and the point of view of psychology can eovercome. This conflict, if what has een said is not mistaken, flowsfrom different methods of classification, and vanishes as soon as itssource is discovered.

In favour of the theory which I have riefly outlined, I do not claimthat it is _certainly_ true. Apart from the likelihood of mistakes,much of it is avowedly hypothetical. What I do claim for the theory isthat it _may_ e true, and that this is more than can e said for anyother theory except the closely analogous theory of Lei niz. Thedifficulties esetting realism, the confusions o structing anyphilosophical account of physics, the dilemma resulting fromdiscrediting sense-data, which yet remain the sole source of ourknowledge of the outer world--all these are avoided y the theorywhich I advocate. This does not prove the theory to e true, sincepro a ly many other theories might e invented which would have thesame merits. But it does prove that the theory has a etter chance of

eing true than any of its present competitors, and it suggests thatwhat can e known with certainty is likely to e discovera le ytaking our theory as a starting-point, and gradually freeing it fromall such assumptions as seem irrelevant, unnecessary, or unfounded. Onthese grounds, I recommend it to attention as a hypothesis and a asisfor further work, though not as itself a finished or adequate solutionof the pro lem with which it deals.

FOOTNOTES:

[23] An address delivered to the Philosophical Society of Manchesterin Fe ruary, 1915. Reprinted from _The Monist_, July, 1915.

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

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

[26] First dialogue etween Hylas and Philonous, _Works_ (Fraser'sedition 1901). I. p. 384.

[27] This point has een well urged y the American realists.



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

VIII

THE RELATION OF SENSE-DATA TO PHYSICS

I. THE PROBLEM STATED

Physics is said to e an empirical science, ased upon o servation andexperiment.

It is supposed to e verifia le, i.e. capa le of calculating eforehand results su sequently confirmed y o servation andexperiment.

What can we learn y o servation and experiment?

Nothing, so far as physics is concerned, except immediate data of

sense: certain patches of colour, sounds, tastes, smells, etc., withcertain spatio-temporal relations.

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

If such o jects are to e verified, it must e solely through theirrelation to sense-data: they must have some kind of correlation withsense-data, and must e verifia le through their correlation _alone_.

But how is the correlation itself ascertained? A correlation can only e ascertained empirically y the correlated o jects eing constantly

_found_ together. But in our case, only one term of the correlation,namely, the sensi le term, is ever _found_: the other term seemsessentially incapa le of eing found. Therefore, it would seem, thecorrelation with o jects of sense, y which physics was to everified, is itself utterly and for ever unverifia le.

There are two ways of avoiding this result.

(1) We may say that we know some principle _a priori_, without theneed of empirical verification, e.g. that our sense-data have _causes_ other than themselves, and that something can e known a out thesecauses y inference from their effects. This way has een oftenadopted y philosophers. It may e necessary to adopt this way to some

extent,

ut in so far as it is adopted physics ceases to

e empiricalor ased upon experiment and o servation alone. Therefore this way isto e avoided as much as possi le.

(2) We may succeed in actually defining the o jects of physics asfunctions of sense-data. Just in so far as physics leads toexpectations, this _must_ e possi le, since we can only _expect_ whatcan e experienced. And in so far as the physical state of affairs isinferred from sense-data, it must e capa le of expression as afunction of sense-data. The pro lem of accomplishing this expression



leads to much interesting logico-mathematical work.

In physics as commonly set forth, sense-data appear as functions ofphysical o jects: when such-and-such waves impinge upon the eye, wesee such-and-such colours, and so on. But the waves are in factinferred from the colours, not vice versa. Physics cannot e regardedas validly ased upon empirical data until the waves have eenexpressed as functions of the colours and other sense-data.

Thus if physics is to e verifia le we are faced with the followingpro lem: Physics exhi its sense-data as functions of physical o jects, ut verification is only possi le if physical o jects can e exhi itedas functions of sense-data. We have therefore to solve the equationsgiving sense-data in terms of physical o jects, so as to make theminstead give physical o jects 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 isgiven in sense at one time. I mean rather such a part of the whole asmight e singled out y attention: particular patches of colour,particular noises, and so on. There is some difficulty in decidingwhat is to e considered _one_ sense-datum: often attention causes

divisions to appear where, so far as can

e discovered, there were nodivisions efore. An o served complex fact, such as that this patch ofred is to the left of that patch of lue, is also to e regarded as adatum from our present point of view: epistemologically, it does notdiffer greatly from a simple sense-datum as regards its function ingiving knowledge. Its _logical_ structure is very different, however,from that of sense: _sense_ gives acquaintance with particulars, andis thus a two-term relation in which the o ject can e _named_ ut not _asserted_, and is inherently incapa le of truth or falsehood, whereasthe o servation of a complex fact, which may e suita ly calledperception, is not a two-term relation, ut involves the propositionalform on the o ject-side, and gives knowledge of a truth, not mereacquaintance with a particular. This logical difference, important as

it is, is not very relevant to our present pro

lem; and it will

econvenient to regard data of perception as included among sense-datafor the purposes of this paper. It is to e o served that theparticulars which are constituents of a datum of perception are alwayssense-data in the strict sense.

Concerning sense-data, we know that they are there while they aredata, and this is the epistemological asis of all our knowledge ofexternal particulars. (The meaning of the word "external" of courseraises pro lems which will concern us later.) We do not know, except y means of more or less precarious inferences, whether the o jectswhich are at one time sense-data continue to exist at times when theyare not data. Sense-data at the times when they are data are all that

we directly and primitively know of the external world; hence inepistemology the fact that they are _data_ is all-important. But thefact that they are all that we directly know gives, of course, nopresumption that they are all that there is. If we could construct animpersonal metaphysic, independent of the accidents of our knowledgeand ignorance, the privileged position of the actual data wouldpro a ly disappear, and they would pro a ly appear as a ratherhaphazard selection from a mass of o jects more or less like them. Insaying this, I assume only that it is pro a le that there areparticulars with which we are not acquainted. Thus the special



importance of sense-data is in relation to epistemology, not tometaphysics. In this respect, physics is to e reckoned asmetaphysics: it is impersonal, and nominally pays no special attentionto sense-data. It is only when we ask how physics can e _known_ thatthe importance of sense-data re-emerges.

III. SENSIBILIA

I shall give the name _sensi ilia_ to those o jects which have thesame metaphysical and physical status as sense-data, withoutnecessarily eing data to any mind. Thus the relation of a _sensi ile_ to a sense-datum is like that of a man to a hus and: a man ecomes ahus and y entering into the relation of marriage, and similarly a _sensi ile_ ecomes a sense-datum y entering into the relation ofacquaintance. It is important to have oth terms; for we wish todiscuss whether an o ject which is at one time a sense-datum can stillexist at a time when it is not a sense-datum. We cannot ask "Cansense-data exist without eing given?" for that is like asking "Canhus ands exist without eing married?" We must ask "Can _sensi ilia_ exist without eing given?" and also "Can a particular _sensi ile_ eat one time a sense-datum, and at another not?" Unless we have theword _sensi ile_ as well as the word "sense-datum," such questions areapt to entangle us in trivial logical puzzles.

It will e seen that all sense-data are _sensi ilia_. It is ametaphysical question whether all _sensi ilia_ are sense-data, and anepistemological question whether there exist means of inferring _sensi ilia_ which are not data from those that are.

A few preliminary remarks, to e amplified as we proceed, will serveto elucidate the use which I propose to make of _sensi ilia_.

I regard sense-data as not mental, and as eing, in fact, part of theactual su ject-matter of physics. There are arguments, shortly to eexamined, for their su jectivity, ut these arguments seem to me onlyto prove _physiological_ su jectivity, i.e. causal dependence on the

sense-organs, nerves, and

rain. The appearance which a thing presentsto us is causally dependent upon these, in exactly the same way as itis dependent upon intervening fog or smoke or coloured glass. Bothdependences are contained in the statement that the appearance which apiece of matter presents when viewed from a given place is a functionnot only of the piece of matter, ut also of the intervening medium.(The terms used in this statement--"matter," "view from a givenplace," "appearance," "intervening medium"--will all e defined in thecourse of the present paper.) We have not the means of ascertaininghow things appear from places not surrounded y rain and nerves andsense-organs, ecause we cannot leave the ody; ut continuity makesit not unreasona le to suppose that they present _some_ appearance atsuch places. Any such appearance would e included among _sensi ilia_.

If--_per impossi

ile_--there were a complete human

ody with no mindinside it, all those _sensi ilia_ would exist, in relation to that ody, which would e sense-data if there were a mind in the ody. Whatthe mind adds to _sensi ilia_, in fact, is _merely_ awareness:everything else is physical or physiological.

IV. SENSE-DATA ARE PHYSICAL

Before discussing this question it will e well to define the sense in



which the terms "mental" and "physical" are to e used. The word"physical," in all preliminary discussions, is to e understood asmeaning "what is dealt with y physics." Physics, it is plain, tellsus something a out some of the constituents of the actual world; whatthese constituents are may e dou tful, ut it is they that are to ecalled physical, whatever their nature may prove to e.

The definition of the term "mental" is more difficult, and can only esatisfactorily given after many difficult controversies have eendiscussed and decided. For present purposes therefore I must contentmyself with assuming a dogmatic answer to these controversies. I shallcall a particular "mental" when it is aware of something, and I shallcall a fact "mental" when it contains a mental particular as aconstituent.

It will e seen that the mental and the physical are not necessarilymutually exclusive, although I know of no reason to suppose that theyoverlap.

The dou t as to the correctness of our definition of the "mental" isof little importance in our present discussion. For what I amconcerned to maintain is that sense-data are physical, and this einggranted it is a matter of indifference in our present inquiry whetheror not they are also mental. Although I do not hold, with Mach and

James and the "new realists," that the difference

etween the mentaland the physical is _merely_ one of arrangement, yet what I have tosay in the present paper is compati le with their doctrine and mighthave een reached from their standpoint.

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

(1) Do sensi le o jects persist when we are not sensi le of them? inother words, do _sensi ilia_ which are data at a certain timesometimes 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 yetmaintaining that they pro a ly never persist unchanged after ceasingto e data. The view that they do not persist is often thought, quiteerroneously in my opinion, to imply that they are mental; and thishas, I elieve, een a potent source of confusion in regard to ourpresent pro lem. If there were, as some have held, a _logicalimpossi ility_ in sense-data persisting after ceasing to e data, thatcertainly would tend to show that they were mental; ut if, as Icontend, their non-persistence is merely a pro a le inference fromempirically ascertained causal laws, then it carries no suchimplication with it, and we are quite free to treat them as part ofthe su ject-matter of physics.

Logically a sense-datum is an o

ject, a particular of which thesu ject is aware. It does not contain the su ject as a part, as forexample eliefs and volitions do. The existence of the sense-datum istherefore not logically dependent upon that of the su ject; for theonly way, so far as I know, in which the existence of _A_ can e _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 isa sense-datum should not persist after it has ceased to e a datum,nor why other similar particulars should not exist without ever eingdata. The view that sense-data are mental is derived, no dou t, in



part from their physiological su jectivity, ut in part also from afailure to distinguish etween sense-data and "sensations." By asensation I mean the fact consisting in the su ject's awareness of thesense-datum. Thus a sensation is a complex of which the su ject is aconstituent and which therefore is mental. The sense-datum, on theother hand, stands over against the su ject as that external o ject ofwhich in sensation the su ject is aware. It is true that thesense-datum is in many cases in the su ject's ody, ut the su ject's ody is as distinct from the su ject as ta les and chairs are, and isin fact merely a part of the material world. So soon, therefore, assense-data are clearly distinguished from sensations, and as theirsu jectivity is recognised to e physiological not psychical, thechief o stacles in the way of regarding them as physical are removed.

V. "SENSIBILIA" AND "THINGS"

But if "sensi ilia" are to e recognised as the ultimate constituentsof the physical world, a long and difficult journey is to e performed efore we can arrive either at the "thing" of common sense or at the"matter" of physics. The supposed impossi ility of com ining thedifferent sense-data which are regarded as appearances of the same"thing" to different people has made it seem as though these"sensi ilia" must e regarded as mere su jective phantasms. A given

ta

le will present to one man a rectangular appearance, while toanother it appears to have two acute angles and two o tuse angles; toone man it appears rown, while to another, towards whom it reflectsthe light, it appears white and shiny. It is said, not wholly withoutplausi ility, that these different shapes and different colours cannotco-exist simultaneously in the same place, and cannot therefore oth e constituents of the physical world. This argument I must confessappeared to me until recently to e irrefuta le. The contrary opinionhas, however, een a ly maintained y Dr. T.P. Nunn in an articleentitled: "Are Secondary Qualities Independent of Perception?"[29] Thesupposed impossi ility derives its apparent force from the phrase:"_in the same place_," and it is precisely in this phrase that itsweakness lies. The conception of space is too often treated in

philosophy--even

y those who on reflection would not defend suchtreatment--as though it were as given, simple, and unam iguous asKant, in his psychological innocence, supposed. It is the unperceivedam iguity of the word "place" which, as we shall shortly see, hascaused the difficulties to realists and given an undeserved advantageto their opponents. Two "places" of different kinds are involved inevery sense-datum, namely the place _at_ which it appears and theplace _from_ which it appears. These elong to different spaces,although, as we shall see, it is possi le, with certain limitations,to esta lish a correlation etween them. What we call the differentappearances of the same thing to different o servers are each in aspace private to the o server concerned. No place in the private worldof one o server is identical with a place in the private world of

another o

server. There is therefore no question of com

ining thedifferent appearances in the one place; and the fact that they cannotall exist in one place affords accordingly no ground whatever forquestioning their physical reality. The "thing" of common sense may infact e identified with the whole class of its appearances--where,however, we must include among appearances not only those which areactual sense-data, ut also those "sensi ilia," if any, which, ongrounds of continuity and resem lance, are to e regarded as elongingto the same system of appearances, although there happen to e noo servers to whom they are data.



An example may make this clearer. Suppose there are a num er of peoplein a room, all seeing, as they say, the same ta les and chairs, wallsand pictures. No two of these people have exactly the same sense-data,yet there is sufficient similarity among their data to ena le them togroup together certain of these data as appearances of one "thing" tothe several spectators, and others as appearances of another "thing."Besides the appearances which a given thing in the room presents tothe actual spectators, there are, we may suppose, other appearanceswhich it would present to other possi le spectators. If a man were tosit down etween two others, the appearance which the room wouldpresent to him would e intermediate etween the appearances which itpresents to the two others: and although this appearance would notexist as it is without the sense organs, nerves and rain, of thenewly arrived spectator, still it is not unnatural to suppose that,from the position which he now occupies, _some_ appearance of theroom existed efore his arrival. This supposition, however, needmerely e noticed and not insisted upon.

Since the "thing" cannot, without indefensi le partiality, eidentified with any single one of its appearances, it came to ethought of as something distinct from all of them and underlying them.But y the principle of Occam's razor, if the class of appearanceswill fulfil the purposes for the sake of which the thing was invented

y the prehistoric metaphysicians to whom common sense is due, economydemands that we should identify the thing with the class of itsappearances. It is not necessary to _deny_ a su stance or su stratumunderlying these appearances; it is merely expedient to a stain fromasserting this unnecessary entity. Our procedure here is preciselyanalogous to that which has swept away from the philosophy ofmathematics the useless menagerie of metaphysical monsters with whichit used to e infested.

VI. CONSTRUCTIONS VERSUS INFERENCES

Before proceeding to analyse and explain the am iguities of the word

"place," a few general remarks on method are desira

le. The suprememaxim in scientific philosophising is this:

_Wherever possi le, logical constructions are to e su stituted for inferred entities._

Some examples of the su stitution of construction for inference in therealm of mathematical philosophy may serve to elucidate the uses ofthis maxim. Take first the case of irrationals. In old days,irrationals were inferred as the supposed limits of series ofrationals which had no rational limit; ut the o jection to thisprocedure was that it left the existence of irrationals merelyoptative, and for this reason the stricter methods of the present day

no longer tolerate such a definition. We now define an irrationalnum er as a certain class of ratios, thus constructing it logically ymeans of ratios, instead of arriving at it y a dou tful inferencefrom them. Take again the case of cardinal num ers. Two equallynumerous collections appear to have something in common: thissomething is supposed to e their cardinal num er. But so long as thecardinal num er is inferred from the collections, not constructed interms of them, its existence must remain in dou t, unless in virtue ofa metaphysical postulate _ad hoc_. By defining the cardinal num er ofa given collection as the class of all equally numerous collections,



we avoid the necessity of this metaphysical postulate, and there yremove a needless element of dou t from the philosophy of arithmetic.A similar method, as I have shown elsewhere, can e applied to classesthemselves, which need not e supposed to have any metaphysicalreality, ut can e regarded as sym olically constructed fictions.

The method y which the construction proceeds is closely analogous inthese and all similar cases. Given a set of propositions nominallydealing with the supposed inferred entities, we o serve the propertieswhich are required of the supposed entities in order to make thesepropositions true. By dint of a little logical ingenuity, we thenconstruct some logical function of less hypothetical entities whichhas the requisite properties. This constructed function we su stitutefor the supposed inferred entities, and there y o tain a new and lessdou tful interpretation of the ody of propositions in question. Thismethod, so fruitful in the philosophy of mathematics, will e foundequally applica le in the philosophy of physics, where, I do notdou t, it would have een applied long ago ut for the fact that allwho have studied this su ject hitherto have een completely ignorantof mathematical logic. I myself cannot claim originality in theapplication of this method to physics, since I owe the suggestion andthe stimulus for its application entirely to my friend andcolla orator Dr. Whitehead, who is engaged in applying it to the moremathematical portions of the region intermediate etween sense-data

and the points, instants and particles of physics.

A complete application of the method which su stitutes constructionsfor inferences would exhi it matter wholly in terms of sense-data, andeven, we may add, of the sense-data of a single person, since thesense-data of others cannot e known without some element ofinference. This, however, must remain for the present an ideal, to eapproached as nearly as possi le, ut to e reached, if at all, onlyafter a long preliminary la our of which as yet we can only see thevery eginning. The inferences which are unavoida le can, however, esu jected to certain guiding principles. In the first place theyshould always e made perfectly explicit, and should e formulated inthe most general manner possi le. In the second place the inferred

entities should, whenever this can

e done,

e similar to those whoseexistence is given, rather than, like the Kantian _Ding an sich_,something wholly remote from the data which nominally support theinference. The inferred entities which I shall allow myself are of twokinds: (_a_) the sense-data of other people, in favour of which thereis the evidence of testimony, resting ultimately upon the analogicalargument in favour of minds other than my own; (_ _) the "sensi ilia"which would appear from places where there happen to e no minds, andwhich I suppose to e real although they are no one's data. Of thesetwo classes of inferred entities, the first will pro a ly e allowedto pass unchallenged. It would give me the greatest satisfaction to ea le to dispense with it, and thus esta lish physics upon asolipsistic asis; ut those--and I fear they are the majority--in

whom the human affections are stronger than the desire for logicaleconomy, will, no dou t, not share my desire to render solipsismscientifically satisfactory. The second class of inferred entitiesraises much more serious questions. It may e thought monstrous tomaintain that a thing can present any appearance at all in a placewhere no sense organs and nervous structure exist through which itcould appear. I do not myself feel the monstrosity; nevertheless Ishould regard these supposed appearances only in the light of ahypothetical scaffolding, to e used while the edifice of physics is eing raised, though possi ly capa le of eing removed as soon as the



edifice is completed. These "sensi ilia" which are not data to anyoneare therefore to e taken rather as an illustrative hypothesis and asan aid in preliminary statement than as a dogmatic part of thephilosophy of physics in its final form.

VII. PRIVATE SPACE AND THE SPACE OF PERSPECTIVES

We have now to explain the am iguity in the word "place," and how itcomes that two places of different sorts are associated with everysense-datum, namely the place _at_ which it is and the place _from_ which it is perceived. The theory to e advocated is closely analogousto Lei niz's monadology, from which it differs chiefly in eing lesssmooth and tidy.

The first fact to notice is that, so far as can e discovered, nosensi ile is ever a datum to two people at once. The things seen ytwo different people are often closely similar, so similar that thesame _words_ can e used to denote them, without which communicationwith others concerning sensi le o jects would e impossi le. But, inspite of this similarity, it would seem that some difference alwaysarises from difference in the point of view. Thus each person, so faras his sense-data are concerned, lives in a private world. Thisprivate world contains its own space, or rather spaces, for it would

seem that only experience teaches us to correlate the space of sightwith the space of touch and with the various other spaces of othersenses. This multiplicity of private spaces, however, thoughinteresting to the psychologist, is of no great importance in regardto our present pro lem, since a merely solipsistic experience ena lesus to correlate them into the one private space which em races all ourown sense-data. The place _at_ which a sense-datum is, is a place inprivate space. This place therefore is different from any place in theprivate space of another percipient. For if we assume, as logicaleconomy demands, that all position is relative, a place is onlydefina le y the things in or around it, and therefore the same placecannot occur in two private worlds which have no common constituent.The question, therefore, of com ining what we call different

appearances of the same thing in the same place does not arise, andthe fact that a given o ject appears to different spectators to havedifferent shapes and colours affords no argument against the physicalreality of all these shapes and colours.

In addition to the private spaces elonging to the private worlds ofdifferent percipients, there is, however, another space, in which onewhole private world counts as a point, or at least as a spatial unit.This might e descri ed as the space of points of view, since eachprivate world may e regarded as the appearance which the universepresents from a certain point of view. I prefer, however, to speak ofit as the space of _perspectives_, in order to o viate the suggestionthat 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 assuminga percipient, I shall call it a "perspective."

We have now to explain how the different perspectives are ordered inone space. This is effected y means of the correlated "sensi ilia"which are regarded as the appearances, in different perspectives, ofone and the same thing. By moving, and y testimony, we discover thattwo different perspectives, though they cannot oth contain the same"sensi ilia," may nevertheless contain very similar ones; and thespatial order of a certain group of "sensi ilia" in a private space of



one perspective is found to e identical with, or very similar to, thespatial order of the correlated "sensi ilia" in the private space ofanother perspective. In this way one "sensi ile" in one perspective iscorrelated with one "sensi ile" in another. Such correlated"sensi ilia" will e called "appearances of one thing." In Lei niz'smonadology, since each monad mirrored the whole universe, there was ineach perspective a "sensi ile" which was an appearance of each thing.In our system of perspectives, we make no such assumption ofcompleteness. A given thing will have appearances in someperspectives, ut presuma ly not in certain others. The "thing" eingdefined as the class of its appearances, if k is the cl

ss ofperspectives in which

cert

in thing q

ppe

rs, then q is

member ofthe multiplic

tive cl

ss of k, k being

cl

ss of mutu

lly exclusivecl

sses of "sensibili

." And simil

rly

perspective is

member ofthe multiplic

tive cl

ss of the things which

ppe

r in it.

The

rr

ngement of perspectives in

sp

ce is effected by me

ns of thedifferences between the

ppe

r

nces of

given thing in the v

riousperspectives. Suppose, s

y, th

t

cert

in penny

ppe

rs in

numberof different perspectives; in some it loo

s l

rger

nd in somesm

ller, in some it loo

s circul

r, in others it presents the

ppe

r

nce of

n ellipse of v

rying eccentricity. We m

y collecttogether

ll those perspectives in which the

ppe

r

nce of the pennyis circul

r. These we will pl

ce on one str

ight line, ordering them

in

series by the v

ri

tions in the

pp

rent size of the penny. Thoseperspectives in which the penny

ppe

rs

s

str

ight line of

cert

in thic

ness will simil

rly be pl

ced upon

pl

ne (though inthis c

se there will be m

ny different perspectives in which the pennyis of the s

me size; when one

rr

ngement is completed these will form

circle concentric with the penny),

nd ordered

s before by the

pp

rent size of the penny. By such me

ns,

ll those perspectives inwhich the penny presents

visu

l

ppe

r

nce c

n be

rr

nged in

three-dimension

l sp

ti

l order. Experience shows th

t the s

mesp

ti

l order of perspectives would h

ve resulted if, inste

d of thepenny, we h

d chosen

ny other thing which

ppe

red in

ll theperspectives in

uestion, or

ny other method of utilising thedifferences between the

ppe

r

nces of the s

me things in different

perspectives. It is this empiric

l f

ct which h

s m

de it possible toconstruct the one

ll-embr

cing sp

ce of physics.

The sp

ce whose construction h

s just been expl

ined,

nd whoseelements

re whole perspectives, will be c

lled "perspective-sp

ce."

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

The world which we h

ve so f

r constructed is

world of sixdimensions, since it is

three-dimension

l series of perspectives,e

ch of which is itself three-dimension

l. We h

ve now to expl

in thecorrel

tion between the perspective sp

ce

nd the v

rious priv

te

sp

ces cont

ined within the v

rious perspectives sever

lly. It is byme

ns of this correl

tion th

t the one three-dimension

l sp

ce ofphysics is constructed;

nd it is bec

use of the unconsciousperform

nce of this correl

tion th

t the distinction betweenperspective sp

ce

nd the percipient's priv

te sp

ce h

s been blurred,with dis

strous results for the philosophy of physics. Let us revertto our penny: the perspectives in which the penny

ppe

rs l

rger

rereg

rded

s being ne

rer to the penny th

n those in which it

ppe

rssm

ller, but

s f

r

s experience goes the

pp

rent size of the pennywill not grow beyond

cert

in limit, n

mely, th

t where (

s we s

y)



the penny is so ne

r the eye th

t if it were

ny ne

rer it could notbe seen. By touch we m

y prolong the series until the penny touchesthe eye, but no further. If we h

ve been tr

velling

long

line ofperspectives in the previously defined sense, we m

y, however, byim

gining the penny removed, prolong the line of perspectives byme

ns, s

y, of

nother penny;

nd the s

me m

y be done with

ny otherline of perspectives defined by me

ns of the penny. All these linesmeet in

cert

in pl

ce, th

t is, in

cert

in perspective. Thisperspective will be defined

s "the pl

ce where the penny is."

It is now evident in wh

t sense two pl

ces in constructed physic

lsp

ce

re

ssoci

ted with

given "sensibile." There is first thepl

ce which is the perspective of which the "sensibile" is

member.This is the pl

ce _from_ which the "sensibile"

ppe

rs. Secondly thereis the pl

ce where the thing is of which the "sensibile" is

member,in other words

n

ppe

r

nce; this is the pl

ce _

t_ which the"sensibile"

ppe

rs. The "sensibile" which is

member of oneperspective is correl

ted with

nother perspective, n

mely, th

t whichis the pl

ce where the thing is of which the "sensibile" is

n

ppe

r

nce. To the psychologist the "pl

ce from which" is the moreinteresting,

nd the "sensibile"

ccordingly

ppe

rs to him subjective

nd where the percipient is. To the physicist the "pl

ce

t which" isthe more interesting,

nd the "sensibile"

ccordingly

ppe

rs to himphysic

l

nd extern

l. The c

uses, limits

nd p

rti

l justific

tion of

e

ch of these two

pp

rently incomp

tible views

re evident from the

bove duplicity of pl

ces

ssoci

ted with

given "sensibile."

We h

ve seen th

t we c

n

ssign to

physic

l thing

pl

ce in theperspective sp

ce. In this w

y different p

rts of our body

c

uirepositions in perspective sp

ce,

nd therefore there is

me

ning(whether true or f

lse need not much concern us) in s

ying th

t theperspective to which our sense-d

t

belong is inside our he

d. Sinceour mind is correl

ted with the perspective to which our sense-d

t

belong, we m

y reg

rd this perspective

s being the position of ourmind in perspective sp

ce. If, therefore, this perspective is, in the

bove defined sense, inside our he

d, there is

good me

ning for thest

tement th

t the mind is in the he

d. We c

n now s

y of the v

rious

ppe

r

nces of

given thing th

t some of them

re ne

rer to the thingth

n others; those

re ne

rer which belong to perspectives th

t

rene

rer to "the pl

ce where the thing is." We c

n thus find

me

ning,true or f

lse, for the st

tement th

t more is to be le

rnt

bout

thing by ex

mining it close to th

n by viewing it from

dist

nce. Wec

n

lso find

me

ning for the phr

se "the things which intervenebetween the subject

nd

thing of which

n

ppe

r

nce is

d

tum tohim." One re

son often

lleged for the subjectivity of sense-d

t

isth

t the

ppe

r

nce of

thing m

y ch

nge when we find it h

rd tosuppose th

t the thing itself h

s ch

nged--for ex

mple, when thech

nge is due to our shutting our eyes, or to our screwing them up so

s to m

e the thing loo

double. If the thing is defined

s the cl

ssof its

ppe

r

nces (which is the definition

dopted

bove), there is

of course necess

rily _some_ ch

nge in the thing whenever

ny one ofits

ppe

r

nces ch

nges. Nevertheless there is

very import

ntdistinction between two different w

ys in which the

ppe

r

nces m

ych

nge. If

fter loo

ing

t

thing I shut my eyes, the

ppe

r

nce ofmy eyes ch

nges in every perspective in which there is such

n

ppe

r

nce, where

s most of the

ppe

r

nces of the thing will rem

inunch

nged. We m

y s

y,

s

m

tter of definition, th

t

thing ch

ngeswhen, however ne

r to the thing

n

ppe

r

nce of it m

y be, there

rech

nges in

ppe

r

nces

s ne

r

s, or still ne

rer to, the thing. Onthe other h

nd we sh

ll s

y th

t the ch

nge is in some other thing if



ll

ppe

r

nces of the thing which

re

t not more th

n

cert

indist

nce from the thing rem

in unch

nged, while only comp

r

tivelydist

nt

ppe

r

nces of the thing

re

ltered. From this consider

tionwe

re n

tur

lly led to the consider

tion of _m

tter_, which must beour next topic.

IX. THE DEFINITION OF MATTER

We defined the "physic

l thing"

s the cl

ss of its

ppe

r

nces, butthis c

n h

rdly be t

en

s

definition of m

tter. We w

nt to be

bleto express the f

ct th

t the

ppe

r

nce of

thing in

givenperspective is c

us

lly

ffected by the m

tter between the thing

ndthe perspective. We h

ve found

me

ning for "between

thing

nd

perspective." But we w

nt m

tter to be something other th

n the wholecl

ss of

ppe

r

nces of

thing, in order to st

te the influence ofm

tter on

ppe

r

nces.

We commonly

ssume th

t the inform

tion we get

bout

thing is more

ccur

te when the thing is ne

rer. F

r off, we see it is

m

n; thenwe see it is Jones; then we see he is smiling. Complete

ccur

cy wouldonly be

tt

in

ble

s

limit: if the

ppe

r

nces of Jones

s we

ppro

ch him tend tow

rds

limit, th

t limit m

y be t

en to be wh

tJones re

lly is. It is obvious th

t from the point of view of physics

the

ppe

r

nces of

thing close to "count" more th

n the

ppe

r

ncesf

r off. We m

y therefore set up the following tent

tive definition:

The _m

tter_ of

given thing is the limit of its

ppe

r

nces

s theirdist

nce from the thing diminishes.

It seems prob

ble th

t there is something in this definition, but itis not

uite s

tisf

ctory, bec

use empiric

lly there is no such limitto be obt

ined from sense-d

t

. The definition will h

ve to be e

edout by constructions

nd definitions. But prob

bly it suggests theright direction in which to loo

.

We

re now in

position to underst

nd in outline the reverse journey

from m

tter to sense-d

t

which is performed by physics. The

ppe

r

nce of

thing in

given perspective is

function of them

tter composing the thing

nd of the intervening m

tter. The

ppe

r

nce of

thing is

ltered by intervening smo

e or mist, by bluespect

cles or by

lter

tions in the sense-org

ns or nerves of thepercipient (which

lso must be rec

oned

s p

rt of the interveningmedium). The ne

rer we

ppro

ch to the thing, the less its

ppe

r

nceis

ffected by the intervening m

tter. As we tr

vel further

ndfurther from the thing, its

ppe

r

nces diverge more

nd more fromtheir initi

l ch

r

cter;

nd the c

us

l l

ws of their divergence

reto be st

ted in terms of the m

tter which lies between them

nd thething. Since the

ppe

r

nces

t very sm

ll dist

nces

re less

ffectedby c

uses other th

n the thing itself, we come to thin

th

t the limit

tow

rds which these

ppe

r

nces tend

s the dist

nce diminishes iswh

t the thing "re

lly is,"

s opposed to wh

t it merely seems to be.This, together with its necessity for the st

tement of c

us

l l

ws,seems to be the source of the entirely erroneous feeling th

t m

tteris more "re

l" th

n sense-d

t

.

Consider for ex

mple the infinite divisibility of m

tter. In loo

ing

t

given thing

nd

ppro

ching it, one sense-d

tum will becomesever

l,

nd e

ch of these will

g

in divide. Thus _one_

ppe

r

ncem

y represent _m

ny_ things,

nd to this process there seems no end.



which

re to be grouped together

s constituting wh

t

cert

in thingis

t

cert

in moment,

re not to be

ll reg

rded

s being

t th

tmoment. On the contr

ry they spre

d outw

rd from the thing withv

rious velocities

ccording to the n

ture of the

ppe

r

nces. Sinceno _direct_ me

ns exist of correl

ting the time in one biogr

phy withthe time in

nother, this tempor

l grouping of the

ppe

r

ncesbelonging to

given thing

t

given moment is in p

rt convention

l.Its motive is p

rtly to secure the verific

tion of such m

xims

s th

tevents which

re ex

ctly simult

neous with the s

me event

re ex

ctlysimult

neous with one

nother, p

rtly to secure convenience in theformul

tion of c

us

l l

ws.

XI. THE PERSISTENCE OF THINGS AND MATTER

Ap

rt from

ny of the fluctu

ting hypotheses of physics, three m

inproblems

rise in connecting the world of physics with the world ofsense, n

mely:

1. the construction of

single sp

ce; 2. the construction of

single time; 3. the construction of perm

nent things or m

tter.

We h

ve

lre

dy considered the first

nd second of these problems; it

rem

ins to consider the third.

We h

ve seen how correl

ted

ppe

r

nces in different perspectives

recombined to form one "thing"

t one moment in the

ll-embr

cing timeof physics. We h

ve now to consider how

ppe

r

nces

t different times

re combined

s belonging to one "thing,"

nd how we

rrive

t thepersistent "m

tter" of physics. The

ssumption of perm

nent subst

nce,which technic

lly underlies the procedure of physics, c

nnot of coursebe reg

rded

s met

physic

lly legitim

te: just

s the one thingsimult

neously seen by m

ny people is

construction, so the one thingseen

t different times by the s

me or different people must be

construction, being in f

ct nothing but

cert

in grouping of cert

in"sensibili

."

We h

ve seen th

t the moment

ry st

te of

"thing" is

n

ssembl

ge of"sensibili

," in different perspectives, not

ll simult

neous in theone constructed time, but spre

ding out from "the pl

ce where thething is" with velocities depending upon the n

ture of the"sensibili

." The time _

t_ which the "thing" is in this st

te is thelower limit of the times

t which these

ppe

r

nces occur. We h

ve nowto consider wh

t le

ds us to spe

of

nother set of

ppe

r

nces

sbelonging to the s

me "thing"

t

different time.

For this purpose, we m

y,

t le

st to begin with, confine ourselveswithin

single biogr

phy. If we c

n

lw

ys s

y when two "sensibili

"in

given biogr

phy

re

ppe

r

nces of one thing, then, since we h

ve

seen how to connect "sensibili

" in different biogr

phies

s

ppe

r

nces of the s

me moment

ry st

te of

thing, we sh

ll h

ve

llth

t is necess

ry for the complete construction of the history of

thing.

It is to be observed, to begin with, th

t the identity of

thing forcommon sense is not

lw

ys correl

ted with the identity of m

tter forphysics. A hum

n body is one persisting thing for common sense, butfor physics its m

tter is const

ntly ch

nging. We m

y s

y, bro

dly,th

t the common-sense conception is b

sed upon continuity in



ppe

r

nces

t the ordin

ry dist

nces of sense-d

t

, while thephysic

l conception is b

sed upon the continuity of

ppe

r

nces

tvery sm

ll dist

nces from the thing. It is prob

ble th

t thecommon-sense conception is not c

p

ble of complete precision. Let ustherefore concentr

te our

ttention upon the conception of thepersistence of m

tter in physics.

The first ch

r

cteristic of two

ppe

r

nces of the s

me piece ofm

tter

t different times is _continuity_. The two

ppe

r

nces must beconnected by

series of intermedi

ries, which, if time

nd sp

ce formcomp

ct series, must themselves form

comp

ct series. The colour ofthe le

ves is different in

utumn from wh

t it is in summer; but webelieve th

t the ch

nge occurs gr

du

lly,

nd th

t, if the colours

redifferent

t two given times, there

re intermedi

te times

t whichthe colours

re intermedi

te between those

t the given times.

But there

re two consider

tions th

t

re import

nt

s reg

rdscontinuity.

First, it is l

rgely hypothetic

l. We do not observe

ny one thingcontinuously,

nd it is merely

hypothesis to

ssume th

t, while we

re not observing it, it p

sses through conditions intermedi

tebetween those in which it is perceived. During uninterruptedobserv

tion, it is true, continuity is ne

rly verified; but even here,

when motions

re very r

pid,

s in the c

se of explosions, thecontinuity is not

ctu

lly c

p

ble of direct verific

tion. Thus we c

nonly s

y th

t the sense-d

t

re found to _permit_

hypothetic

lcomplement of "sensibili

" such

s will preserve continuity,

nd th

ttherefore there _m

y_ be such

complement. Since, however, we h

ve

lre

dy m

de such use of hypothetic

l "sensibili

," we will let thispoint p

ss,

nd

dmit such "sensibili

"

s

re re

uired to preservecontinuity.

Secondly, continuity is not

sufficient criterion of m

teri

lidentity. It is true th

t in m

ny c

ses, such

s roc

s, mount

ins,t

bles, ch

irs, etc., where the

ppe

r

nces ch

nge slowly, continuityis sufficient, but in other c

ses, such

s the p

rts of

n

pproxim

tely homogeneous fluid, it f

ils us utterly. We c

n tr

vel bysensibly continuous gr

d

tions from

ny one drop of the se

t

ny onetime to

ny other drop

t

ny other time. We infer the motions ofse

-w

ter from the effects of the current, but they c

nnot be inferredfrom direct sensible observ

tion together with the

ssumption ofcontinuity.

The ch

r

cteristic re

uired in

ddition to continuity is conformitywith the l

ws of dyn

mics. St

rting from wh

t common sense reg

rds

spersistent things,

nd m

ing only such modific

tions

s from time totime seem re

son

ble, we

rrive

t

ssembl

ges of "sensibili

" which

re found to obey cert

in simple l

ws, n

mely those of dyn

mics. Byreg

rding "sensibili

"

t different times

s belonging to the s

me

piece of m

tter, we

re

ble to define _motion_, which presupposes the

ssumption or construction of something persisting throughout thetime of the motion. The motions which

re reg

rded

s occurring,during

period in which

ll the "sensibili

"

nd the times of their

ppe

r

nce

re given, will be different

ccording to the m

nner inwhich we combine "sensibili

"

t different times

s belonging to thes

me piece of m

tter. Thus even when the whole history of the world isgiven in every p

rticul

r, the

uestion wh

t motions t

e pl

ce isstill to

cert

in extent

rbitr

ry even

fter the

ssumption ofcontinuity. Experience shows th

t it is possible to determine motions



in such

w

y

s to s

tisfy the l

ws of dyn

mics,

nd th

t thisdetermin

tion, roughly

nd on the whole, is f

irly in

greement withthe common-sense opinions

bout persistent things. This determin

tion,therefore, is

dopted,

nd le

ds to

criterion by which we c

ndetermine, sometimes pr

ctic

lly, sometimes only theoretic

lly,whether two

ppe

r

nces

t different times

re to be reg

rded

sbelonging to the s

me piece of m

tter. The persistence of

ll m

tterthroughout

ll time c

n, I im

gine, be secured by definition.

To recommend this conclusion, we must consider wh

t it is th

t isproved by the empiric

l success of physics. Wh

t is proved is th

t itshypotheses, though unverifi

ble where they go beyond sense-d

t

,

re

t no point in contr

diction with sense-d

t

, but, on the contr

ry,

re ide

lly such

s to render

ll sense-d

t

c

lcul

ble when

sufficient collection of "sensibili

" is given. Now physics h

s foundit empiric

lly possible to collect sense-d

t

into series, e

ch seriesbeing reg

rded

s belonging to one "thing,"

nd beh

ving, with reg

rdto the l

ws of physics, in

w

y in which series not belonging to onething would in gener

l not beh

ve. If it is to be un

mbiguous whethertwo

ppe

r

nces belong to the s

me thing or not, there must be onlyone w

y of grouping

ppe

r

nces so th

t the resulting things obey thel

ws of physics. It would be very difficult to prove th

t this is thec

se, but for our present purposes we m

y let this point p

ss,

nd

ssume th

t there is only one w

y. Thus we m

y l

y down the following

definition: _Physic

l things

re those series of

ppe

r

nces whosem

tter obeys the l

ws of physics_. Th

t such series exist is

nempiric

l f

ct, which constitutes the verifi

bility of physics.

XII. ILLUSIONS, HALLUCINATIONS, AND DREAMS

It rem

ins to

s

how, in our system, we

re to find

pl

ce forsense-d

t

which

pp

rently f

il to h

ve the usu

l connection with theworld of physics. Such sense-d

t

re of v

rious

inds, re

uiringsomewh

t different tre

tment. But

ll

re of the sort th

t would bec

lled "unre

l,"

nd therefore, before emb

r

ing upon the discussion,cert

in logic

l rem

r

s must be m

de upon the conceptions of re

lity

nd unre

lity.

Mr. A. Wolf[31] s

ys:

"The conception of mind

s

system of tr

nsp

rent

ctivities is, I thin

,

lso unten

ble bec

use of its f

ilure to

ccount for the very possibility of dre

ms

nd h

llucin

tions. It seems impossible to re

lise how

b

re, tr

nsp

rent

ctivity c

n be directed to wh

t is not there, to

pprehend wh

t is not given."

This st

tement is one which, prob

bly, most people would endorse. Butit is open to two objections. First it is difficult to see how

n

ctivity, however un-"tr

nsp

rent," c

n be directed tow

rds

nothing:

term of

rel

tion c

nnot be

mere nonentity. Secondly, no re

sonis given,

nd I

m convinced th

t none c

n be given, for the

ssertionth

t dre

m-objects

re not "there"

nd not "given." Let us t

e thesecond point first.

(1) The belief th

t dre

m-objects

re not given comes, I thin

, fromf

ilure to distinguish,

s reg

rds w

ing life, between thesense-d

tum

nd the corresponding "thing." In dre

ms, there is no suchcorresponding "thing"

s the dre

mer supposes; if, therefore, the"thing" were given in w

ing life,

s e.g. Meinong m

int

ins,[32] then



there would be

difference in respect of givenness between dre

ms

ndw

ing life. But if,

s we h

ve m

int

ined, wh

t is given is never thething, but merely one of the "sensibili

" which compose the thing,then wh

t we

pprehend in

dre

m is just

s much given

s wh

t we

pprehend in w

ing life.

Ex

ctly the s

me

rgument

pplies

s to the dre

m-objects being"there." They h

ve their position in the priv

te sp

ce of theperspective of the dre

mer; where they f

il is in their correl

tionwith other priv

te sp

ces

nd therefore with perspective sp

ce. But inthe only sense in which "there" c

n be

d

tum, they

re "there" just

s truly

s

ny of the sense-d

t

of w

ing life.

(2) The conception of "illusion" or "unre

lity,"

nd the correl

tiveconception of "re

lity,"

re gener

lly used in

w

y which embodiesprofound logic

l confusions. Words th

t go in p

irs, such

s "re

l"

nd "unre

l," "existent"

nd "non-existent," "v

lid"

nd "inv

lid,"etc.,

re

ll derived from the one fund

ment

l p

ir, "true"

nd"f

lse." Now "true"

nd "f

lse"

re

pplic

ble only--except inderiv

tive signific

tions--to _propositions_. Thus wherever the

bovep

irs c

n be signific

ntly

pplied, we must be de

ling either withpropositions or with such incomplete phr

ses

s only

c

uire me

ningwhen put into

context which, with them, forms

proposition. Thussuch p

irs of words c

n be

pplied to _descriptions_,[33] but not to

proper n

mes: in other words, they h

ve no

pplic

tion wh

tever tod

t

, but only to entities or non-entities described in terms of d

t

.

Let us illustr

te by the terms "existence"

nd "non-existence." Given

ny d

tum _x_, it is me

ningless either to

ssert or to deny th

t _x_ "exists." We might be tempted to s

y: "Of course _x_ exists, forotherwise it could not be

d

tum." But such

st

tement is re

llyme

ningless,

lthough it is signific

nt

nd true to s

y "My presentsense-d

tum exists,"

nd it m

y

lso be true th

t "_x_ is my presentsense-d

tum." The inference from these two propositions to "_x_ exists" is one which seems irresistible to people un

ccustomed tologic; yet the

pp

rent proposition inferred is not merely f

lse, butstrictly me

ningless. To s

y "My present sense-d

tum exists" is to s

y

(roughly): "There is

n object of which 'my present sense-d

tum' is

description." But we c

nnot s

y: "There is

n object of which '_x_' is

description," bec

use '_x_' is (in the c

se we

re supposing)

n

me, not

description. Dr. Whitehe

d

nd I h

ve expl

ined this pointfully elsewhere (_loc. cit._) with the help of symbols, without whichit is h

rd to underst

nd; I sh

ll not therefore here repe

t thedemonstr

tion of the

bove propositions, but sh

ll proceed with their

pplic

tion to our present problem.

The f

ct th

t "existence" is only

pplic

ble to descriptions isconce

led by the use of wh

t

re gr

mm

tic

lly proper n

mes in

w

ywhich re

lly tr

nsforms them into descriptions. It is, for ex

mple,

legitim

te

uestion whether Homer existed; but here "Homer" me

ns

"the

uthor of the Homeric poems,"

nd is

description. Simil

rly wem

y

s

whether God exists; but then "God" me

ns "the Supreme Being"or "the _ens re

lissimum_" or wh

tever other description we m

yprefer. If "God" were

proper n

me, God would h

ve to be

d

tum;

ndthen no

uestion could

rise

s to His existence. The distinctionbetween existence

nd other predic

tes, which K

nt obscurely felt, isbrought to light by the theory of descriptions,

nd is seen to remove"existence"

ltogether from the fund

ment

l notions of met

physics.

Wh

t h

s been s

id

bout "existence"

pplies e

u

lly to "re

lity,"



which m

y, in f

ct, be t

en

s synonymous with "existence."Concerning the immedi

te objects in illusions, h

llucin

tions,

nddre

ms, it is me

ningless to

s

whether they "exist" or

re "re

l."There they

re,

nd th

t ends the m

tter. But we m

y legitim

telyin

uire

s to the existence or re

lity of "things" or other"sensibili

" inferred from such objects. It is the unre

lity of these"things"

nd other "sensibili

," together with

f

ilure to noticeth

t they

re not d

t

, which h

s led to the view th

t the objects ofdre

ms

re unre

l.

We m

y now

pply these consider

tions in det

il to the stoc

rguments

g

inst re

lism, though wh

t is to be s

id will be m

inly

repetitionof wh

t others h

ve s

id before.

(1) We h

ve first the v

riety of norm

l

ppe

r

nces, supposed to beincomp

tible. This is the c

se of the different sh

pes

nd colourswhich

given thing presents to different spect

tors. Loc

e's w

terwhich seems both hot

nd cold belongs to this cl

ss of c

ses. Oursystem of different perspectives fully

ccounts for these c

ses,

ndshows th

t they

fford no

rgument

g

inst re

lism.

(2) We h

ve c

ses where the correl

tion between different senses isunusu

l. The bent stic

in w

ter belongs here. People s

y it loo

sbent but is str

ight: this only me

ns th

t it is str

ight to the

touch, though bent to sight. There is no "illusion," but only

f

lseinference, if we thin

th

t the stic

would feel bent to the touch.The stic

would loo

just

s bent in

photogr

ph,

nd,

s Mr.Gl

dstone used to s

y, "the photogr

ph c

nnot lie."[34] The c

se ofseeing double

lso belongs here, though in this c

se the c

use of theunusu

l correl

tion is physiologic

l,

nd would therefore not oper

tein

photogr

ph. It is

mist

e to

s

whether the "thing" isduplic

ted when we see it double. The "thing" is

whole system of"sensibili

,"

nd it is only those visu

l "sensibili

" which

re d

t

to the percipient th

t

re duplic

ted. The phenomenon h

s

purelyphysiologic

l expl

n

tion; indeed, in view of our h

ving two eyes, itis in less need of expl

n

tion th

n the single visu

l sense-d

tumwhich we norm

lly obt

in from the things on which we focus.

(3) We come now to c

ses li

e dre

ms, which m

y,

t the moment ofdre

ming, cont

in nothing to

rouse suspicion, but

re condemned on theground of their supposed incomp

tibility with e

rlier

nd l

ter d

t

. Ofcourse it often h

ppens th

t dre

m-objects f

il to beh

ve in the

ccustomed m

nner: he

vy objects fly, solid objects melt, b

bies turninto pigs or undergo even gre

ter ch

nges. But none of these unusu

loccurrences _need_ h

ppen in

dre

m,

nd it is not on

ccount of suchoccurrences th

t dre

m-objects

re c

lled "unre

l." It is their l

c

ofcontinuity with the dre

mer's p

st

nd future th

t m

es him, when hew

es, condemn them;

nd it is their l

c

of correl

tion with otherpriv

te worlds th

t m

es others condemn them. Omitting the l

tterground, our re

son for condemning them is th

t the "things" which we

infer from them c

nnot be combined

ccording to the l

ws of physics withthe "things" inferred from w

ing sense-d

t

. This might be used tocondemn the "things" inferred from the d

t

of dre

ms. Dre

m-d

t

re nodoubt

ppe

r

nces of "things," but not of such "things"

s the dre

mersupposes. I h

ve no wish to comb

t psychologic

l theories of dre

ms,such

s those of the psycho-

n

lysts. But there cert

inly

re c

seswhere (wh

tever psychologic

l c

uses m

y contribute) the presence ofphysic

l c

uses

lso is very evident. For inst

nce,

door b

nging m

yproduce

dre

m of

n

v

l eng

gement, with im

ges of b

ttleships

ndse

nd smo

e. The whole dre

m will be

n

ppe

r

nce of the door



b

nging, but owing to the peculi

r condition of the body (especi

lly thebr

in) during sleep, this

ppe

r

nce is not th

t expected to be producedby

door b

nging,

nd thus the dre

mer is led to entert

in f

lsebeliefs. But his sense-d

t

re still physic

l,

nd

re such

s

completed physics would include

nd c

lcul

te.

(4) The l

st cl

ss of illusions

re those which c

nnot be discoveredwithin one person's experience, except through the discovery ofdiscrep

ncies with the experiences of others. Dre

ms might conceiv

blybelong to this cl

ss, if they were jointed sufficiently ne

tly intow

ing life; but the chief inst

nces

re recurrent sensoryh

llucin

tions of the

ind th

t le

d to ins

nity. Wh

t m

es thep

tient, in such c

ses, become wh

t others c

ll ins

ne is the f

ctth

t, within his own experience, there is nothing to show th

t theh

llucin

tory sense-d

t

do not h

ve the usu

l

ind of connection with"sensibili

" in other perspectives. Of course he m

y le

rn thisthrough testimony, but he prob

bly finds it simpler to suppose th

tthe testimony is untrue

nd th

t he is being wilfully deceived. Thereis, so f

r

s I c

n see, no theoretic

l criterion by which the p

tientc

n decide, in such

c

se, between the two e

u

lly s

tisf

ctoryhypotheses of his m

dness

nd of his friends' mend

city.

From the

bove inst

nces it would

ppe

r th

t

bnorm

l sense-d

t

, ofthe

ind which we reg

rd

s deceptive, h

ve intrinsic

lly just the

s

me st

tus

s

ny others, but differ

s reg

rds their correl

tions orc

us

l connections with other "sensibili

"

nd with "things." Sincethe usu

l correl

tions

nd connections become p

rt of our unreflectiveexpect

tions,

nd even seem, except to the psychologist, to form p

rtof our d

t

, it comes to be thought, mist

enly, th

t in such c

sesthe d

t

re unre

l, where

s they

re merely the c

uses of f

lseinferences. The f

ct th

t correl

tions

nd connections of unusu

l

inds occur

dds to the difficulty of inferring things from sense

ndof expressing physics in terms of sense-d

t

. But the unusu

lnesswould seem to be

lw

ys physic

lly or physiologic

lly explic

ble,

ndtherefore r

ises only

complic

tion, not

philosophic

l objection.

I conclude, therefore, th

t no v

lid objection exists to the view

which reg

rds sense-d

t

s p

rt of the

ctu

l subst

nce of thephysic

l world,

nd th

t, on the other h

nd, this view is the only onewhich

ccounts for the empiric

l verifi

bility of physics. In thepresent p

per, I h

ve given only

rough prelimin

ry s

etch. Inp

rticul

r, the p

rt pl

yed by _time_ in the construction of thephysic

l world is, I thin

, more fund

ment

l th

n would

ppe

r fromthe

bove

ccount. I should hope th

t, with further el

bor

tion, thep

rt pl

yed by unperceived "sensibili

" could be indefinitelydiminished, prob

bly by invo

ing the history of

"thing" to e

e outthe inferences deriv

ble from its moment

ry

ppe

r

nce.

FOOTNOTES:

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

[30] On this subject, comp

re _A Theory of Time

nd Sp

ce_, by Mr.A.A. Robb (C

mb. Univ. Press), which first suggested to me the views

dvoc

ted here, though I h

ve, for present purposes, omitted wh

t ismost interesting

nd novel in his theory. Mr. Robb h

s given

s

etchof his theory in

p

mphlet with the s

me title (Heffer

nd Sons,C

mbridge, 1913).

[31] "N

tur

l Re

lism

nd Present Tendencies in Philosophy," _Proc.



Arist. Soc._, 1908-1909, p. 165.

[32] _Die Erf

hrungsgrundl

gen unseres Wissens_, p. 28.

[33] Cf. _Principi

M

them

tic

_, Vol. I, * 14,

nd Introduction,Ch

p. III. For the definition of _existence_, cf. * 14. 02.

[34] Cf. Edwin B. Holt, _The Pl

ce of Illusory Experience in

Re

listic World._ "The New Re

lism," p. 303, both on this point

nd

sreg

rds _seeing double_.

IX

ON THE NOTION OF CAUSE

In the following p

per I wish, first, to m

int

in th

t the word"c

use" is so inextric

bly bound up with misle

ding

ssoci

tions

s tom

e its complete extrusion from the philosophic

l voc

bul

rydesir

ble; secondly, to in

uire wh

t principle, if

ny, is employed inscience in pl

ce of the supposed "l

w of c

us

lity" which philosophers

im

gine to be employed; thirdly, to exhibit cert

in confusions,especi

lly in reg

rd to teleology

nd determinism, which

ppe

r to meto be connected with erroneous notions

s to c

us

lity.

All philosophers, of every school, im

gine th

t c

us

tion is one ofthe fund

ment

l

xioms or postul

tes of science, yet, oddly enough, in

dv

nced sciences such

s gr

vit

tion

l

stronomy, the word "c

use"never occurs. Dr. J

mes W

rd, in his _N

tur

lism

nd Agnosticism_,m

es this

ground of compl

int

g

inst physics: the business ofthose who wish to

scert

in the ultim

te truth

bout the world, he

pp

rently thin

s, should be the discovery of c

uses, yet physicsnever even see

s them. To me it seems th

t philosophy ought not to

ssume such legisl

tive functions,

nd th

t the re

son why physics h

s

ce

sed to loo

for c

uses is th

t, in f

ct, there

re no such things.The l

w of c

us

lity, I believe, li

e much th

t p

sses muster

mongphilosophers, is

relic of

bygone

ge, surviving, li

e themon

rchy, only bec

use it is erroneously supposed to do no h

rm. Inorder to find out wh

t philosophers commonly underst

nd by "c

use," Iconsulted B

ldwin's _Diction

ry_,

nd w

s rew

rded beyond myexpect

tions, for I found the following three mutu

lly incomp

tibledefinitions:--

"CAUSALITY. (1) The necess

ry connection of events in the time-series....

"CAUSE (notion of). Wh

tever m

y be included in the thought or

perception of

process

s t

ing pl

ce in conse

uence of

nother process....

"CAUSE AND EFFECT. (1) C

use

nd effect ...

re correl

tive terms denoting

ny two distinguish

ble things, ph

ses, or

spects of re

lity, which

re so rel

ted to e

ch other th

t whenever the first ce

ses to exist the second comes into existence immedi

tely

fter,

nd whenever the second comes into existence the first h

s ce

sed to exist immedi

tely before."



Let us consider these three definitions in turn. The first, obviously,is unintelligible without

definition of "necess

ry." Under thishe

d, B

ldwin's _Diction

ry_ gives the following:--

"NECESSARY. Th

t is necess

ry which not only is true, but would be true under

ll circumst

nces. Something more th

n brute compulsion is, therefore, involved in the conception; there is

gener

l l

w under which the thing t

es pl

ce."

The notion of c

use is so intim

tely connected with th

t of necessityth

t it will be no digression to linger over the

bove definition,with

view to discovering, if possible, _some_ me

ning of which it isc

p

ble; for,

s it st

nds, it is very f

r from h

ving

ny definitesignific

tion.

The first point to notice is th

t, if

ny me

ning is to be given tothe phr

se "would be true under

ll circumst

nces," the subject of itmust be

proposition

l function, not

proposition.[35] Aproposition is simply true or f

lse,

nd th

t ends the m

tter: therec

n be no

uestion of "circumst

nces." "Ch

rles I's he

d w

s cut off"is just

s true in summer

s in winter, on Sund

ys

s on Mond

ys. Thuswhen it is worth s

ying th

t something "would be true under

llcircumst

nces," the something in

uestion must be

proposition

lfunction, i.e.

n expression cont

ining

v

ri

ble,

nd becoming

proposition when

v

lue is

ssigned to the v

ri

ble; the v

rying"circumst

nces"

lluded to

re then the different v

lues of which thev

ri

ble is c

p

ble. Thus if "necess

ry" me

ns "wh

t is true under

llcircumst

nces," then "if _x_ is

m

n, _x_ is mort

l" is necess

ry,bec

use it is true for

ny possible v

lue of _x_. Thus we should beled to the following definition:--

"NECESSARY is

predic

te of

proposition

l function, me

ning th

t it is true for

ll possible v

lues of its

rgument or

rguments."

Unfortun

tely, however, the definition in B

ldwin's _Diction

ry_ s

ysth

t wh

t is necess

ry is not only "true under

ll circumst

nces" but

is

lso "true." Now these two

re incomp

tible. Only propositions c

nbe "true,"

nd only proposition

l functions c

n be "true under

llcircumst

nces." Hence the definition

s it st

nds is nonsense. Wh

t isme

nt seems to be this: "A proposition is necess

ry when it is

v

lueof

proposition

l function which is true under

ll circumst

nces,i.e. for

ll v

lues of its

rgument or

rguments." But if we

doptthis definition, the s

me proposition will be necess

ry or contingent

ccording

s we choose one or other of its terms

s the

rgument toour proposition

l function. For ex

mple, "if Socr

tes is

m

n,Socr

tes is mort

l," is necess

ry if Socr

tes is chosen

s

rgument,but not if _m

n_ or _mort

l_ is chosen. Ag

in, "if Socr

tes is

m

n,Pl

to is mort

l," will be necess

ry if either Socr

tes or _m

n_ ischosen

s

rgument, but not if Pl

to or _mort

l_ is chosen. However,

this difficulty c

n be overcome by specifying the constituent which isto be reg

rded

s

rgument,

nd we thus

rrive

t the followingdefinition:

"A proposition is _necess

ry_ with respect to

given constituent ifit rem

ins true when th

t constituent is

ltered in

ny w

y comp

tiblewith the proposition rem

ining signific

nt."

We m

y now

pply this definition to the definition of c

us

lity

uoted

bove. It is obvious th

t the

rgument must be the time

t which the



e

rlier event occurs. Thus

n inst

nce of c

us

lity will be such

s:"If the event [M

th: e_{1}] occurs

t the time [M

th: t_{1}], it willbe followed by the event [M

th: e_{2}]." This proposition is intendedto be necess

ry with respect to [M

th: t_{1}], i.e. to rem

in truehowever [M

th: t_{1}] m

y be v

ried. C

us

lity,

s

univers

l l

w,will then be the following: "Given

ny event [M

th: t_{1}], there is

n event [M

th: e_{2}] such th

t, whenever [M

th: t_{1}] occurs,[M

th: e_{2}] occurs l

ter." But before this c

n be consideredprecise, we must specify how much l

ter [M

th: e_{2}] is to occur.Thus the principle becomes:--

"Given

ny event [M

th: e_{1}], there is

n event [M

th: e_{2}]

nd

time-interv

l t such ha , whenever [Ma h: e_{1}] occurs, [Ma h:e_{2}] follows af er an in erval t."

I am no concerned as ye o consider whe her his law is rue orfalse. For he presen , I am merely concerned o discover wha he lawof causali y is supposed o be. I pass, herefore, o he o herdefini ions quo ed above.

The second defini ion need no de ain us long, for wo reasons. Firs ,because i is psychological: no he " hough or percep ion" of aprocess, bu he process i self, mus be wha concerns us inconsidering causali y. Secondly, because i is circular: in speaking

of a process as "

aking place in consequence of" ano

her process, i

in roduces he very no ion of cause which was o be defined.

The hird defini ion is by far he mos precise; indeed as regardsclearness i leaves no hing o be desired. Bu a grea difficul y iscaused by he emporal con igui y of cause and effec which hedefini ion asser s. No wo ins an s are con iguous, since he

ime-series is compac ; hence ei her he cause or he effec or bo hmus , if he defini ion is correc , endure for a fini e ime; indeed,by he wording of he defini ion i is plain ha bo h are assumed oendure for a fini e ime. Bu hen we are faced wi h a dilemma: if hecause is a process involving change wi hin i self, we shall require(if causali y is universal) causal rela ions be ween i s earlier and

la

er par

s; moreover, i

would seem

ha

only

he la

er par

s can berelevan o he effec , since he earlier par s are no con iguous o he effec , and herefore (by he defini ion) canno influence heeffec . Thus we shall be led o diminish he dura ion of he causewi hou limi , and however much we may diminish i , here will s illremain an earlier par which migh be al ered wi hou al ering heeffec , so ha he rue cause, as defined, will no have beenreached, for i will be observed ha he defini ion excludesplurali y of causes. If, on he o her hand, he cause is purelys a ic, involving no change wi hin i self, hen, in he firs place,no such cause is o be found in na ure, and in he second place, i

seems s range-- oo s range o be accep ed, in spi e of bare logicalpossibili y-- ha he cause, af er exis ing placidly for some ime,

should suddenly explode in

o

he effec

, when i

migh

jus

as wellhave done so a any earlier ime, or have gone on unchanged wi hou

producing i s effec . This dilemma, herefore, is fa al o he view ha cause and effec can be con iguous in ime; if here are causesand effec s, hey mus be separa ed by a fini e ime-in erval t, aswas assumed in he above in erpre a ion of he firs defini ion.

Wha is essen ially he same s a emen of he law of causali y as heone elici ed above from he firs of Baldwin's defini ions is given byo her philosophers. Thus John S uar Mill says:--



"The Law of Causa ion, he recogni ion of which is he main pillar ofinduc ive science, is bu he familiar ru h, ha invariabili y ofsuccession is found by observa ion o ob ain be ween every fac inna ure and some o her fac which has preceded i ."[36]

And Bergson, who has righ ly perceived ha he law as s a ed byphilosophers is wor hless, never heless con inues o suppose ha i

is used in science. Thus he says:--

"Now, i is argued, his law [ he law of causali y] means ha everyphenomenon is de ermined by i s condi ions, or, in o her words, ha

he same causes produce he same effec s."[37]

And again:--

"We perceive physical phenomena, and hese phenomena obey laws. Thismeans: (1) Tha phenomena _a_, _b_, _c_, _d_, previously perceived,can occur again in he same shape; (2) ha a cer ain phenomenon P,which appeared af er he condi ions _a_, _b_, _c_, _d_, and af er

hese condi ions only, will no fail o recur as soon as he samecondi ions are again presen ."[38]

A grea par of Bergson's a ack on science res s on he assump ion

ha

i

employs

his principle. In fac

, i

employs no such principle,bu philosophers--even Bergson--are oo ap o ake heir views onscience from each o her, no from science. As o wha he principleis, here is a fair consensus among philosophers of differen schools.There are, however, a number of difficul ies which a once arise. Iomi he ques ion of plurali y of causes for he presen , since o hergraver ques ions have o be considered. Two of hese, which are forcedon our a en ion by he above s a emen of he law, are hefollowing:--

(1) Wha is mean by an "even "?

(2) How long may he ime-in erval be be ween cause and effec ?

(1) An "even ," in he s a emen of he law, is obviously in ended obe some hing ha is likely o recur since o herwise he law becomes

rivial. I follows ha an "even " is no a par icular, bu someuniversal of which here may be many ins ances. I follows also ha

an "even " mus be some hing shor of he whole s a e of he universe,since i is highly improbable ha his will recur. Wha is mean byan "even " is some hing like s riking a ma ch, or dropping a pennyin o he slo of an au oma ic machine. If such an even is o recur,i mus no be defined oo narrowly: we mus no s a e wi h wha

degree of force he ma ch is o be s ruck, nor wha is o be he empera ure of he penny. For if such considera ions were relevan ,our "even " would occur a mos once, and he law would cease o give

informa

ion. An "even

,"

hen, is a universal defined sufficien

lywidely o admi of many par icular occurrences in ime being ins ancesof i .

(2) The nex ques ion concerns he ime-in erval. Philosophers, nodoub , hink of cause and effec as con iguous in ime, bu his, forreasons already given, is impossible. Hence, since here are noinfini esimal ime-in ervals, here mus be some fini e lapse of imet be ween cause and effec . This, however, a once raises insuperabledifficul ies. However shor we make he in erval t, some hing may



happen during his in erval which preven s he expec ed resul . I pu

my penny in he slo , bu before I can draw ou my icke here is anear hquake which upse s he machine and my calcula ions. In order obe sure of he expec ed effec , we mus know ha here is no hing in

he environmen o in erfere wi h i . Bu his means ha he supposedcause is no , by i self, adequa e o insure he effec . And as soon aswe include he environmen , he probabili y of repe i ion isdiminished, un il a las , when he whole environmen is included, heprobabili y of repe i ion becomes almos _nil_.

In spi e of hese difficul ies, i mus , of course, be admi ed ha

many fairly dependable regulari ies of sequence occur in daily life.I is hese regulari ies ha have sugges ed he supposed law ofcausali y; where hey are found o fail, i is hough ha a be erformula ion could have been found which would have never failed. I amfar from denying ha here may be such sequences which in fac neverdo fail. I may be ha here will never be an excep ion o he rule

ha when a s one of more han a cer ain mass, moving wi h more han acer ain veloci y, comes in con ac wi h a pane of glass of less hana cer ain hickness, he glass breaks. I also do no deny ha heobserva ion of such regulari ies, even when hey are no wi hou

excep ions, is useful in he infancy of a science: he observa ion ha unsuppor ed bodies in air usually fall was a s age on he way o he law of gravi a ion. Wha I deny is ha science assumes he

exis

ence of invariable uniformi

ies of sequence of

his kind, or

ha

i aims a discovering hem. All such uniformi ies, as we saw, dependupon a cer ain vagueness in he defini ion of he "even s." Tha

bodies fall is a vague quali a ive s a emen ; science wishes o knowhow fas hey fall. This depends upon he shape of he bodies and hedensi y of he air. I is rue ha here is more nearly uniformi ywhen hey fall in a vacuum; so far as Galileo could observe, heuniformi y is hen comple e. Bu la er i appeared ha even here hela i ude made a difference, and he al i ude. Theore ically, heposi ion of he sun and moon mus make a difference. In shor , everyadvance in a science akes us far her away from he crude uniformi ieswhich are firs observed, in o grea er differen ia ion of an eceden

and consequen , and in o a con inually wider circle of an eceden s

recognised as relevan

.

The principle "same cause, same effec ," which philosophers imagine obe vi al o science, is herefore u erly o iose. As soon as hean eceden s have been given sufficien ly fully o enable heconsequen o be calcula ed wi h some exac i ude, he an eceden s havebecome so complica ed ha i is very unlikely hey will ever recur.Hence, if his were he principle involved, science would remainu erly s erile.

The impor ance of hese considera ions lies par ly in he fac ha

hey lead o a more correc accoun of scien ific procedure, par ly in he fac ha hey remove he analogy wi h human voli ion which makes

he concep

ion of cause such a frui

ful source of fallacies. Thela er poin will become clearer by he help of some illus ra ions.For his purpose I shall consider a few maxims which have played agrea par in he his ory of philosophy.

(1) "Cause and effec mus more or less resemble each o her." Thisprinciple was prominen in he philosophy of occasionalism, and iss ill by no means ex inc . I is s ill of en hough , for example,

ha mind could no have grown up in a universe which previouslycon ained no hing men al, and one ground for his belief is ha



ma er is oo dissimilar from mind o have been able o cause i . Or,more par icularly, wha are ermed he nobler par s of our na ure aresupposed o be inexplicable, unless he universe always con ainedsome hing a leas equally noble which could cause hem. All suchviews seem o depend upon assuming some unduly simplified law ofcausali y; for, in any legi ima e sense of "cause" and "effec ,"science seems o show ha hey are usually very widely dissimilar,

he "cause" being, in fac , wo s a es of he whole universe, and he"effec " some par icular even .

(2) "Cause is analogous o voli ion, since here mus be anin elligible _nexus_ be ween cause and effec ." This maxim is, I

hink, of en unconsciously in he imagina ions of philosophers whowould rejec i when explici ly s a ed. I is probably opera ive in

he view we have jus been considering, ha mind could no haveresul ed from a purely ma erial world. I do no profess o know wha

is mean by "in elligible"; i seems o mean "familiar oimagina ion." No hing is less "in elligible," in any o her sense, han

he connec ion be ween an ac of will and i s fulfilmen . Bu

obviously he sor of nexus desired be ween cause and effec is suchas could only hold be ween he "even s" which he supposed law ofcausali y con empla es; he laws which replace causali y in such ascience as physics leave no room for any wo even s be ween which anexus could be sough .

(3) "The cause _compels_ he effec in some sense in which he effec

does no compel he cause." This belief seems largely opera ive in hedislike of de erminism; bu , as a ma er of fac , i is connec ed wi hour second maxim, and falls as soon as ha is abandoned. We maydefine "compulsion" as follows: "Any se of circums ances is said ocompel A when A desires o do some hing which he circums ancespreven , or o abs ain from some hing which he circums ances cause."This presupposes ha some meaning has been found for he word"cause"--a poin o which I shall re urn la er. Wha I wan o makeclear a presen is ha compulsion is a very complex no ion,involving hwar ed desire. So long as a person does wha he wishes odo, here is no compulsion, however much his wishes may be calculable

by

he help of earlier even

s. And where desire does no

come in, here can be no ques ion of compulsion. Hence i is, in general,misleading o regard he cause as compelling he effec .

A vaguer form of he same maxim subs i u es he word "de ermine" for he word "compel"; we are old ha he cause _de ermines_ he effec

in a sense in which he effec does no _de ermine_ he cause. I isno qui e clear wha is mean by "de ermining"; he only precisesense, so far as I know, is ha of a func ion or one-many rela ion.If we admi plurali y of causes, bu no of effec s, ha is, if wesuppose ha , given he cause, he effec mus be such and such, bu ,given he effec , he cause may have been one of many al erna ives,

hen we may say ha he cause de ermines he effec , bu no he

effec

he cause. Plurali

y of causes, however, resul

s only fromconceiving he effec vaguely and narrowly and he cause precisely andwidely. Many an eceden s may "cause" a man's dea h, because his dea his vague and narrow. Bu if we adop he opposi e course, aking as

he "cause" he drinking of a dose of arsenic, and as he "effec " hewhole s a e of he world five minu es la er, we shall have plurali yof effec s ins ead of plurali y of causes. Thus he supposed lack ofsymme ry be ween "cause" and "effec " is illusory.

(4) "A cause canno opera e when i has ceased o exis , because wha



has ceased o exis is no hing." This is a common maxim, and a s illmore common unexpressed prejudice. I has, I fancy, a good deal o dowi h he a rac iveness of Bergson's "_durée_": since he pas haseffec s now, i mus s ill exis in some sense. The mis ake in hismaxim consis s in he supposi ion ha causes "opera e" a all. Avoli ion "opera es" when wha i wills akes place; bu no hing canopera e excep a voli ion. The belief ha causes "opera e" resul sfrom assimila ing hem, consciously or unconsciously, o voli ions. Wehave already seen ha , if here are causes a all, hey mus besepara ed by a fini e in erval of ime from heir effec s, and huscause heir effec s af er hey have ceased o exis .

I may be objec ed o he above defini ion of a voli ion "opera ing" ha i only opera es when i "causes" wha i wills, no when i

merely happens o be followed by wha i wills. This cer ainlyrepresen s he usual view of wha is mean by a voli ion "opera ing,"bu as i involves he very view of causa ion which we are engaged incomba ing, i is no open o us as a defini ion. We may say ha avoli ion "opera es" when here is some law in vir ue of which asimilar voli ion in ra her similar circums ances will usually befollowed by wha i wills. Bu his is a vague concep ion, andin roduces ideas which we have no ye considered. Wha is chieflyimpor an o no ice is ha he usual no ion of "opera ing" is no

open o us if we rejec , as I con end ha we should, he usual no ion

of causa

ion.

(5) "A cause canno opera e excep where i is." This maxim is verywidespread; i was urged agains New on, and has remained a source ofprejudice agains "ac ion a a dis ance." In philosophy i has led oa denial of ransien ac ion, and hence o monism or Leibnizianmonadism. Like he analogous maxim concerning emporal con igui y, i

res s upon he assump ion ha causes "opera e," i.e. ha hey are insome obscure way analogous o voli ions. And, as in he case of

emporal con igui y, he inferences drawn from his maxim are whollygroundless.

I re urn now o he ques ion, Wha law or laws can be found o ake

he place of

he supposed law of causali

y?

Firs , wi hou passing beyond such uniformi ies of sequence as arecon empla ed by he radi ional law, we may admi ha , if any suchsequence has been observed in a grea many cases, and has never beenfound o fail, here is an induc ive probabili y ha i will be found

o hold in fu ure cases. If s ones have hi her o been found o breakwindows, i is probable ha hey will con inue o do so. This, ofcourse, assumes he induc ive principle, of which he ru h mayreasonably be ques ioned; bu as his principle is no our presen

concern, I shall in his discussion rea i as indubi able. We may hen say, in he case of any such frequen ly observed sequence, ha

he earlier even is he _cause_ and he la er even he _effec _.

Several considera ions, however, make such special sequences verydifferen from he radi ional rela ion of cause and effec . In hefirs place, he sequence, in any hi her o unobserved ins ance, is nomore han probable, whereas he rela ion of cause and effec wassupposed o be necessary. I do no mean by his merely ha we are no

sure of having discovered a rue case of cause and effec ; I mean ha , even when we have a case of cause and effec in our presen

sense, all ha is mean is ha on grounds of observa ion, i isprobable ha when one occurs he o her will also occur. Thus in our



change is de ermina e when he s a e of he universe is given." If he"law of causali y" is o be some hing ac ually discoverable in heprac ice of science, he above proposi ion has a be er righ o hename han any "law of causali y" o be found in he books ofphilosophers.

In regard o he above principle, several observa ions mus be made--

(1) No one can pre end ha he above principle is _a priori_ orself-eviden or a "necessi y of hough ." Nor is i , in any sense, apremiss of science: i is an empirical generalisa ion from a number oflaws which are hemselves empirical generalisa ions.

(2) The law makes no difference be ween pas and fu ure: he fu ure"de ermines" he pas in exac ly he same sense in which he pas

"de ermines" he fu ure. The word "de ermine," here, has a purelylogical significance: a cer ain number of variables "de ermine"ano her variable if ha o her variable is a func ion of hem.

(3) The law will no be empirically verifiable unless he course ofeven s wi hin some sufficien ly small volume will be approxima ely

he same in any wo s a es of he universe which only differ in regard o wha is a a considerable dis ance from he small volume inques ion. For example, mo ions of plane s in he solar sys em mus be

approxima

ely

he same however

he fixed s

ars may be dis

ribu

ed,provided ha all he fixed s ars are very much far her from he sun han he plane s are. If gravi a ion varied direc ly as he dis ance,so ha he mos remo e s ars made he mos difference o he mo ionsof he plane s, he world migh be jus as regular and jus as muchsubjec o ma hema ical laws as i is a presen , bu we could neverdiscover he fac .

(4) Al hough he old "law of causali y" is no assumed by science,some hing which we may call he "uniformi y of na ure" is assumed, orra her is accep ed on induc ive grounds. The uniformi y of na ure doesno asser he rivial principle "same cause, same effec ," bu heprinciple of he permanence of laws. Tha is o say, when a law

exhibi

ing, e.g. an accelera

ion as a func

ion of

he configura

ionhas been found o hold hroughou he observable pas , i is expec ed ha i will con inue o hold in he fu ure, or ha , if i does no

i self hold, here is some o her law, agreeing wi h he supposed lawas regards he pas , which will hold for he fu ure. The ground of

his principle is simply he induc ive ground ha i has been found o be rue in very many ins ances; hence he principle canno beconsidered cer ain, bu only probable o a degree which canno beaccura ely es ima ed.

The uniformi y of na ure, in he above sense, al hough i is assumedin he prac ice of science, mus no , in i s generali y, be regardedas a kind of major premiss, wi hou which all scien ific reasoning

would be in error. The assump

ion

ha

_all_ laws of na

ure arepermanen has, of course, less probabili y han he assump ion ha

his or ha par icular law is permanen ; and he assump ion ha apar icular law is permanen for all ime has less probabili y han heassump ion ha i will be valid up o such and such a da e. Science,in any given case, will assume wha he case requires, bu no more. Incons ruc ing he _Nau ical Almanac_ for 1915 i will assume ha helaw of gravi a ion will remain rue up o he end of ha year; bu i

will make no assump ion as o 1916 un il i comes o he nex volumeof he almanac. This procedure is, of course, dic a ed by he fac



generally, even s a any ime from even s a cer ain assigned imes.Any sys em in which such inference is possible may be called a"de erminis ic" sys em. We may define a de erminis ic sys em asfollows:--

A sys em is said o be "de erminis ic" when, given cer ain da a, [Ma h: e_{1}, e_{2}, ..., e_{n}, a imes _{1}, _{2}, ..., _{n}] respec ively, concerning his sys em, if [Ma h: E_{ }] is he s a e of he sys em a any ime _ _, here is a func ional rela ion of he form

[Ma h: E_{ } = f (e_{1}, _{1}, e_{2}, _{2}, ..., e_{n}, _{n}, )]. (A)

The sys em will be "de erminis ic hroughou a given period" if _ _, in he above formula, may be any ime wi hin ha period, hough ou side ha period he formula may be no longer rue. If he universe, as a whole, is such a sys em, de erminism is rue of he universe; if no , no . A sys em which is par of a de erminis ic sys em I shall call "de ermined"; one which is no

par of any such sys em I shall call "capricious."

The even s [Ma h: e_{1}, e_{2}, ..., e_{n}] I shall call "de erminan s"of he sys em. I is o be observed ha a sys em which has one se ofde erminan s will in general have many. In he case of he mo ions of

he plane

s, for example,

he configura

ions of

he solar sys

em a

any wo given imes will be de erminan s.

We may ake ano her illus ra ion from he hypo hesis ofpsycho-physical parallelism. Le us assume, for he purposes of hisillus ra ion, ha o a given s a e of brain a given s a e of mindalways corresponds, and vice versa, i.e. ha here is a one-onerela ion be ween hem, so ha each is a func ion of he o her. We mayalso assume, wha is prac ically cer ain, ha o a given s a e of acer ain brain a given s a e of he whole ma erial universecorresponds, since i is highly improbable ha a given brain is ever

wice in exac ly he same s a e. Hence here will be a one-onerela ion be ween he s a e of a given person's mind and he s a e of

he whole ma

erial universe. I

follows

ha

, if _n_ s

a

es of

hema erial universe are de erminan s of he ma erial universe, hen _n_ s a es of a given man's mind are de erminan s of he whole ma erialand men al universe--assuming, ha is o say, ha psycho-physicalparallelism is rue.

The above illus ra ion is impor an in connec ion wi h a cer ainconfusion which seems o have bese hose who have philosophised on

he rela ion of mind and ma er. I is of en hough ha , if hes a e of he mind is de ermina e when he s a e of he brain is given,and if he ma erial world forms a de erminis ic sys em, hen mind is"subjec " o ma er in some sense in which ma er is no "subjec " omind. Bu if he s a e of he brain is also de ermina e when he s a e

of

he mind is given, i

mus

be exac

ly as

rue

o regard ma

er assubjec o mind as i would be o regard mind as subjec o ma er. Wecould, heore ically, work ou he his ory of mind wi hou evermen ioning ma er, and hen, a he end, deduce ha ma er mus

meanwhile have gone hrough he corresponding his ory. I is rue ha

if he rela ion of brain o mind were many-one, no one-one, herewould be a one-sided dependence of mind on brain, while conversely, if

he rela ion were one-many, as Bergson supposes, here would be aone-aided dependence of brain on mind. Bu he dependence involved is,in any case, only logical; i does no mean ha we shall be



compelled o do hings we desire no o do, which is wha peopleins inc ively imagine i o mean.

As ano her illus ra ion we may ake he case of mechanism and eleology. A sys em may be defined as "mechanical" when i has a se

of de erminan s ha are purely ma erial, such as he posi ions ofcer ain pieces of ma er a cer ain imes. I is an open ques ionwhe her he world of mind and ma er, as we know i , is a mechanicalsys em or no ; le us suppose, for he sake of argumen , ha i is amechanical sys em. This supposi ion--so I con end-- hrows no ligh

wha ever on he ques ion whe her he universe is or is no a" eleological" sys em. I is difficul o define accura ely wha ismean by a " eleological" sys em, bu he argumen is no muchaffec ed by he par icular defini ion we adop . Broadly, a

eleological sys em is one in which purposes are realised, i.e. inwhich cer ain desires-- hose ha are deeper or nobler or morefundamen al or more universal or wha no --are followed by heirrealisa ion. Now he fac --if i be a fac -- ha he universe ismechanical has no bearing wha ever on he ques ion whe her i is

eleological in he above sense. There migh be a mechanical sys em inwhich all wishes were realised, and here migh be one in which allwishes were hwar ed. The ques ion whe her, or how far, our ac ualworld is eleological, canno , herefore, be se led by proving ha

i is mechanical, and he desire ha i should be eleological is no

ground for wishing i

o be no

mechanical.

There is, in all hese ques ions, a very grea difficul y in avoidingconfusion be ween wha we can infer and wha is in fac de ermined.Le us consider, for a momen , he various senses in which he fu uremay be "de ermined." There is one sense--and a very impor an one--inwhich i is de ermined qui e independen ly of scien ific laws, namely,

he sense ha i will be wha i will be. We all regard he pas asde ermined simply by he fac ha i has happened; bu for heacciden ha memory works backward and no forward, we should regard

he fu ure as equally de ermined by he fac ha i will happen."Bu ," we are old, "you canno al er he pas , while you can o someex en al er he fu ure." This view seems o me o res upon jus

hose errors in regard

o causa

ion which i

has been my objec

oremove. You canno make he pas o her han i was-- rue, bu his isa mere applica ion of he law of con radic ion. If you already knowwha he pas was, obviously i is useless o wish i differen . Bu

also you canno make he fu ure o her han i will be; his again isan applica ion of he law of con radic ion. And if you happen o know

he fu ure--e.g. in he case of a for hcoming eclipse--i is jus asuseless o wish i differen as o wish he pas differen . "Bu ," i

will be rejoined, "our wishes can _cause_ he fu ure, some imes, o bedifferen from wha i would be if hey did no exis , and hey canhave no such effec upon he pas ." This, again, is a mere au ology.An effec being _defined_ as some hing subsequen o i s cause,obviously we can have no _effec _ upon he pas . Bu ha does no

mean

ha

he pas

would no

have been differen

if our presen

wisheshad been differen . Obviously, our presen wishes are condi ioned by

he pas , and herefore could no have been differen unless he pas

had been differen ; herefore, if our presen wishes were differen , he pas would be differen . Of course, he pas canno be differen

from wha i was, bu no more can our presen wishes be differen fromwha hey are; his again is merely he law of con radic ion. Thefac s seem o be merely (1) ha wishing generally depends uponignorance, and is herefore commoner in regard o he fu ure han inregard o he pas ; (2) ha where a wish concerns he fu ure, i and



i s realisa ion very of en form a "prac ically independen sys em,"i.e. many wishes regarding he fu ure are realised. Bu here seems nodoub ha he main difference in our feelings arises from heacciden al fac ha he pas bu no he fu ure can be known bymemory.

Al hough he sense of "de ermined" in which he fu ure is de erminedby he mere fac ha i will be wha i will be is sufficien (a

leas so i seems o me) o refu e some opponen s of de erminism,no ably M. Bergson and he pragma is s, ye i is no wha mos peoplehave in mind when hey speak of he fu ure as de ermined. Wha heyhave in mind is a formula by means of which he fu ure can beexhibi ed, and a leas heore ically calcula ed, as a func ion of hepas . Bu a his poin we mee wi h a grea difficul y, which bese swha has been said above abou de erminis ic sys ems, as well as wha

is said by o hers.

If formulæ of any degree of complexi y, however grea , are admi ed,i would seem ha any sys em, whose s a e a a given momen is afunc ion of cer ain measurable quan i ies, mus be a de erminis icsys em. Le us consider, in illus ra ion, a single ma erial par icle,whose co-ordina es a ime _ _ are [Ma h: x_{ }, y_{ }, z_{ }]. Then,however, he par icle moves, here mus be, heore ically, func ions[Ma h: f_{1}, f_{2}, f_{3}], such ha

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

I follows ha , heore ically, he whole s a e of he ma erialuniverse a ime _ _ mus be capable of being exhibi ed as a func ionof _ _. Hence our universe will be de erminis ic in he sense definedabove. Bu if his be rue, no informa ion is conveyed abou heuniverse in s a ing ha i is de erminis ic. I is rue ha heformulæ involved may be of s ric ly infini e complexi y, and hereforeno prac ically capable of being wri en down or apprehended. Bu

excep from he poin of view of our knowledge, his migh seem o bea de ail: in i self, if he above considera ions are sound, hema erial universe _mus _ be de erminis ic, _mus _ be subjec o laws.

This, however, is plainly no wha was in ended. The differencebe ween his view and he view in ended may be seen as follows. Givensome formula which fi s he fac s hi her o--say he law ofgravi a ion-- here will be an infini e number of o her formulæ, no

empirically dis inguishable from i in he pas , bu diverging from i

more and more in he fu ure. Hence, even assuming ha here arepersis en laws, we shall have no reason for assuming ha he law of

he inverse square will hold in fu ure; i may be some o her hi her oindis inguishable law ha will hold. We canno say ha _every_ lawwhich has held hi her o mus hold in he fu ure, because pas fac swhich obey one law will also obey o hers, hi her o indis inguishablebu diverging in fu ure. Hence here mus , a every momen , be laws

hi

her

o unbroken which are now broken for

he firs

ime. Wha

science does, in fac , is o selec he _simples _ formula ha willfi he fac s. Bu his, qui e obviously, is merely a me hodologicalprecep , no a law of Na ure. If he simples formula ceases, af er a

ime, o be applicable, he simples formula ha remains applicableis selec ed, and science has no sense ha an axiom has beenfalsified. We are hus lef wi h he bru e fac ha , in manydepar men s of science, qui e simple laws have hi her o been found ohold. This fac canno be regarded as having any _a priori_ ground,nor can i be used o suppor induc ively he opinion ha he same



laws will con inue; for a every momen laws hi her o rue are beingfalsified, hough in he advanced sciences hese laws are less simple

han hose ha have remained rue. Moreover i would be fallacious oargue induc ively from he s a e of he advanced sciences o hefu ure s a e of he o hers, for i may well be ha he advancedsciences are advanced simply because, hi her o, heir subjec -ma erhas obeyed simple and easily ascer ainable laws, while hesubjec -ma er of o her sciences has no done so.

The difficul y we have been considering seems o be me par ly, if no

wholly, by he principle ha he _ ime_ mus no en er explici lyin o our formulæ. All mechanical laws exhibi accelera ion as afunc ion of configura ion, no of configura ion and ime join ly; and

his principle of he irrelevance of he ime may be ex ended o allscien ific laws. In fac we migh in erpre he "uniformi y of na ure"as meaning jus his, ha no scien ific law involves he ime as anargumen , unless, of course, i is given in an in egra ed form, inwhich case _lapse_ of ime, hough no absolu e ime, may appear inour formulæ. Whe her his considera ion suffices o overcome ourdifficul y comple ely, I do no know; bu in any case i does much odiminish i .

I will serve o illus ra e wha has been said if we apply i o heques ion of free will.

(1) De erminism in regard o he will is he doc rine ha ourvoli ions belong o some de erminis ic sys em, i.e. are "de ermined"in he sense defined above. Whe her his doc rine is rue or false, isa mere ques ion of fac ; no _a priori_ considera ions (if our previousdiscussions have been correc ) can exis on ei her side. On he onehand, here is no _a priori_ ca egory of causali y, bu merely cer ainobserved uniformi ies. As a ma er of fac , here are observeduniformi ies in regard o voli ions; hus here is some empiricalevidence ha voli ions are de ermined. Bu i would be very rash omain ain ha he evidence is overwhelming, and i is qui e possible

ha some voli ions, as well as some o her hings, are no de ermined,excep in he sense in which we found ha every hing mus be

de

ermined.

(2) Bu , on he o her hand, he subjec ive sense of freedom, some imesalleged agains de erminism, has no bearing on he ques ion wha ever.The view ha i has a bearing res s upon he belief ha causescompel heir effec s, or ha na ure enforces obedience o i s laws asgovernmen s do. These are mere an hropomorphic supers i ions, due oassimila ion of causes wi h voli ions and of na ural laws wi h humanedic s. We feel ha our will is no compelled, bu ha only means

ha i is no o her han we choose i o be. I is one of hedemeri s of he radi ional heory of causali y ha i has crea ed anar ificial opposi ion be ween de erminism and he freedom of which weare in rospec ively conscious.

(3) Besides he general ques ion whe her voli ions are de ermined, here is he fur her ques ion whe her hey are _mechanically_ de ermined, i.e. whe her hey are par of wha was above defined as amechanical sys em. This is he ques ion whe her hey form par of asys em wi h purely ma erial de erminan s, i.e. whe her here are lawswhich, given cer ain ma erial da a, make all voli ions func ions of

hose da a. Here again, here is empirical evidence up o a poin , bu

i is no conclusive in regard o all voli ions. I is impor an oobserve, however ha even if voli ions are par of a mechanical



sys em, his by no means implies any supremacy of ma er over mind. I

may well be ha he same sys em which is suscep ible of ma erialde erminan s is also suscep ible of men al de erminan s; hus amechanical sys em may be de ermined by se s of voli ions, as well asby se s of ma erial fac s. I would seem, herefore, ha he reasonswhich make people dislike he view ha voli ions are mechanicallyde ermined are fallacious.

(4) The no ion of _necessi y_, which is of en associa ed wi hde erminism, is a confused no ion no legi ima ely deducible fromde erminism. Three meanings are commonly confounded when necessi y isspoken of:--

(a) An _

ction_ is necess

ry when it will be performed however muchthe

gent m

y wish to do otherwise. Determinism does not imply th

t

ctions

re necess

ry in this sense.

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

(g) A _proposition_ is necessary with respect to a iven constituentwhen it is the value, with that constituent as ar

ument, of anecessary propositional function, in other words, when it remains truehowever that constituent may be varied. In this sense, in a

deterministic system, the connection of a volition with itsdeterminants is necessary, if the time at which the determinants occurbe taken as the constituent to be varied, the time-interval betweenthe determinants and the volition bein

kept constant. But this senseof necessity is purely lo ical, and has no emotional importance.

We may now sum up our discussion of causality. We found first that thelaw of causality, as usually stated by philosophers, is false, and isnot employed in science. We then considered the nature of scientificlaws, and found that, instead of statin

that one event A is alwaysfollowed by another event B, they stated functional relations betweencertain events at certain times, which we called determinants, andother events at earlier or later times or at the same time. We were

unable to find any _a priori_ cate

ory involved: the existence ofscientific laws appeared as a purely empirical fact, not necessarilyuniversal, except in a trivial and scientifically useless form. Wefound that a system with one set of determinants may very likely haveother sets of a quite different kind, that, for example, amechanically determined system may also be teleolo ically orvolitionally determined. Finally we considered the problem of freewill: here we found that the reasons for supposin

volitions to bedetermined are stron but not conclusive, and we decided that even ifvolitions are mechanically determined, that is no reason for denyin

freedom in the sense revealed by introspection, or for supposin thatmechanical events are not determined by volitions. The problem of freewill _versus_ determinism is therefore, if we were ri ht, mainly

illusory, but in part not yet capable of bein

decisively solved.

FOOTNOTES:

[35] A propositional function is an expression containin a variable,or undetermined constituent, and becomin a proposition as soon as adefinite value is assi

ned to the variable. Examples are: "A is A,""_x_ is a number." The variable is called the _ar ument_ of thefunction.



acquainted, the first and most obvious example is _sense-data_. When Isee a colour or hear a noise, I have direct acquaintance with thecolour or the noise. The sense-datum with which I am acquainted inthese cases is enerally, if not always, complex. This isparticularly obvious in the case of si ht. I do not mean, of course,merely that the supposed physical object is complex, but that thedirect sensible object is complex and contains parts with spatialrelations. Whether it is possible to be aware of a complex withoutbein aware of its constituents is not an easy question, but on thewhole it would seem that there is no reason why it should not bepossible. This question arises in an acute form in connection withself-consciousness, which we must now briefly consider.

In introspection, we seem to be immediately aware of varyin

complexes, consistin of objects in various co nitive and conativerelations to ourselves. When I see the sun, it often happens that Iam aware of my seein the sun, in addition to bein aware of the sun;and when I desire food, it often happens that I am aware of my desirefor food. But it is hard to discover any state of mind in which I amaware of myself alone, as opposed to a complex of which I am aconstituent. The question of the nature of self-consciousness is toolar

e and too sli

htly connected with our subject, to be ar

ued atlen th here. It is difficult, but probably not impossible, to accountfor plain facts if we assume that we do not have acquaintance with

ourselves. It is plain that we are not only _acquainted_ with thecomplex "Self-acquainted-with-A," but we also _know_ the proposition"I am acquainted with A." Now here the complex has been analysed, andif "I" does not stand for somethin

which is a direct object ofacquaintance, we shall have to suppose that "I" is somethin known bydescription. If we wished to maintain the view that there is noacquaintance with Self, we mi ht ar ue as follows: We are acquaintedwith _acquaintance_, and we know that it is a relation. Also we areacquainted with a complex in which we perceive that acquaintance isthe relatin

relation. Hence we know that this complex must have aconstituent which is that which is acquainted, i.e. must have asubject-term as well as an object-term. This subject-term we defineas "I." Thus "I" means "the subject-term in awarenesses of which _I_

am aware." But as a definition this cannot be re

arded as a happyeffort. It would seem necessary, therefore, either to suppose that Iam acquainted with myself, and that "I," therefore, requires nodefinition, bein

merely the proper name of a certain object, or tofind some other analysis of self-consciousness. Thus self-consciousnesscannot be re arded as throwin li ht on the question whether we canknow a complex without knowin its constituents. This question,however, is not important for our present purposes, and I shalltherefore not discuss it further.

The awarenesses we have considered so far have all been awarenesses ofparticular existents, and mi

ht all in a lar

e sense be calledsense-data. For, from the point of view of theory of knowled e,

introspective knowled

e is exactly on a level with knowled

e derivedfrom si ht or hearin . But, in addition to awareness of the above kindof objects, which may be called awareness of _particulars_; we havealso (thou h not quite in the same sense) what may be called awarenessof _universals_. Awareness of universals is called _conceivin _, and auniversal of which we are aware is called a _concept_. Not only are weaware of particular yellows, but if we have seen a sufficient numberof yellows and have sufficient intelli ence, we are aware of theuniversal _yellow_; this universal is the subject in such jud ments as"yellow differs from blue" or "yellow resembles blue less than reen



does." And the universal yellow is the predicate in such jud ments as"this is yellow," where "this" is a particular sense-datum. Anduniversal 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 re ard to relations, it mi ht be ur ed that we are never aware ofthe universal relation itself, but only of complexes in which it is aconstituent. For example, it may be said that we do not know directlysuch a relation as _before_, thou

h we understand such a propositionas "this is before that," and may be directly aware of such a complexas "this bein

before that." This view, however, is difficult toreconcile with the fact that we often know propositions in which therelation is the subject, or in which the relata are not definite

ivenobjects, but "anythin ." For example, we know that if one thin isbefore another, and the other before a third, then the first is beforethe third; and here the thin s concerned are not definite thin s, but"anythin

." It is hard to see how we could know such a fact about"before" unless we were acquainted with "before," and not merely withactual particular cases of one

iven object bein

before another

ivenobject. And more directly: A jud ment such as "this is before that,"where this jud

ment is derived from awareness of a complex,constitutes an analysis, and we should not understand the analysis ifwe were not acquainted with the meanin

of the terms employed. Thus we

must suppose that we are acquainted with the meanin

of "before," andnot merely with instances of it.

There are thus at least two sorts of objects of which we are aware,namely, particulars and universals. Amon particulars I include allexistents, and all complexes of which one or more constituents areexistents, such as this-before-that, this-above-that,the-yellowness-of-this. Amon

universals I include all objects ofwhich no particular is a constituent. Thus the disjunction"universal-particular" includes all objects. We mi

ht also call it thedisjunction "abstract-concrete." It is not quite parallel with theopposition "concept-percept," because thin

s remembered or ima

inedbelon with particulars, but can hardly be called percepts. (On the

other hand, universals with which we are acquainted may be identifiedwith concepts.)

It will be seen that amon

the objects with which we are acquaintedare not included physical objects (as opposed to sense-data), norother people's minds. These thin s are known to us by what I call"knowled e by description," which we must now consider.

By a "description" I mean any phrase of the form "a so-and-so" or "theso-and-so." A phrase of the form "a so-and-so" I shall call an"ambi uous" description; a phrase of the form "the so-and-so" (in thesin

ular) I shall call a "definite" description. Thus "a man" is anambi uous description, and "the man with the iron mask" is a definite

description. There are various problems connected with ambi

uousdescriptions, but I pass them by, since they do not directly concernthe matter I wish to discuss. What I wish to discuss is the nature ofour knowled e concernin objects in cases where we know that there isan object answerin to a definite description, thou h we are not

_acquainted_ with any such object. This is a matter which is concernedexclusively with _definite_ descriptions. I shall, therefore, in thesequel, speak simply of "descriptions" when I mean "definitedescriptions." Thus a description will mean any phrase of the form"the so-and-so" in the sin ular.



I shall say that an object is "known by description" when we know thatit is "_the_ so-and-so," i.e. when we know that there is one object,and no more, havin a certain property; and it will enerally beimplied that we do not have knowled e of the same object byacquaintance. We know that the man with the iron mask existed, andmany propositions are known about him; but we do not know who he was.We know that the candidate who

ets most votes will be elected, and inthis case we are very likely also acquainted (in the only sense inwhich one can be acquainted with some one else) with the man who is,in fact, the candidate who will et most votes, but we do not knowwhich of the candidates he is, i.e. we do not know any proposition ofthe form "A is the candidate who will et most votes" where A is oneof the candidates by name. We shall say that we have "_merely_ descriptive knowled e" of the so-and-so when, althou h we know thatthe so-and-so exists, and althou

h we may possibly be acquainted withthe object which is, in fact, the so-and-so, yet we do not know anyproposition "_a_ is the so-and-so," where _a_ is somethin

with whichwe are acquainted.

When we say "the so-and-so exists," we mean that there is just oneobject which is the so-and-so. The proposition "_a_ is the so-and-so"means that _a_ has the property so-and-so, and nothin else has. "SirJoseph Larmor is the Unionist candidate" means "Sir Joseph Larmor is a

Unionist candidate, and no one else is." "The Unionist candidateexists" 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 theso-and-so, we know that the so-and-so exists but we may know that theso-and-so exists when we are not acquainted with any object which weknow to be the so-and-so, and even when we are not acquainted with anyobject which, in fact, is the so-and-so.

Common words, even proper names, are usually really descriptions. Thatis to say, the thou

ht in the mind of a person usin

a proper namecorrectly can enerally only be expressed explicitly if we replace theproper name by a description. Moreover, the description required toexpress the thou ht will vary for different people, or for the same

person at different times. The only thin

constant (so lon

as thename is ri htly used) is the object to which the name applies. But solon

as this remains constant, the particular description involvedusually makes no difference to the truth or falsehood of theproposition in which the name appears.

Let us take some illustrations. Suppose some statement made aboutBismarck. Assumin

that there is such a thin

as direct acquaintancewith oneself, Bismarck himself mi ht have used his name directly todesi nate the particular person with whom he was acquainted. In thiscase, if he made a jud ment about himself, he himself mi ht be aconstituent of the jud

ment. Here the proper name has the direct usewhich it always wishes to have, as simply standin for a certain

object, and not for a description of the object. But if a person whoknew Bismarck made a jud ment about him, the case is different. Whatthis person was acquainted with were certain sense-data which heconnected (ri htly, we will suppose) with Bismarck's body. His body asa physical object, and still more his mind, were only known as thebody and the mind connected with these sense-data. That is, they wereknown by description. It is, of course, very much a matter of chancewhich characteristics of a man's appearance will come into a friend'smind when he thinks of him; thus the description actually in thefriend's mind is accidental. The essential point is that he knows that



the various descriptions all apply to the same entity, in spite ofnot bein

acquainted with the entity in question.

When we, who did not know Bismarck, make a jud ment about him, thedescription in our minds will probably be some more or less va ue massof historical knowled

e--far more, in most cases, than is required toidentify him. But, for the sake of illustration, let us assume that wethink of him as "the first Chancellor of the German Empire." Here allthe words are abstract except "German." The word "German" will a ainhave different meanin

s for different people. To some it will recalltravels 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 brin in a reference to aparticular with which we are acquainted. Such reference is involved inany mention of past, present, and future (as opposed to definitedates), or of here and there, or of what others have told us. Thus itwould seem that, in some way or other, a description known to beapplicable to a particular must involve some reference to a particularwith which we are acquainted, if our knowled e about the thin

described is not to be merely what follows lo

ically from thedescription. For example, "the most lon -lived of men" is adescription which must apply to some man, but we can make no jud

mentsconcernin this man which involve knowled e about him beyond what thedescription

ives. If, however, we say, "the first Chancellor of the

German Empire was an astute diplomatist," we can only be assured ofthe truth of our jud

ment in virtue of somethin

with which we areacquainted--usually a testimony heard or read. Consideredpsycholo

ically, apart from the information we convey to others, apartfrom the fact about the actual Bismarck, which ives importance toour jud

ment, the thou

ht we really have contains the one or moreparticulars involved, and otherwise consists wholly of concepts. Allnames of places--London, En

land, Europe, the earth, the SolarSystem--similarly involve, when used, descriptions which start fromsome one or more particulars with which we are acquainted. I suspectthat even the Universe, as considered by metaphysics, involves such aconnection with particulars. In lo

ic, on the contrary, where we areconcerned not merely with what does exist, but with whatever mi ht or

could exist or be, no reference to actual particulars is involved.

It would seem that, when we make a statement about somethin

onlyknown by description, we often _intend_ to make our statement, not inthe form involvin the description, but about the actual thin

described. That is to say, when we say anythin about Bismarck, weshould like, if we could, to make the jud ment which Bismarck alonecan make, namely, the jud

ment of which he himself is a constituent.In this we are necessarily defeated, since the actual Bismarck isunknown to us. But we know that there is an object B called Bismarck,and that B was an astute diplomatist. We can thus _describe_ theproposition we should like to affirm, namely, "B was an astutediplomatist," where B is the object which was Bismarck. What enables

us to communicate in spite of the varyin

descriptions we employ isthat we know there is a true proposition concernin the actualBismarck, and that, however we may vary the description (so lon

asthe description is correct), the proposition described is still thesame. This proposition, which is described and is known to be true, iswhat interests us; but we are not acquainted with the propositionitself, and do not know _it_, thou

h we know it is true.

It will be seen that there are various sta es in the removal fromacquaintance with particulars: there is Bismarck to people who knew



him, Bismarck to those who only know of him throu h history, the manwith the iron mask, the lon

est-lived of men. These are pro

ressivelyfurther removed from acquaintance with particulars, and there is asimilar hierarchy in the re ion of universals. Many universals, likemany particulars, are only known to us by description. But here, as inthe case of particulars, knowled

e concernin

what is known bydescription is ultimately reducible to knowled e concernin what isknown by acquaintance.

The fundamental epistemolo

ical principle in the analysis ofpropositions containin descriptions is this: _Every proposition whichwe can understand must be composed wholly of constituents with whichwe are acquainted._ From what has been said already, it will be plainwhy I advocate this principle, and how I propose to meet the case ofpropositions which at first si ht contravene it. Let us be in with thereasons for supposin

the principle true.

The chief reason for supposin

the principle true is that it seemsscarcely possible to believe that we can make a jud ment or entertaina supposition without knowin

what it is that we are jud

in

orsupposin about. If we make a jud ment about (say) Julius Cæsar, it isplain that the actual person who was Julius Cæsar is not a constituentof the jud ment. But before oin further, it may be well to explainwhat I mean when I say that this or that is a constituent of a

jud

ment, or of a proposition which we understand. To be

in withjud

ments: a jud

ment, as an occurrence, I take to be a relation of amind to several entities, namely, the entities which compose what isjud

ed. If, e.

. I jud

e that A loves B, the jud

ment as an eventconsists in the existence, at a certain moment, of a specificfour-term relation, called _jud

in

_, between me and A and love and B.That is to say, at the time when I jud e, there is a certain complexwhose terms are myself and A and love and B, and whose relatin

relation is _jud in _. My reasons for this view have been set forthelsewhere,[41] and I shall not repeat them here. Assumin

this view ofjud ment, the constituents of the jud ment are simply the constituentsof the complex which is the jud

ment. Thus, in the above case, theconstituents are myself and A and love and B and jud in . But myself

and jud

in

are constituents shared by all my jud

ments; thus the _distinctive_ constituents of the particular jud ment in question areA and love and B. Comin

now to what is meant by "understandin

aproposition," I should say that there is another relation possiblebetween me and A and love and B, which is called my _supposin _ that Aloves B.[42] When we can _suppose_ that A loves B, we "understand theproposition" _A loves B_. Thus we often understand a proposition incases where we have not enou

h knowled

e to make a jud

ment.Supposin , like jud in , is a many-term relation, of which a mind isone term. The other terms of the relation are called the constituentsof the proposition supposed. Thus the principle which I enunciated maybe re-stated as follows: _Whenever a relation of supposin

or jud

in

occurs, the terms to which the supposin or jud in mind is related by

the relation of supposin

or jud

in

must be terms with which the mindin question is acquainted._ This is merely to say that we cannot makea jud

ment or a supposition without knowin

what it is that we aremakin our jud ment or supposition about. It seems to me that thetruth of this principle is evident as soon as the principle isunderstood; I shall, therefore, in what follows, assume the principle,and use it as a

uide in analysin

jud

ments that containdescriptions.

Returnin now to Julius Cæsar, I assume that it will be admitted that



he himself is not a constituent of any jud ment which I can make. Butat this point it is necessary to examine the view that jud

ments arecomposed of somethin called "ideas," and that it is the "idea" ofJulius Cæsar that is a constituent of my jud ment. I believe theplausibility of this view rests upon a failure to form a ri ht theoryof descriptions. We may mean by my "idea" of Julius Cæsar the thin

sthat I know about him, e. . that he conquered Gaul, was assassinatedon the Ides of March, and is a pla

ue to schoolboys. Now I amadmittin , and indeed contendin , that in order to discover what isactually in my mind when I jud

e about Julius Cæsar, we mustsubstitute for the proper name a description made up of some of thethin

s I know about him. (A description which will often serve toexpress my thou ht is "the man whose name was _Julius Cæsar_." Forwhatever else I may have for

otten about him, it is plain that when Imention him I have not for otten that that was his name.) But althou hI think the theory that jud

ments consist of ideas may have beensu ested in some such way, yet I think the theory itself isfundamentally mistaken. The view seems to be that there is somemental existent which may be called the "idea" of somethin outsidethe mind of the person who has the idea, and that, since jud

ment is amental event, its constituents must be constituents of the mind of theperson jud

in

. But in this view ideas become a veil between us andoutside thin s--we never really, in knowled e, attain to the thin s weare supposed to be knowin

about, but only to the ideas of those

thin

s. The relation of mind, idea, and object, on this view, isutterly obscure, and, so far as I can see, nothin

discoverable byinspection warrants the intrusion of the idea between the mind and theobject. I suspect that the view is fostered by the dislike ofrelations, and that it is felt the mind could not know objects unlessthere were somethin

"in" the mind which could be called the state ofknowin the object. Such a view, however, leads at once to a viciousendless re

ress, since the relation of idea to object will have to beexplained by supposin 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 somethin whichcan be called the "idea" of the object. On the contrary, I hold thatacquaintance is wholly a relation, not demandin any such constituent

of the mind as is supposed by advocates of "ideas." This is, ofcourse, a lar e question, and one which would take us far from oursubject if it were adequately discussed. I therefore content myselfwith the above indications, and with the corollary that, in jud

in

,the actual objects concernin which we jud e, rather than any supposedpurely mental entities, are constituents of the complex which is thejud ment.

When, therefore, I say that we must substitute for "Julius Cæsar" somedescription of Julius Cæsar, in order to discover the meanin of ajud ment nominally about him, I am not sayin that we must substitutean idea. Suppose our description is "the man whose name was _JuliusCæsar_." Let our jud ment 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, andall the other constituents of the jud

ment (ne

lectin

the tense in"was") are _concepts_ with which we are acquainted. Thus our jud mentis wholly reduced to constituents with which we are acquainted, butJulius Cæsar himself has ceased to be a constituent of our jud ment.This, however, requires a proviso, to be further explained shortly,namely that "the man whose name was _Julius Cæsar_" must not, as awhole, be a constituent of our jud ment, that is to say, this phrasemust not, as a whole, have a meanin which enters into the jud ment.



Any ri ht analysis of the jud ment, therefore, must break up thisphrase, and not treat it as a subordinate complex which is part of thejud ment. The jud ment "the man whose name was _Julius Cæsar_ wasassassinated" may be interpreted as meanin "one and only one man wascalled _Julius Cæsar_, and that one was assassinated." Here it isplain that there is no constituent correspondin

to the phrase "theman whose name was _Julius Cæsar_." Thus there is no reason to re ardthis phrase as expressin

a constituent of the jud

ment, and we haveseen that this phrase must be broken up if we are to be acquaintedwith all the constituents of the jud

ment. This conclusion, which wehave reached from considerations concerned with the theory ofknowled

e, is also forced upon us by lo

ical considerations, whichmust now be briefly reviewed.

It is common to distin uish two aspects, _meanin _ and _denotation_,such phrases as "the author of Waverley." The meanin

will be acertain complex, consistin (at least) of authorship and Waverley withsome relation; the denotation will be Scott. Similarly "featherlessbipeds" will have a complex meanin , containin as constituents thepresence of two feet and the absence of feathers, while its denotationwill be the class of men. Thus when we say "Scott is the author ofWaverley" or "men are the same as featherless bipeds," we areassertin an identity of denotation, and this assertion is worthmakin

because of the diversity of meanin

.[43] I believe that the

duality of meanin

and denotation, thou

h capable of a trueinterpretation, is misleadin

if taken as fundamental. The denotation,I believe, is not a constituent of the proposition, except in the caseof proper names, i.e. of words which do not assi

n a property to anobject, but merely and solely name it. And I should hold further that,in this sense, there are only two words which are strictly propernames of particulars, namely, "I" and "this."[44]

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

sufficiently emphasised.

A second reason is that propositions concernin

"the so-and-so" arepossible even when "the so-and-so" has no denotation. Take, e.

. "the olden mountain does not exist" or "the round square isself-contradictory." If we are to preserve the duality of meanin anddenotation, we have to say, with Meinon , that there are such objectsas the

olden mountain and the round square, althou

h these objects donot have bein . We even have to admit that the existent round squareis existent, but does not exist.[45] Meinon does not re ard this as acontradiction, but I fail to see that it is not one. Indeed, it seemsto me evident that the jud

ment "there is no such object as the roundsquare" does not presuppose that there is such an object. If this is

admitted, however, we are led to the conclusion that, by parity ofform, no jud ment concernin "the so-and-so" actually involves theso-and-so as a constituent.

Miss Jones[46] contends that there is no difficulty in admittin

contradictory predicates concernin such an object as "the presentKin

of France," on the

round that this object is in itselfcontradictory. Now it mi ht, of course, be ar ued that this object,unlike the round square, is not self-contradictory, but merelynon-existent. This, however, would not o to the root of the matter.



The real objection to such an ar ument is that the law ofcontradiction ou

ht not to be stated in the traditional form "A is notboth B and not B," but in the form "no proposition is both true andfalse." The traditional form only applies to certain propositions,namely, to those which attribute a predicate to a subject. When thelaw is stated of propositions, instead of bein

stated concernin

subjects and predicates, it is at once evident that propositions aboutthe present Kin

of France or the round square can form no exception,but are just as incapable of bein both true and false as otherpropositions. Miss Jones[47] ar

ues that "Scott is the author ofWaverley" asserts identity of denotation between _Scott_ and _theauthor of Waverley_. But there is some difficulty in choosin

amon

alternative meanin s of this contention. In the first place, it shouldbe observed that _the author of Waverley_ is not a _mere_ name, like

_Scott_. _Scott_ is merely a noise or shape conventionally used todesi

nate a certain person; it

ives us no information about thatperson, and has nothin that can be called meanin as opposed todenotation. (I ne

lect the fact, considered above, that even propernames, as a rule, really stand for descriptions.) But _the author ofWaverley_ is not merely conventionally a name for Scott; the elementof mere convention belon s here to the separate words, _the_ and

_author_ and _of_ and _Waverley_. Given what these words stand for, _the author of Waverley_ is no lon er arbitrary. When it is said thatScott is the author of Waverley, we are not statin

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 beencalled the author of Waverley, that would not have made him be theauthor; it was necessary for him actually to write Waverley, which wasa fact havin nothin to do with names.

If, then, we are assertin identity of denotation, we must not mean by _denotation_ the mere relation of a name to the thin

named. In fact,it would be nearer to the truth to say that the _meanin _ 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 thisconcept. Thus if we distin uish meanin and denotation in "the author

of Waverley," we shall have to say that "Scott" has meanin

but notdenotation. Also when we say "Scott is the author of Waverley," the

_meanin

_ of "the author of Waverley" is relevant to our assertion.For if the denotation alone were relevant, any other phrase with thesame denotation would ive the same proposition. Thus "Scott is theauthor of Marmion" would be the same proposition as "Scott is theauthor of Waverley." But this is plainly not the case, since from thefirst we learn that Scott wrote Marmion and from the second we learnthat he wrote Waverley, but the first tells us nothin about Waverleyand the second nothin about Marmion. Hence the meanin of "the authorof Waverley," as opposed to the denotation, is certainly relevant to"Scott is the author of Waverley."

We have thus a

reed that "the author of Waverley" is not a mere name,and that its meanin is relevant in propositions in which it occurs.Thus if we are to say, as Miss Jones does, that "Scott is the authorof Waverley" asserts an identity of denotation, we must re ard thedenotation of "the author of Waverley" as the denotation of what is

_meant_ by "the author of Waverley." Let us call the meanin of "theauthor 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 explainin ourproposition by another of the same form, and thus we have made no



pro ress towards a real explanation. "The denotation of M," like "theauthor of Waverley," has both meanin

and denotation, on the theory weare examinin . If we call its meanin M', our proposition becomes"Scott is the denotation of M'." But this leads at once to an endlessre ress. Thus the attempt to re ard our proposition as assertin

identity of denotation breaks down, and it becomes imperative to findsome other analysis. When this analysis has been completed, we shallbe able to reinterpret the phrase "identity of denotation," whichremains obscure so lon as it is taken as fundamental.

The first point to observe is that, in any proposition about "theauthor of Waverley," provided Scott is not explicitly mentioned, thedenotation itself, i.e. Scott, does not occur, but only the concept ofdenotation, which will be represented by a variable. Suppose we say"the author of Waverley was the author of Marmion," we are certainlynot sayin

that both were Scott--we may have for

otten that there wassuch a person as Scott. We are sayin that there is some man who wasthe author of Waverley and the author of Marmion. That is to say,there is some one who wrote Waverley and Marmion, and no one elsewrote them. Thus the identity is that of a variable, i.e. of anindefinite subject, "some one." This is why we can understandpropositions about "the author of Waverley," without knowin

who hewas. When we say "the author of Waverley was a poet," we mean "one andonly 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 wroteWaverley, and he was Scott." Here the identity is between a variable,i.e. an indeterminate subject ("he"), and Scott; "the author ofWaverley" has been analysed away, and no lon

er appears as aconstituent of the proposition.[48]

The reason why it is imperative to analyse away the phrase "the authorof Waverley" may be stated as follows. It is plain that when we say"the author of Waverley is the author of Marmion," the _is_ expressesidentity. We have seen also that the common _denotation_, namelyScott, is not a constituent of this proposition, while the _meanin s_ (if any) of "the author of Waverley" and "the author of Marmion" arenot identical. We have seen also that, in any sense in which the

meanin

of a word is a constituent of a proposition in whose verbalexpression the word occurs, "Scott" means the actual man Scott, in thesame 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 _meanin _ would have to be what was said to be identical with the _meanin _ of"the author of Marmion." This is plainly not the case; and the onlyescape is to say that "the author of Waverley" does not, by itself,have a meanin , thou h phrases of which it is part do have a meanin .That is, in a ri ht analysis of the above proposition, "the author ofWaverley" must disappear. This is effected when the above propositionis analysed as meanin

: "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 sayin

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. Thusthe true subject of our jud ment is a propositional function, i.e. acomplex containin an undetermined constituent, and becomin aproposition as soon as this constituent is determined.

We may now define the denotation of a phrase. If we know that theproposition "_a_ is the so-and-so" is true, i.e. that _a_ is so-and-soand nothin else is, we call _a_ the denotation of the phrase "the



so-and-so." A very reat many of the propositions we naturally makeabout "the so-and-so" will remain true or remain false if wesubstitute _a_ for "the so-and-so," where _a_ is the denotation of"the so-and-so." Such propositions will also remain true or remainfalse if we substitute for "the so-and-so" any other phrase havin thesame denotation. Hence, as practical men, we become interested in thedenotation more than in the description, since the denotation decidesas to the truth or falsehood of so many statements in which thedescription occurs. Moreover, as we saw earlier in considerin therelations of description and acquaintance, we often wish to reach thedenotation, and are only hindered by lack of acquaintance: in suchcases the description is merely the means we employ to

et as near aspossible to the denotation. Hence it naturally comes to be supposedthat the denotation is part of the proposition in which thedescription occurs. But we have seen, both on lo ical and onepistemolo

ical

rounds, that this is an error. The actual object (ifany) which is the denotation is not (unless it is explicitlymentioned) 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 donot need acquaintance with its denotation. The first result ofanalysis, when applied to propositions whose

rammatical subject is"the so-and-so," is to substitute a variable as subject; i.e. weobtain a proposition of the form: "There is _somethin

_ which alone is

so-and-so, and that _somethin

_ is such-and-such." The furtheranalysis of propositions concernin

"the so-and-so" is thus mer

ed inthe problem of the nature of the variable, i.e. of the meanin s of

_some_, _any_, and _all_. This is a difficult problem, concernin

which I do not intend to say anythin at present.

To sum up our whole discussion. We be an by distin uishin two sortsof knowled

e of objects, namely, knowled

e by _acquaintance_ andknowled e by _description_. Of these it is only the former that brin sthe object itself before the mind. We have acquaintance withsense-data, with many universals, and possibly with ourselves, but notwith physical objects or other minds. We have _descriptive_ knowled

eof an object when we know that it is _the_ object havin some property

or properties with which we are acquainted; that is to say, when weknow that the property or properties in question belon to one objectand no more, we are said to have knowled

e of that one object bydescription, whether or not we are acquainted with the object. Ourknowled e of physical objects and of other minds is only knowled e bydescription, the descriptions involved bein usually such as involvesense-data. All propositions intelli ible to us, whether or not theyprimarily concern thin

s only known to us by description, are composedwholly of constituents with which we are acquainted, for a constituentwith which we are not acquainted is unintelli ible to us. A jud ment,we found, is not composed of mental constituents called "ideas," butconsists of an occurrence whose constituents are a mind[49] andcertain objects, particulars or universals. (One at least must be a

universal.) When a jud

ment is ri

htly analysed, the objects which areconstituents of it must all be objects with which the mind which is aconstituent of it is acquainted. This conclusion forces us to analysedescriptive phrases occurrin in propositions, and to say that theobjects denoted by such phrases are not constituents of jud ments inwhich such phrases occur (unless these objects are explicitlymentioned). This leads us to the view (recommended also on purelylo ical rounds) that when we say "the author of Marmion was theauthor of Waverley," Scott himself is not a constituent of ourjud ment, and that the jud ment cannot be explained by sayin that it



affirms identity of denotation with diversity of meanin . It also,plainly, does not assert identity of meanin

. Such jud

ments,therefore, can only be analysed by breakin up the descriptivephrases, introducin a variable, and makin propositional functionsthe ultimate subjects. In fact, "the so-and-so is such-and-such" willmean that "_x_ is so-and-so and nothin

else is, and _x_ issuch-and-such" is capable of truth. The analysis of such jud mentsinvolves many fresh problems, but the discussion of these problems isnot undertaken in the present paper.

FOOTNOTES:

[40] See references later.

[41] _Philosophical Essays_, "The Nature of Truth." I have beenpersuaded by Mr. Witt

enstein that this theory is somewhat undulysimple, but the modification which I believe it to require does notaffect the above ar

ument [1917].

[42] Cf. Meinon

, _Ueber Annahmen_, _passim_. I formerly supposed,contrary to Meinon 's view, that the relationship of supposin mi htbe merely that of presentation. In this view I now think I wasmistaken, and Meinon is ri ht. But my present view depends upon thetheory that both in jud

ment and in assumption there is no sin

le

Objective, but the several constituents of the jud

ment or assumptionare in a many-term relation to the mind.

[43] This view has been recently advocated by Miss E.E.C. Jones. "ANew Law of Thou ht 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] Meinon

, _Ueber Annahmen_, 2nd ed., Leipzi

, 1910, p. 141.

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

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

[48] The theory which I am advocatin

is set forth fully, with thelo

ical

rounds 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 somethin psycholo icalwhich enters into jud

ment, without intendin

to prejud

e the questionas to what this somethin 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

Ber son, 14 ff, 22, 105, 128, 185 ff, 203

Berkeley, 97, 132

Blake, 1

Bosanquet, 99

Broad, 89 _n_

Calculus, the, 82

Cantor, Geor

, 64, 81 ff, 85, 91

Carlyle, 50, 82

Cause, the conception of, 135 _n_, 180 ff

Christianity and renunciation, 51

Chuan

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;



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mysticism and logic and other essays

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