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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Mysticism and Logic and Other Essays + +Author: Bertrand Russell + +Release Date: May 12, 2008 [EBook #25447] +[Last updated: May 5, 2012] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK MYSTICISM AND LOGIC *** + + + + +Produced by Jeannie Howse, Adrian Mastronardi and the +Online Distributed Proofreading Team at http://www.pgdp.net +(This file was produced from images generously made +available by The Internet Archive/Canadian Libraries) + + + + + + + * * * * * + + +-----------------------------------------------------------+ + | Transcriber's Note: | + | | + | Inconsistent hyphenation in the original document has | + | been preserved. | + | | + | Mathematical equations and variables are marked with | + | [Math:] to reduce confusion between underscores used for | + | italics and underscores used to indicate subscripts. | + | Subscripts are indicated with underscores, with the | + | subscripted character marked with {}'s, i.e.: x_{t} | + | | + | Greek letters have been transliterated and marked with | + | []'s, i.e.: [alpha] | + | | + | Obvious typographical errors have been corrected. For | + | a complete list, please see the end of this document. | + | | + +-----------------------------------------------------------+ + + * * * * * + + + + +BERTRAND RUSSELL + +MYSTICISM AND +LOGIC + +AND OTHER ESSAYS + + + + +_LONDON_ +GEORGE ALLEN & UNWIN LTD +RUSKIN HOUSE MUSEUM STREET + + + + +MYSTICISM AND LOGIC +AND OTHER ESSAYS + + + + +BY BERTRAND RUSSELL + +_The ABC of Relativity_ +_The Analysis of Matter_ +_Human Society in Ethics and Politics_ +_The Impact of Science on Society_ +_New Hopes for a Changing World_ +_Authority and the Individual_ +_Human Knowledge_ +_History of Western Philosophy_ +_The Principles of Mathematics_ +_Introduction to Mathematical Philosophy_ +_The Analysis of Mind_ +_Our Knowledge of the External World_ +_An Outline of Philosophy_ +_The Philosophy of Leibniz_ +_An Inquiry into Meaning and Truth_ +_Logic and Knowledge_ +_The Problems of Philosophy_ +_Principia Mathematica_ + +_Common Sense and Nuclear Warfare_ +_Why I am Not a Christian_ +_Portraits from Memory_ +_My Philosophical Development_ +_Unpopular Essays_ +_Power_ +_In Praise of Idleness_ +_The Conquest of Happiness_ +_Sceptical Essays_ +_The Scientific Outlook_ +_Marriage and Morals_ +_Education and the Social Order_ +_On Education_ + +_Freedom and Organization_ +_Principles of Social Reconstruction_ +_Roads to Freedom_ +_Practice and Theory of Bolshevism_ + +_Satan in The Suburbs_ +_Nightmares of Eminent Persons_ + + + + +_First published as "Philosophical Essays"_ _October 1910_ +_Second Edition as "Mysticism and Logic"_ _December 1917_ +_Third Impression_ _April 1918_ +_Fourth Impression_ _February 1919_ +_Fifth Impression_ _October 1921_ +_Sixth Impression_ _August 1925_ +_Seventh Impression_ _January 1932_ +_Eighth Impression_ _1949_ +_Ninth Impression_ _1950_ +_Tenth Impression_ _1951_ +_Eleventh Impression_ _1959_ + + _This book is copyright under the Berne Convention. Apart from any + fair dealing for the purpose of private study, research, criticism + or review, as permitted under the Copyright Act, 1956, no portion + may be reproduced by any process without written permission. + Enquiry should be made to the publisher._ + + +PRINTED IN GREAT BRITAIN +_by Taylor Garnett Evans & Co. Ltd., +Watford, Herts._ + + + + +PREFACE + + +The following essays have been written and published at various times, +and my thanks are due to the previous publishers for the permission to +reprint them. + +The essay on "Mysticism and Logic" appeared in the _Hibbert Journal_ +for July, 1914. "The Place of Science in a Liberal Education" appeared +in two numbers of _The New Statesman_, May 24 and 31, 1913. "The Free +Man's Worship" and "The Study of Mathematics" were included in a +former collection (now out of print), _Philosophical Essays_, also +published by Messrs. Longmans, Green & Co. Both were written in 1902; +the first appeared originally in the _Independent Review_ for 1903, +the second in the _New Quarterly_, November, 1907. In theoretical +Ethics, the position advocated in "The Free Man's Worship" is not +quite identical with that which I hold now: I feel less convinced than +I did then of the objectivity of good and evil. But the general +attitude towards life which is suggested in that essay still seems to +me, in the main, the one which must be adopted in times of stress and +difficulty by those who have no dogmatic religious beliefs, if inward +defeat is to be avoided. + +The essay on "Mathematics and the Metaphysicians" was written in 1901, +and appeared in an American magazine, _The International Monthly_, +under the title "Recent Work in the Philosophy of Mathematics." Some +points in this essay require modification in view of later work. +These are indicated in footnotes. Its tone is partly explained by the +fact that the editor begged me to make the article "as romantic as +possible." + +All the above essays are entirely popular, but those that follow are +somewhat more technical. "On Scientific Method in Philosophy" was the +Herbert Spencer lecture at Oxford in 1914, and was published by the +Clarendon Press, which has kindly allowed me to include it in this +collection. "The Ultimate Constituents of Matter" was an address to +the Manchester Philosophical Society, early in 1915, and was published +in the _Monist_ in July of that year. The essay on "The Relation of +Sense-data to Physics" was written in January, 1914, and first +appeared in No. 4 of that year's volume of _Scientia_, an +International Review of Scientific Synthesis, edited by M. Eugenio +Rignano, published monthly by Messrs. Williams and Norgate, London, +Nicola Zanichelli, Bologna, and Félix Alcan, Paris. The essay "On the +Notion of Cause" was the presidential address to the Aristotelian +Society in November, 1912, and was published in their _Proceedings_ +for 1912-13. "Knowledge by Acquaintance and Knowledge by Description" +was also a paper read before the Aristotelian Society, and published +in their _Proceedings_ for 1910-11. + + LONDON, + _September, 1917_ + + + + +CONTENTS + + +_Chapter_ _Page_ + + I. _Mysticism and Logic_ 1 + + II. _The Place of Science in a Liberal Education_ 33 + + III. _A Free Man's Worship_ 46 + + IV. _The Study of Mathematics_ 58 + + V. _Mathematics and the Metaphysicians_ 74 + + VI. _On Scientific Method in Philosophy_ 97 + + VII. _The Ultimate Constituents of Matter_ 125 + +VIII. _The Relation of Sense-data to Physics_ 145 + + IX. _On the Notion of Cause_ 180 + + X. _Knowledge by Acquaintance and Knowledge by + Description_ 209 + + + + +MYSTICISM AND LOGIC AND OTHER ESSAYS + + + + +I + +MYSTICISM AND LOGIC + + +Metaphysics, or the attempt to conceive the world as a whole by means +of thought, has been developed, from the first, by the union and +conflict of two very different human impulses, the one urging men +towards mysticism, the other urging them towards science. Some men +have achieved greatness through one of these impulses alone, others +through the other alone: in Hume, for example, the scientific impulse +reigns quite unchecked, while in Blake a strong hostility to science +co-exists with profound mystic insight. But the greatest men who have +been philosophers have felt the need both of science and of mysticism: +the attempt to harmonise the two was what made their life, and what +always must, for all its arduous uncertainty, make philosophy, to some +minds, a greater thing than either science or religion. + +Before attempting an explicit characterisation of the scientific and +the mystical impulses, I will illustrate them by examples from two +philosophers whose greatness lies in the very intimate blending which +they achieved. The two philosophers I mean are Heraclitus and Plato. + +Heraclitus, as every one knows, was a believer in universal flux: time +builds and destroys all things. From the few fragments that remain, it +is not easy to discover how he arrived at his opinions, but there are +some sayings that strongly suggest scientific observation as the +source. + +"The things that can be seen, heard, and learned," he says, "are what +I prize the most." This is the language of the empiricist, to whom +observation is the sole guarantee of truth. "The sun is new every +day," is another fragment; and this opinion, in spite of its +paradoxical character, is obviously inspired by scientific reflection, +and no doubt seemed to him to obviate the difficulty of understanding +how the sun can work its way underground from west to east during the +night. Actual observation must also have suggested to him his central +doctrine, that Fire is the one permanent substance, of which all +visible things are passing phases. In combustion we see things change +utterly, while their flame and heat rise up into the air and vanish. + +"This world, which is the same for all," he says, "no one of gods or +men has made; but it was ever, is now, and ever shall be, an +ever-living Fire, with measures kindling, and measures going out." + +"The transformations of Fire are, first of all, sea; and half of the +sea is earth, half whirlwind." + +This theory, though no longer one which science can accept, is +nevertheless scientific in spirit. Science, too, might have inspired +the famous saying to which Plato alludes: "You cannot step twice into +the same rivers; for fresh waters are ever flowing in upon you." But +we find also another statement among the extant fragments: "We step +and do not step into the same rivers; we are and are not." + +The comparison of this statement, which is mystical, with the one +quoted by Plato, which is scientific, shows how intimately the two +tendencies are blended in the system of Heraclitus. Mysticism is, in +essence, little more than a certain intensity and depth of feeling in +regard to what is believed about the universe; and this kind of +feeling leads Heraclitus, on the basis of his science, to strangely +poignant sayings concerning life and the world, such as: + +"Time is a child playing draughts, the kingly power is a child's." + +It is poetic imagination, not science, which presents Time as despotic +lord of the world, with all the irresponsible frivolity of a child. It +is mysticism, too, which leads Heraclitus to assert the identity of +opposites: "Good and ill are one," he says; and again: "To God all +things are fair and good and right, but men hold some things wrong and +some right." + +Much of mysticism underlies the ethics of Heraclitus. It is true that +a scientific determinism alone might have inspired the statement: +"Man's character is his fate"; but only a mystic would have said: + +"Every beast is driven to the pasture with blows"; and again: + +"It is hard to fight with one's heart's desire. Whatever it wishes to +get, it purchases at the cost of soul"; and again: + +"Wisdom is one thing. It is to know the thought by which all things +are steered through all things."[1] + +Examples might be multiplied, but those that have been given are +enough to show the character of the man: the facts of science, as they +appeared to him, fed the flame in his soul, and in its light he saw +into the depths of the world by the reflection of his own dancing +swiftly penetrating fire. In such a nature we see the true union of +the mystic and the man of science--the highest eminence, as I think, +that it is possible to achieve in the world of thought. + +In Plato, the same twofold impulse exists, though the mystic impulse +is distinctly the stronger of the two, and secures ultimate victory +whenever the conflict is sharp. His description of the cave is the +classical statement of belief in a knowledge and reality truer and +more real than that of the senses: + + "Imagine[2] a number of men living in an underground cavernous + chamber, with an entrance open to the light, extending along the + entire length of the cavern, in which they have been confined, from + their childhood, with their legs and necks so shackled that they + are obliged to sit still and look straight forwards, because their + chains render it impossible for them to turn their heads round: and + imagine a bright fire burning some way off, above and behind them, + and an elevated roadway passing between the fire and the prisoners, + with a low wall built along it, like the screens which conjurors + put up in front of their audience, and above which they exhibit + their wonders. + + I have it, he replied. + + Also figure to yourself a number of persons walking behind this + wall, and carrying with them statues of men, and images of other + animals, wrought in wood and stone and all kinds of materials, + together with various other articles, which overtop the wall; and, + as you might expect, let some of the passers-by be talking, and + others silent. + + You are describing a strange scene, and strange prisoners. + + They resemble us, I replied. + + Now consider what would happen if the course of nature brought them + a release from their fetters, and a remedy for their foolishness, + in the following manner. Let us suppose that one of them has been + released, and compelled suddenly to stand up, and turn his neck + round and walk with open eyes towards the light; and let us suppose + that he goes through all these actions with pain, and that the + dazzling splendour renders him incapable of discerning those + objects of which he used formerly to see the shadows. What answer + should you expect him to make, if some one were to tell him that in + those days he was watching foolish phantoms, but that now he is + somewhat nearer to reality, and is turned towards things more real, + and sees more correctly; above all, if he were to point out to him + the several objects that are passing by, and question him, and + compel him to answer what they are? Should you not expect him to be + puzzled, and to regard his old visions as truer than the objects + now forced upon his notice? + + Yes, much truer.... + + Hence, I suppose, habit will be necessary to enable him to perceive + objects in that upper world. At first he will be most successful in + distinguishing shadows; then he will discern the reflections of men + and other things in water, and afterwards the realities; and after + this he will raise his eyes to encounter the light of the moon and + stars, finding it less difficult to study the heavenly bodies and + the heaven itself by night, than the sun and the sun's light by + day. + + Doubtless. + + Last of all, I imagine, he will be able to observe and contemplate + the nature of the sun, not as it _appears_ in water or on alien + ground, but as it is in itself in its own territory. + + Of course. + + His next step will be to draw the conclusion, that the sun is the + author of the seasons and the years, and the guardian of all things + in the visible world, and in a manner the cause of all those things + which he and his companions used to see. + + Obviously, this will be his next step.... + + Now this imaginary case, my dear Glancon, you must apply in all its + parts to our former statements, by comparing the region which the + eye reveals to the prison house, and the light of the fire therein + to the power of the sun: and if, by the upward ascent and the + contemplation of the upper world, you understand the mounting of + the soul into the intellectual region, you will hit the tendency of + my own surmises, since you desire to be told what they are; though, + indeed, God only knows whether they are correct. But, be that as it + may, the view which I take of the subject is to the following + effect. In the world of knowledge, the essential Form of Good is + the limit of our enquiries, and can barely be perceived; but, when + perceived, we cannot help concluding that it is in every case the + source of all that is bright and beautiful,--in the visible world + giving birth to light and its master, and in the intellectual world + dispensing, immediately and with full authority, truth and + reason;--and that whosoever would act wisely, either in private or + in public, must set this Form of Good before his eyes." + +But in this passage, as throughout most of Plato's teaching, there is +an identification of the good with the truly real, which became +embodied in the philosophical tradition, and is still largely +operative in our own day. In thus allowing a legislative function to +the good, Plato produced a divorce between philosophy and science, +from which, in my opinion, both have suffered ever since and are still +suffering. The man of science, whatever his hopes may be, must lay +them aside while he studies nature; and the philosopher, if he is to +achieve truth, must do the same. Ethical considerations can only +legitimately appear when the truth has been ascertained: they can and +should appear as determining our feeling towards the truth, and our +manner of ordering our lives in view of the truth, but not as +themselves dictating what the truth is to be. + +There are passages in Plato--among those which illustrate the +scientific side of his mind--where he seems clearly aware of this. The +most noteworthy is the one in which Socrates, as a young man, is +explaining the theory of ideas to Parmenides. + +After Socrates has explained that there is an idea of the good, but +not of such things as hair and mud and dirt, Parmenides advises him +"not to despise even the meanest things," and this advice shows the +genuine scientific temper. It is with this impartial temper that the +mystic's apparent insight into a higher reality and a hidden good has +to be combined if philosophy is to realise its greatest possibilities. +And it is failure in this respect that has made so much of idealistic +philosophy thin, lifeless, and insubstantial. It is only in marriage +with the world that our ideals can bear fruit: divorced from it, they +remain barren. But marriage with the world is not to be achieved by an +ideal which shrinks from fact, or demands in advance that the world +shall conform to its desires. + +Parmenides himself is the source of a peculiarly interesting strain +of mysticism which pervades Plato's thought--the mysticism which may +be called "logical" because it is embodied in theories on logic. This +form of mysticism, which appears, so far as the West is concerned, to +have originated with Parmenides, dominates the reasonings of all the +great mystical metaphysicians from his day to that of Hegel and his +modern disciples. Reality, he says, is uncreated, indestructible, +unchanging, indivisible; it is "immovable in the bonds of mighty +chains, without beginning and without end; since coming into being and +passing away have been driven afar, and true belief has cast them +away." The fundamental principle of his inquiry is stated in a +sentence which would not be out of place in Hegel: "Thou canst not +know what is not--that is impossible--nor utter it; for it is the same +thing that can be thought and that can be." And again: "It needs must +be that what can be thought and spoken of is; for it is possible for +it to be, and it is not possible for what is nothing to be." The +impossibility of change follows from this principle; for what is past +can be spoken of, and therefore, by the principle, still is. + +Mystical philosophy, in all ages and in all parts of the world, is +characterised by certain beliefs which are illustrated by the +doctrines we have been considering. + +There is, first, the belief in insight as against discursive analytic +knowledge: the belief in a way of wisdom, sudden, penetrating, +coercive, which is contrasted with the slow and fallible study of +outward appearance by a science relying wholly upon the senses. All +who are capable of absorption in an inward passion must have +experienced at times the strange feeling of unreality in common +objects, the loss of contact with daily things, in which the solidity +of the outer world is lost, and the soul seems, in utter loneliness, +to bring forth, out of its own depths, the mad dance of fantastic +phantoms which have hitherto appeared as independently real and +living. This is the negative side of the mystic's initiation: the +doubt concerning common knowledge, preparing the way for the reception +of what seems a higher wisdom. Many men to whom this negative +experience is familiar do not pass beyond it, but for the mystic it is +merely the gateway to an ampler world. + +The mystic insight begins with the sense of a mystery unveiled, of a +hidden wisdom now suddenly become certain beyond the possibility of a +doubt. The sense of certainty and revelation comes earlier than any +definite belief. The definite beliefs at which mystics arrive are the +result of reflection upon the inarticulate experience gained in the +moment of insight. Often, beliefs which have no real connection with +this moment become subsequently attracted into the central nucleus; +thus in addition to the convictions which all mystics share, we find, +in many of them, other convictions of a more local and temporary +character, which no doubt become amalgamated with what was essentially +mystical in virtue of their subjective certainty. We may ignore such +inessential accretions, and confine ourselves to the beliefs which all +mystics share. + +The first and most direct outcome of the moment of illumination is +belief in the possibility of a way of knowledge which may be called +revelation or insight or intuition, as contrasted with sense, reason, +and analysis, which are regarded as blind guides leading to the morass +of illusion. Closely connected with this belief is the conception of a +Reality behind the world of appearance and utterly different from it. +This Reality is regarded with an admiration often amounting to +worship; it is felt to be always and everywhere close at hand, thinly +veiled by the shows of sense, ready, for the receptive mind, to shine +in its glory even through the apparent folly and wickedness of Man. +The poet, the artist, and the lover are seekers after that glory: the +haunting beauty that they pursue is the faint reflection of its sun. +But the mystic lives in the full light of the vision: what others +dimly seek he knows, with a knowledge beside which all other knowledge +is ignorance. + +The second characteristic of mysticism is its belief in unity, and its +refusal to admit opposition or division anywhere. We found Heraclitus +saying "good and ill are one"; and again he says, "the way up and the +way down is one and the same." The same attitude appears in the +simultaneous assertion of contradictory propositions, such as: "We +step and do not step into the same rivers; we are and are not." The +assertion of Parmenides, that reality is one and indivisible, comes +from the same impulse towards unity. In Plato, this impulse is less +prominent, being held in check by his theory of ideas; but it +reappears, so far as his logic permits, in the doctrine of the primacy +of the Good. + +A third mark of almost all mystical metaphysics is the denial of the +reality of Time. This is an outcome of the denial of division; if all +is one, the distinction of past and future must be illusory. We have +seen this doctrine prominent in Parmenides; and among moderns it is +fundamental in the systems of Spinoza and Hegel. + +The last of the doctrines of mysticism which we have to consider is +its belief that all evil is mere appearance, an illusion produced by +the divisions and oppositions of the analytic intellect. Mysticism +does not maintain that such things as cruelty, for example, are good, +but it denies that they are real: they belong to that lower world of +phantoms from which we are to be liberated by the insight of the +vision. Sometimes--for example in Hegel, and at least verbally in +Spinoza--not only evil, but good also, is regarded as illusory, though +nevertheless the emotional attitude towards what is held to be Reality +is such as would naturally be associated with the belief that Reality +is good. What is, in all cases, ethically characteristic of mysticism +is absence of indignation or protest, acceptance with joy, disbelief +in the ultimate truth of the division into two hostile camps, the good +and the bad. This attitude is a direct outcome of the nature of the +mystical experience: with its sense of unity is associated a feeling +of infinite peace. Indeed it may be suspected that the feeling of +peace produces, as feelings do in dreams, the whole system of +associated beliefs which make up the body of mystic doctrine. But this +is a difficult question, and one on which it cannot be hoped that +mankind will reach agreement. + +Four questions thus arise in considering the truth or falsehood of +mysticism, namely: + + I. Are there two ways of knowing, which may be called respectively + reason and intuition? And if so, is either to be preferred to the + other? + + II. Is all plurality and division illusory? + + III. Is time unreal? + + IV. What kind of reality belongs to good and evil? + +On all four of these questions, while fully developed mysticism seems +to me mistaken, I yet believe that, by sufficient restraint, there is +an element of wisdom to be learned from the mystical way of feeling, +which does not seem to be attainable in any other manner. If this is +the truth, mysticism is to be commended as an attitude towards life, +not as a creed about the world. The meta-physical creed, I shall +maintain, is a mistaken outcome of the emotion, although this emotion, +as colouring and informing all other thoughts and feelings, is the +inspirer of whatever is best in Man. Even the cautious and patient +investigation of truth by science, which seems the very antithesis of +the mystic's swift certainty, may be fostered and nourished by that +very spirit of reverence in which mysticism lives and moves. + + +I. REASON AND INTUITION[3] + +Of the reality or unreality of the mystic's world I know nothing. I +have no wish to deny it, nor even to declare that the insight which +reveals it is not a genuine insight. What I do wish to maintain--and +it is here that the scientific attitude becomes imperative--is that +insight, untested and unsupported, is an insufficient guarantee of +truth, in spite of the fact that much of the most important truth is +first suggested by its means. It is common to speak of an opposition +between instinct and reason; in the eighteenth century, the opposition +was drawn in favour of reason, but under the influence of Rousseau and +the romantic movement instinct was given the preference, first by +those who rebelled against artificial forms of government and thought, +and then, as the purely rationalistic defence of traditional theology +became increasingly difficult, by all who felt in science a menace to +creeds which they associated with a spiritual outlook on life and the +world. Bergson, under the name of "intuition," has raised instinct to +the position of sole arbiter of metaphysical truth. But in fact the +opposition of instinct and reason is mainly illusory. Instinct, +intuition, or insight is what first leads to the beliefs which +subsequent reason confirms or confutes; but the confirmation, where it +is possible, consists, in the last analysis, of agreement with other +beliefs no less instinctive. Reason is a harmonising, controlling +force rather than a creative one. Even in the most purely logical +realm, it is insight that first arrives at what is new. + +Where instinct and reason do sometimes conflict is in regard to single +beliefs, held instinctively, and held with such determination that no +degree of inconsistency with other beliefs leads to their abandonment. +Instinct, like all human faculties, is liable to error. Those in whom +reason is weak are often unwilling to admit this as regards +themselves, though all admit it in regard to others. Where instinct is +least liable to error is in practical matters as to which right +judgment is a help to survival: friendship and hostility in others, +for instance, are often felt with extraordinary discrimination through +very careful disguises. But even in such matters a wrong impression +may be given by reserve or flattery; and in matters less directly +practical, such as philosophy deals with, very strong instinctive +beliefs are sometimes wholly mistaken, as we may come to know through +their perceived inconsistency with other equally strong beliefs. It is +such considerations that necessitate the harmonising mediation of +reason, which tests our beliefs by their mutual compatibility, and +examines, in doubtful cases, the possible sources of error on the one +side and on the other. In this there is no opposition to instinct as a +whole, but only to blind reliance upon some one interesting aspect of +instinct to the exclusion of other more commonplace but not less +trustworthy aspects. It is such one-sidedness, not instinct itself, +that reason aims at correcting. + +These more or less trite maxims may be illustrated by application to +Bergson's advocacy of "intuition" as against "intellect." There are, +he says, "two profoundly different ways of knowing a thing. The first +implies that we move round the object: the second that we enter into +it. The first depends on the point of view at which we are placed and +on the symbols by which we express ourselves. The second neither +depends on a point of view nor relies on any symbol. The first kind of +knowledge may be said to stop at the _relative_; the second, in those +cases where it is possible, to attain the _absolute_."[4] The second +of these, which is intuition, is, he says, "the kind of _intellectual +sympathy_ by which one places oneself within an object in order to +coincide with what is unique in it and therefore inexpressible" (p. +6). In illustration, he mentions self-knowledge: "there is one +reality, at least, which we all seize from within, by intuition and +not by simple analysis. It is our own personality in its flowing +through time--our self which endures" (p. 8). The rest of Bergson's +philosophy consists in reporting, through the imperfect medium of +words, the knowledge gained by intuition, and the consequent complete +condemnation of all the pretended knowledge derived from science and +common sense. + +This procedure, since it takes sides in a conflict of instinctive +beliefs, stands in need of justification by proving the greater +trustworthiness of the beliefs on one side than of those on the other. +Bergson attempts this justification in two ways, first by explaining +that intellect is a purely practical faculty to secure biological +success, secondly by mentioning remarkable feats of instinct in +animals and by pointing out characteristics of the world which, though +intuition can apprehend them, are baffling to intellect as he +interprets it. + +Of Bergson's theory that intellect is a purely practical faculty, +developed in the struggle for survival, and not a source of true +beliefs, we may say, first, that it is only through intellect that we +know of the struggle for survival and of the biological ancestry of +man: if the intellect is misleading, the whole of this merely inferred +history is presumably untrue. If, on the other hand, we agree with him +in thinking that evolution took place as Darwin believed, then it is +not only intellect, but all our faculties, that have been developed +under the stress of practical utility. Intuition is seen at its best +where it is directly useful, for example in regard to other people's +characters and dispositions. Bergson apparently holds that capacity +for this kind of knowledge is less explicable by the struggle for +existence than, for example, capacity for pure mathematics. Yet the +savage deceived by false friendship is likely to pay for his mistake +with his life; whereas even in the most civilised societies men are +not put to death for mathematical incompetence. All the most striking +of his instances of intuition in animals have a very direct survival +value. The fact is, of course, that both intuition and intellect have +been developed because they are useful, and that, speaking broadly, +they are useful when they give truth and become harmful when they give +falsehood. Intellect, in civilised man, like artistic capacity, has +occasionally been developed beyond the point where it is useful to the +individual; intuition, on the other hand, seems on the whole to +diminish as civilisation increases. It is greater, as a rule, in +children than in adults, in the uneducated than in the educated. +Probably in dogs it exceeds anything to be found in human beings. But +those who see in these facts a recommendation of intuition ought to +return to running wild in the woods, dyeing themselves with woad and +living on hips and haws. + +Let us next examine whether intuition possesses any such infallibility +as Bergson claims for it. The best instance of it, according to him, +is our acquaintance with ourselves; yet self-knowledge is proverbially +rare and difficult. Most men, for example, have in their nature +meannesses, vanities, and envies of which they are quite unconscious, +though even their best friends can perceive them without any +difficulty. It is true that intuition has a convincingness which is +lacking to intellect: while it is present, it is almost impossible to +doubt its truth. But if it should appear, on examination, to be at +least as fallible as intellect, its greater subjective certainty +becomes a demerit, making it only the more irresistibly deceptive. +Apart from self-knowledge, one of the most notable examples of +intuition is the knowledge people believe themselves to possess of +those with whom they are in love: the wall between different +personalities seems to become transparent, and people think they see +into another soul as into their own. Yet deception in such cases is +constantly practised with success; and even where there is no +intentional deception, experience gradually proves, as a rule, that +the supposed insight was illusory, and that the slower more groping +methods of the intellect are in the long run more reliable. + +Bergson maintains that intellect can only deal with things in so far +as they resemble what has been experienced in the past, while +intuition has the power of apprehending the uniqueness and novelty +that always belong to each fresh moment. That there is something +unique and new at every moment, is certainly true; it is also true +that this cannot be fully expressed by means of intellectual concepts. +Only direct acquaintance can give knowledge of what is unique and new. +But direct acquaintance of this kind is given fully in sensation, and +does not require, so far as I can see, any special faculty of +intuition for its apprehension. It is neither intellect nor intuition, +but sensation, that supplies new data; but when the data are new in +any remarkable manner, intellect is much more capable of dealing with +them than intuition would be. The hen with a brood of ducklings no +doubt has intuition which seems to place her inside them, and not +merely to know them analytically; but when the ducklings take to the +water, the whole apparent intuition is seen to be illusory, and the +hen is left helpless on the shore. Intuition, in fact, is an aspect +and development of instinct, and, like all instinct, is admirable in +those customary surroundings which have moulded the habits of the +animal in question, but totally incompetent as soon as the +surroundings are changed in a way which demands some non-habitual mode +of action. + +The theoretical understanding of the world, which is the aim of +philosophy, is not a matter of great practical importance to animals, +or to savages, or even to most civilised men. It is hardly to be +supposed, therefore, that the rapid, rough and ready methods of +instinct or intuition will find in this field a favourable ground for +their application. It is the older kinds of activity, which bring out +our kinship with remote generations of animal and semi-human +ancestors, that show intuition at its best. In such matters as +self-preservation and love, intuition will act sometimes (though not +always) with a swiftness and precision which are astonishing to the +critical intellect. But philosophy is not one of the pursuits which +illustrate our affinity with the past: it is a highly refined, highly +civilised pursuit, demanding, for its success, a certain liberation +from the life of instinct, and even, at times, a certain aloofness +from all mundane hopes and fears. It is not in philosophy, therefore, +that we can hope to see intuition at its best. On the contrary, since +the true objects of philosophy, and the habit of thought demanded for +their apprehension, are strange, unusual, and remote, it is here, more +almost than anywhere else, that intellect proves superior to +intuition, and that quick unanalysed convictions are least deserving +of uncritical acceptance. + +In advocating the scientific restraint and balance, as against the +self-assertion of a confident reliance upon intuition, we are only +urging, in the sphere of knowledge, that largeness of contemplation, +that impersonal disinterestedness, and that freedom from practical +preoccupations which have been inculcated by all the great religions +of the world. Thus our conclusion, however it may conflict with the +explicit beliefs of many mystics, is, in essence, not contrary to the +spirit which inspires those beliefs, but rather the outcome of this +very spirit as applied in the realm of thought. + + +II. UNITY AND PLURALITY + +One of the most convincing aspects of the mystic illumination is the +apparent revelation of the oneness of all things, giving rise to +pantheism in religion and to monism in philosophy. An elaborate logic, +beginning with Parmenides, and culminating in Hegel and his followers, +has been gradually developed, to prove that the universe is one +indivisible Whole, and that what seem to be its parts, if considered +as substantial and self-existing, are mere illusion. The conception +of a Reality quite other than the world of appearance, a reality one, +indivisible, and unchanging, was introduced into Western philosophy by +Parmenides, not, nominally at least, for mystical or religious +reasons, but on the basis of a logical argument as to the +impossibility of not-being, and most subsequent metaphysical systems +are the outcome of this fundamental idea. + +The logic used in defence of mysticism seems to be faulty as logic, +and open to technical criticisms, which I have explained elsewhere. I +shall not here repeat these criticisms, since they are lengthy and +difficult, but shall instead attempt an analysis of the state of mind +from which mystical logic has arisen. + +Belief in a reality quite different from what appears to the senses +arises with irresistible force in certain moods, which are the source +of most mysticism, and of most metaphysics. While such a mood is +dominant, the need of logic is not felt, and accordingly the more +thoroughgoing mystics do not employ logic, but appeal directly to the +immediate deliverance of their insight. But such fully developed +mysticism is rare in the West. When the intensity of emotional +conviction subsides, a man who is in the habit of reasoning will +search for logical grounds in favour of the belief which he finds in +himself. But since the belief already exists, he will be very +hospitable to any ground that suggests itself. The paradoxes +apparently proved by his logic are really the paradoxes of mysticism, +and are the goal which he feels his logic must reach if it is to be in +accordance with insight. The resulting logic has rendered most +philosophers incapable of giving any account of the world of science +and daily life. If they had been anxious to give such an account, they +would probably have discovered the errors of their logic; but most of +them were less anxious to understand the world of science and daily +life than to convict it of unreality in the interests of a +super-sensible "real" world. + +It is in this way that logic has been pursued by those of the great +philosophers who were mystics. But since they usually took for granted +the supposed insight of the mystic emotion, their logical doctrines +were presented with a certain dryness, and were believed by their +disciples to be quite independent of the sudden illumination from +which they sprang. Nevertheless their origin clung to them, and they +remained--to borrow a useful word from Mr. Santayana--"malicious" in +regard to the world of science and common sense. It is only so that we +can account for the complacency with which philosophers have accepted +the inconsistency of their doctrines with all the common and +scientific facts which seem best established and most worthy of +belief. + +The logic of mysticism shows, as is natural, the defects which are +inherent in anything malicious. The impulse to logic, not felt while +the mystic mood is dominant, reasserts itself as the mood fades, but +with a desire to retain the vanishing insight, or at least to prove +that it _was_ insight, and that what seems to contradict it is +illusion. The logic which thus arises is not quite disinterested or +candid, and is inspired by a certain hatred of the daily world to +which it is to be applied. Such an attitude naturally does not tend to +the best results. Everyone knows that to read an author simply in +order to refute him is not the way to understand him; and to read the +book of Nature with a conviction that it is all illusion is just as +unlikely to lead to understanding. If our logic is to find the common +world intelligible, it must not be hostile, but must be inspired by a +genuine acceptance such as is not usually to be found among +metaphysicians. + + +III. TIME + +The unreality of time is a cardinal doctrine of many metaphysical +systems, often nominally based, as already by Parmenides, upon logical +arguments, but originally derived, at any rate in the founders of new +systems, from the certainty which is born in the moment of mystic +insight. As a Persian Sufi poet says: + + "Past and future are what veil God from our sight. + Burn up both of them with fire! How long + Wilt thou be partitioned by these segments as a reed?"[5] + +The belief that what is ultimately real must be immutable is a very +common one: it gave rise to the metaphysical notion of substance, and +finds, even now, a wholly illegitimate satisfaction in such scientific +doctrines as the conservation of energy and mass. + +It is difficult to disentangle the truth and the error in this view. +The arguments for the contention that time is unreal and that the +world of sense is illusory must, I think, be regarded as fallacious. +Nevertheless there is some sense--easier to feel than to state--in +which time is an unimportant and superficial characteristic of +reality. Past and future must be acknowledged to be as real as the +present, and a certain emancipation from slavery to time is essential +to philosophic thought. The importance of time is rather practical +than theoretical, rather in relation to our desires than in relation +to truth. A truer image of the world, I think, is obtained by +picturing things as entering into the stream of time from an eternal +world outside, than from a view which regards time as the devouring +tyrant of all that is. Both in thought and in feeling, even though +time be real, to realise the unimportance of time is the gate of +wisdom. + +That this is the case may be seen at once by asking ourselves why our +feelings towards the past are so different from our feelings towards +the future. The reason for this difference is wholly practical: our +wishes can affect the future but not the past, the future is to some +extent subject to our power, while the past is unalterably fixed. But +every future will some day be past: if we see the past truly now, it +must, when it was still future, have been just what we now see it to +be, and what is now future must be just what we shall see it to be +when it has become past. The felt difference of quality between past +and future, therefore, is not an intrinsic difference, but only a +difference in relation to us: to impartial contemplation, it ceases to +exist. And impartiality of contemplation is, in the intellectual +sphere, that very same virtue of disinterestedness which, in the +sphere of action, appears as justice and unselfishness. Whoever wishes +to see the world truly, to rise in thought above the tyranny of +practical desires, must learn to overcome the difference of attitude +towards past and future, and to survey the whole stream of time in one +comprehensive vision. + +The kind of way in which, as it seems to me, time ought not to enter +into our theoretic philosophical thought, may be illustrated by the +philosophy which has become associated with the idea of evolution, and +which is exemplified by Nietzsche, pragmatism, and Bergson. This +philosophy, on the basis of the development which has led from the +lowest forms of life up to man, sees in _progress_ the fundamental law +of the universe, and thus admits the difference between _earlier_ and +_later_ into the very citadel of its contemplative outlook. With its +past and future history of the world, conjectural as it is, I do not +wish to quarrel. But I think that, in the intoxication of a quick +success, much that is required for a true understanding of the +universe has been forgotten. Something of Hellenism, something, too, +of Oriental resignation, must be combined with its hurrying Western +self-assertion before it can emerge from the ardour of youth into the +mature wisdom of manhood. In spite of its appeals to science, the true +scientific philosophy, I think, is something more arduous and more +aloof, appealing to less mundane hopes, and requiring a severer +discipline for its successful practice. + +Darwin's _Origin of Species_ persuaded the world that the difference +between different species of animals and plants is not the fixed +immutable difference that it appears to be. The doctrine of natural +kinds, which had rendered classification easy and definite, which was +enshrined in the Aristotelian tradition, and protected by its supposed +necessity for orthodox dogma, was suddenly swept away for ever out of +the biological world. The difference between man and the lower +animals, which to our human conceit appears enormous, was shown to be +a gradual achievement, involving intermediate being who could not with +certainty be placed either within or without the human family. The sun +and the planets had already been shown by Laplace to be very probably +derived from a primitive more or less undifferentiated nebula. Thus +the old fixed landmarks became wavering and indistinct, and all sharp +outlines were blurred. Things and species lost their boundaries, and +none could say where they began or where they ended. + +But if human conceit was staggered for a moment by its kinship with +the ape, it soon found a way to reassert itself, and that way is the +"philosophy" of evolution. A process which led from the amoeba to Man +appeared to the philosophers to be obviously a progress--though +whether the amoeba would agree with this opinion is not known. Hence +the cycle of changes which science had shown to be the probable +history of the past was welcomed as revealing a law of development +towards good in the universe--an evolution or unfolding of an idea +slowly embodying itself in the actual. But such a view, though it +might satisfy Spencer and those whom we may call Hegelian +evolutionists, could not be accepted as adequate by the more +whole-hearted votaries of change. An ideal to which the world +continuously approaches is, to these minds, too dead and static to be +inspiring. Not only the aspiration, but the ideal too, must change and +develop with the course of evolution: there must be no fixed goal, but +a continual fashioning of fresh needs by the impulse which is life and +which alone gives unity to the process. + +Life, in this philosophy, is a continuous stream, in which all +divisions are artificial and unreal. Separate things, beginnings and +endings, are mere convenient fictions: there is only smooth unbroken +transition. The beliefs of to-day may count as true to-day, if they +carry us along the stream; but to-morrow they will be false, and must +be replaced by new beliefs to meet the new situation. All our thinking +consists of convenient fictions, imaginary congealings of the stream: +reality flows on in spite of all our fictions, and though it can be +lived, it cannot be conceived in thought. Somehow, without explicit +statement, the assurance is slipped in that the future, though we +cannot foresee it, will be better than the past or the present: the +reader is like the child which expects a sweet because it has been +told to open its mouth and shut its eyes. Logic, mathematics, physics +disappear in this philosophy, because they are too "static"; what is +real is no impulse and movement towards a goal which, like the +rainbow, recedes as we advance, and makes every place different when +it reaches it from what it appeared to be at a distance. + +I do not propose to enter upon a technical examination of this +philosophy. I wish only to maintain that the motives and interests +which inspire it are so exclusively practical, and the problems with +which it deals are so special, that it can hardly be regarded as +touching any of the questions that, to my mind, constitute genuine +philosophy. + +The predominant interest of evolutionism is in the question of human +destiny, or at least of the destiny of Life. It is more interested in +morality and happiness than in knowledge for its own sake. It must be +admitted that the same may be said of many other philosophies, and +that a desire for the kind of knowledge which philosophy can give is +very rare. But if philosophy is to attain truth, it is necessary first +and foremost that philosophers should acquire the disinterested +intellectual curiosity which characterises the genuine man of science. +Knowledge concerning the future--which is the kind of knowledge that +must be sought if we are to know about human destiny--is possible +within certain narrow limits. It is impossible to say how much the +limits may be enlarged with the progress of science. But what is +evident is that any proposition about the future belongs by its +subject-matter to some particular science, and is to be ascertained, +if at all, by the methods of that science. Philosophy is not a short +cut to the same kind of results as those of the other sciences: if it +is to be a genuine study, it must have a province of its own, and aim +at results which the other sciences can neither prove nor disprove. + +Evolutionism, in basing itself upon the notion of _progress_, which is +change from the worse to the better, allows the notion of time, as it +seems to me, to become its tyrant rather than its servant, and thereby +loses that impartiality of contemplation which is the source of all +that is best in philosophic thought and feeling. Metaphysicians, as we +saw, have frequently denied altogether the reality of time. I do not +wish to do this; I wish only to preserve the mental outlook which +inspired the denial, the attitude which, in thought, regards the past +as having the same reality as the present and the same importance as +the future. "In so far," says Spinoza,[6] "as the mind conceives a +thing according to the dictate of reason, it will be equally affected +whether the idea is that of a future, past, or present thing." It is +this "conceiving according to the dictate of reason" that I find +lacking in the philosophy which is based on evolution. + + +IV. GOOD AND EVIL + +Mysticism maintains that all evil is illusory, and sometimes maintains +the same view as regards good, but more often holds that all Reality +is good. Both views are to be found in Heraclitus: "Good and ill are +one," he says, but again, "To God all things are fair and good and +right, but men hold some things wrong and some right." A similar +twofold position is to be found in Spinoza, but he uses the word +"perfection" when he means to speak of the good that is not merely +human. "By reality and perfection I mean the same thing," he says;[7] +but elsewhere we find the definition: "By _good_ I shall mean that +which we certainly know to be useful to us."[8] Thus perfection +belongs to Reality in its own nature, but goodness is relative to +ourselves and our needs, and disappears in an impartial survey. Some +such distinction, I think, is necessary in order to understand the +ethical outlook of mysticism: there is a lower mundane kind of good +and evil, which divides the world of appearance into what seem to be +conflicting parts; but there is also a higher, mystical kind of good, +which belongs to Reality and is not opposed by any correlative kind of +evil. + +It is difficult to give a logically tenable account of this position +without recognising that good and evil are subjective, that what is +good is merely that towards which we have one kind of feeling, and +what is evil is merely that towards which we have another kind of +feeling. In our active life, where we have to exercise choice, and to +prefer this to that of two possible acts, it is necessary to have a +distinction of good and evil, or at least of better and worse. But +this distinction, like everything pertaining to action, belongs to +what mysticism regards as the world of illusion, if only because it is +essentially concerned with time. In our contemplative life, where +action is not called for, it is possible to be impartial, and to +overcome the ethical dualism which action requires. So long as we +remain _merely_ impartial, we may be content to say that both the good +and the evil of action are illusions. But if, as we must do if we have +the mystic vision, we find the whole world worthy of love and worship, +if we see + + "The earth, and every common sight.... + Apparell'd in celestial light," + +we shall say that there is a higher good than that of action, and that +this higher good belongs to the whole world as it is in reality. In +this way the twofold attitude and the apparent vacillation of +mysticism are explained and justified. + +The possibility of this universal love and joy in all that exists is +of supreme importance for the conduct and happiness of life, and gives +inestimable value to the mystic emotion, apart from any creeds which +may be built upon it. But if we are not to be led into false beliefs, +it is necessary to realise exactly _what_ the mystic emotion reveals. +It reveals a possibility of human nature--a possibility of a nobler, +happier, freer life than any that can be otherwise achieved. But it +does not reveal anything about the non-human, or about the nature of +the universe in general. Good and bad, and even the higher good that +mysticism finds everywhere, are the reflections of our own emotions on +other things, not part of the substance of things as they are in +themselves. And therefore an impartial contemplation, freed from all +pre-occupation with Self, will not judge things good or bad, although +it is very easily combined with that feeling of universal love which +leads the mystic to say that the whole world is good. + +The philosophy of evolution, through the notion of progress, is bound +up with the ethical dualism of the worse and the better, and is thus +shut out, not only from the kind of survey which discards good and +evil altogether from its view, but also from the mystical belief in +the goodness of everything. In this way the distinction of good and +evil, like time, becomes a tyrant in this philosophy, and introduces +into thought the restless selectiveness of action. Good and evil, like +time, are, it would seem, not general or fundamental in the world of +thought, but late and highly specialised members of the intellectual +hierarchy. + +Although, as we saw, mysticism can be interpreted so as to agree with +the view that good and evil are not intellectually fundamental, it +must be admitted that here we are no longer in verbal agreement with +most of the great philosophers and religious teachers of the past. I +believe, however, that the elimination of ethical considerations from +philosophy is both scientifically necessary and--though this may seem +a paradox--an ethical advance. Both these contentions must be briefly +defended. + +The hope of satisfaction to our more human desires--the hope of +demonstrating that the world has this or that desirable ethical +characteristic--is not one which, so far as I can see, a scientific +philosophy can do anything whatever to satisfy. The difference between +a good world and a bad one is a difference in the particular +characteristics of the particular things that exist in these worlds: +it is not a sufficiently abstract difference to come within the +province of philosophy. Love and hate, for example, are ethical +opposites, but to philosophy they are closely analogous attitudes +towards objects. The general form and structure of those attitudes +towards objects which constitute mental phenomena is a problem for +philosophy, but the difference between love and hate is not a +difference of form or structure, and therefore belongs rather to the +special science of psychology than to philosophy. Thus the ethical +interests which have often inspired philosophers must remain in the +background: some kind of ethical interest may inspire the whole study, +but none must obtrude in the detail or be expected in the special +results which are sought. + +If this view seems at first sight disappointing, we may remind +ourselves that a similar change has been found necessary in all the +other sciences. The physicist or chemist is not now required to prove +the ethical importance of his ions or atoms; the biologist is not +expected to prove the utility of the plants or animals which he +dissects. In pre-scientific ages this was not the case. Astronomy, for +example, was studied because men believed in astrology: it was thought +that the movements of the planets had the most direct and important +bearing upon the lives of human beings. Presumably, when this belief +decayed and the disinterested study of astronomy began, many who had +found astrology absorbingly interesting decided that astronomy had too +little human interest to be worthy of study. Physics, as it appears in +Plato's Timæus for example, is full of ethical notions: it is an +essential part of its purpose to show that the earth is worthy of +admiration. The modern physicist, on the contrary, though he has no +wish to deny that the earth is admirable, is not concerned, as +physicist, with its ethical attributes: he is merely concerned to find +out facts, not to consider whether they are good or bad. In +psychology, the scientific attitude is even more recent and more +difficult than in the physical sciences: it is natural to consider +that human nature is either good or bad, and to suppose that the +difference between good and bad, so all-important in practice, must be +important in theory also. It is only during the last century that an +ethically neutral psychology has grown up; and here too, ethical +neutrality has been essential to scientific success. + +In philosophy, hitherto, ethical neutrality has been seldom sought and +hardly ever achieved. Men have remembered their wishes, and have +judged philosophies in relation to their wishes. Driven from the +particular sciences, the belief that the notions of good and evil must +afford a key to the understanding of the world has sought a refuge in +philosophy. But even from this last refuge, if philosophy is not to +remain a set of pleasing dreams, this belief must be driven forth. It +is a commonplace that happiness is not best achieved by those who +seek it directly; and it would seem that the same is true of the good. +In thought, at any rate, those who forget good and evil and seek only +to know the facts are more likely to achieve good than those who view +the world through the distorting medium of their own desires. + +We are thus brought back to our seeming paradox, that a philosophy +which does not seek to impose upon the world its own conceptions of +good and evil is not only more likely to achieve truth, but is also +the outcome of a higher ethical standpoint than one which, like +evolutionism and most traditional systems, is perpetually appraising +the universe and seeking to find in it an embodiment of present +ideals. In religion, and in every deeply serious view of the world and +of human destiny, there is an element of submission, a realisation of +the limits of human power, which is somewhat lacking in the modern +world, with its quick material successes and its insolent belief in +the boundless possibilities of progress. "He that loveth his life +shall lose it"; and there is danger lest, through a too confident love +of life, life itself should lose much of what gives it its highest +worth. The submission which religion inculcates in action is +essentially the same in spirit as that which science teaches in +thought; and the ethical neutrality by which its victories have been +achieved is the outcome of that submission. + +The good which it concerns us to remember is the good which it lies in +our power to create--the good in our own lives and in our attitude +towards the world. Insistence on belief in an external realisation of +the good is a form of self-assertion, which, while it cannot secure +the external good which it desires, can seriously impair the inward +good which lies within our power, and destroy that reverence towards +fact which constitutes both what is valuable in humility and what is +fruitful in the scientific temper. + +Human beings cannot, of course, wholly transcend human nature; +something subjective, if only the interest that determines the +direction of our attention, must remain in all our thought. But +scientific philosophy comes nearer to objectivity than any other human +pursuit, and gives us, therefore, the closest constant and the most +intimate relation with the outer world that it is possible to achieve. +To the primitive mind, everything is either friendly or hostile; but +experience has shown that friendliness and hostility are not the +conceptions by which the world is to be understood. Scientific +philosophy thus represents, though as yet only in a nascent condition, +a higher form of thought than any pre-scientific belief or +imagination, and, like every approach to self-transcendence, it brings +with it a rich reward in increase of scope and breadth and +comprehension. Evolutionism, in spite of its appeals to particular +scientific facts, fails to be a truly scientific philosophy because of +its slavery to time, its ethical preoccupations, and its predominant +interest in our mundane concerns and destiny. A truly scientific +philosophy will be more humble, more piecemeal, more arduous, offering +less glitter of outward mirage to flatter fallacious hopes, but more +indifferent to fate, and more capable of accepting the world without +the tyrannous imposition of our human and temporary demands. + +FOOTNOTES: + +[1] All the above quotations are from Burnet's _Early Greek +Philosophy_, (2nd ed., 1908), pp. 146-156. + +[2] _Republic_, 514, translated by Davies and Vaughan. + +[3] This section, and also one or two pages in later sections, have +been printed in a course of Lowell lectures _On our knowledge of the +external world_, published by the Open Court Publishing Company. But I +have left them here, as this is the context for which they were +originally written. + +[4] _Introduction to Metaphysics_, p. 1. + +[5] Whinfield's translation of the _Masnavi_ (Trübner, 1887), p. 34. + +[6] _Ethics_, Bk. IV, Prop. LXII. + +[7] Ib., Pt. IV, Df. I. + +[8] _Ethics_. Pt. II. Df. VI. + + + + +II + +THE PLACE OF SCIENCE IN A LIBERAL EDUCATION + + +I + +Science, to the ordinary reader of newspapers, is represented by a +varying selection of sensational triumphs, such as wireless telegraphy +and aeroplanes, radio-activity and the marvels of modern alchemy. It is +not of this aspect of science that I wish to speak. Science, in this +aspect, consists of detached up-to-date fragments, interesting only +until they are replaced by something newer and more up-to-date, +displaying nothing of the systems of patiently constructed knowledge +out of which, almost as a casual incident, have come the practically +useful results which interest the man in the street. The increased +command over the forces of nature which is derived from science is +undoubtedly an amply sufficient reason for encouraging scientific +research, but this reason has been so often urged and is so easily +appreciated that other reasons, to my mind quite as important, are apt +to be overlooked. It is with these other reasons, especially with the +intrinsic value of a scientific habit of mind in forming our outlook +on the world, that I shall be concerned in what follows. + +The instance of wireless telegraphy will serve to illustrate the +difference between the two points of view. Almost all the serious +intellectual labour required for the possibility of this invention is +due to three men--Faraday, Maxwell, and Hertz. In alternating layers +of experiment and theory these three men built up the modern theory of +electromagnetism, and demonstrated the identity of light with +electromagnetic waves. The system which they discovered is one of +profound intellectual interest, bringing together and unifying an +endless variety of apparently detached phenomena, and displaying a +cumulative mental power which cannot but afford delight to every +generous spirit. The mechanical details which remained to be adjusted +in order to utilise their discoveries for a practical system of +telegraphy demanded, no doubt, very considerable ingenuity, but had +not that broad sweep and that universality which could give them +intrinsic interest as an object of disinterested contemplation. + +From the point of view of training the mind, of giving that +well-informed, impersonal outlook which constitutes culture in the +good sense of this much-misused word, it seems to be generally held +indisputable that a literary education is superior to one based on +science. Even the warmest advocates of science are apt to rest their +claims on the contention that culture ought to be sacrificed to +utility. Those men of science who respect culture, when they associate +with men learned in the classics, are apt to admit, not merely +politely, but sincerely, a certain inferiority on their side, +compensated doubtless by the services which science renders to +humanity, but none the less real. And so long as this attitude exists +among men of science, it tends to verify itself: the intrinsically +valuable aspects of science tend to be sacrificed to the merely +useful, and little attempt is made to preserve that leisurely, +systematic survey by which the finer quality of mind is formed and +nourished. + +But even if there be, in present fact, any such inferiority as is +supposed in the educational value of science, this is, I believe, not +the fault of science itself, but the fault of the spirit in which +science is taught. If its full possibilities were realised by those +who teach it, I believe that its capacity of producing those habits of +mind which constitute the highest mental excellence would be at least +as great as that of literature, and more particularly of Greek and +Latin literature. In saying this I have no wish whatever to disparage +a classical education. I have not myself enjoyed its benefits, and my +knowledge of Greek and Latin authors is derived almost wholly from +translations. But I am firmly persuaded that the Greeks fully deserve +all the admiration that is bestowed upon them, and that it is a very +great and serious loss to be unacquainted with their writings. It is +not by attacking them, but by drawing attention to neglected +excellences in science, that I wish to conduct my argument. + +One defect, however, does seem inherent in a purely classical +education--namely, a too exclusive emphasis on the past. By the study +of what is absolutely ended and can never be renewed, a habit of +criticism towards the present and the future is engendered. The +qualities in which the present excels are qualities to which the study +of the past does not direct attention, and to which, therefore, the +student of Greek civilisation may easily become blind. In what is new +and growing there is apt to be something crude, insolent, even a +little vulgar, which is shocking to the man of sensitive taste; +quivering from the rough contact, he retires to the trim gardens of a +polished past, forgetting that they were reclaimed from the wilderness +by men as rough and earth-soiled as those from whom he shrinks in his +own day. The habit of being unable to recognise merit until it is +dead is too apt to be the result of a purely bookish life, and a +culture based wholly on the past will seldom be able to pierce through +everyday surroundings to the essential splendour of contemporary +things, or to the hope of still greater splendour in the future. + + "My eyes saw not the men of old; + And now their age away has rolled. + I weep--to think I shall not see + The heroes of posterity." + +So says the Chinese poet; but such impartiality is rare in the more +pugnacious atmosphere of the West, where the champions of past and +future fight a never-ending battle, instead of combining to seek out +the merits of both. + +This consideration, which militates not only against the exclusive +study of the classics, but against every form of culture which has +become static, traditional, and academic, leads inevitably to the +fundamental question: What is the true end of education? But before +attempting to answer this question it will be well to define the sense +in which we are to use the word "education." For this purpose I shall +distinguish the sense in which I mean to use it from two others, both +perfectly legitimate, the one broader and the other narrower than the +sense in which I mean to use the word. + +In the broader sense, education will include not only what we learn +through instruction, but all that we learn through personal +experience--the formation of character through the education of life. +Of this aspect of education, vitally important as it is, I will say +nothing, since its consideration would introduce topics quite foreign +to the question with which we are concerned. + +In the narrower sense, education may be confined to instruction, the +imparting of definite information on various subjects, because such +information, in and for itself, is useful in daily life. Elementary +education--reading, writing, and arithmetic--is almost wholly of this +kind. But instruction, necessary as it is, does not _per se_ +constitute education in the sense in which I wish to consider it. + +Education, in the sense in which I mean it, may be defined as _the +formation, by means of instruction, of certain mental habits and a +certain outlook on life and the world_. It remains to ask ourselves, +what mental habits, and what sort of outlook, can be hoped for as the +result of instruction? When we have answered this question we can +attempt to decide what science has to contribute to the formation of +the habits and outlook which we desire. + +Our whole life is built about a certain number--not a very small +number--of primary instincts and impulses. Only what is in some way +connected with these instincts and impulses appears to us desirable or +important; there is no faculty, whether "reason" or "virtue" or +whatever it may be called, that can take our active life and our hopes +and fears outside the region controlled by these first movers of all +desire. Each of them is like a queen-bee, aided by a hive of workers +gathering honey; but when the queen is gone the workers languish and +die, and the cells remain empty of their expected sweetness. So with +each primary impulse in civilised man: it is surrounded and protected +by a busy swarm of attendant derivative desires, which store up in its +service whatever honey the surrounding world affords. But if the +queen-impulse dies, the death-dealing influence, though retarded a +little by habit, spreads slowly through all the subsidiary impulses, +and a whole tract of life becomes inexplicably colourless. What was +formerly full of zest, and so obviously worth doing that it raised no +questions, has now grown dreary and purposeless: with a sense of +disillusion we inquire the meaning of life, and decide, perhaps, that +all is vanity. The search for an outside meaning that can _compel_ an +inner response must always be disappointed: all "meaning" must be at +bottom related to our primary desires, and when they are extinct no +miracle can restore to the world the value which they reflected upon +it. + +The purpose of education, therefore, cannot be to create any primary +impulse which is lacking in the uneducated; the purpose can only be to +enlarge the scope of those that human nature provides, by increasing +the number and variety of attendant thoughts, and by showing where the +most permanent satisfaction is to be found. Under the impulse of a +Calvinistic horror of the "natural man," this obvious truth has been +too often misconceived in the training of the young; "nature" has been +falsely regarded as excluding all that is best in what is natural, and +the endeavour to teach virtue has led to the production of stunted and +contorted hypocrites instead of full-grown human beings. From such +mistakes in education a better psychology or a kinder heart is +beginning to preserve the present generation; we need, therefore, +waste no more words on the theory that the purpose of education is to +thwart or eradicate nature. + +But although nature must supply the initial force of desire, nature is +not, in the civilised man, the spasmodic, fragmentary, and yet violent +set of impulses that it is in the savage. Each impulse has its +constitutional ministry of thought and knowledge and reflection, +through which possible conflicts of impulses are foreseen, and +temporary impulses are controlled by the unifying impulse which may be +called wisdom. In this way education destroys the crudity of +instinct, and increases through knowledge the wealth and variety of +the individual's contacts with the outside world, making him no longer +an isolated fighting unit, but a citizen of the universe, embracing +distant countries, remote regions of space, and vast stretches of past +and future within the circle of his interests. It is this simultaneous +softening in the insistence of desire and enlargement of its scope +that is the chief moral end of education. + +Closely connected with this moral end is the more purely intellectual +aim of education, the endeavour to make us see and imagine the world +in an objective manner, as far as possible as it is in itself, and not +merely through the distorting medium of personal desire. The complete +attainment of such an objective view is no doubt an ideal, +indefinitely approachable, but not actually and fully realisable. +Education, considered as a process of forming our mental habits and +our outlook on the world, is to be judged successful in proportion as +its outcome approximates to this ideal; in proportion, that is to say, +as it gives us a true view of our place in society, of the relation of +the whole human society to its non-human environment, and of the +nature of the non-human world as it is in itself apart from our +desires and interests. If this standard is admitted, we can return to +the consideration of science, inquiring how far science contributes to +such an aim, and whether it is in any respect superior to its rivals +in educational practice. + + +II + +Two opposite and at first sight conflicting merits belong to science +as against literature and art. The one, which is not inherently +necessary, but is certainly true at the present day, is hopefulness +as to the future of human achievement, and in particular as to the +useful work that may be accomplished by any intelligent student. This +merit and the cheerful outlook which it engenders prevent what might +otherwise be the depressing effect of another aspect of science, to my +mind also a merit, and perhaps its greatest merit--I mean the +irrelevance of human passions and of the whole subjective apparatus +where scientific truth is concerned. Each of these reasons for +preferring the study of science requires some amplification. Let us +begin with the first. + +In the study of literature or art our attention is perpetually riveted +upon the past: the men of Greece or of the Renaissance did better than +any men do now; the triumphs of former ages, so far from facilitating +fresh triumphs in our own age, actually increase the difficulty of +fresh triumphs by rendering originality harder of attainment; not only +is artistic achievement not cumulative, but it seems even to depend +upon a certain freshness and _naïveté_ of impulse and vision which +civilisation tends to destroy. Hence comes, to those who have been +nourished on the literary and artistic productions of former ages, a +certain peevishness and undue fastidiousness towards the present, from +which there seems no escape except into the deliberate vandalism which +ignores tradition and in the search after originality achieves only +the eccentric. But in such vandalism there is none of the simplicity +and spontaneity out of which great art springs: theory is still the +canker in its core, and insincerity destroys the advantages of a +merely pretended ignorance. + +The despair thus arising from an education which suggests no +pre-eminent mental activity except that of artistic creation is wholly +absent from an education which gives the knowledge of scientific +method. The discovery of scientific method, except in pure +mathematics, is a thing of yesterday; speaking broadly, we may say +that it dates from Galileo. Yet already it has transformed the world, +and its success proceeds with ever-accelerating velocity. In science +men have discovered an activity of the very highest value in which +they are no longer, as in art, dependent for progress upon the +appearance of continually greater genius, for in science the +successors stand upon the shoulders of their predecessors; where one +man of supreme genius has invented a method, a thousand lesser men can +apply it. No transcendent ability is required in order to make useful +discoveries in science; the edifice of science needs its masons, +bricklayers, and common labourers as well as its foremen, +master-builders, and architects. In art nothing worth doing can be +done without genius; in science even a very moderate capacity can +contribute to a supreme achievement. + +In science the man of real genius is the man who invents a new method. +The notable discoveries are often made by his successors, who can +apply the method with fresh vigour, unimpaired by the previous labour +of perfecting it; but the mental calibre of the thought required for +their work, however brilliant, is not so great as that required by the +first inventor of the method. There are in science immense numbers of +different methods, appropriate to different classes of problems; but +over and above them all, there is something not easily definable, +which may be called _the_ method of science. It was formerly customary +to identify this with the inductive method, and to associate it with +the name of Bacon. But the true inductive method was not discovered by +Bacon, and the true method of science is something which includes +deduction as much as induction, logic and mathematics as much as +botany and geology. I shall not attempt the difficult task of stating +what the scientific method is, but I will try to indicate the temper +of mind out of which the scientific method grows, which is the second +of the two merits that were mentioned above as belonging to a +scientific education. + +The kernel of the scientific outlook is a thing so simple, so obvious, +so seemingly trivial, that the mention of it may almost excite +derision. The kernel of the scientific outlook is the refusal to +regard our own desires, tastes, and interests as affording a key to +the understanding of the world. Stated thus baldly, this may seem no +more than a trite truism. But to remember it consistently in matters +arousing our passionate partisanship is by no means easy, especially +where the available evidence is uncertain and inconclusive. A few +illustrations will make this clear. + +Aristotle, I understand, considered that the stars must move in +circles because the circle is the most perfect curve. In the absence +of evidence to the contrary, he allowed himself to decide a question +of fact by an appeal to æsthetico-moral considerations. In such a case +it is at once obvious to us that this appeal was unjustifiable. We +know now how to ascertain as a fact the way in which the heavenly +bodies move, and we know that they do not move in circles, or even in +accurate ellipses, or in any other kind of simply describable curve. +This may be painful to a certain hankering after simplicity of pattern +in the universe, but we know that in astronomy such feelings are +irrelevant. Easy as this knowledge seems now, we owe it to the courage +and insight of the first inventors of scientific method, and more +especially of Galileo. + +We may take as another illustration Malthus's doctrine of population. +This illustration is all the better for the fact that his actual +doctrine is now known to be largely erroneous. It is not his +conclusions that are valuable, but the temper and method of his +inquiry. As everyone knows, it was to him that Darwin owed an +essential part of his theory of natural selection, and this was only +possible because Malthus's outlook was truly scientific. His great +merit lies in considering man not as the object of praise or blame, +but as a part of nature, a thing with a certain characteristic +behaviour from which certain consequences must follow. If the +behaviour is not quite what Malthus supposed, if the consequences are +not quite what he inferred, that may falsify his conclusions, but does +not impair the value of his method. The objections which were made +when his doctrine was new--that it was horrible and depressing, that +people ought not to act as he said they did, and so on--were all such +as implied an unscientific attitude of mind; as against all of them, +his calm determination to treat man as a natural phenomenon marks an +important advance over the reformers of the eighteenth century and the +Revolution. + +Under the influence of Darwinism the scientific attitude towards man +has now become fairly common, and is to some people quite natural, +though to most it is still a difficult and artificial intellectual +contortion. There is however, one study which is as yet almost wholly +untouched by the scientific spirit--I mean the study of philosophy. +Philosophers and the public imagine that the scientific spirit must +pervade pages that bristle with allusions to ions, germ-plasms, and +the eyes of shell-fish. But as the devil can quote Scripture, so the +philosopher can quote science. The scientific spirit is not an affair +of quotation, of externally acquired information, any more than +manners are an affair of the etiquette-book. The scientific attitude +of mind involves a sweeping away of all other desires in the interests +of the desire to know--it involves suppression of hopes and fears, +loves and hates, and the whole subjective emotional life, until we +become subdued to the material, able to see it frankly, without +preconceptions, without bias, without any wish except to see it as it +is, and without any belief that what it is must be determined by some +relation, positive or negative, to what we should like it to be, or to +what we can easily imagine it to be. + +Now in philosophy this attitude of mind has not as yet been achieved. +A certain self-absorption, not personal, but human, has marked almost +all attempts to conceive the universe as a whole. Mind, or some aspect +of it--thought or will or sentience--has been regarded as the pattern +after which the universe is to be conceived, for no better reason, at +bottom, than that such a universe would not seem strange, and would +give us the cosy feeling that every place is like home. To conceive +the universe as essentially progressive or essentially deteriorating, +for example, is to give to our hopes and fears a cosmic importance +which _may_, of course, be justified, but which we have as yet no +reason to suppose justified. Until we have learnt to think of it in +ethically neutral terms, we have not arrived at a scientific attitude +in philosophy; and until we have arrived at such an attitude, it is +hardly to be hoped that philosophy will achieve any solid results. + +I have spoken so far largely of the negative aspect of the scientific +spirit, but it is from the positive aspect that its value is derived. +The instinct of constructiveness, which is one of the chief incentives +to artistic creation, can find in scientific systems a satisfaction +more massive than any epic poem. Disinterested curiosity, which is the +source of almost all intellectual effort, finds with astonished +delight that science can unveil secrets which might well have seemed +for ever undiscoverable. The desire for a larger life and wider +interests, for an escape from private circumstances, and even from the +whole recurring human cycle of birth and death, is fulfilled by the +impersonal cosmic outlook of science as by nothing else. To all these +must be added, as contributing to the happiness of the man of science, +the admiration of splendid achievement, and the consciousness of +inestimable utility to the human race. A life devoted to science is +therefore a happy life, and its happiness is derived from the very +best sources that are open to dwellers on this troubled and passionate +planet. + + + + +III + +A FREE MAN'S WORSHIP[9] + + +To Dr. Faustus in his study Mephistopheles told the history of the +Creation, saying: + +"The endless praises of the choirs of angels had begun to grow +wearisome; for, after all, did he not deserve their praise? Had he not +given them endless joy? Would it not be more amusing to obtain +undeserved praise, to be worshipped by beings whom he tortured? He +smiled inwardly, and resolved that the great drama should be +performed. + +"For countless ages the hot nebula whirled aimlessly through space. At +length it began to take shape, the central mass threw off planets, the +planets cooled, boiling seas and burning mountains heaved and tossed, +from black masses of cloud hot sheets of rain deluged the barely solid +crust. And now the first germ of life grew in the depths of the ocean, +and developed rapidly in the fructifying warmth into vast forest +trees, huge ferns springing from the damp mould, sea monsters +breeding, fighting, devouring, and passing away. And from the +monsters, as the play unfolded itself, Man was born, with the power of +thought, the knowledge of good and evil, and the cruel thirst for +worship. And Man saw that all is passing in this mad, monstrous world, +that all is struggling to snatch, at any cost, a few brief moments of +life before Death's inexorable decree. And Man said: 'There is a +hidden purpose, could we but fathom it, and the purpose is good; for +we must reverence something, and in the visible world there is nothing +worthy of reverence.' And Man stood aside from the struggle, resolving +that God intended harmony to come out of chaos by human efforts. And +when he followed the instincts which God had transmitted to him from +his ancestry of beasts of prey, he called it Sin, and asked God to +forgive him. But he doubted whether he could be justly forgiven, until +he invented a divine Plan by which God's wrath was to have been +appeased. And seeing the present was bad, he made it yet worse, that +thereby the future might be better. And he gave God thanks for the +strength that enabled him to forgo even the joys that were possible. +And God smiled; and when he saw that Man had become perfect in +renunciation and worship, he sent another sun through the sky, which +crashed into Man's sun; and all returned again to nebula. + +"'Yes,' he murmured, 'it was a good play; I will have it performed +again.'" + +Such, in outline, but even more purposeless, more void of meaning, is +the world which Science presents for our belief. Amid such a world, if +anywhere, our ideals henceforward must find a home. That Man is the +product of causes which had no prevision of the end they were +achieving; that his origin, his growth, his hopes and fears, his loves +and his beliefs, are but the outcome of accidental collocations of +atoms; that no fire, no heroism, no intensity of thought and feeling, +can preserve an individual life beyond the grave; that all the labours +of the ages, all the devotion, all the inspiration, all the noonday +brightness of human genius, are destined to extinction in the vast +death of the solar system, and that the whole temple of Man's +achievement must inevitably be buried beneath the débris of a universe +in ruins--all these things, if not quite beyond dispute, are yet so +nearly certain, that no philosophy which rejects them can hope to +stand. Only within the scaffolding of these truths, only on the firm +foundation of unyielding despair, can the soul's habitation henceforth +be safely built. + +How, in such an alien and inhuman world, can so powerless a creature +as Man preserve his aspirations untarnished? A strange mystery it is +that Nature, omnipotent but blind, in the revolutions of her secular +hurryings through the abysses of space, has brought forth at last a +child, subject still to her power, but gifted with sight, with +knowledge of good and evil, with the capacity of judging all the works +of his unthinking Mother. In spite of Death, the mark and seal of the +parental control, Man is yet free, during his brief years, to examine, +to criticise, to know, and in imagination to create. To him alone, in +the world with which he is acquainted, this freedom belongs; and in +this lies his superiority to the resistless forces that control his +outward life. + +The savage, like ourselves, feels the oppression of his impotence +before the powers of Nature; but having in himself nothing that he +respects more than Power, he is willing to prostrate himself before +his gods, without inquiring whether they are worthy of his worship. +Pathetic and very terrible is the long history of cruelty and torture, +of degradation and human sacrifice, endured in the hope of placating +the jealous gods: surely, the trembling believer thinks, when what is +most precious has been freely given, their lust for blood must be +appeased, and more will not be required. The religion of Moloch--as +such creeds may be generically called--is in essence the cringing +submission of the slave, who dare not, even in his heart, allow the +thought that his master deserves no adulation. Since the independence +of ideals is not yet acknowledged, Power may be freely worshipped, and +receive an unlimited respect, despite its wanton infliction of pain. + +But gradually, as morality grows bolder, the claim of the ideal world +begins to be felt; and worship, if it is not to cease, must be given +to gods of another kind than those created by the savage. Some, though +they feel the demands of the ideal, will still consciously reject +them, still urging that naked Power is worthy of worship. Such is the +attitude inculcated in God's answer to Job out of the whirlwind: the +divine power and knowledge are paraded, but of the divine goodness +there is no hint. Such also is the attitude of those who, in our own +day, base their morality upon the struggle for survival, maintaining +that the survivors are necessarily the fittest. But others, not +content with an answer so repugnant to the moral sense, will adopt the +position which we have become accustomed to regard as specially +religious, maintaining that, in some hidden manner, the world of fact +is really harmonious with the world of ideals. Thus Man creates God, +all-powerful and all-good, the mystic unity of what is and what should +be. + +But the world of fact, after all, is not good; and, in submitting our +judgment to it, there is an element of slavishness from which our +thoughts must be purged. For in all things it is well to exalt the +dignity of Man, by freeing him as far as possible from the tyranny of +non-human Power. When we have realised that Power is largely bad, that +man, with his knowledge of good and evil, is but a helpless atom in a +world which has no such knowledge, the choice is again presented to +us: Shall we worship Force, or shall we worship Goodness? Shall our +God exist and be evil, or shall he be recognised as the creation of +our own conscience? + +The answer to this question is very momentous, and affects profoundly +our whole morality. The worship of Force, to which Carlyle and +Nietzsche and the creed of Militarism have accustomed us, is the +result of failure to maintain our own ideals against a hostile +universe: it is itself a prostrate submission to evil, a sacrifice of +our best to Moloch. If strength indeed is to be respected, let us +respect rather the strength of those who refuse that false +"recognition of facts" which fails to recognise that facts are often +bad. Let us admit that, in the world we know, there are many things +that would be better otherwise, and that the ideals to which we do and +must adhere are not realised in the realm of matter. Let us preserve +our respect for truth, for beauty, for the ideal of perfection which +life does not permit us to attain, though none of these things meet +with the approval of the unconscious universe. If Power is bad, as it +seems to be, let us reject it from our hearts. In this lies Man's true +freedom: in determination to worship only the God created by our own +love of the good, to respect only the heaven which inspires the +insight of our best moments. In action, in desire, we must submit +perpetually to the tyranny of outside forces; but in thought, in +aspiration, we are free, free from our fellow-men, free from the petty +planet on which our bodies impotently crawl, free even, while we live, +from the tyranny of death. Let us learn, then, that energy of faith +which enables us to live constantly in the vision of the good; and let +us descend, in action, into the world of fact, with that vision always +before us. + +When first the opposition of fact and ideal grows fully visible, a +spirit of fiery revolt, of fierce hatred of the gods, seems necessary +to the assertion of freedom. To defy with Promethean constancy a +hostile universe, to keep its evil always in view, always actively +hated, to refuse no pain that the malice of Power can invent, appears +to be the duty of all who will not bow before the inevitable. But +indignation is still a bondage, for it compels our thoughts to be +occupied with an evil world; and in the fierceness of desire from +which rebellion springs there is a kind of self-assertion which it is +necessary for the wise to overcome. Indignation is a submission of our +thoughts, but not of our desires; the Stoic freedom in which wisdom +consists is found in the submission of our desires, but not of our +thoughts. From the submission of our desires springs the virtue of +resignation; from the freedom of our thoughts springs the whole world +of art and philosophy, and the vision of beauty by which, at last, we +half reconquer the reluctant world. But the vision of beauty is +possible only to unfettered contemplation, to thoughts not weighted by +the load of eager wishes; and thus Freedom comes only to those who no +longer ask of life that it shall yield them any of those personal +goods that are subject to the mutations of Time. + +Although the necessity of renunciation is evidence of the existence of +evil, yet Christianity, in preaching it, has shown a wisdom exceeding +that of the Promethean philosophy of rebellion. It must be admitted +that, of the things we desire, some, though they prove impossible, are +yet real goods; others, however, as ardently longed for, do not form +part of a fully purified ideal. The belief that what must be renounced +is bad, though sometimes false, is far less often false than untamed +passion supposes; and the creed of religion, by providing a reason +for proving that it is never false, has been the means of purifying +our hopes by the discovery of many austere truths. + +But there is in resignation a further good element: even real goods, +when they are unattainable, ought not to be fretfully desired. To +every man comes, sooner or later, the great renunciation. For the +young, there is nothing unattainable; a good thing desired with the +whole force of a passionate will, and yet impossible, is to them not +credible. Yet, by death, by illness, by poverty, or by the voice of +duty, we must learn, each one of us, that the world was not made for +us, and that, however beautiful may be the things we crave, Fate may +nevertheless forbid them. It is the part of courage, when misfortune +comes, to bear without repining the ruin of our hopes, to turn away +our thoughts from vain regrets. This degree of submission to Power is +not only just and right: it is the very gate of wisdom. + +But passive renunciation is not the whole of wisdom; for not by +renunciation alone can we build a temple for the worship of our own +ideals. Haunting foreshadowings of the temple appear in the realm of +imagination, in music, in architecture, in the untroubled kingdom of +reason, and in the golden sunset magic of lyrics, where beauty shines +and glows, remote from the touch of sorrow, remote from the fear of +change, remote from the failures and disenchantments of the world of +fact. In the contemplation of these things the vision of heaven will +shape itself in our hearts, giving at once a touchstone to judge the +world about us, and an inspiration by which to fashion to our needs +whatever is not incapable of serving as a stone in the sacred temple. + +Except for those rare spirits that are born without sin, there is a +cavern of darkness to be traversed before that temple can be entered. +The gate of the cavern is despair, and its floor is paved with the +gravestones of abandoned hopes. There Self must die; there the +eagerness, the greed of untamed desire must be slain, for only so can +the soul be freed from the empire of Fate. But out of the cavern the +Gate of Renunciation leads again to the daylight of wisdom, by whose +radiance a new insight, a new joy, a new tenderness, shine forth to +gladden the pilgrim's heart. + +When, without the bitterness of impotent rebellion, we have learnt +both to resign ourselves to the outward rule of Fate and to recognise +that the non-human world is unworthy of our worship, it becomes +possible at last so to transform and refashion the unconscious +universe, so to transmute it in the crucible of imagination, that a +new image of shining gold replaces the old idol of clay. In all the +multiform facts of the world--in the visual shapes of trees and +mountains and clouds, in the events of the life of man, even in the +very omnipotence of Death--the insight of creative idealism can find +the reflection of a beauty which its own thoughts first made. In this +way mind asserts its subtle mastery over the thoughtless forces of +Nature. The more evil the material with which it deals, the more +thwarting to untrained desire, the greater is its achievement in +inducing the reluctant rock to yield up its hidden treasures, the +prouder its victory in compelling the opposing forces to swell the +pageant of its triumph. Of all the arts, Tragedy is the proudest, the +most triumphant; for it builds its shining citadel in the very centre +of the enemy's country, on the very summit of his highest mountain; +from its impregnable watchtowers, his camps and arsenals, his columns +and forts, are all revealed; within its walls the free life continues, +while the legions of Death and Pain and Despair, and all the servile +captains of tyrant Fate, afford the burghers of that dauntless city +new spectacles of beauty. Happy those sacred ramparts, thrice happy +the dwellers on that all-seeing eminence. Honour to those brave +warriors who, through countless ages of warfare, have preserved for us +the priceless heritage of liberty, and have kept undefiled by +sacrilegious invaders the home of the unsubdued. + +But the beauty of Tragedy does but make visible a quality which, in +more or less obvious shapes, is present always and everywhere in life. +In the spectacle of Death, in the endurance of intolerable pain, and +in the irrevocableness of a vanished past, there is a sacredness, an +overpowering awe, a feeling of the vastness, the depth, the +inexhaustible mystery of existence, in which, as by some strange +marriage of pain, the sufferer is bound to the world by bonds of +sorrow. In these moments of insight, we lose all eagerness of +temporary desire, all struggling and striving for petty ends, all care +for the little trivial things that, to a superficial view, make up the +common life of day by day; we see, surrounding the narrow raft +illumined by the flickering light of human comradeship, the dark ocean +on whose rolling waves we toss for a brief hour; from the great night +without, a chill blast breaks in upon our refuge; all the loneliness +of humanity amid hostile forces is concentrated upon the individual +soul, which must struggle alone, with what of courage it can command, +against the whole weight of a universe that cares nothing for its +hopes and fears. Victory, in this struggle with the powers of +darkness, is the true baptism into the glorious company of heroes, the +true initiation into the overmastering beauty of human existence. From +that awful encounter of the soul with the outer world, renunciation, +wisdom, and charity are born; and with their birth a new life begins. +To take into the inmost shrine of the soul the irresistible forces +whose puppets we seem to be--Death and change, the irrevocableness of +the past, and the powerlessness of man before the blind hurry of the +universe from vanity to vanity--to feel these things and know them is +to conquer them. + +This is the reason why the Past has such magical power. The beauty of +its motionless and silent pictures is like the enchanted purity of +late autumn, when the leaves, though one breath would make them fall, +still glow against the sky in golden glory. The Past does not change +or strive; like Duncan, after life's fitful fever it sleeps well; what +was eager and grasping, what was petty and transitory, has faded away, +the things that were beautiful and eternal shine out of it like stars +in the night. Its beauty, to a soul not worthy of it, is unendurable; +but to a soul which has conquered Fate it is the key of religion. + +The life of Man, viewed outwardly, is but a small thing in comparison +with the forces of Nature. The slave is doomed to worship Time and +Fate and Death, because they are greater than anything he finds in +himself, and because all his thoughts are of things which they devour. +But, great as they are, to think of them greatly, to feel their +passionless splendour, is greater still. And such thought makes us +free men; we no longer bow before the inevitable in Oriental +subjection, but we absorb it, and make it a part of ourselves. To +abandon the struggle for private happiness, to expel all eagerness of +temporary desire, to burn with passion for eternal things--this is +emancipation, and this is the free man's worship. And this liberation +is effected by a contemplation of Fate; for Fate itself is subdued by +the mind which leaves nothing to be purged by the purifying fire of +Time. + +United with his fellow-men by the strongest of all ties, the tie of a +common doom, the free man finds that a new vision is with him always, +shedding over every daily task the light of love. The life of Man is a +long march through the night, surrounded by invisible foes, tortured +by weariness and pain, towards a goal that few can hope to reach, and +where none may tarry long. One by one, as they march, our comrades +vanish from our sight, seized by the silent orders of omnipotent +Death. Very brief is the time in which we can help them, in which +their happiness or misery is decided. Be it ours to shed sunshine on +their path, to lighten their sorrows by the balm of sympathy, to give +them the pure joy of a never-tiring affection, to strengthen failing +courage, to instil faith in hours of despair. Let us not weigh in +grudging scales their merits and demerits, but let us think only of +their need--of the sorrows, the difficulties, perhaps the blindnesses, +that make the misery of their lives; let us remember that they are +fellow-sufferers in the same darkness, actors in the same tragedy with +ourselves. And so, when their day is over, when their good and their +evil have become eternal by the immortality of the past, be it ours to +feel that, where they suffered, where they failed, no deed of ours was +the cause; but wherever a spark of the divine fire kindled in their +hearts, we were ready with encouragement, with sympathy, with brave +words in which high courage glowed. + +Brief and powerless is Man's life; on him and all his race the slow, +sure doom falls pitiless and dark. Blind to good and evil, reckless of +destruction, omnipotent matter rolls on its relentless way; for Man, +condemned to-day to lose his dearest, to-morrow himself to pass +through the gate of darkness, it remains only to cherish, ere yet the +blow falls, the lofty thoughts that ennoble his little day; disdaining +the coward terrors of the slave of Fate, to worship at the shrine that +his own hands have built; undismayed by the empire of chance, to +preserve a mind free from the wanton tyranny that rules his outward +life; proudly defiant of the irresistible forces that tolerate, for a +moment, his knowledge and his condemnation, to sustain alone, a weary +but unyielding Atlas, the world that his own ideals have fashioned +despite the trampling march of unconscious power. + +FOOTNOTES: + +[9] Reprinted from the _Independent Review_, December, 1903. + + + + +IV + +THE STUDY OF MATHEMATICS + + +In regard to every form of human activity it is necessary that the +question should be asked from time to time, What is its purpose and +ideal? In what way does it contribute to the beauty of human +existence? As respects those pursuits which contribute only remotely, +by providing the mechanism of life, it is well to be reminded that not +the mere fact of living is to be desired, but the art of living in the +contemplation of great things. Still more in regard to those +avocations which have no end outside themselves, which are to be +justified, if at all, as actually adding to the sum of the world's +permanent possessions, it is necessary to keep alive a knowledge of +their aims, a clear prefiguring vision of the temple in which creative +imagination is to be embodied. + +The fulfilment of this need, in what concerns the studies forming the +material upon which custom has decided to train the youthful mind, is +indeed sadly remote--so remote as to make the mere statement of such a +claim appear preposterous. Great men, fully alive to the beauty of the +contemplations to whose service their lives are devoted, desiring that +others may share in their joys, persuade mankind to impart to the +successive generations the mechanical knowledge without which it is +impossible to cross the threshold. Dry pedants possess themselves of +the privilege of instilling this knowledge: they forget that it is to +serve but as a key to open the doors of the temple; though they spend +their lives on the steps leading up to those sacred doors, they turn +their backs upon the temple so resolutely that its very existence is +forgotten, and the eager youth, who would press forward to be +initiated to its domes and arches, is bidden to turn back and count +the steps. + +Mathematics, perhaps more even than the study of Greece and Rome, has +suffered from this oblivion of its due place in civilisation. Although +tradition has decreed that the great bulk of educated men shall know +at least the elements of the subject, the reasons for which the +tradition arose are forgotten, buried beneath a great rubbish-heap of +pedantries and trivialities. To those who inquire as to the purpose of +mathematics, the usual answer will be that it facilitates the making +of machines, the travelling from place to place, and the victory over +foreign nations, whether in war or commerce. If it be objected that +these ends--all of which are of doubtful value--are not furthered by +the merely elementary study imposed upon those who do not become +expert mathematicians, the reply, it is true, will probably be that +mathematics trains the reasoning faculties. Yet the very men who make +this reply are, for the most part, unwilling to abandon the teaching +of definite fallacies, known to be such, and instinctively rejected by +the unsophisticated mind of every intelligent learner. And the +reasoning faculty itself is generally conceived, by those who urge its +cultivation, as merely a means for the avoidance of pitfalls and a +help in the discovery of rules for the guidance of practical life. All +these are undeniably important achievements to the credit of +mathematics; yet it is none of these that entitles mathematics to a +place in every liberal education. Plato, we know, regarded the +contemplation of mathematical truths as worthy of the Deity; and +Plato realised, more perhaps than any other single man, what those +elements are in human life which merit a place in heaven. There is in +mathematics, he says, "something which is _necessary_ and cannot be +set aside ... and, if I mistake not, of divine necessity; for as to +the human necessities of which the Many talk in this connection, +nothing can be more ridiculous than such an application of the words. +_Cleinias._ And what are these necessities of knowledge, Stranger, +which are divine and not human? _Athenian._ Those things without some +use or knowledge of which a man cannot become a God to the world, nor +a spirit, nor yet a hero, nor able earnestly to think and care for +man" (_Laws_, p. 818).[10] Such was Plato's judgment of mathematics; +but the mathematicians do not read Plato, while those who read him +know no mathematics, and regard his opinion upon this question as +merely a curious aberration. + +Mathematics, rightly viewed, possesses not only truth, but supreme +beauty--a beauty cold and austere, like that of sculpture, without +appeal to any part of our weaker nature, without the gorgeous +trappings of painting or music, yet sublimely pure, and capable of a +stern perfection such as only the greatest art can show. The true +spirit of delight, the exaltation, the sense of being more than man, +which is the touchstone of the highest excellence, is to be found in +mathematics as surely as in poetry. What is best in mathematics +deserves not merely to be learnt as a task, but to be assimilated as a +part of daily thought, and brought again and again before the mind +with ever-renewed encouragement. Real life is, to most men, a long +second-best, a perpetual compromise between the ideal and the +possible; but the world of pure reason knows no compromise, no +practical limitations, no barrier to the creative activity embodying +in splendid edifices the passionate aspiration after the perfect from +which all great work springs. Remote from human passions, remote even +from the pitiful facts of nature, the generations have gradually +created an ordered cosmos, where pure thought can dwell as in its +natural home, and where one, at least, of our nobler impulses can +escape from the dreary exile of the actual world. + +So little, however, have mathematicians aimed at beauty, that hardly +anything in their work has had this conscious purpose. Much, owing to +irrepressible instincts, which were better than avowed beliefs, has +been moulded by an unconscious taste; but much also has been spoilt by +false notions of what was fitting. The characteristic excellence of +mathematics is only to be found where the reasoning is rigidly +logical: the rules of logic are to mathematics what those of structure +are to architecture. In the most beautiful work, a chain of argument +is presented in which every link is important on its own account, in +which there is an air of ease and lucidity throughout, and the +premises achieve more than would have been thought possible, by means +which appear natural and inevitable. Literature embodies what is +general in particular circumstances whose universal significance +shines through their individual dress; but mathematics endeavours to +present whatever is most general in its purity, without any irrelevant +trappings. + +How should the teaching of mathematics be conducted so as to +communicate to the learner as much as possible of this high ideal? +Here experience must, in a great measure, be our guide; but some +maxims may result from our consideration of the ultimate purpose to be +achieved. + +One of the chief ends served by mathematics, when rightly taught, is +to awaken the learner's belief in reason, his confidence in the truth +of what has been demonstrated, and in the value of demonstration. This +purpose is not served by existing instruction; but it is easy to see +ways in which it might be served. At present, in what concerns +arithmetic, the boy or girl is given a set of rules, which present +themselves as neither true nor false, but as merely the will of the +teacher, the way in which, for some unfathomable reason, the teacher +prefers to have the game played. To some degree, in a study of such +definite practical utility, this is no doubt unavoidable; but as soon +as possible, the reasons of rules should be set forth by whatever +means most readily appeal to the childish mind. In geometry, instead +of the tedious apparatus of fallacious proofs for obvious truisms +which constitutes the beginning of Euclid, the learner should be +allowed at first to assume the truth of everything obvious, and should +be instructed in the demonstrations of theorems which are at once +startling and easily verifiable by actual drawing, such as those in +which it is shown that three or more lines meet in a point. In this +way belief is generated; it is seen that reasoning may lead to +startling conclusions, which nevertheless the facts will verify; and +thus the instinctive distrust of whatever is abstract or rational is +gradually overcome. Where theorems are difficult, they should be first +taught as exercises in geometrical drawing, until the figure has +become thoroughly familiar; it will then be an agreeable advance to be +taught the logical connections of the various lines or circles that +occur. It is desirable also that the figure illustrating a theorem +should be drawn in all possible cases and shapes, that so the abstract +relations with which geometry is concerned may of themselves emerge +as the residue of similarity amid such great apparent diversity. In +this way the abstract demonstrations should form but a small part of +the instruction, and should be given when, by familiarity with +concrete illustrations, they have come to be felt as the natural +embodiment of visible fact. In this early stage proofs should not be +given with pedantic fullness; definitely fallacious methods, such as +that of superposition, should be rigidly excluded from the first, but +where, without such methods, the proof would be very difficult, the +result should be rendered acceptable by arguments and illustrations +which are explicitly contrasted with demonstrations. + +In the beginning of algebra, even the most intelligent child finds, as +a rule, very great difficulty. The use of letters is a mystery, which +seems to have no purpose except mystification. It is almost +impossible, at first, not to think that every letter stands for some +particular number, if only the teacher would reveal _what_ number it +stands for. The fact is, that in algebra the mind is first taught to +consider general truths, truths which are not asserted to hold only of +this or that particular thing, but of any one of a whole group of +things. It is in the power of understanding and discovering such +truths that the mastery of the intellect over the whole world of +things actual and possible resides; and ability to deal with the +general as such is one of the gifts that a mathematical education +should bestow. But how little, as a rule, is the teacher of algebra +able to explain the chasm which divides it from arithmetic, and how +little is the learner assisted in his groping efforts at +comprehension! Usually the method that has been adopted in arithmetic +is continued: rules are set forth, with no adequate explanation of +their grounds; the pupil learns to use the rules blindly, and +presently, when he is able to obtain the answer that the teacher +desires, he feels that he has mastered the difficulties of the +subject. But of inner comprehension of the processes employed he has +probably acquired almost nothing. + +When algebra has been learnt, all goes smoothly until we reach those +studies in which the notion of infinity is employed--the infinitesimal +calculus and the whole of higher mathematics. The solution of the +difficulties which formerly surrounded the mathematical infinite is +probably the greatest achievement of which our own age has to boast. +Since the beginnings of Greek thought these difficulties have been +known; in every age the finest intellects have vainly endeavoured to +answer the apparently unanswerable questions that had been asked by +Zeno the Eleatic. At last Georg Cantor has found the answer, and has +conquered for the intellect a new and vast province which had been +given over to Chaos and old Night. It was assumed as self-evident, +until Cantor and Dedekind established the opposite, that if, from any +collection of things, some were taken away, the number of things left +must always be less than the original number of things. This +assumption, as a matter of fact, holds only of finite collections; and +the rejection of it, where the infinite is concerned, has been shown +to remove all the difficulties that had hitherto baffled human reason +in this matter, and to render possible the creation of an exact +science of the infinite. This stupendous fact ought to produce a +revolution in the higher teaching of mathematics; it has itself added +immeasurably to the educational value of the subject, and it has at +last given the means of treating with logical precision many studies +which, until lately, were wrapped in fallacy and obscurity. By those +who were educated on the old lines, the new work is considered to be +appallingly difficult, abstruse, and obscure; and it must be confessed +that the discoverer, as is so often the case, has hardly himself +emerged from the mists which the light of his intellect is dispelling. +But inherently, the new doctrine of the infinite, to all candid and +inquiring minds, has facilitated the mastery of higher mathematics; +for hitherto, it has been necessary to learn, by a long process of +sophistication, to give assent to arguments which, on first +acquaintance, were rightly judged to be confused and erroneous. So far +from producing a fearless belief in reason, a bold rejection of +whatever failed to fulfil the strictest requirements of logic, a +mathematical training, during the past two centuries, encouraged the +belief that many things, which a rigid inquiry would reject as +fallacious, must yet be accepted because they work in what the +mathematician calls "practice." By this means, a timid, compromising +spirit, or else a sacerdotal belief in mysteries not intelligible to +the profane, has been bred where reason alone should have ruled. All +this it is now time to sweep away; let those who wish to penetrate +into the arcana of mathematics be taught at once the true theory in +all its logical purity, and in the concatenation established by the +very essence of the entities concerned. + +If we are considering mathematics as an end in itself, and not as a +technical training for engineers, it is very desirable to preserve the +purity and strictness of its reasoning. Accordingly those who have +attained a sufficient familiarity with its easier portions should be +led backward from propositions to which they have assented as +self-evident to more and more fundamental principles from which what +had previously appeared as premises can be deduced. They should be +taught--what the theory of infinity very aptly illustrates--that many +propositions seem self-evident to the untrained mind which, +nevertheless, a nearer scrutiny shows to be false. By this means they +will be led to a sceptical inquiry into first principles, an +examination of the foundations upon which the whole edifice of +reasoning is built, or, to take perhaps a more fitting metaphor, the +great trunk from which the spreading branches spring. At this stage, +it is well to study afresh the elementary portions of mathematics, +asking no longer merely whether a given proposition is true, but also +how it grows out of the central principles of logic. Questions of this +nature can now be answered with a precision and certainty which were +formerly quite impossible; and in the chains of reasoning that the +answer requires the unity of all mathematical studies at last unfolds +itself. + +In the great majority of mathematical text-books there is a total lack +of unity in method and of systematic development of a central theme. +Propositions of very diverse kinds are proved by whatever means are +thought most easily intelligible, and much space is devoted to mere +curiosities which in no way contribute to the main argument. But in +the greatest works, unity and inevitability are felt as in the +unfolding of a drama; in the premisses a subject is proposed for +consideration, and in every subsequent step some definite advance is +made towards mastery of its nature. The love of system, of +interconnection, which is perhaps the inmost essence of the +intellectual impulse, can find free play in mathematics as nowhere +else. The learner who feels this impulse must not be repelled by an +array of meaningless examples or distracted by amusing oddities, but +must be encouraged to dwell upon central principles, to become +familiar with the structure of the various subjects which are put +before him, to travel easily over the steps of the more important +deductions. In this way a good tone of mind is cultivated, and +selective attention is taught to dwell by preference upon what is +weighty and essential. + +When the separate studies into which mathematics is divided have each +been viewed as a logical whole, as a natural growth from the +propositions which constitute their principles, the learner will be +able to understand the fundamental science which unifies and +systematises the whole of deductive reasoning. This is symbolic +logic--a study which, though it owes its inception to Aristotle, is +yet, in its wider developments, a product, almost wholly, of the +nineteenth century, and is indeed, in the present day, still growing +with great rapidity. The true method of discovery in symbolic logic, +and probably also the best method for introducing the study to a +learner acquainted with other parts of mathematics, is the analysis of +actual examples of deductive reasoning, with a view to the discovery +of the principles employed. These principles, for the most part, are +so embedded in our ratiocinative instincts, that they are employed +quite unconsciously, and can be dragged to light only by much patient +effort. But when at last they have been found, they are seen to be few +in number, and to be the sole source of everything in pure +mathematics. The discovery that all mathematics follows inevitably +from a small collection of fundamental laws is one which immeasurably +enhances the intellectual beauty of the whole; to those who have been +oppressed by the fragmentary and incomplete nature of most existing +chains of deduction this discovery comes with all the overwhelming +force of a revelation; like a palace emerging from the autumn mist as +the traveller ascends an Italian hill-side, the stately storeys of the +mathematical edifice appear in their due order and proportion, with a +new perfection in every part. + +Until symbolic logic had acquired its present development, the +principles upon which mathematics depends were always supposed to be +philosophical, and discoverable only by the uncertain, unprogressive +methods hitherto employed by philosophers. So long as this was +thought, mathematics seemed to be not autonomous, but dependent upon a +study which had quite other methods than its own. Moreover, since the +nature of the postulates from which arithmetic, analysis, and geometry +are to be deduced was wrapped in all the traditional obscurities of +metaphysical discussion, the edifice built upon such dubious +foundations began to be viewed as no better than a castle in the air. +In this respect, the discovery that the true principles are as much a +part of mathematics as any of their consequences has very greatly +increased the intellectual satisfaction to be obtained. This +satisfaction ought not to be refused to learners capable of enjoying +it, for it is of a kind to increase our respect for human powers and +our knowledge of the beauties belonging to the abstract world. + +Philosophers have commonly held that the laws of logic, which underlie +mathematics, are laws of thought, laws regulating the operations of +our minds. By this opinion the true dignity of reason is very greatly +lowered: it ceases to be an investigation into the very heart and +immutable essence of all things actual and possible, becoming, +instead, an inquiry into something more or less human and subject to +our limitations. The contemplation of what is non-human, the discovery +that our minds are capable of dealing with material not created by +them, above all, the realisation that beauty belongs to the outer +world as to the inner, are the chief means of overcoming the terrible +sense of impotence, of weakness, of exile amid hostile powers, which +is too apt to result from acknowledging the all-but omnipotence of +alien forces. To reconcile us, by the exhibition of its awful beauty, +to the reign of Fate--which is merely the literary personification of +these forces--is the task of tragedy. But mathematics takes us still +further from what is human, into the region of absolute necessity, to +which not only the actual world, but every possible world, must +conform; and even here it builds a habitation, or rather finds a +habitation eternally standing, where our ideals are fully satisfied +and our best hopes are not thwarted. It is only when we thoroughly +understand the entire independence of ourselves, which belongs to this +world that reason finds, that we can adequately realise the profound +importance of its beauty. + +Not only is mathematics independent of us and our thoughts, but in +another sense we and the whole universe of existing things are +independent of mathematics. The apprehension of this purely ideal +character is indispensable, if we are to understand rightly the place +of mathematics as one among the arts. It was formerly supposed that +pure reason could decide, in some respects, as to the nature of the +actual world: geometry, at least, was thought to deal with the space +in which we live. But we now know that pure mathematics can never +pronounce upon questions of actual existence: the world of reason, in +a sense, controls the world of fact, but it is not at any point +creative of fact, and in the application of its results to the world +in time and space, its certainty and precision are lost among +approximations and working hypotheses. The objects considered by +mathematicians have, in the past, been mainly of a kind suggested by +phenomena; but from such restrictions the abstract imagination should +be wholly free. A reciprocal liberty must thus be accorded: reason +cannot dictate to the world of facts, but the facts cannot restrict +reason's privilege of dealing with whatever objects its love of beauty +may cause to seem worthy of consideration. Here, as elsewhere, we +build up our own ideals out of the fragments to be found in the world; +and in the end it is hard to say whether the result is a creation or a +discovery. + +It is very desirable, in instruction, not merely to persuade the +student of the accuracy of important theorems, but to persuade him in +the way which itself has, of all possible ways, the most beauty. The +true interest of a demonstration is not, as traditional modes of +exposition suggest, concentrated wholly in the result; where this does +occur, it must be viewed as a defect, to be remedied, if possible, by +so generalising the steps of the proof that each becomes important in +and for itself. An argument which serves only to prove a conclusion is +like a story subordinated to some moral which it is meant to teach: +for æsthetic perfection no part of the whole should be merely a means. +A certain practical spirit, a desire for rapid progress, for conquest +of new realms, is responsible for the undue emphasis upon results +which prevails in mathematical instruction. The better way is to +propose some theme for consideration--in geometry, a figure having +important properties; in analysis, a function of which the study is +illuminating, and so on. Whenever proofs depend upon some only of the +marks by which we define the object to be studied, these marks should +be isolated and investigated on their own account. For it is a defect, +in an argument, to employ more premisses than the conclusion demands: +what mathematicians call elegance results from employing only the +essential principles in virtue of which the thesis is true. It is a +merit in Euclid that he advances as far as he is able to go without +employing the axiom of parallels--not, as is often said, because this +axiom is inherently objectionable, but because, in mathematics, every +new axiom diminishes the generality of the resulting theorems, and the +greatest possible generality is before all things to be sought. + +Of the effects of mathematics outside its own sphere more has been +written than on the subject of its own proper ideal. The effect upon +philosophy has, in the past, been most notable, but most varied; in +the seventeenth century, idealism and rationalism, in the eighteenth, +materialism and sensationalism, seemed equally its offspring. Of the +effect which it is likely to have in the future it would be very rash +to say much; but in one respect a good result appears probable. +Against that kind of scepticism which abandons the pursuit of ideals +because the road is arduous and the goal not certainly attainable, +mathematics, within its own sphere, is a complete answer. Too often it +is said that there is no absolute truth, but only opinion and private +judgment; that each of us is conditioned, in his view of the world, by +his own peculiarities, his own taste and bias; that there is no +external kingdom of truth to which, by patience and discipline, we may +at last obtain admittance, but only truth for me, for you, for every +separate person. By this habit of mind one of the chief ends of human +effort is denied, and the supreme virtue of candour, of fearless +acknowledgment of what is, disappears from our moral vision. Of such +scepticism mathematics is a perpetual reproof; for its edifice of +truths stands unshakable and inexpungable to all the weapons of +doubting cynicism. + +The effects of mathematics upon practical life, though they should not +be regarded as the motive of our studies, may be used to answer a +doubt to which the solitary student must always be liable. In a world +so full of evil and suffering, retirement into the cloister of +contemplation, to the enjoyment of delights which, however noble, must +always be for the few only, cannot but appear as a somewhat selfish +refusal to share the burden imposed upon others by accidents in which +justice plays no part. Have any of us the right, we ask, to withdraw +from present evils, to leave our fellow-men unaided, while we live a +life which, though arduous and austere, is yet plainly good in its own +nature? When these questions arise, the true answer is, no doubt, that +some must keep alive the sacred fire, some must preserve, in every +generation, the haunting vision which shadows forth the goal of so +much striving. But when, as must sometimes occur, this answer seems +too cold, when we are almost maddened by the spectacle of sorrows to +which we bring no help, then we may reflect that indirectly the +mathematician often does more for human happiness than any of his more +practically active contemporaries. The history of science abundantly +proves that a body of abstract propositions--even if, as in the case +of conic sections, it remains two thousand years without effect upon +daily life--may yet, at any moment, be used to cause a revolution in +the habitual thoughts and occupations of every citizen. The use of +steam and electricity--to take striking instances--is rendered +possible only by mathematics. In the results of abstract thought the +world possesses a capital of which the employment in enriching the +common round has no hitherto discoverable limits. Nor does experience +give any means of deciding what parts of mathematics will be found +useful. Utility, therefore, can be only a consolation in moments of +discouragement, not a guide in directing our studies. + +For the health of the moral life, for ennobling the tone of an age or +a nation, the austerer virtues have a strange power, exceeding the +power of those not informed and purified by thought. Of these austerer +virtues the love of truth is the chief, and in mathematics, more than +elsewhere, the love of truth may find encouragement for waning faith. +Every great study is not only an end in itself, but also a means of +creating and sustaining a lofty habit of mind; and this purpose should +be kept always in view throughout the teaching and learning of +mathematics. + +FOOTNOTES: + +[10] This passage was pointed out to me by Professor Gilbert Murray. + + + + +V + +MATHEMATICS AND THE METAPHYSICIANS + + +The nineteenth century, which prided itself upon the invention of +steam and evolution, might have derived a more legitimate title to +fame from the discovery of pure mathematics. This science, like most +others, was baptised long before it was born; and thus we find writers +before the nineteenth century alluding to what they called pure +mathematics. But if they had been asked what this subject was, they +would only have been able to say that it consisted of Arithmetic, +Algebra, Geometry, and so on. As to what these studies had in common, +and as to what distinguished them from applied mathematics, our +ancestors were completely in the dark. + +Pure mathematics was discovered by Boole, in a work which he called +the _Laws of Thought_ (1854). This work abounds in asseverations that +it is not mathematical, the fact being that Boole was too modest to +suppose his book the first ever written on mathematics. He was also +mistaken in supposing that he was dealing with the laws of thought: +the question how people actually think was quite irrelevant to him, +and if his book had really contained the laws of thought, it was +curious that no one should ever have thought in such a way before. His +book was in fact concerned with formal logic, and this is the same +thing as mathematics. + +Pure mathematics consists entirely of assertions to the effect that, +if such and such a proposition is true of _anything_, then such and +such another proposition is true of that thing. It is essential not to +discuss whether the first proposition is really true, and not to +mention what the anything is, of which it is supposed to be true. Both +these points would belong to applied mathematics. We start, in pure +mathematics, from certain rules of inference, by which we can infer +that _if_ one proposition is true, then so is some other proposition. +These rules of inference constitute the major part of the principles +of formal logic. We then take any hypothesis that seems amusing, and +deduce its consequences. _If_ our hypothesis is about _anything_, and +not about some one or more particular things, then our deductions +constitute mathematics. Thus mathematics may be defined as the subject +in which we never know what we are talking about, nor whether what we +are saying is true. People who have been puzzled by the beginnings of +mathematics will, I hope, find comfort in this definition, and will +probably agree that it is accurate. + +As one of the chief triumphs of modern mathematics consists in having +discovered what mathematics really is, a few more words on this +subject may not be amiss. It is common to start any branch of +mathematics--for instance, Geometry--with a certain number of +primitive ideas, supposed incapable of definition, and a certain +number of primitive propositions or axioms, supposed incapable of +proof. Now the fact is that, though there are indefinables and +indemonstrables in every branch of applied mathematics, there are none +in pure mathematics except such as belong to general logic. Logic, +broadly speaking, is distinguished by the fact that its propositions +can be put into a form in which they apply to anything whatever. All +pure mathematics--Arithmetic, Analysis, and Geometry--is built up by +combinations of the primitive ideas of logic, and its propositions are +deduced from the general axioms of logic, such as the syllogism and +the other rules of inference. And this is no longer a dream or an +aspiration. On the contrary, over the greater and more difficult part +of the domain of mathematics, it has been already accomplished; in the +few remaining cases, there is no special difficulty, and it is now +being rapidly achieved. Philosophers have disputed for ages whether +such deduction was possible; mathematicians have sat down and made the +deduction. For the philosophers there is now nothing left but graceful +acknowledgments. + +The subject of formal logic, which has thus at last shown itself to be +identical with mathematics, was, as every one knows, invented by +Aristotle, and formed the chief study (other than theology) of the +Middle Ages. But Aristotle never got beyond the syllogism, which is a +very small part of the subject, and the schoolmen never got beyond +Aristotle. If any proof were required of our superiority to the +mediæval doctors, it might be found in this. Throughout the Middle +Ages, almost all the best intellects devoted themselves to formal +logic, whereas in the nineteenth century only an infinitesimal +proportion of the world's thought went into this subject. +Nevertheless, in each decade since 1850 more has been done to advance +the subject than in the whole period from Aristotle to Leibniz. People +have discovered how to make reasoning symbolic, as it is in Algebra, +so that deductions are effected by mathematical rules. They have +discovered many rules besides the syllogism, and a new branch of +logic, called the Logic of Relatives,[11] has been invented to deal +with topics that wholly surpassed the powers of the old logic, though +they form the chief contents of mathematics. + +It is not easy for the lay mind to realise the importance of symbolism +in discussing the foundations of mathematics, and the explanation may +perhaps seem strangely paradoxical. The fact is that symbolism is +useful because it makes things difficult. (This is not true of the +advanced parts of mathematics, but only of the beginnings.) What we +wish to know is, what can be deduced from what. Now, in the +beginnings, everything is self-evident; and it is very hard to see +whether one self-evident proposition follows from another or not. +Obviousness is always the enemy to correctness. Hence we invent some +new and difficult symbolism, in which nothing seems obvious. Then we +set up certain rules for operating on the symbols, and the whole thing +becomes mechanical. In this way we find out what must be taken as +premiss and what can be demonstrated or defined. For instance, the +whole of Arithmetic and Algebra has been shown to require three +indefinable notions and five indemonstrable propositions. But without +a symbolism it would have been very hard to find this out. It is so +obvious that two and two are four, that we can hardly make ourselves +sufficiently sceptical to doubt whether it can be proved. And the same +holds in other cases where self-evident things are to be proved. + +But the proof of self-evident propositions may seem, to the +uninitiated, a somewhat frivolous occupation. To this we might reply +that it is often by no means self-evident that one obvious proposition +follows from another obvious proposition; so that we are really +discovering new truths when we prove what is evident by a method which +is not evident. But a more interesting retort is, that since people +have tried to prove obvious propositions, they have found that many of +them are false. Self-evidence is often a mere will-o'-the-wisp, which +is sure to lead us astray if we take it as our guide. For instance, +nothing is plainer than that a whole always has more terms than a +part, or that a number is increased by adding one to it. But these +propositions are now known to be usually false. Most numbers are +infinite, and if a number is infinite you may add ones to it as long +as you like without disturbing it in the least. One of the merits of a +proof is that it instils a certain doubt as to the result proved; and +when what is obvious can be proved in some cases, but not in others, +it becomes possible to suppose that in these other cases it is false. + +The great master of the art of formal reasoning, among the men of our +own day, is an Italian, Professor Peano, of the University of +Turin.[12] He has reduced the greater part of mathematics (and he or +his followers will, in time, have reduced the whole) to strict +symbolic form, in which there are no words at all. In the ordinary +mathematical books, there are no doubt fewer words than most readers +would wish. Still, little phrases occur, such as _therefore, let us +assume, consider_, or _hence it follows_. All these, however, are a +concession, and are swept away by Professor Peano. For instance, if we +wish to learn the whole of Arithmetic, Algebra, the Calculus, and +indeed all that is usually called pure mathematics (except Geometry), +we must start with a dictionary of three words. One symbol stands for +_zero_, another for _number_, and a third for _next after_. What these +ideas mean, it is necessary to know if you wish to become an +arithmetician. But after symbols have been invented for these three +ideas, not another word is required in the whole development. All +future symbols are symbolically explained by means of these three. +Even these three can be explained by means of the notions of +_relation_ and _class_; but this requires the Logic of Relations, +which Professor Peano has never taken up. It must be admitted that +what a mathematician has to know to begin with is not much. There are +at most a dozen notions out of which all the notions in all pure +mathematics (including Geometry) are compounded. Professor Peano, who +is assisted by a very able school of young Italian disciples, has +shown how this may be done; and although the method which he has +invented is capable of being carried a good deal further than he has +carried it, the honour of the pioneer must belong to him. + +Two hundred years ago, Leibniz foresaw the science which Peano has +perfected, and endeavoured to create it. He was prevented from +succeeding by respect for the authority of Aristotle, whom he could +not believe guilty of definite, formal fallacies; but the subject +which he desired to create now exists, in spite of the patronising +contempt with which his schemes have been treated by all superior +persons. From this "Universal Characteristic," as he called it, he +hoped for a solution of all problems, and an end to all disputes. "If +controversies were to arise," he says, "there would be no more need of +disputation between two philosophers than between two accountants. For +it would suffice to take their pens in their hands, to sit down to +their desks, and to say to each other (with a friend as witness, if +they liked), 'Let us calculate.'" This optimism has now appeared to be +somewhat excessive; there still are problems whose solution is +doubtful, and disputes which calculation cannot decide. But over an +enormous field of what was formerly controversial, Leibniz's dream has +become sober fact. In the whole philosophy of mathematics, which used +to be at least as full of doubt as any other part of philosophy, order +and certainty have replaced the confusion and hesitation which +formerly reigned. Philosophers, of course, have not yet discovered +this fact, and continue to write on such subjects in the old way. But +mathematicians, at least in Italy, have now the power of treating the +principles of mathematics in an exact and masterly manner, by means of +which the certainty of mathematics extends also to mathematical +philosophy. Hence many of the topics which used to be placed among the +great mysteries--for example, the natures of infinity, of continuity, +of space, time and motion--are now no longer in any degree open to +doubt or discussion. Those who wish to know the nature of these things +need only read the works of such men as Peano or Georg Cantor; they +will there find exact and indubitable expositions of all these quondam +mysteries. + +In this capricious world, nothing is more capricious than posthumous +fame. One of the most notable examples of posterity's lack of judgment +is the Eleatic Zeno. This man, who may be regarded as the founder of +the philosophy of infinity, appears in Plato's Parmenides in the +privileged position of instructor to Socrates. He invented four +arguments, all immeasurably subtle and profound, to prove that motion +is impossible, that Achilles can never overtake the tortoise, and that +an arrow in flight is really at rest. After being refuted by +Aristotle, and by every subsequent philosopher from that day to our +own, these arguments were reinstated, and made the basis of a +mathematical renaissance, by a German professor, who probably never +dreamed of any connection between himself and Zeno. Weierstrass,[13] +by strictly banishing from mathematics the use of infinitesimals, has +at last shown that we live in an unchanging world, and that the arrow +in its flight is truly at rest. Zeno's only error lay in inferring (if +he did infer) that, because there is no such thing as a state of +change, therefore the world is in the same state at any one time as at +any other. This is a consequence which by no means follows; and in +this respect, the German mathematician is more constructive than the +ingenious Greek. Weierstrass has been able, by embodying his views in +mathematics, where familiarity with truth eliminates the vulgar +prejudices of common sense, to invest Zeno's paradoxes with the +respectable air of platitudes; and if the result is less delightful to +the lover of reason than Zeno's bold defiance, it is at any rate more +calculated to appease the mass of academic mankind. + +Zeno was concerned, as a matter of fact, with three problems, each +presented by motion, but each more abstract than motion, and capable +of a purely arithmetical treatment. These are the problems of the +infinitesimal, the infinite, and continuity. To state clearly the +difficulties involved, was to accomplish perhaps the hardest part of +the philosopher's task. This was done by Zeno. From him to our own +day, the finest intellects of each generation in turn attacked the +problems, but achieved, broadly speaking, nothing. In our own time, +however, three men--Weierstrass, Dedekind, and Cantor--have not merely +advanced the three problems, but have completely solved them. The +solutions, for those acquainted with mathematics, are so clear as to +leave no longer the slightest doubt or difficulty. This achievement is +probably the greatest of which our age has to boast; and I know of no +age (except perhaps the golden age of Greece) which has a more +convincing proof to offer of the transcendent genius of its great men. +Of the three problems, that of the infinitesimal was solved by +Weierstrass; the solution of the other two was begun by Dedekind, and +definitively accomplished by Cantor. + +The infinitesimal played formerly a great part in mathematics. It was +introduced by the Greeks, who regarded a circle as differing +infinitesimally from a polygon with a very large number of very small +equal sides. It gradually grew in importance, until, when Leibniz +invented the Infinitesimal Calculus, it seemed to become the +fundamental notion of all higher mathematics. Carlyle tells, in his +_Frederick the Great_, how Leibniz used to discourse to Queen Sophia +Charlotte of Prussia concerning the infinitely little, and how she +would reply that on that subject she needed no instruction--the +behaviour of courtiers had made her thoroughly familiar with it. But +philosophers and mathematicians--who for the most part had less +acquaintance with courts--continued to discuss this topic, though +without making any advance. The Calculus required continuity, and +continuity was supposed to require the infinitely little; but nobody +could discover what the infinitely little might be. It was plainly not +quite zero, because a sufficiently large number of infinitesimals, +added together, were seen to make up a finite whole. But nobody could +point out any fraction which was not zero, and yet not finite. Thus +there was a deadlock. But at last Weierstrass discovered that the +infinitesimal was not needed at all, and that everything could be +accomplished without it. Thus there was no longer any need to suppose +that there was such a thing. Nowadays, therefore, mathematicians are +more dignified than Leibniz: instead of talking about the infinitely +small, they talk about the infinitely great--a subject which, however +appropriate to monarchs, seems, unfortunately, to interest them even +less than the infinitely little interested the monarchs to whom +Leibniz discoursed. + +The banishment of the infinitesimal has all sorts of odd consequences, +to which one has to become gradually accustomed. For example, there is +no such thing as the next moment. The interval between one moment and +the next would have to be infinitesimal, since, if we take two moments +with a finite interval between them, there are always other moments in +the interval. Thus if there are to be no infinitesimals, no two +moments are quite consecutive, but there are always other moments +between any two. Hence there must be an infinite number of moments +between any two; because if there were a finite number one would be +nearest the first of the two moments, and therefore next to it. This +might be thought to be a difficulty; but, as a matter of fact, it is +here that the philosophy of the infinite comes in, and makes all +straight. + +The same sort of thing happens in space. If any piece of matter be cut +in two, and then each part be halved, and so on, the bits will become +smaller and smaller, and can theoretically be made as small as we +please. However small they may be, they can still be cut up and made +smaller still. But they will always have _some_ finite size, however +small they may be. We never reach the infinitesimal in this way, and +no finite number of divisions will bring us to points. Nevertheless +there _are_ points, only these are not to be reached by successive +divisions. Here again, the philosophy of the infinite shows us how +this is possible, and why points are not infinitesimal lengths. + +As regards motion and change, we get similarly curious results. People +used to think that when a thing changes, it must be in a state of +change, and that when a thing moves, it is in a state of motion. This +is now known to be a mistake. When a body moves, all that can be said +is that it is in one place at one time and in another at another. We +must not say that it will be in a neighbouring place at the next +instant, since there is no next instant. Philosophers often tell us +that when a body is in motion, it changes its position within the +instant. To this view Zeno long ago made the fatal retort that every +body always is where it is; but a retort so simple and brief was not +of the kind to which philosophers are accustomed to give weight, and +they have continued down to our own day to repeat the same phrases +which roused the Eleatic's destructive ardour. It was only recently +that it became possible to explain motion in detail in accordance with +Zeno's platitude, and in opposition to the philosopher's paradox. We +may now at last indulge the comfortable belief that a body in motion +is just as truly where it is as a body at rest. Motion consists merely +in the fact that bodies are sometimes in one place and sometimes in +another, and that they are at intermediate places at intermediate +times. Only those who have waded through the quagmire of philosophic +speculation on this subject can realise what a liberation from antique +prejudices is involved in this simple and straightforward commonplace. + +The philosophy of the infinitesimal, as we have just seen, is mainly +negative. People used to believe in it, and now they have found out +their mistake. The philosophy of the infinite, on the other hand, is +wholly positive. It was formerly supposed that infinite numbers, and +the mathematical infinite generally, were self-contradictory. But as +it was obvious that there were infinities--for example, the number of +numbers--the contradictions of infinity seemed unavoidable, and +philosophy seemed to have wandered into a "cul-de-sac." This +difficulty led to Kant's antinomies, and hence, more or less +indirectly, to much of Hegel's dialectic method. Almost all current +philosophy is upset by the fact (of which very few philosophers are as +yet aware) that all the ancient and respectable contradictions in the +notion of the infinite have been once for all disposed of. The method +by which this has been done is most interesting and instructive. In +the first place, though people had talked glibly about infinity ever +since the beginnings of Greek thought, nobody had ever thought of +asking, What is infinity? If any philosopher had been asked for a +definition of infinity, he might have produced some unintelligible +rigmarole, but he would certainly not have been able to give a +definition that had any meaning at all. Twenty years ago, roughly +speaking, Dedekind and Cantor asked this question, and, what is more +remarkable, they answered it. They found, that is to say, a perfectly +precise definition of an infinite number or an infinite collection of +things. This was the first and perhaps the greatest step. It then +remained to examine the supposed contradictions in this notion. Here +Cantor proceeded in the only proper way. He took pairs of +contradictory propositions, in which both sides of the contradiction +would be usually regarded as demonstrable, and he strictly examined +the supposed proofs. He found that all proofs adverse to infinity +involved a certain principle, at first sight obviously true, but +destructive, in its consequences, of almost all mathematics. The +proofs favourable to infinity, on the other hand, involved no +principle that had evil consequences. It thus appeared that common +sense had allowed itself to be taken in by a specious maxim, and that, +when once this maxim was rejected, all went well. + +The maxim in question is, that if one collection is part of another, +the one which is a part has fewer terms than the one of which it is a +part. This maxim is true of finite numbers. For example, Englishmen +are only some among Europeans, and there are fewer Englishmen than +Europeans. But when we come to infinite numbers, this is no longer +true. This breakdown of the maxim gives us the precise definition of +infinity. A collection of terms is infinite when it contains as parts +other collections which have just as many terms as it has. If you can +take away some of the terms of a collection, without diminishing the +number of terms, then there are an infinite number of terms in the +collection. For example, there are just as many even numbers as there +are numbers altogether, since every number can be doubled. This may be +seen by putting odd and even numbers together in one row, and even +numbers alone in a row below:-- + + 1, 2, 3, 4, 5, _ad infinitum_. + 2, 4, 6, 8, 10, _ad infinitum_. + +There are obviously just as many numbers in the row below as in the +row above, because there is one below for each one above. This +property, which was formerly thought to be a contradiction, is now +transformed into a harmless definition of infinity, and shows, in the +above case, that the number of finite numbers is infinite. + +But the uninitiated may wonder how it is possible to deal with a +number which cannot be counted. It is impossible to count up _all_ the +numbers, one by one, because, however many we may count, there are +always more to follow. The fact is that counting is a very vulgar and +elementary way of finding out how many terms there are in a +collection. And in any case, counting gives us what mathematicians +call the _ordinal_ number of our terms; that is to say, it arranges +our terms in an order or series, and its result tells us what type of +series results from this arrangement. In other words, it is impossible +to count things without counting some first and others afterwards, so +that counting always has to do with order. Now when there are only a +finite number of terms, we can count them in any order we like; but +when there are an infinite number, what corresponds to counting will +give us quite different results according to the way in which we carry +out the operation. Thus the ordinal number, which results from what, +in a general sense may be called counting, depends not only upon how +many terms we have, but also (where the number of terms is infinite) +upon the way in which the terms are arranged. + +The fundamental infinite numbers are not ordinal, but are what is +called _cardinal_. They are not obtained by putting our terms in order +and counting them, but by a different method, which tells us, to begin +with, whether two collections have the same number of terms, or, if +not, which is the greater.[14] It does not tell us, in the way in +which counting does, _what_ number of terms a collection has; but if +we define a number as the number of terms in such and such a +collection, then this method enables us to discover whether some other +collection that may be mentioned has more or fewer terms. An +illustration will show how this is done. If there existed some country +in which, for one reason or another, it was impossible to take a +census, but in which it was known that every man had a wife and every +woman a husband, then (provided polygamy was not a national +institution) we should know, without counting, that there were exactly +as many men as there were women in that country, neither more nor +less. This method can be applied generally. If there is some relation +which, like marriage, connects the things in one collection each with +one of the things in another collection, and vice versa, then the two +collections have the same number of terms. This was the way in which +we found that there are as many even numbers as there are numbers. +Every number can be doubled, and every even number can be halved, and +each process gives just one number corresponding to the one that is +doubled or halved. And in this way we can find any number of +collections each of which has just as many terms as there are finite +numbers. If every term of a collection can be hooked on to a number, +and all the finite numbers are used once, and only once, in the +process, then our collection must have just as many terms as there are +finite numbers. This is the general method by which the numbers of +infinite collections are defined. + +But it must not be supposed that all infinite numbers are equal. On +the contrary, there are infinitely more infinite numbers than finite +ones. There are more ways of arranging the finite numbers in different +types of series than there are finite numbers. There are probably more +points in space and more moments in time than there are finite +numbers. There are exactly as many fractions as whole numbers, +although there are an infinite number of fractions between any two +whole numbers. But there are more irrational numbers than there are +whole numbers or fractions. There are probably exactly as many points +in space as there are irrational numbers, and exactly as many points +on a line a millionth of an inch long as in the whole of infinite +space. There is a greatest of all infinite numbers, which is the +number of things altogether, of every sort and kind. It is obvious +that there cannot be a greater number than this, because, if +everything has been taken, there is nothing left to add. Cantor has a +proof that there is no greatest number, and if this proof were valid, +the contradictions of infinity would reappear in a sublimated form. +But in this one point, the master has been guilty of a very subtle +fallacy, which I hope to explain in some future work.[15] + +We can now understand why Zeno believed that Achilles cannot overtake +the tortoise and why as a matter of fact he can overtake it. We shall +see that all the people who disagreed with Zeno had no right to do so, +because they all accepted premises from which his conclusion followed. +The argument is this: Let Achilles and the tortoise start along a road +at the same time, the tortoise (as is only fair) being allowed a +handicap. Let Achilles go twice as fast as the tortoise, or ten times +or a hundred times as fast. Then he will never reach the tortoise. For +at every moment the tortoise is somewhere and Achilles is somewhere; +and neither is ever twice in the same place while the race is going +on. Thus the tortoise goes to just as many places as Achilles does, +because each is in one place at one moment, and in another at any +other moment. But if Achilles were to catch up with the tortoise, the +places where the tortoise would have been would be only part of the +places where Achilles would have been. Here, we must suppose, Zeno +appealed to the maxim that the whole has more terms than the part.[16] +Thus if Achilles were to overtake the tortoise, he would have been in +more places than the tortoise; but we saw that he must, in any period, +be in exactly as many places as the tortoise. Hence we infer that he +can never catch the tortoise. This argument is strictly correct, if we +allow the axiom that the whole has more terms than the part. As the +conclusion is absurd, the axiom must be rejected, and then all goes +well. But there is no good word to be said for the philosophers of the +past two thousand years and more, who have all allowed the axiom and +denied the conclusion. + +The retention of this axiom leads to absolute contradictions, while +its rejection leads only to oddities. Some of these oddities, it must +be confessed, are very odd. One of them, which I call the paradox of +Tristram Shandy, is the converse of the Achilles, and shows that the +tortoise, if you give him time, will go just as far as Achilles. +Tristram Shandy, as we know, employed two years in chronicling the +first two days of his life, and lamented that, at this rate, material +would accumulate faster than he could deal with it, so that, as years +went by, he would be farther and farther from the end of his history. +Now I maintain that, if he had lived for ever, and had not wearied of +his task, then, even if his life had continued as event fully as it +began, no part of his biography would have remained unwritten. For +consider: the hundredth day will be described in the hundredth year, +the thousandth in the thousandth year, and so on. Whatever day we may +choose as so far on that he cannot hope to reach it, that day will be +described in the corresponding year. Thus any day that may be +mentioned will be written up sooner or later, and therefore no part of +the biography will remain permanently unwritten. This paradoxical but +perfectly true proposition depends upon the fact that the number of +days in all time is no greater than the number of years. + +Thus on the subject of infinity it is impossible to avoid conclusions +which at first sight appear paradoxical, and this is the reason why so +many philosophers have supposed that there were inherent +contradictions in the infinite. But a little practice enables one to +grasp the true principles of Cantor's doctrine, and to acquire new and +better instincts as to the true and the false. The oddities then +become no odder than the people at the antipodes, who used to be +thought impossible because they would find it so inconvenient to stand +on their heads. + +The solution of the problems concerning infinity has enabled Cantor to +solve also the problems of continuity. Of this, as of infinity, he has +given a perfectly precise definition, and has shown that there are no +contradictions in the notion so defined. But this subject is so +technical that it is impossible to give any account of it here. + +The notion of continuity depends upon that of _order_, since +continuity is merely a particular type of order. Mathematics has, in +modern times, brought order into greater and greater prominence. In +former days, it was supposed (and philosophers are still apt to +suppose) that quantity was the fundamental notion of mathematics. But +nowadays, quantity is banished altogether, except from one little +corner of Geometry, while order more and more reigns supreme. The +investigation of different kinds of series and their relations is now +a very large part of mathematics, and it has been found that this +investigation can be conducted without any reference to quantity, and, +for the most part, without any reference to number. All types of +series are capable of formal definition, and their properties can be +deduced from the principles of symbolic logic by means of the Algebra +of Relatives. The notion of a limit, which is fundamental in the +greater part of higher mathematics, used to be defined by means of +quantity, as a term to which the terms of some series approximate as +nearly as we please. But nowadays the limit is defined quite +differently, and the series which it limits may not approximate to it +at all. This improvement also is due to Cantor, and it is one which +has revolutionised mathematics. Only order is now relevant to limits. +Thus, for instance, the smallest of the infinite integers is the limit +of the finite integers, though all finite integers are at an infinite +distance from it. The study of different types of series is a general +subject of which the study of ordinal numbers (mentioned above) is a +special and very interesting branch. But the unavoidable +technicalities of this subject render it impossible to explain to any +but professed mathematicians. + +Geometry, like Arithmetic, has been subsumed, in recent times, under +the general study of order. It was formerly supposed that Geometry was +the study of the nature of the space in which we live, and accordingly +it was urged, by those who held that what exists can only be known +empirically, that Geometry should really be regarded as belonging to +applied mathematics. But it has gradually appeared, by the increase of +non-Euclidean systems, that Geometry throws no more light upon the +nature of space than Arithmetic throws upon the population of the +United States. Geometry is a whole collection of deductive sciences +based on a corresponding collection of sets of axioms. One set of +axioms is Euclid's; other equally good sets of axioms lead to other +results. Whether Euclid's axioms are true, is a question as to which +the pure mathematician is indifferent; and, what is more, it is a +question which it is theoretically impossible to answer with certainty +in the affirmative. It might possibly be shown, by very careful +measurements, that Euclid's axioms are false; but no measurements +could ever assure us (owing to the errors of observation) that they +are exactly true. Thus the geometer leaves to the man of science to +decide, as best he may, what axioms are most nearly true in the actual +world. The geometer takes any set of axioms that seem interesting, and +deduces their consequences. What defines Geometry, in this sense, is +that the axioms must give rise to a series of more than one dimension. +And it is thus that Geometry becomes a department in the study of +order. + +In Geometry, as in other parts of mathematics, Peano and his disciples +have done work of the very greatest merit as regards principles. +Formerly, it was held by philosophers and mathematicians alike that +the proofs in Geometry depended on the figure; nowadays, this is known +to be false. In the best books there are no figures at all. The +reasoning proceeds by the strict rules of formal logic from a set of +axioms laid down to begin with. If a figure is used, all sorts of +things seem obviously to follow, which no formal reasoning can prove +from the explicit axioms, and which, as a matter of fact, are only +accepted because they are obvious. By banishing the figure, it becomes +possible to discover _all_ the axioms that are needed; and in this way +all sorts of possibilities, which would have otherwise remained +undetected, are brought to light. + +One great advance, from the point of view of correctness, has been +made by introducing points as they are required, and not starting, as +was formerly done, by assuming the whole of space. This method is due +partly to Peano, partly to another Italian named Fano. To those +unaccustomed to it, it has an air of somewhat wilful pedantry. In this +way, we begin with the following axioms: (1) There is a class of +entities called _points_. (2) There is at least one point. (3) If _a_ +be a point, there is at least one other point besides _a_. Then we +bring in the straight line joining two points, and begin again with +(4), namely, on the straight line joining _a_ and _b_, there is at +least one other point besides _a_ and _b_. (5) There is at least one +point not on the line _ab_. And so we go on, till we have the means of +obtaining as many points as we require. But the word _space_, as Peano +humorously remarks, is one for which Geometry has no use at all. + +The rigid methods employed by modern geometers have deposed Euclid +from his pinnacle of correctness. It was thought, until recent times, +that, as Sir Henry Savile remarked in 1621, there were only two +blemishes in Euclid, the theory of parallels and the theory of +proportion. It is now known that these are almost the only points in +which Euclid is free from blemish. Countless errors are involved in +his first eight propositions. That is to say, not only is it doubtful +whether his axioms are true, which is a comparatively trivial matter, +but it is certain that his propositions do not follow from the axioms +which he enunciates. A vastly greater number of axioms, which Euclid +unconsciously employs, are required for the proof of his propositions. +Even in the first proposition of all, where he constructs an +equilateral triangle on a given base, he uses two circles which are +assumed to intersect. But no explicit axiom assures us that they do +so, and in some kinds of spaces they do not always intersect. It is +quite doubtful whether our space belongs to one of these kinds or not. +Thus Euclid fails entirely to prove his point in the very first +proposition. As he is certainly not an easy author, and is terribly +long-winded, he has no longer any but an historical interest. Under +these circumstances, it is nothing less than a scandal that he should +still be taught to boys in England.[17] A book should have either +intelligibility or correctness; to combine the two is impossible, but +to lack both is to be unworthy of such a place as Euclid has occupied +in education. + +The most remarkable result of modern methods in mathematics is the +importance of symbolic logic and of rigid formalism. Mathematicians, +under the influence of Weierstrass, have shown in modern times a care +for accuracy, and an aversion to slipshod reasoning, such as had not +been known among them previously since the time of the Greeks. The +great inventions of the seventeenth century--Analytical Geometry and +the Infinitesimal Calculus--were so fruitful in new results that +mathematicians had neither time nor inclination to examine their +foundations. Philosophers, who should have taken up the task, had too +little mathematical ability to invent the new branches of mathematics +which have now been found necessary for any adequate discussion. Thus +mathematicians were only awakened from their "dogmatic slumbers" when +Weierstrass and his followers showed that many of their most cherished +propositions are in general false. Macaulay, contrasting the certainty +of mathematics with the uncertainty of philosophy, asks who ever heard +of a reaction against Taylor's theorem? If he had lived now, he +himself might have heard of such a reaction, for this is precisely one +of the theorems which modern investigations have overthrown. Such rude +shocks to mathematical faith have produced that love of formalism +which appears, to those who are ignorant of its motive, to be mere +outrageous pedantry. + +The proof that all pure mathematics, including Geometry, is nothing +but formal logic, is a fatal blow to the Kantian philosophy. Kant, +rightly perceiving that Euclid's propositions could not be deduced +from Euclid's axioms without the help of the figures, invented a +theory of knowledge to account for this fact; and it accounted so +successfully that, when the fact is shown to be a mere defect in +Euclid, and not a result of the nature of geometrical reasoning, +Kant's theory also has to be abandoned. The whole doctrine of _a +priori_ intuitions, by which Kant explained the possibility of pure +mathematics, is wholly inapplicable to mathematics in its present +form. The Aristotelian doctrines of the schoolmen come nearer in +spirit to the doctrines which modern mathematics inspire; but the +schoolmen were hampered by the fact that their formal logic was very +defective, and that the philosophical logic based upon the syllogism +showed a corresponding narrowness. What is now required is to give the +greatest possible development to mathematical logic, to allow to the +full the importance of relations, and then to found upon this secure +basis a new philosophical logic, which may hope to borrow some of the +exactitude and certainty of its mathematical foundation. If this can +be successfully accomplished, there is every reason to hope that the +near future will be as great an epoch in pure philosophy as the +immediate past has been in the principles of mathematics. Great +triumphs inspire great hopes; and pure thought may achieve, within our +generation, such results as will place our time, in this respect, on a +level with the greatest age of Greece.[18] + +FOOTNOTES: + +[11] This subject is due in the main to Mr. C.S. Peirce. + +[12] I ought to have added Frege, but his writings were unknown to me +when this article was written. [Note added in 1917.] + +[13] Professor of Mathematics in the University of Berlin. He died in +1897. + +[14] [Note added in 1917.] Although some infinite numbers are greater +than some others, it cannot be proved that of any two infinite numbers +one must be the greater. + +[15] Cantor was not guilty of a fallacy on this point. His proof that +there is no greatest number is valid. The solution of the puzzle is +complicated and depends upon the theory of types, which is explained +in _Principia Mathematica_, Vol. I (Camb. Univ. Press, 1910). [Note +added in 1917.] + +[16] This must not be regarded as a historically correct account of +what Zeno actually had in mind. It is a new argument for his +conclusion, not the argument which influenced him. On this point, see +e.g. C.D. Broad, "Note on Achilles and the Tortoise," _Mind_, N.S., +Vol. XXII, pp. 318-19. Much valuable work on the interpretation of +Zeno has been done since this article was written. [Note added in +1917.] + +[17] Since the above was written, he has ceased to be used as a +textbook. But I fear many of the books now used are so bad that the +change is no great improvement. [Note added in 1917.] + +[18] The greatest age of Greece was brought to an end by the +Peloponnesian War. [Note added in 1917.] + + + + +VI + +ON SCIENTIFIC METHOD IN PHILOSOPHY + + +When we try to ascertain the motives which have led men to the +investigation of philosophical questions, we find that, broadly +speaking, they can be divided into two groups, often antagonistic, and +leading to very divergent systems. These two groups of motives are, on +the one hand, those derived from religion and ethics, and, on the +other hand, those derived from science. Plato, Spinoza, and Hegel may +be taken as typical of the philosophers whose interests are mainly +religious and ethical, while Leibniz, Locke, and Hume may be taken as +representatives of the scientific wing. In Aristotle, Descartes, +Berkeley, and Kant we find both groups of motives strongly present. + +Herbert Spencer, in whose honour we are assembled to-day, would +naturally be classed among scientific philosophers: it was mainly from +science that he drew his data, his formulation of problems, and his +conception of method. But his strong religious sense is obvious in +much of his writing, and his ethical pre-occupations are what make him +value the conception of evolution--that conception in which, as a +whole generation has believed, science and morals are to be united in +fruitful and indissoluble marriage. + +It is my belief that the ethical and religious motives in spite of +the splendidly imaginative systems to which they have given rise, have +been on the whole a hindrance to the progress of philosophy, and ought +now to be consciously thrust aside by those who wish to discover +philosophical truth. Science, originally, was entangled in similar +motives, and was thereby hindered in its advances. It is, I maintain, +from science, rather than from ethics and religion, that philosophy +should draw its inspiration. + +But there are two different ways in which a philosophy may seek to +base itself upon science. It may emphasise the most general _results_ +of science, and seek to give even greater generality and unity to +these results. Or it may study the _methods_ of science, and seek to +apply these methods, with the necessary adaptations, to its own +peculiar province. Much philosophy inspired by science has gone astray +through preoccupation with the _results_ momentarily supposed to have +been achieved. It is not results, but _methods_ that can be +transferred with profit from the sphere of the special sciences to the +sphere of philosophy. What I wish to bring to your notice is the +possibility and importance of applying to philosophical problems +certain broad principles of method which have been found successful in +the study of scientific questions. + +The opposition between a philosophy guided by scientific method and a +philosophy dominated by religious and ethical ideas may be illustrated +by two notions which are very prevalent in the works of philosophers, +namely the notion of _the universe_, and the notion of _good and +evil_. A philosopher is expected to tell us something about the nature +of the universe as a whole, and to give grounds for either optimism or +pessimism. Both these expectations seem to me mistaken. I believe the +conception of "the universe" to be, as its etymology indicates, a +mere relic of pre-Copernican astronomy: and I believe the question of +optimism and pessimism to be one which the philosopher will regard as +outside his scope, except, possibly, to the extent of maintaining that +it is insoluble. + +In the days before Copernicus, the conception of the "universe" was +defensible on scientific grounds: the diurnal revolution of the +heavenly bodies bound them together as all parts of one system, of +which the earth was the centre. Round this apparent scientific fact, +many human desires rallied: the wish to believe Man important in the +scheme of things, the theoretical desire for a comprehensive +understanding of the Whole, the hope that the course of nature might +be guided by some sympathy with our wishes. In this way, an ethically +inspired system of metaphysics grew up, whose anthropocentrism was +apparently warranted by the geocentrism of astronomy. When Copernicus +swept away the astronomical basis of this system of thought, it had +grown so familiar, and had associated itself so intimately with men's +aspirations, that it survived with scarcely diminished force--survived +even Kant's "Copernican revolution," and is still now the unconscious +premiss of most metaphysical systems. + +The oneness of the world is an almost undiscussed postulate of most +metaphysics. "Reality is not merely one and self-consistent, but is a +system of reciprocally determinate parts"[19]--such a statement would +pass almost unnoticed as a mere truism. Yet I believe that it embodies +a failure to effect thoroughly the "Copernican revolution," and that +the apparent oneness of the world is merely the oneness of what is +seen by a single spectator or apprehended by a single mind. The +Critical Philosophy, although it intended to emphasise the subjective +element in many apparent characteristics of the world, yet, by +regarding the world in itself as unknowable, so concentrated attention +upon the subjective representation that its subjectivity was soon +forgotten. Having recognised the categories as the work of the mind, +it was paralysed by its own recognition, and abandoned in despair the +attempt to undo the work of subjective falsification. In part, no +doubt, its despair was well founded, but not, I think, in any absolute +or ultimate sense. Still less was it a ground for rejoicing, or for +supposing that the nescience to which it ought to have given rise +could be legitimately exchanged for a metaphysical dogmatism. + + +I + +As regards our present question, namely, the question of the unity of +the world, the right method, as I think, has been indicated by William +James.[20] "Let us now turn our backs upon ineffable or unintelligible +ways of accounting for the world's oneness, and inquire whether, +instead of being a principle, the 'oneness' affirmed may not merely be +a name like 'substance' descriptive of the fact that certain _specific +and verifiable connections_ are found among the parts of the +experiential flux.... We can easily conceive of things that shall have +no connection whatever with each other. We may assume them to inhabit +different times and spaces, as the dreams of different persons do even +now. They may be so unlike and incommensurable, and so inert towards +one another, as never to jostle or interfere. Even now there may +actually be whole universes so disparate from ours that we who know +ours have no means of perceiving that they exist. We conceive their +diversity, however; and by that fact the whole lot of them form what +is known in logic as 'a universe of discourse.' To form a universe of +discourse argues, as this example shows, no further kind of connexion. +The importance attached by certain monistic writers to the fact that +any chaos may become a universe by merely being named, is to me +incomprehensible." We are thus left with two kinds of unity in the +experienced world; the one what we may call the epistemological unity, +due merely to the fact that my experienced world is what _one_ +experience selects from the sum total of existence: the other that +tentative and partial unity exhibited in the prevalence of scientific +laws in those portions of the world which science has hitherto +mastered. Now a generalisation based upon either of these kinds of +unity would be fallacious. That the things which we experience have +the common property of being experienced by us is a truism from which +obviously nothing of importance can be deducible: it is clearly +fallacious to draw from the fact that whatever we experience is +experienced the conclusion that therefore everything must be +experienced. The generalisation of the second kind of unity, namely, +that derived from scientific laws, would be equally fallacious, though +the fallacy is a trifle less elementary. In order to explain it let us +consider for a moment what is called the reign of law. People often +speak as though it were a remarkable fact that the physical world is +subject to invariable laws. In fact, however, it is not easy to see +how such a world could fail to obey general laws. Taking any arbitrary +set of points in space, there is a function of the time corresponding +to these points, i.e. expressing the motion of a particle which +traverses these points: this function may be regarded as a general law +to which the behaviour of such a particle is subject. Taking all such +functions for all the particles in the universe, there will be +theoretically some one formula embracing them all, and this formula +may be regarded as the single and supreme law of the spatio-temporal +world. Thus what is surprising in physics is not the existence of +general laws, but their extreme simplicity. It is not the uniformity +of nature that should surprise us, for, by sufficient analytic +ingenuity, any conceivable course of nature might be shown to exhibit +uniformity. What should surprise us is the fact that the uniformity is +simple enough for us to be able to discover it. But it is just this +characteristic of simplicity in the laws of nature hitherto discovered +which it would be fallacious to generalise, for it is obvious that +simplicity has been a part cause of their discovery, and can, +therefore, give no ground for the supposition that other undiscovered +laws are equally simple. + +The fallacies to which these two kinds of unity have given rise +suggest a caution as regards all use in philosophy of general +_results_ that science is supposed to have achieved. In the first +place, in generalising these results beyond past experience, it is +necessary to examine very carefully whether there is not some reason +making it more probable that these results should hold of all that has +been experienced than that they should hold of things universally. The +sum total of what is experienced by mankind is a selection from the +sum total of what exists, and any general character exhibited by this +selection may be due to the manner of selecting rather than to the +general character of that from which experience selects. In the second +place, the most general results of science are the least certain and +the most liable to be upset by subsequent research. In utilizing these +results as the basis of a philosophy, we sacrifice the most valuable +and remarkable characteristic of scientific method, namely, that, +although almost everything in science is found sooner or later to +require some correction, yet this correction is almost always such as +to leave untouched, or only slightly modified, the greater part of the +results which have been deduced from the premiss subsequently +discovered to be faulty. The prudent man of science acquires a certain +instinct as to the kind of uses which may be made of present +scientific beliefs without incurring the danger of complete and utter +refutation from the modifications likely to be introduced by +subsequent discoveries. Unfortunately the use of scientific +generalisations of a sweeping kind as the basis of philosophy is just +that kind of use which an instinct of scientific caution would avoid, +since, as a rule, it would only lead to true results if the +generalisation upon which it is based stood in _no_ need of +correction. + +We may illustrate these general considerations by means of two +examples, namely, the conservation of energy and the principle of +evolution. + +(1) Let us begin with the conservation of energy, or, as Herbert +Spencer used to call it, the persistence of force. He says:[21] + + "Before taking a first step in the rational interpretation of + Evolution, it is needful to recognise, not only the facts that + Matter is indestructible and Motion continuous, but also the fact + that Force persists. An attempt to assign the _causes_ of + Evolution would manifestly be absurd if that agency to which the + metamorphosis in general and in detail is due, could either come + into existence or cease to exist. The succession of phenomena + would in such case be altogether arbitrary, and deductive Science + impossible." + +This paragraph illustrates the kind of way in which the philosopher is +tempted to give an air of absoluteness and necessity to empirical +generalisations, of which only the approximate truth in the regions +hitherto investigated can be guaranteed by the unaided methods of +science. It is very often said that the persistence of something or +other is a necessary presupposition of all scientific investigation, +and this presupposition is then thought to be exemplified in some +quantity which physics declares to be constant. There are here, as it +seems to me, three distinct errors. First, the detailed scientific +investigation of nature does not _presuppose_ any such general laws as +its results are found to verify. Apart from particular observations, +science need presuppose nothing except the general principles of +logic, and these principles are not laws of nature, for they are +merely hypothetical, and apply not only to the actual world but to +whatever is _possible_. The second error consists in the +identification of a constant quantity with a persistent entity. Energy +is a certain function of a physical system, but is not a thing or +substance persisting throughout the changes of the system. The same is +true of mass, in spite of the fact that mass has often been defined as +_quantity of matter_. The whole conception of quantity, involving, as +it does, numerical measurement based largely upon conventions, is far +more artificial, far more an embodiment of mathematical convenience, +than is commonly believed by those who philosophise on physics. Thus +even if (which I cannot for a moment admit) the persistence of some +entity were among the necessary postulates of science, it would be a +sheer error to infer from this the constancy of any physical quantity, +or the _a priori_ necessity of any such constancy which may be +empirically discovered. In the third place, it has become more and +more evident with the progress of physics that large generalisations, +such as the conservation of energy or mass, are far from certain and +are very likely only approximate. Mass, which used to be regarded as +the most indubitable of physical quantities, is now generally believed +to vary according to velocity, and to be, in fact, a vector quantity +which at a given moment is different in different directions. The +detailed conclusions deduced from the supposed constancy of mass for +such motions as used to be studied in physics will remain very nearly +exact, and therefore over the field of the older investigations very +little modification of the older results is required. But as soon as +such a principle as the conservation of mass or of energy is erected +into a universal _a priori_ law, the slightest failure in absolute +exactness is fatal, and the whole philosophic structure raised upon +this foundation is necessarily ruined. The prudent philosopher, +therefore, though he may with advantage study the methods of physics, +will be very chary of basing anything upon what happen at the moment +to be the most general results apparently obtained by those methods. + +(2) The philosophy of evolution, which was to be our second example, +illustrates the same tendency to hasty generalisation, and also +another sort, namely, the undue preoccupation with ethical notions. +There are two kinds of evolutionist philosophy, of which both Hegel +and Spencer represent the older and less radical kind, while +Pragmatism and Bergson represent the more modern and revolutionary +variety. But both these sorts of evolutionism have in common the +emphasis on _progress_, that is, upon a continual change from the +worse to the better, or from the simpler to the more complex. It +would be unfair to attribute to Hegel any scientific motive or +foundation, but all the other evolutionists, including Hegel's modern +disciples, have derived their impetus very largely from the history of +biological development. To a philosophy which derives a law of +universal progress from this history there are two objections. First, +that this history itself is concerned with a very small selection of +facts confined to an infinitesimal fragment of space and time, and +even on scientific grounds probably not an average sample of events in +the world at large. For we know that decay as well as growth is a +normal occurrence in the world. An extra-terrestrial philosopher, who +had watched a single youth up to the age of twenty-one and had never +come across any other human being, might conclude that it is the +nature of human beings to grow continually taller and wiser in an +indefinite progress towards perfection; and this generalisation would +be just as well founded as the generalisation which evolutionists base +upon the previous history of this planet. Apart, however, from this +scientific objection to evolutionism, there is another, derived from +the undue admixture of ethical notions in the very idea of progress +from which evolutionism derives its charm. Organic life, we are told, +has developed gradually from the protozoon to the philosopher, and +this development, we are assured, is indubitably an advance. +Unfortunately it is the philosopher, not the protozoon, who gives us +this assurance, and we can have no security that the impartial +outsider would agree with the philosopher's self-complacent +assumption. This point has been illustrated by the philosopher Chuang +Tz[)u] in the following instructive anecdote: + + "The Grand Augur, in his ceremonial robes, approached the shambles + and thus addressed the pigs: 'How can you object to die? I shall + fatten you for three months. I shall discipline myself for ten + days and fast for three. I shall strew fine grass, and place you + bodily upon a carved sacrificial dish. Does not this satisfy you?' + + Then, speaking from the pigs' point of view, he continued: 'It is + better, perhaps, after all, to live on bran and escape the + shambles....' + + 'But then,' added he, speaking from his own point of view, 'to + enjoy honour when alive one would readily die on a war-shield or + in the headsman's basket.' + + So he rejected the pigs' point of view and adopted his own point + of view. In what sense, then, was he different from the pigs?" + +I much fear that the evolutionists too often resemble the Grand Augur +and the pigs. + +The ethical element which has been prominent in many of the most +famous systems of philosophy is, in my opinion, one of the most +serious obstacles to the victory of scientific method in the +investigation of philosophical questions. Human ethical notions, as +Chuang Tz[)u] perceived, are essentially anthropocentric, and involve, +when used in metaphysics, an attempt, however veiled, to legislate for +the universe on the basis of the present desires of men. In this way +they interfere with that receptivity to fact which is the essence of +the scientific attitude towards the world. To regard ethical notions +as a key to the understanding of the world is essentially +pre-Copernican. It is to make man, with the hopes and ideals which he +happens to have at the present moment, the centre of the universe and +the interpreter of its supposed aims and purposes. Ethical metaphysics +is fundamentally an attempt, however disguised, to give legislative +force to our own wishes. This may, of course, be questioned, but I +think that it is confirmed by a consideration of the way in which +ethical notions arise. Ethics is essentially a product of the +gregarious instinct, that is to say, of the instinct to co-operate +with those who are to form our own group against those who belong to +other groups. Those who belong to our own group are good; those who +belong to hostile groups are wicked. The ends which are pursued by our +own group are desirable ends, the ends pursued by hostile groups are +nefarious. The subjectivity of this situation is not apparent to the +gregarious animal, which feels that the general principles of justice +are on the side of its own herd. When the animal has arrived at the +dignity of the metaphysician, it invents ethics as the embodiment of +its belief in the justice of its own herd. So the Grand Augur invokes +ethics as the justification of Augurs in their conflicts with pigs. +But, it may be said, this view of ethics takes no account of such +truly ethical notions as that of self-sacrifice. This, however, would +be a mistake. The success of gregarious animals in the struggle for +existence depends upon co-operation within the herd, and co-operation +requires sacrifice, to some extent, of what would otherwise be the +interest of the individual. Hence arises a conflict of desires and +instincts, since both self-preservation and the preservation of the +herd are biological ends to the individual. Ethics is in origin the +art of recommending to others the sacrifices required for co-operation +with oneself. Hence, by reflexion, it comes, through the operation of +social justice, to recommend sacrifices by oneself, but all ethics, +however refined, remains more or less subjective. Even vegetarians do +not hesitate, for example, to save the life of a man in a fever, +although in doing so they destroy the lives of many millions of +microbes. The view of the world taken by the philosophy derived from +ethical notions is thus never impartial and therefore never fully +scientific. As compared with science, it fails to achieve the +imaginative liberation from self which is necessary to such +understanding of the world as man can hope to achieve, and the +philosophy which it inspires is always more or less parochial, more or +less infected with the prejudices of a time and a place. + +I do not deny the importance or value, within its own sphere, of the +kind of philosophy which is inspired by ethical notions. The ethical +work of Spinoza, for example, appears to me of the very highest +significance, but what is valuable in such work is not any +metaphysical theory as to the nature of the world to which it may give +rise, nor indeed anything which can be proved or disproved by +argument. What is valuable is the indication of some new way of +feeling towards life and the world, some way of feeling by which our +own existence can acquire more of the characteristics which we must +deeply desire. The value of such work, however immeasurable it is, +belongs with practice and not with theory. Such theoretic importance +as it may possess is only in relation to human nature, not in relation +to the world at large. The scientific philosophy, therefore, which +aims only at understanding the world and not directly at any other +improvement of human life, cannot take account of ethical notions +without being turned aside from that submission to fact which is the +essence of the scientific temper. + + +II + +If the notion of the universe and the notion of good and evil are +extruded from scientific philosophy, it may be asked what specific +problems remain for the philosopher as opposed to the man of science? +It would be difficult to give a precise answer to this question, but +certain characteristics may be noted as distinguishing the province of +philosophy from that of the special sciences. + +In the first place a philosophical proposition must be general. It +must not deal specially with things on the surface of the earth, or +with the solar system, or with any other portion of space and time. It +is this need of generality which has led to the belief that philosophy +deals with the universe as a whole. I do not believe that this belief +is justified, but I do believe that a philosophical proposition must +be applicable to everything that exists or may exist. It might be +supposed that this admission would be scarcely distinguishable from +the view which I wish to reject. This, however, would be an error, and +an important one. The traditional view would make the universe itself +the subject of various predicates which could not be applied to any +particular thing in the universe, and the ascription of such peculiar +predicates to the universe would be the special business of +philosophy. I maintain, on the contrary, that there are no +propositions of which the "universe" is the subject; in other words, +that there is no such thing as the "universe." What I do maintain is +that there are general propositions which may be asserted of each +individual thing, such as the propositions of logic. This does not +involve that all the things there are form a whole which could be +regarded as another thing and be made the subject of predicates. It +involves only the assertion that there are properties which belong to +each separate thing, not that there are properties belonging to the +whole of things collectively. The philosophy which I wish to advocate +may be called logical atomism or absolute pluralism, because, while +maintaining that there are many things, it denies that there is a +whole composed of those things. We shall see, therefore, that +philosophical propositions, instead of being concerned with the whole +of things collectively, are concerned with all things distributively; +and not only must they be concerned with all things, but they must be +concerned with such properties of all things as do not depend upon the +accidental nature of the things that there happen to be, but are true +of any possible world, independently of such facts as can only be +discovered by our senses. + +This brings us to a second characteristic of philosophical +propositions, namely, that they must be _a priori_. A philosophical +proposition must be such as can be neither proved nor disproved by +empirical evidence. Too often we find in philosophical books arguments +based upon the course of history, or the convolutions of the brain, or +the eyes of shell-fish. Special and accidental facts of this kind are +irrelevant to philosophy, which must make only such assertions as +would be equally true however the actual world were constituted. + +We may sum up these two characteristics of philosophical propositions +by saying that _philosophy is the science of the possible_. But this +statement unexplained is liable to be misleading, since it may be +thought that the possible is something other than the general, whereas +in fact the two are indistinguishable. + +Philosophy, if what has been said is correct, becomes +indistinguishable from logic as that word has now come to be used. +The study of logic consists, broadly speaking, of two not very sharply +distinguished portions. On the one hand it is concerned with those +general statements which can be made concerning everything without +mentioning any one thing or predicate or relation, such for example as +"if _x_ is a member of the class [alpha] and every member of [alpha] +is a member of [beta], then _x_ is a member of the class [beta], +whatever _x_, [alpha], and [beta] may be." On the other hand, it is +concerned with the analysis and enumeration of logical _forms_, i.e. +with the kinds of propositions that may occur, with the various types +of facts, and with the classification of the constituents of facts. In +this way logic provides an inventory of possibilities, a repertory of +abstractly tenable hypotheses. + +It might be thought that such a study would be too vague and too +general to be of any very great importance, and that, if its problems +became at any point sufficiently definite, they would be merged in the +problems of some special science. It appears, however, that this is +not the case. In some problems, for example, the analysis of space and +time, the nature of perception, or the theory of judgment, the +discovery of the logical form of the facts involved is the hardest +part of the work and the part whose performance has been most lacking +hitherto. It is chiefly for want of the right logical hypothesis that +such problems have hitherto been treated in such an unsatisfactory +manner, and have given rise to those contradictions or antinomies in +which the enemies of reason among philosophers have at all times +delighted. + +By concentrating attention upon the investigation of logical forms, it +becomes possible at last for philosophy to deal with its problems +piecemeal, and to obtain, as the sciences do, such partial and +probably not wholly correct results as subsequent investigation can +utilise even while it supplements and improves them. Most +philosophies hitherto have been constructed all in one block, in such +a way that, if they were not wholly correct, they were wholly +incorrect, and could not be used as a basis for further +investigations. It is chiefly owing to this fact that philosophy, +unlike science, has hitherto been unprogressive, because each original +philosopher has had to begin the work again from the beginning, +without being able to accept anything definite from the work of his +predecessors. A scientific philosophy such as I wish to recommend will +be piecemeal and tentative like other sciences; above all, it will be +able to invent hypotheses which, even if they are not wholly true, +will yet remain fruitful after the necessary corrections have been +made. This possibility of successive approximations to the truth is, +more than anything else, the source of the triumphs of science, and to +transfer this possibility to philosophy is to ensure a progress in +method whose importance it would be almost impossible to exaggerate. + +The essence of philosophy as thus conceived is analysis, not +synthesis. To build up systems of the world, like Heine's German +professor who knit together fragments of life and made an intelligible +system out of them, is not, I believe, any more feasible than the +discovery of the philosopher's stone. What is feasible is the +understanding of general forms, and the division of traditional +problems into a number of separate and less baffling questions. +"Divide and conquer" is the maxim of success here as elsewhere. + +Let us illustrate these somewhat general maxims by examining their +application to the philosophy of space, for it is only in application +that the meaning or importance of a method can be understood. Suppose +we are confronted with the problem of space as presented in Kant's +Transcendental Æsthetic, and suppose we wish to discover what are the +elements of the problem and what hope there is of obtaining a solution +of them. It will soon appear that three entirely distinct problems, +belonging to different studies, and requiring different methods for +their solution, have been confusedly combined in the supposed single +problem with which Kant is concerned. There is a problem of logic, a +problem of physics, and a problem of theory of knowledge. Of these +three, the problem of logic can be solved exactly and perfectly; the +problem of physics can probably be solved with as great a degree of +certainty and as great an approach to exactness as can be hoped in an +empirical region; the problem of theory of knowledge, however, remains +very obscure and very difficult to deal with. Let us see how these +three problems arise. + +(1) The logical problem has arisen through the suggestions of +non-Euclidean geometry. Given a body of geometrical propositions, it +is not difficult to find a minimum statement of the axioms from which +this body of propositions can be deduced. It is also not difficult, by +dropping or altering some of these axioms, to obtain a more general or +a different geometry, having, from the point of view of pure +mathematics, the same logical coherence and the same title to respect +as the more familiar Euclidean geometry. The Euclidean geometry itself +is true perhaps of actual space (though this is doubtful), but +certainly of an infinite number of purely arithmetical systems, each +of which, from the point of view of abstract logic, has an equal and +indefeasible right to be called a Euclidean space. Thus space as an +object of logical or mathematical study loses its uniqueness; not only +are there many kinds of spaces, but there are an infinity of examples +of each kind, though it is difficult to find any kind of which the +space of physics may be an example, and it is impossible to find any +kind of which the space of physics is certainly an example. As an +illustration of one possible logical system of geometry we may +consider all relations of three terms which are analogous in certain +formal respects to the relation "between" as it appears to be in +actual space. A space is then defined by means of one such three-term +relation. The points of the space are all the terms which have this +relation to something or other, and their order in the space in +question is determined by this relation. The points of one space are +necessarily also points of other spaces, since there are necessarily +other three-term relations having those same points for their field. +The space in fact is not determined by the class of its points, but by +the ordering three-term relation. When enough abstract logical +properties of such relations have been enumerated to determine the +resulting kind of geometry, say, for example, Euclidean geometry, it +becomes unnecessary for the pure geometer in his abstract capacity to +distinguish between the various relations which have all these +properties. He considers the whole class of such relations, not any +single one among them. Thus in studying a given kind of geometry the +pure mathematician is studying a certain class of relations defined by +means of certain abstract logical properties which take the place of +what used to be called axioms. The nature of geometrical _reasoning_ +therefore is purely deductive and purely logical; if any special +epistemological peculiarities are to be found in geometry, it must not +be in the reasoning, but in our knowledge concerning the axioms in +some given space. + +(2) The physical problem of space is both more interesting and more +difficult than the logical problem. The physical problem may be +stated as follows: to find in the physical world, or to construct from +physical materials, a space of one of the kinds enumerated by the +logical treatment of geometry. This problem derives its difficulty +from the attempt to accommodate to the roughness and vagueness of the +real world some system possessing the logical clearness and exactitude +of pure mathematics. That this can be done with a certain degree of +approximation is fairly evident If I see three people _A_, _B_, and +_C_ sitting in a row, I become aware of the fact which may be +expressed by saying that _B_ is between _A_ and _C_ rather than that +_A_ is between _B_ and _C_, or _C_ is between _A_ and _B_. This +relation of "between" which is thus perceived to hold has some of the +abstract logical properties of those three-term relations which, we +saw, give rise to a geometry, but its properties fail to be exact, and +are not, as empirically given, amenable to the kind of treatment at +which geometry aims. In abstract geometry we deal with points, +straight lines, and planes; but the three people _A_, _B_, and _C_ +whom I see sitting in a row are not exactly points, nor is the row +exactly a straight line. Nevertheless physics, which formally assumes +a space containing points, straight lines, and planes, is found +empirically to give results applicable to the sensible world. It must +therefore be possible to find an interpretation of the points, +straight lines, and planes of physics in terms of physical data, or at +any rate in terms of data together with such hypothetical additions as +seem least open to question. Since all data suffer from a lack of +mathematical precision through being of a certain size and somewhat +vague in outline, it is plain that if such a notion as that of a point +is to find any application to empirical material, the point must be +neither a datum nor a hypothetical addition to data, but a +_construction_ by means of data with their hypothetical additions. It +is obvious that any hypothetical filling out of data is less dubious +and unsatisfactory when the additions are closely analogous to data +than when they are of a radically different sort. To assume, for +example, that objects which we see continue, after we have turned away +our eyes, to be more or less analogous to what they were while we were +looking, is a less violent assumption than to assume that such objects +are composed of an infinite number of mathematical points. Hence in +the physical study of the geometry of physical space, points must not +be assumed _ab initio_ as they are in the logical treatment of +geometry, but must be constructed as systems composed of data and +hypothetical analogues of data. We are thus led naturally to define a +physical point as a certain class of those objects which are the +ultimate constituents of the physical world. It will be the class of +all those objects which, as one would naturally say, _contain_ the +point. To secure a definition giving this result, without previously +assuming that physical objects are composed of points, is an agreeable +problem in mathematical logic. The solution of this problem and the +perception of its importance are due to my friend Dr. Whitehead. The +oddity of regarding a point as a class of physical entities wears off +with familiarity, and ought in any case not to be felt by those who +maintain, as practically every one does, that points are mathematical +fictions. The word "fiction" is used glibly in such connexions by many +men who seem not to feel the necessity of explaining how it can come +about that a fiction can be so useful in the study of the actual world +as the points of mathematical physics have been found to be. By our +definition, which regards a point as a class of physical objects, it +is explained both how the use of points can lead to important +physical results, and how we can nevertheless avoid the assumption +that points are themselves entities in the physical world. + +Many of the mathematically convenient properties of abstract logical +spaces cannot be either known to belong or known not to belong to the +space of physics. Such are all the properties connected with continuity. +For to know that actual space has these properties would require an +infinite exactness of sense-perception. If actual space is continuous, +there are nevertheless many possible non-continuous spaces which will be +empirically indistinguishable from it; and, conversely, actual space may +be non-continuous and yet empirically indistinguishable from a possible +continuous space. Continuity, therefore, though obtainable in the _a +priori_ region of arithmetic, is not with certainty obtainable in the +space or time of the physical world: whether these are continuous or not +would seem to be a question not only unanswered but for ever +unanswerable. From the point of view of philosophy, however, the +discovery that a question is unanswerable is as complete an answer as +any that could possibly be obtained. And from the point of view of +physics, where no empirical means of distinction can be found, there can +be no empirical objection to the mathematically simplest assumption, +which is that of continuity. + +The subject of the physical theory of space is a very large one, +hitherto little explored. It is associated with a similar theory of +time, and both have been forced upon the attention of philosophically +minded physicists by the discussions which have raged concerning the +theory of relativity. + +(3) The problem with which Kant is concerned in the Transcendental +Æsthetic is primarily the epistemological problem: "How do we come to +have knowledge of geometry _a priori_?" By the distinction between the +logical and physical problems of geometry, the bearing and scope of +this question are greatly altered. Our knowledge of pure geometry is +_a priori_ but is wholly logical. Our knowledge of physical geometry +is synthetic, but is not _a priori_. Our knowledge of pure geometry is +hypothetical, and does not enable us to assert, for example, that the +axiom of parallels is true in the physical world. Our knowledge of +physical geometry, while it does enable us to assert that this axiom +is approximately verified, does not, owing to the inevitable +inexactitude of observation, enable us to assert that it is verified +_exactly_. Thus, with the separation which we have made between pure +geometry and the geometry of physics, the Kantian problem collapses. +To the question, "How is synthetic _a priori_ knowledge possible?" we +can now reply, at any rate so far as geometry is concerned, "It is not +possible," if "synthetic" means "not deducible from logic alone." Our +knowledge of geometry, like the rest of our knowledge, is derived +partly from logic, partly from sense, and the peculiar position which +in Kant's day geometry appeared to occupy is seen now to be a +delusion. There are still some philosophers, it is true, who maintain +that our knowledge that the axiom of parallels, for example, is true +of actual space, is not to be accounted for empirically, but is as +Kant maintained derived from an _a priori_ intuition. This position is +not logically refutable, but I think it loses all plausibility as soon +as we realise how complicated and derivative is the notion of physical +space. As we have seen, the application of geometry to the physical +world in no way demands that there should really be points and +straight lines among physical entities. The principle of economy, +therefore, demands that we should abstain from assuming the existence +of points and straight lines. As soon, however, as we accept the view +that points and straight lines are complicated constructions by means +of classes of physical entities, the hypothesis that we have an _a +priori_ intuition enabling us to know what happens to straight lines +when they are produced indefinitely becomes extremely strained and +harsh; nor do I think that such an hypothesis would ever have arisen +in the mind of a philosopher who had grasped the nature of physical +space. Kant, under the influence of Newton, adopted, though with some +vacillation, the hypothesis of absolute space, and this hypothesis, +though logically unobjectionable, is removed by Occam's razor, since +absolute space is an unnecessary entity in the explanation of the +physical world. Although, therefore, we cannot refute the Kantian +theory of an _a priori_ intuition, we can remove its grounds one by +one through an analysis of the problem. Thus, here as in many other +philosophical questions, the analytic method, while not capable of +arriving at a demonstrative result, is nevertheless capable of showing +that all the positive grounds in favour of a certain theory are +fallacious and that a less unnatural theory is capable of accounting +for the facts. + +Another question by which the capacity of the analytic method can be +shown is the question of realism. Both those who advocate and those +who combat realism seem to me to be far from clear as to the nature of +the problem which they are discussing. If we ask: "Are our objects of +perception _real_ and are they _independent_ of the percipient?" it +must be supposed that we attach some meaning to the words "real" and +"independent," and yet, if either side in the controversy of realism +is asked to define these two words, their answer is pretty sure to +embody confusions such as logical analysis will reveal. + +Let us begin with the word "real." There certainly are objects of +perception, and therefore, if the question whether these objects are +real is to be a substantial question, there must be in the world two +sorts of objects, namely, the real and the unreal, and yet the unreal +is supposed to be essentially what there is not. The question what +properties must belong to an object in order to make it real is one to +which an adequate answer is seldom if ever forthcoming. There is of +course the Hegelian answer, that the real is the self-consistent and +that nothing is self-consistent except the Whole; but this answer, +true or false, is not relevant in our present discussion, which moves +on a lower plane and is concerned with the status of objects of +perception among other objects of equal fragmentariness. Objects of +perception are contrasted, in the discussions concerning realism, +rather with psychical states on the one hand and matter on the other +hand than with the all-inclusive whole of things. The question we have +therefore to consider is the question as to what can be meant by +assigning "reality" to some but not all of the entities that make up +the world. Two elements, I think, make up what is felt rather than +thought when the word "reality" is used in this sense. A thing is real +if it persists at times when it is not perceived; or again, a thing is +real when it is correlated with other things in a way which experience +has led us to expect. It will be seen that reality in either of these +senses is by no means necessary to a thing, and that in fact there +might be a whole world in which nothing was real in either of these +senses. It might turn out that the objects of perception failed of +reality in one or both of these respects, without its being in any way +deducible that they are not parts of the external world with which +physics deals. Similar remarks will apply to the word "independent." +Most of the associations of this word are bound up with ideas as to +causation which it is not now possible to maintain. _A_ is independent +of _B_ when _B_ is not an indispensable part of the _cause_ of _A_. +But when it is recognised that causation is nothing more than +correlation, and that there are correlations of simultaneity as well +as of succession, it becomes evident that there is no uniqueness in a +series of casual antecedents of a given event, but that, at any point +where there is a correlation of simultaneity, we can pass from one +line of antecedents to another in order to obtain a new series of +causal antecedents. It will be necessary to specify the causal law +according to which the antecedents are to be considered. I received a +letter the other day from a correspondent who had been puzzled by +various philosophical questions. After enumerating them he says: +"These questions led me from Bonn to Strassburg, where I found +Professor Simmel." Now, it would be absurd to deny that these +questions caused his body to move from Bonn to Strassburg, and yet it +must be supposed that a set of purely mechanical antecedents could +also be found which would account for this transfer of matter from one +place to another. Owing to this plurality of causal series antecedent +to a given event, the notion of _the_ cause becomes indefinite, and +the question of independence becomes correspondingly ambiguous. Thus, +instead of asking simply whether _A_ is independent of _B_, we ought +to ask whether there is a series determined by such and such causal +laws leading from _B_ to _A_. This point is important in connexion +with the particular question of objects of perception. It may be that +no objects quite like those which we perceive ever exist unperceived; +in this case there will be a causal law according to which objects of +perception are not independent of being perceived. But even if this be +the case, it may nevertheless also happen that there are purely +physical causal laws determining the occurrence of objects which are +perceived by means of other objects which perhaps are not perceived. +In that case, in regard to such causal laws objects of perception will +be independent of being perceived. Thus the question whether objects +of perception are independent of being perceived is, as it stands, +indeterminate, and the answer will be yes or no according to the +method adopted of making it determinate. I believe that this confusion +has borne a very large part in prolonging the controversies on this +subject, which might well have seemed capable of remaining for ever +undecided. The view which I should wish to advocate is that objects of +perception do not persist unchanged at times when they are not +perceived, although probably objects more or less resembling them do +exist at such times; that objects of perception are part, and the only +empirically knowable part, of the actual subject-matter of physics, +and are themselves properly to be called physical; that purely +physical laws exist determining the character and duration of objects +of perception without any reference to the fact that they are +perceived; and that in the establishment of such laws the propositions +of physics do not presuppose any propositions of psychology or even +the existence of mind. I do not know whether realists would recognise +such a view as realism. All that I should claim for it is, that it +avoids difficulties which seem to me to beset both realism and +idealism as hitherto advocated, and that it avoids the appeal which +they have made to ideas which logical analysis shows to be ambiguous. +A further defence and elaboration of the positions which I advocate, +but for which time is lacking now, will be found indicated in my book +on _Our Knowledge of the External World_.[22] + +The adoption of scientific method in philosophy, if I am not mistaken, +compels us to abandon the hope of solving many of the more ambitious +and humanly interesting problems of traditional philosophy. Some of +these it relegates, though with little expectation of a successful +solution, to special sciences, others it shows to be such as our +capacities are essentially incapable of solving. But there remain a +large number of the recognised problems of philosophy in regard to +which the method advocated gives all those advantages of division into +distinct questions, of tentative, partial, and progressive advance, +and of appeal to principles with which, independently of temperament, +all competent students must agree. The failure of philosophy hitherto +has been due in the main to haste and ambition: patience and modesty, +here as in other sciences, will open the road to solid and durable +progress. + +FOOTNOTES: + +[19] Bosanquet, _Logic_, ii, p. 211. + +[20] _Some Problems of Philosophy_, p 124. + +[21] _First Principles_ (1862), Part II, beginning of chap. viii. + +[22] Open Court Company, 1914. + + + + +VII + +THE ULTIMATE CONSTITUENTS OF MATTER[23] + + +I wish to discuss in this article no less a question than the ancient +metaphysical query, "What is matter?" The question, "What is matter?" +in so far as it concerns philosophy, is, I think, already capable of +an answer which in principle will be as complete as an answer can hope +to be; that is to say, we can separate the problem into an essentially +soluble and an essentially insoluble portion, and we can now see how +to solve the essentially soluble portion, at least as regards its main +outlines. It is these outlines which I wish to suggest in the present +article. My main position, which is realistic, is, I hope and believe, +not remote from that of Professor Alexander, by whose writings on this +subject I have profited greatly.[24] It is also in close accord with +that of Dr. Nunn.[25] + +Common sense is accustomed to the division of the world into mind and +matter. It is supposed by all who have never studied philosophy that +the distinction between mind and matter is perfectly clear and easy, +that the two do not at any point overlap, and that only a fool or a +philosopher could be in doubt as to whether any given entity is mental +or material. This simple faith survives in Descartes and in a +somewhat modified form in Spinoza, but with Leibniz it begins to +disappear, and from his day to our own almost every philosopher of +note has criticised and rejected the dualism of common sense. It is my +intention in this article to defend this dualism; but before defending +it we must spend a few moments on the reasons which have prompted its +rejection. + +Our knowledge of the material world is obtained by means of the +senses, of sight and touch and so on. At first it is supposed that +things are just as they seem, but two opposite sophistications soon +destroy this naïve belief. On the one hand the physicists cut up +matter into molecules, atoms, corpuscles, and as many more such +subdivisions as their future needs may make them postulate, and the +units at which they arrive are uncommonly different from the visible, +tangible objects of daily life. A unit of matter tends more and more +to be something like an electromagnetic field filling all space, +though having its greatest intensity in a small region. Matter +consisting of such elements is as remote from daily life as any +metaphysical theory. It differs from the theories of metaphysicians +only in the fact that its practical efficacy proves that it contains +some measure of truth and induces business men to invest money on the +strength of it; but, in spite of its connection with the money market, +it remains a metaphysical theory none the less. + +The second kind of sophistication to which the world of common sense +has been subjected is derived from the psychologists and +physiologists. The physiologists point out that what we see depends +upon the eye, that what we hear depends upon the ear, and that all our +senses are liable to be affected by anything which affects the brain, +like alcohol or hasheesh. Psychologists point out how much of what we +think we see is supplied by association or unconscious inference, how +much is mental interpretation, and how doubtful is the residuum which +can be regarded as crude datum. From these facts it is argued by the +psychologists that the notion of a datum passively received by the +mind is a delusion, and it is argued by the physiologists that even if +a pure datum of sense could be obtained by the analysis of experience, +still this datum could not belong, as common sense supposes, to the +outer world, since its whole nature is conditioned by our nerves and +sense organs, changing as they change in ways which it is thought +impossible to connect with any change in the matter supposed to be +perceived. This physiologist's argument is exposed to the rejoinder, +more specious than solid, that our knowledge of the existence of the +sense organs and nerves is obtained by that very process which the +physiologist has been engaged in discrediting, since the existence of +the nerves and sense organs is only known through the evidence of the +senses themselves. This argument may prove that some reinterpretation +of the results of physiology is necessary before they can acquire +metaphysical validity. But it does not upset the physiological +argument in so far as this constitutes merely a _reductio ad absurdum_ +of naïve realism. + +These various lines of argument prove, I think, that some part of the +beliefs of common sense must be abandoned. They prove that, if we take +these beliefs as a whole, we are forced into conclusions which are in +part self-contradictory; but such arguments cannot of themselves +decide what portion of our common-sense beliefs is in need of +correction. Common sense believes that what we see is physical, +outside the mind, and continuing to exist if we shut our eyes or turn +them in another direction. I believe that common sense is right in +regarding what we see as physical and (in one of several possible +senses) outside the mind, but is probably wrong in supposing that it +continues to exist when we are no longer looking at it. It seems to me +that the whole discussion of matter has been obscured by two errors +which support each other. The first of these is the error that what we +see, or perceive through any of our other senses, is subjective: the +second is the belief that what is physical must be persistent. +Whatever physics may regard as the ultimate constituents of matter, it +always supposes these constituents to be indestructible. Since the +immediate data of sense are not indestructible but in a state of +perpetual flux, it is argued that these data themselves cannot be +among the ultimate constituents of matter. I believe this to be a +sheer mistake. The persistent particles of mathematical physics I +regard as logical constructions, symbolic fictions enabling us to +express compendiously very complicated assemblages of facts; and, on +the other hand, I believe that the actual data in sensation, the +immediate objects of sight or touch or hearing, are extra-mental, +purely physical, and among the ultimate constituents of matter. + +My meaning in regard to the impermanence of physical entities may +perhaps be made clearer by the use of Bergson's favourite illustration +of the cinematograph. When I first read Bergson's statement that the +mathematician conceives the world after the analogy of a +cinematograph, I had never seen a cinematograph, and my first visit to +one was determined by the desire to verify Bergson's statement, which +I found to be completely true, at least so far as I am concerned. +When, in a picture palace, we see a man rolling down hill, or running +away from the police, or falling into a river, or doing any of those +other things to which men in such places are addicted, we know that +there is not really only one man moving, but a succession of films, +each with a different momentary man. The illusion of persistence +arises only through the approach to continuity in the series of +momentary men. Now what I wish to suggest is that in this respect the +cinema is a better metaphysician than common sense, physics, or +philosophy. The real man too, I believe, however the police may swear +to his identity, is really a series of momentary men, each different +one from the other, and bound together, not by a numerical identity, +but by continuity and certain intrinsic causal laws. And what applies +to men applies equally to tables and chairs, the sun, moon and stars. +Each of these is to be regarded, not as one single persistent entity, +but as a series of entities succeeding each other in time, each +lasting for a very brief period, though probably not for a mere +mathematical instant. In saying this I am only urging the same kind of +division in time as we are accustomed to acknowledge in the case of +space. A body which fills a cubic foot will be admitted to consist of +many smaller bodies, each occupying only a very tiny volume; similarly +a thing which persists for an hour is to be regarded as composed of +many things of less duration. A true theory of matter requires a +division of things into time-corpuscles as well as into +space-corpuscles. + +The world may be conceived as consisting of a multitude of entities +arranged in a certain pattern. The entities which are arranged I shall +call "particulars." The arrangement or pattern results from relations +among particulars. Classes or series of particulars, collected +together on account of some property which makes it convenient to be +able to speak of them as wholes, are what I call logical constructions +or symbolic fictions. The particulars are to be conceived, not on the +analogy of bricks in a building, but rather on the analogy of notes +in a symphony. The ultimate constituents of a symphony (apart from +relations) are the notes, each of which lasts only for a very short +time. We may collect together all the notes played by one instrument: +these may be regarded as the analogues of the successive particulars +which common sense would regard as successive states of one "thing." +But the "thing" ought to be regarded as no more "real" or +"substantial" than, for example, the rôle of the trombone. As soon as +"things" are conceived in this manner it will be found that the +difficulties in the way of regarding immediate objects of sense as +physical have largely disappeared. + +When people ask, "Is the object of sense mental or physical?" they +seldom have any clear idea either what is meant by "mental" or +"physical," or what criteria are to be applied for deciding whether a +given entity belongs to one class or the other. I do not know how to +give a sharp definition of the word "mental," but something may be +done by enumerating occurrences which are indubitably mental: +believing, doubting, wishing, willing, being pleased or pained, are +certainly mental occurrences; so are what we may call experiences, +seeing, hearing, smelling, perceiving generally. But it does not +follow from this that what is seen, what is heard, what is smelt, what +is perceived, must be mental. When I see a flash of lightning, my +seeing of it is mental, but what I see, although it is not quite the +same as what anybody else sees at the same moment, and although it +seems very unlike what the physicist would describe as a flash of +lightning, is not mental. I maintain, in fact, that if the physicist +could describe truly and fully all that occurs in the physical world +when there is a flash of lightning, it would contain as a constituent +what I see, and also what is seen by anybody else who would commonly +be said to see the same flash. What I mean may perhaps be made plainer +by saying that if my body could remain in exactly the same state in +which it is, although my mind had ceased to exist, precisely that +object which I now see when I see the flash would exist, although of +course I should not see it, since my seeing is mental. The principal +reasons which have led people to reject this view have, I think, been +two: first, that they did not adequately distinguish between my seeing +and what I see; secondly, that the causal dependence of what I see +upon my body has made people suppose that what I see cannot be +"outside" me. The first of these reasons need not detain us, since the +confusion only needs to be pointed out in order to be obviated; but +the second requires some discussion, since it can only be answered by +removing current misconceptions, on the one hand as to the nature of +space, and on the other, as to the meaning of causal dependence. + +When people ask whether colours, for example, or other secondary +qualities are inside or outside the mind, they seem to suppose that +their meaning must be clear, and that it ought to be possible to say +yes or no without any further discussion of the terms involved. In +fact, however, such terms as "inside" or "outside" are very ambiguous. +What is meant by asking whether this or that is "in" the mind? The +mind is not like a bag or a pie; it does not occupy a certain region +in space, or, if (in a sense) it does, what is in that region is +presumably part of the brain, which would not be said to be in the +mind. When people say that sensible qualities are in the mind, they do +not mean "spatially contained in" in the sense in which the blackbirds +were in the pie. We might regard the mind as an assemblage of +particulars, namely, what would be called "states of mind," which +would belong together in virtue of some specific common quality. The +common quality of all states of mind would be the quality designated +by the word "mental"; and besides this we should have to suppose that +each separate person's states of mind have some common characteristic +distinguishing them from the states of mind of other people. Ignoring +this latter point, let us ask ourselves whether the quality designated +by the word "mental" does, as a matter of observation, actually belong +to objects of sense, such as colours or noises. I think any candid +person must reply that, however difficult it may be to know what we +mean by "mental," it is not difficult to see that colours and noises +are not mental in the sense of having that intrinsic peculiarity which +belongs to beliefs and wishes and volitions, but not to the physical +world. Berkeley advances on this subject a plausible argument[26] +which seems to me to rest upon an ambiguity in the word "pain." He +argues that the realist supposes the heat which he feels in +approaching a fire to be something outside his mind, but that as he +approaches nearer and nearer to the fire the sensation of heat passes +imperceptibly into pain, and that no one could regard pain as +something outside the mind. In reply to this argument, it should be +observed in the first place that the heat of which we are immediately +aware is not in the fire but in our own body. It is only by inference +that the fire is judged to be the cause of the heat which we feel in +our body. In the second place (and this is the more important point), +when we speak of pain we may mean one of two things: we may mean the +object of the sensation or other experience which has the quality of +being painful, or we may mean the quality of painfulness itself. When +a man says he has a pain in his great toe, what he means is that he +has a sensation associated with his great toe and having the quality +of painfulness. The sensation itself, like every sensation, consists +in experiencing a sensible object, and the experiencing has that +quality of painfulness which only mental occurrences can have, but +which may belong to thoughts or desires, as well as to sensations. But +in common language we speak of the sensible object experienced in a +painful sensation as a pain, and it is this way of speaking which +causes the confusion upon which the plausibility of Berkeley's +argument depends. It would be absurd to attribute the quality of +painfulness to anything non-mental, and hence it comes to be thought +that what we call a pain in the toe must be mental. In fact, however, +it is not the sensible object in such a case which is painful, but the +sensation, that is to say, the experience of the sensible object. As +the heat which we experience from the fire grows greater, the +experience passes gradually from being pleasant to being painful, but +neither the pleasure nor the pain is a quality of the object +experienced as opposed to the experience, and it is therefore a +fallacy to argue that this object must be mental on the ground that +painfulness can only be attributed to what is mental. + +If, then, when we say that something is in the mind we mean that it +has a certain recognisable intrinsic characteristic such as belongs to +thoughts and desires, it must be maintained on grounds of immediate +inspection that objects of sense are not in any mind. + +A different meaning of "in the mind" is, however, to be inferred from +the arguments advanced by those who regard sensible objects as being +in the mind. The arguments used are, in the main, such as would prove +the causal dependence of objects of sense upon the percipient. Now +the notion of causal dependence is very obscure and difficult, much +more so in fact than is generally realised by philosophers. I shall +return to this point in a moment. For the present, however, accepting +the notion of causal dependence without criticism, I wish to urge that +the dependence in question is rather upon our bodies than upon our +minds. The visual appearance of an object is altered if we shut one +eye, or squint, or look previously at something dazzling; but all +these are bodily acts, and the alterations which they effect are to be +explained by physiology and optics, not by psychology.[27] They are in +fact of exactly the same kind as the alterations effected by +spectacles or a microscope. They belong therefore to the theory of the +physical world, and can have no bearing upon the question whether what +we see is causally dependent upon the mind. What they do tend to +prove, and what I for my part have no wish to deny, is that what we +see is causally dependent upon our body and is not, as crude common +sense would suppose, something which would exist equally if our eyes +and nerves and brain were absent, any more than the visual appearance +presented by an object seen through a microscope would remain if the +microscope were removed. So long as it is supposed that the physical +world is composed of stable and more or less permanent constituents, +the fact that what we see is changed by changes in our body appears to +afford reason for regarding what we see as not an ultimate constituent +of matter. But if it is recognised that the ultimate constituents of +matter are as circumscribed in duration as in spatial extent, the +whole of this difficulty vanishes. + +There remains, however, another difficulty, connected with space. When +we look at the sun we wish to know something about the sun itself, +which is ninety-three million miles away; but what we see is dependent +upon our eyes, and it is difficult to suppose that our eyes can affect +what happens at a distance of ninety-three million miles. Physics +tells us that certain electromagnetic waves start from the sun, and +reach our eyes after about eight minutes. They there produce +disturbances in the rods and cones, thence in the optic nerve, thence +in the brain. At the end of this purely physical series, by some odd +miracle, comes the experience which we call "seeing the sun," and it +is such experiences which form the whole and sole reason for our +belief in the optic nerve, the rods and cones, the ninety-three +million miles, the electromagnetic waves, and the sun itself. It is +this curious oppositeness of direction between the order of causation +as affirmed by physics, and the order of evidence as revealed by +theory of knowledge, that causes the most serious perplexities in +regard to the nature of physical reality. Anything that invalidates +our seeing, as a source of knowledge concerning physical reality, +invalidates also the whole of physics and physiology. And yet, +starting from a common-sense acceptance of our seeing, physics has +been led step by step to the construction of the causal chain in which +our seeing is the last link, and the immediate object which we see +cannot be regarded as that initial cause which we believe to be +ninety-three million miles away, and which we are inclined to regard +as the "real" sun. + +I have stated this difficulty as forcibly as I can, because I believe +that it can only be answered by a radical analysis and reconstruction +of all the conceptions upon whose employment it depends. + +Space, time, matter and cause, are the chief of these conceptions. Let +us begin with the conception of cause. + +Causal dependence, as I observed a moment ago, is a conception which +it is very dangerous to accept at its face value. There exists a +notion that in regard to any event there is something which may be +called _the_ cause of that event--some one definite occurrence, +without which the event would have been impossible and with which it +becomes necessary. An event is supposed to be dependent upon its cause +in some way which in it is not dependent upon other things. Thus men +will urge that the mind is dependent upon the brain, or, with equal +plausibility, that the brain is dependent upon the mind. It seems not +improbable that if we had sufficient knowledge we could infer the +state of a man's mind from the state of his brain, or the state of his +brain from the state of his mind. So long as the usual conception of +causal dependence is retained, this state of affairs can be used by +the materialist to urge that the state of our brain causes our +thoughts, and by the idealist to urge that our thoughts cause the +state of our brain. Either contention is equally valid or equally +invalid. The fact seems to be that there are many correlations of the +sort which may be called causal, and that, for example, either a +physical or a mental event can be predicted, theoretically, either +from a sufficient number of physical antecedents or from a sufficient +number of mental antecedents. To speak of _the_ cause of an event is +therefore misleading. Any set of antecedents from which the event can +theoretically be inferred by means of correlations might be called a +cause of the event. But to speak of _the_ cause is to imply a +uniqueness which does not exist. + +The relevance of this to the experience which we call "seeing the sun" +is obvious. The fact that there exists a chain of antecedents which +makes our seeing dependent upon the eyes and nerves and brain does not +even tend to show that there is not another chain of antecedents in +which the eyes and nerves and brain as physical things are ignored. If +we are to escape from the dilemma which seemed to arise out of the +physiological causation of what we see when we say we see the sun, we +must find, at least in theory, a way of stating causal laws for the +physical world, in which the units are not material things, such as +the eyes and nerves and brain, but momentary particulars of the same +sort as our momentary visual object when we look at the sun. The sun +itself and the eyes and nerves and brain must be regarded as +assemblages of momentary particulars. Instead of supposing, as we +naturally do when we start from an uncritical acceptance of the +apparent dicta of physics, that _matter_ is what is "really real" in +the physical world, and that the immediate objects of sense are mere +phantasms, we must regard matter as a logical construction, of which +the constituents will be just such evanescent particulars as may, when +an observer happens to be present, become data of sense to that +observer. What physics regards as the sun of eight minutes ago will be +a whole assemblage of particulars, existing at different times, +spreading out from a centre with the velocity of light, and containing +among their number all those visual data which are seen by people who +are now looking at the sun. Thus the sun of eight minutes ago is a +class of particulars, and what I see when I now look at the sun is one +member of this class. The various particulars constituting this class +will be correlated with each other by a certain continuity and certain +intrinsic laws of variation as we pass outwards from the centre, +together with certain modifications correlated extrinsically with +other particulars which are not members of this class. It is these +extrinsic modifications which represent the sort of facts that, in our +former account, appeared as the influence of the eyes and nerves in +modifying the appearance of the sun.[28] + +The _prima facie_ difficulties in the way of this view are chiefly +derived from an unduly conventional theory of space. It might seem at +first sight as if we had packed the world much fuller than it could +possibly hold. At every place between us and the sun, we said, there +is to be a particular which is to be a member of the sun as it was a +few minutes ago. There will also, of course, have to be a particular +which is a member of any planet or fixed star that may happen to be +visible from that place. At the place where I am, there will be +particulars which will be members severally of all the "things" I am +now said to be perceiving. Thus throughout the world, everywhere, +there will be an enormous number of particulars coexisting in the same +place. But these troubles result from contenting ourselves too readily +with the merely three-dimensional space to which schoolmasters have +accustomed us. The space of the real world is a space of six +dimensions, and as soon as we realise this we see that there is plenty +of room for all the particulars for which we want to find positions. +In order to realise this we have only to return for a moment from the +polished space of physics to the rough and untidy space of our +immediate sensible experience. The space of one man's sensible objects +is a three-dimensional space. It does not appear probable that two men +ever both perceive at the same time any one sensible object; when they +are said to see the same thing or hear the same noise, there will +always be some difference, however slight, between the actual shapes +seen or the actual sounds heard. If this is so, and if, as is +generally assumed, position in space is purely relative, it follows +that the space of one man's objects and the space of another man's +objects have no place in common, that they are in fact different +spaces, and not merely different parts of one space. I mean by this +that such immediate spatial relations as are perceived to hold +between the different parts of the sensible space perceived by one +man, do not hold between parts of sensible spaces perceived by +different men. There are therefore a multitude of three-dimensional +spaces in the world: there are all those perceived by observers, and +presumably also those which are not perceived, merely because no +observer is suitably situated for perceiving them. + +But although these spaces do not have to one another the same kind of +spatial relations as obtain between the parts of one of them, it is +nevertheless possible to arrange these spaces themselves in a +three-dimensional order. This is done by means of the correlated +particulars which we regard as members (or aspects) of one physical +thing. When a number of people are said to see the same object, those +who would be said to be near to the object see a particular occupying +a larger part of their field of vision than is occupied by the +corresponding particular seen by people who would be said to be +farther from the thing. By means of such considerations it is +possible, in ways which need not now be further specified, to arrange +all the different spaces in a three-dimensional series. Since each of +the spaces is itself three-dimensional, the whole world of particulars +is thus arranged in a six-dimensional space, that is to say, six +co-ordinates will be required to assign completely the position of any +given particular, namely, three to assign its position in its own +space and three more to assign the position of its space among the +other spaces. + +There are two ways of classifying particulars: we may take together +all those that belong to a given "perspective," or all those that are, +as common sense would say, different "aspects" of the same "thing." +For example, if I am (as is said) seeing the sun, what I see belongs +to two assemblages: (1) the assemblage of all my present objects of +sense, which is what I call a "perspective"; (2) the assemblage of +all the different particulars which would be called aspects of the sun +of eight minutes ago--this assemblage is what I define as _being_ the +sun of eight minutes ago. Thus "perspectives" and "things" are merely +two different ways of classifying particulars. It is to be observed +that there is no _a priori_ necessity for particulars to be +susceptible of this double classification. There may be what might be +called "wild" particulars, not having the usual relations by which the +classification is effected; perhaps dreams and hallucinations are +composed of particulars which are "wild" in this sense. + +The exact definition of what is meant by a perspective is not quite +easy. So long as we confine ourselves to visible objects or to objects +of touch we might define the perspective of a given particular as "all +particulars which have a simple (direct) spatial relation to the given +particular." Between two patches of colour which I see now, there is a +direct spatial relation which I equally see. But between patches of +colour seen by different men there is only an indirect constructed +spatial relation by means of the placing of "things" in physical space +(which is the same as the space composed of perspectives). Those +particulars which have direct spatial relations to a given particular +will belong to the same perspective. But if, for example, the sounds +which I hear are to belong to the same perspective with the patches of +colour which I see, there must be particulars which have no direct +spatial relation and yet belong to the same perspective. We cannot +define a perspective as all the data of one percipient at one time, +because we wish to allow the possibility of perspectives which are not +perceived by any one. There will be need, therefore, in defining a +perspective, of some principle derived neither from psychology nor +from space. + +Such a principle may be obtained from the consideration of _time_. +The one all-embracing time, like the one all-embracing space, is a +construction; there is no _direct_ time-relation between particulars +belonging to my perspective and particulars belonging to another +man's. On the other hand, any two particulars of which I am aware are +either simultaneous or successive, and their simultaneity or +successiveness is sometimes itself a datum to me. We may therefore +define the perspective to which a given particular belongs as "all +particulars simultaneous with the given particular," where +"simultaneous" is to be understood as a direct simple relation, not +the derivative constructed relation of physics. It may be observed +that the introduction of "local time" suggested by the principle of +relativity has effected, for purely scientific reasons, much the same +multiplication of times as we have just been advocating. + +The sum-total of all the particulars that are (directly) either +simultaneous with or before or after a given particular may be defined +as the "biography" to which that particular belongs. It will be +observed that, just as a perspective need not be actually perceived by +any one, so a biography need not be actually lived by any one. Those +biographies that are lived by no one are called "official." + +The definition of a "thing" is effected by means of continuity and of +correlations which have a certain differential independence of other +"things." That is to say, given a particular in one perspective, there +will usually in a neighbouring perspective be a very similar +particular, differing from the given particular, to the first order of +small quantities, according to a law involving only the difference of +position of the two perspectives in perspective space, and not any of +the other "things" in the universe. It is this continuity and +differential independence in the law of change as we pass from one +perspective to another that defines the class of particulars which is +to be called "one thing." + +Broadly speaking, we may say that the physicist finds it convenient to +classify particulars into "things," while the psychologist finds it +convenient to classify them into "perspectives" and "biographies," +since one perspective _may_ constitute the momentary data of one +percipient, and one biography _may_ constitute the whole of the data +of one percipient throughout his life. + +We may now sum up our discussion. Our object has been to discover as +far as possible the nature of the ultimate constituents of the +physical world. When I speak of the "physical world," I mean, to begin +with, the world dealt with by physics. It is obvious that physics is +an empirical science, giving us a certain amount of knowledge and +based upon evidence obtained through the senses. But partly through +the development of physics itself, partly through arguments derived +from physiology, psychology or metaphysics, it has come to be thought +that the immediate data of sense could not themselves form part of the +ultimate constituents of the physical world, but were in some sense +"mental," "in the mind," or "subjective." The grounds for this view, +in so far as they depend upon physics, can only be adequately dealt +with by rather elaborate constructions depending upon symbolic logic, +showing that out of such materials as are provided by the senses it is +possible to construct classes and series having the properties which +physics assigns to matter. Since this argument is difficult and +technical, I have not embarked upon it in this article. But in so far +as the view that sense-data are "mental" rests upon physiology, +psychology, or metaphysics, I have tried to show that it rests upon +confusions and prejudices--prejudices in favour of permanence in the +ultimate constituents of matter, and confusions derived from unduly +simple notions as to space, from the causal correlation of sense-data +with sense-organs, and from failure to distinguish between sense-data +and sensations. If what we have said on these subjects is valid, the +existence of sense-data is logically independent of the existence of +mind, and is causally dependent upon the _body_ of the percipient +rather than upon his mind. The causal dependence upon the body of the +percipient, we found, is a more complicated matter than it appears to +be, and, like all causal dependence, is apt to give rise to erroneous +beliefs through misconceptions as to the nature of causal correlation. +If we have been right in our contentions, sense-data are merely those +among the ultimate constituents of the physical world, of which we +happen to be immediately aware; they themselves are purely physical, +and all that is mental in connection with them is our awareness of +them, which is irrelevant to their nature and to their place in +physics. + +Unduly simple notions as to space have been a great stumbling-block to +realists. When two men look at the same table, it is supposed that +what the one sees and what the other sees are in the same place. Since +the shape and colour are not quite the same for the two men, this +raises a difficulty, hastily solved, or rather covered up, by +declaring what each sees to be purely "subjective"--though it would +puzzle those who use this glib word to say what they mean by it. The +truth seems to be that space--and time also--is much more complicated +than it would appear to be from the finished structure of physics, and +that the one all-embracing three-dimensional space is a logical +construction, obtained by means of correlations from a crude space of +six dimensions. The particulars occupying this six-dimensional space, +classified in one way, form "things," from which with certain further +manipulations we can obtain what physics can regard as matter; +classified in another way, they form "perspectives" and "biographies," +which may, if a suitable percipient happens to exist, form +respectively the sense-data of a momentary or of a total experience. +It is only when physical "things" have been dissected into series of +classes of particulars, as we have done, that the conflict between the +point of view of physics and the point of view of psychology can be +overcome. This conflict, if what has been said is not mistaken, flows +from different methods of classification, and vanishes as soon as its +source is discovered. + +In favour of the theory which I have briefly outlined, I do not claim +that it is _certainly_ true. Apart from the likelihood of mistakes, +much of it is avowedly hypothetical. What I do claim for the theory is +that it _may_ be true, and that this is more than can be said for any +other theory except the closely analogous theory of Leibniz. The +difficulties besetting realism, the confusions obstructing any +philosophical account of physics, the dilemma resulting from +discrediting sense-data, which yet remain the sole source of our +knowledge of the outer world--all these are avoided by the theory +which I advocate. This does not prove the theory to be true, since +probably many other theories might be invented which would have the +same merits. But it does prove that the theory has a better chance of +being true than any of its present competitors, and it suggests that +what can be known with certainty is likely to be discoverable by +taking our theory as a starting-point, and gradually freeing it from +all such assumptions as seem irrelevant, unnecessary, or unfounded. On +these grounds, I recommend it to attention as a hypothesis and a basis +for further work, though not as itself a finished or adequate solution +of the problem with which it deals. + +FOOTNOTES: + +[23] An address delivered to the Philosophical Society of Manchester +in February, 1915. Reprinted from _The Monist_, July, 1915. + +[24] Cf. especially Samuel Alexander, "The Basis of Realism," _British +Academy_, Vol. VI. + +[25] "Are Secondary Qualities Independent of Perception?" _Proc. +Arist. Soc._, 1909-10, pp. 191-218. + +[26] First dialogue between Hylas and Philonous, _Works_ (Fraser's +edition 1901). I. p. 384. + +[27] This point has been well urged by the American realists. + +[28] Cf. T.P. Nunn, "Are Secondary Qualities Independent of +Perception?" _Proc. Arist. Soc._, 1909-1910. + + + + +VIII + +THE RELATION OF SENSE-DATA TO PHYSICS + + +I. THE PROBLEM STATED + +Physics is said to be an empirical science, based upon observation and +experiment. + +It is supposed to be verifiable, i.e. capable of calculating +beforehand results subsequently confirmed by observation and +experiment. + +What can we learn by observation and experiment? + +Nothing, so far as physics is concerned, except immediate data of +sense: certain patches of colour, sounds, tastes, smells, etc., with +certain spatio-temporal relations. + +The supposed contents of the physical world are _prima facie_ very +different from these: molecules have no colour, atoms make no noise, +electrons have no taste, and corpuscles do not even smell. + +If such objects are to be verified, it must be solely through their +relation to sense-data: they must have some kind of correlation with +sense-data, and must be verifiable through their correlation _alone_. + +But how is the correlation itself ascertained? A correlation can only +be ascertained empirically by the correlated objects being constantly +_found_ together. But in our case, only one term of the correlation, +namely, the sensible term, is ever _found_: the other term seems +essentially incapable of being found. Therefore, it would seem, the +correlation with objects of sense, by which physics was to be +verified, is itself utterly and for ever unverifiable. + +There are two ways of avoiding this result. + +(1) We may say that we know some principle _a priori_, without the +need of empirical verification, e.g. that our sense-data have _causes_ +other than themselves, and that something can be known about these +causes by inference from their effects. This way has been often +adopted by philosophers. It may be necessary to adopt this way to some +extent, but in so far as it is adopted physics ceases to be empirical +or based upon experiment and observation alone. Therefore this way is +to be avoided as much as possible. + +(2) We may succeed in actually defining the objects of physics as +functions of sense-data. Just in so far as physics leads to +expectations, this _must_ be possible, since we can only _expect_ what +can be experienced. And in so far as the physical state of affairs is +inferred from sense-data, it must be capable of expression as a +function of sense-data. The problem of accomplishing this expression +leads to much interesting logico-mathematical work. + +In physics as commonly set forth, sense-data appear as functions of +physical objects: when such-and-such waves impinge upon the eye, we +see such-and-such colours, and so on. But the waves are in fact +inferred from the colours, not vice versa. Physics cannot be regarded +as validly based upon empirical data until the waves have been +expressed as functions of the colours and other sense-data. + +Thus if physics is to be verifiable we are faced with the following +problem: Physics exhibits sense-data as functions of physical objects, +but verification is only possible if physical objects can be exhibited +as functions of sense-data. We have therefore to solve the equations +giving sense-data in terms of physical objects, so as to make them +instead give physical objects in terms of sense-data. + + +II. CHARACTERISTICS OF SENSE-DATA + +When I speak of a "sense-datum," I do not mean the whole of what is +given in sense at one time. I mean rather such a part of the whole as +might be singled out by attention: particular patches of colour, +particular noises, and so on. There is some difficulty in deciding +what is to be considered _one_ sense-datum: often attention causes +divisions to appear where, so far as can be discovered, there were no +divisions before. An observed complex fact, such as that this patch of +red is to the left of that patch of blue, is also to be regarded as a +datum from our present point of view: epistemologically, it does not +differ greatly from a simple sense-datum as regards its function in +giving knowledge. Its _logical_ structure is very different, however, +from that of sense: _sense_ gives acquaintance with particulars, and +is thus a two-term relation in which the object can be _named_ but not +_asserted_, and is inherently incapable of truth or falsehood, whereas +the observation of a complex fact, which may be suitably called +perception, is not a two-term relation, but involves the propositional +form on the object-side, and gives knowledge of a truth, not mere +acquaintance with a particular. This logical difference, important as +it is, is not very relevant to our present problem; and it will be +convenient to regard data of perception as included among sense-data +for the purposes of this paper. It is to be observed that the +particulars which are constituents of a datum of perception are always +sense-data in the strict sense. + +Concerning sense-data, we know that they are there while they are +data, and this is the epistemological basis of all our knowledge of +external particulars. (The meaning of the word "external" of course +raises problems which will concern us later.) We do not know, except +by means of more or less precarious inferences, whether the objects +which are at one time sense-data continue to exist at times when they +are not data. Sense-data at the times when they are data are all that +we directly and primitively know of the external world; hence in +epistemology the fact that they are _data_ is all-important. But the +fact that they are all that we directly know gives, of course, no +presumption that they are all that there is. If we could construct an +impersonal metaphysic, independent of the accidents of our knowledge +and ignorance, the privileged position of the actual data would +probably disappear, and they would probably appear as a rather +haphazard selection from a mass of objects more or less like them. In +saying this, I assume only that it is probable that there are +particulars with which we are not acquainted. Thus the special +importance of sense-data is in relation to epistemology, not to +metaphysics. In this respect, physics is to be reckoned as +metaphysics: it is impersonal, and nominally pays no special attention +to sense-data. It is only when we ask how physics can be _known_ that +the importance of sense-data re-emerges. + + +III. SENSIBILIA + +I shall give the name _sensibilia_ to those objects which have the +same metaphysical and physical status as sense-data, without +necessarily being data to any mind. Thus the relation of a _sensibile_ +to a sense-datum is like that of a man to a husband: a man becomes a +husband by entering into the relation of marriage, and similarly a +_sensibile_ becomes a sense-datum by entering into the relation of +acquaintance. It is important to have both terms; for we wish to +discuss whether an object which is at one time a sense-datum can still +exist at a time when it is not a sense-datum. We cannot ask "Can +sense-data exist without being given?" for that is like asking "Can +husbands exist without being married?" We must ask "Can _sensibilia_ +exist without being given?" and also "Can a particular _sensibile_ be +at one time a sense-datum, and at another not?" Unless we have the +word _sensibile_ as well as the word "sense-datum," such questions are +apt to entangle us in trivial logical puzzles. + +It will be seen that all sense-data are _sensibilia_. It is a +metaphysical question whether all _sensibilia_ are sense-data, and an +epistemological question whether there exist means of inferring +_sensibilia_ which are not data from those that are. + +A few preliminary remarks, to be amplified as we proceed, will serve +to elucidate the use which I propose to make of _sensibilia_. + +I regard sense-data as not mental, and as being, in fact, part of the +actual subject-matter of physics. There are arguments, shortly to be +examined, for their subjectivity, but these arguments seem to me only +to prove _physiological_ subjectivity, i.e. causal dependence on the +sense-organs, nerves, and brain. The appearance which a thing presents +to us is causally dependent upon these, in exactly the same way as it +is dependent upon intervening fog or smoke or coloured glass. Both +dependences are contained in the statement that the appearance which a +piece of matter presents when viewed from a given place is a function +not only of the piece of matter, but also of the intervening medium. +(The terms used in this statement--"matter," "view from a given +place," "appearance," "intervening medium"--will all be defined in the +course of the present paper.) We have not the means of ascertaining +how things appear from places not surrounded by brain and nerves and +sense-organs, because we cannot leave the body; but continuity makes +it not unreasonable to suppose that they present _some_ appearance at +such places. Any such appearance would be included among _sensibilia_. +If--_per impossibile_--there were a complete human body with no mind +inside it, all those _sensibilia_ would exist, in relation to that +body, which would be sense-data if there were a mind in the body. What +the mind adds to _sensibilia_, in fact, is _merely_ awareness: +everything else is physical or physiological. + + +IV. SENSE-DATA ARE PHYSICAL + +Before discussing this question it will be well to define the sense in +which the terms "mental" and "physical" are to be used. The word +"physical," in all preliminary discussions, is to be understood as +meaning "what is dealt with by physics." Physics, it is plain, tells +us something about some of the constituents of the actual world; what +these constituents are may be doubtful, but it is they that are to be +called physical, whatever their nature may prove to be. + +The definition of the term "mental" is more difficult, and can only be +satisfactorily given after many difficult controversies have been +discussed and decided. For present purposes therefore I must content +myself with assuming a dogmatic answer to these controversies. I shall +call a particular "mental" when it is aware of something, and I shall +call a fact "mental" when it contains a mental particular as a +constituent. + +It will be seen that the mental and the physical are not necessarily +mutually exclusive, although I know of no reason to suppose that they +overlap. + +The doubt as to the correctness of our definition of the "mental" is +of little importance in our present discussion. For what I am +concerned to maintain is that sense-data are physical, and this being +granted it is a matter of indifference in our present inquiry whether +or not they are also mental. Although I do not hold, with Mach and +James and the "new realists," that the difference between the mental +and the physical is _merely_ one of arrangement, yet what I have to +say in the present paper is compatible with their doctrine and might +have been reached from their standpoint. + +In discussions on sense-data, two questions are commonly confused, +namely: + +(1) Do sensible objects persist when we are not sensible of them? in +other words, do _sensibilia_ which are data at a certain time +sometimes continue to exist at times when they are not data? And (2) +are sense-data mental or physical? + +I propose to assert that sense-data are physical, while yet +maintaining that they probably never persist unchanged after ceasing +to be data. The view that they do not persist is often thought, quite +erroneously in my opinion, to imply that they are mental; and this +has, I believe, been a potent source of confusion in regard to our +present problem. If there were, as some have held, a _logical +impossibility_ in sense-data persisting after ceasing to be data, that +certainly would tend to show that they were mental; but if, as I +contend, their non-persistence is merely a probable inference from +empirically ascertained causal laws, then it carries no such +implication with it, and we are quite free to treat them as part of +the subject-matter of physics. + +Logically a sense-datum is an object, a particular of which the +subject is aware. It does not contain the subject as a part, as for +example beliefs and volitions do. The existence of the sense-datum is +therefore not logically dependent upon that of the subject; for the +only way, so far as I know, in which the existence of _A_ can be +_logically_ dependent upon the existence of _B_ is when _B_ is part of +_A_. There is therefore no _a priori_ reason why a particular which is +a sense-datum should not persist after it has ceased to be a datum, +nor why other similar particulars should not exist without ever being +data. The view that sense-data are mental is derived, no doubt, in +part from their physiological subjectivity, but in part also from a +failure to distinguish between sense-data and "sensations." By a +sensation I mean the fact consisting in the subject's awareness of the +sense-datum. Thus a sensation is a complex of which the subject is a +constituent and which therefore is mental. The sense-datum, on the +other hand, stands over against the subject as that external object of +which in sensation the subject is aware. It is true that the +sense-datum is in many cases in the subject's body, but the subject's +body is as distinct from the subject as tables and chairs are, and is +in fact merely a part of the material world. So soon, therefore, as +sense-data are clearly distinguished from sensations, and as their +subjectivity is recognised to be physiological not psychical, the +chief obstacles in the way of regarding them as physical are removed. + + +V. "SENSIBILIA" AND "THINGS" + +But if "sensibilia" are to be recognised as the ultimate constituents +of the physical world, a long and difficult journey is to be performed +before we can arrive either at the "thing" of common sense or at the +"matter" of physics. The supposed impossibility of combining the +different sense-data which are regarded as appearances of the same +"thing" to different people has made it seem as though these +"sensibilia" must be regarded as mere subjective phantasms. A given +table will present to one man a rectangular appearance, while to +another it appears to have two acute angles and two obtuse angles; to +one man it appears brown, while to another, towards whom it reflects +the light, it appears white and shiny. It is said, not wholly without +plausibility, that these different shapes and different colours cannot +co-exist simultaneously in the same place, and cannot therefore both +be constituents of the physical world. This argument I must confess +appeared to me until recently to be irrefutable. The contrary opinion +has, however, been ably maintained by Dr. T.P. Nunn in an article +entitled: "Are Secondary Qualities Independent of Perception?"[29] The +supposed impossibility derives its apparent force from the phrase: +"_in the same place_," and it is precisely in this phrase that its +weakness lies. The conception of space is too often treated in +philosophy--even by those who on reflection would not defend such +treatment--as though it were as given, simple, and unambiguous as +Kant, in his psychological innocence, supposed. It is the unperceived +ambiguity of the word "place" which, as we shall shortly see, has +caused the difficulties to realists and given an undeserved advantage +to their opponents. Two "places" of different kinds are involved in +every sense-datum, namely the place _at_ which it appears and the +place _from_ which it appears. These belong to different spaces, +although, as we shall see, it is possible, with certain limitations, +to establish a correlation between them. What we call the different +appearances of the same thing to different observers are each in a +space private to the observer concerned. No place in the private world +of one observer is identical with a place in the private world of +another observer. There is therefore no question of combining the +different appearances in the one place; and the fact that they cannot +all exist in one place affords accordingly no ground whatever for +questioning their physical reality. The "thing" of common sense may in +fact be identified with the whole class of its appearances--where, +however, we must include among appearances not only those which are +actual sense-data, but also those "sensibilia," if any, which, on +grounds of continuity and resemblance, are to be regarded as belonging +to the same system of appearances, although there happen to be no +observers to whom they are data. + +An example may make this clearer. Suppose there are a number of people +in a room, all seeing, as they say, the same tables and chairs, walls +and pictures. No two of these people have exactly the same sense-data, +yet there is sufficient similarity among their data to enable them to +group together certain of these data as appearances of one "thing" to +the several spectators, and others as appearances of another "thing." +Besides the appearances which a given thing in the room presents to +the actual spectators, there are, we may suppose, other appearances +which it would present to other possible spectators. If a man were to +sit down between two others, the appearance which the room would +present to him would be intermediate between the appearances which it +presents to the two others: and although this appearance would not +exist as it is without the sense organs, nerves and brain, of the +newly arrived spectator, still it is not unnatural to suppose that, +from the position which he now occupies, _some_ appearance of the +room existed before his arrival. This supposition, however, need +merely be noticed and not insisted upon. + +Since the "thing" cannot, without indefensible partiality, be +identified with any single one of its appearances, it came to be +thought of as something distinct from all of them and underlying them. +But by the principle of Occam's razor, if the class of appearances +will fulfil the purposes for the sake of which the thing was invented +by the prehistoric metaphysicians to whom common sense is due, economy +demands that we should identify the thing with the class of its +appearances. It is not necessary to _deny_ a substance or substratum +underlying these appearances; it is merely expedient to abstain from +asserting this unnecessary entity. Our procedure here is precisely +analogous to that which has swept away from the philosophy of +mathematics the useless menagerie of metaphysical monsters with which +it used to be infested. + + +VI. CONSTRUCTIONS VERSUS INFERENCES + +Before proceeding to analyse and explain the ambiguities of the word +"place," a few general remarks on method are desirable. The supreme +maxim in scientific philosophising is this: + + _Wherever possible, logical constructions are to be substituted + for inferred entities._ + +Some examples of the substitution of construction for inference in the +realm of mathematical philosophy may serve to elucidate the uses of +this maxim. Take first the case of irrationals. In old days, +irrationals were inferred as the supposed limits of series of +rationals which had no rational limit; but the objection to this +procedure was that it left the existence of irrationals merely +optative, and for this reason the stricter methods of the present day +no longer tolerate such a definition. We now define an irrational +number as a certain class of ratios, thus constructing it logically by +means of ratios, instead of arriving at it by a doubtful inference +from them. Take again the case of cardinal numbers. Two equally +numerous collections appear to have something in common: this +something is supposed to be their cardinal number. But so long as the +cardinal number is inferred from the collections, not constructed in +terms of them, its existence must remain in doubt, unless in virtue of +a metaphysical postulate _ad hoc_. By defining the cardinal number of +a given collection as the class of all equally numerous collections, +we avoid the necessity of this metaphysical postulate, and thereby +remove a needless element of doubt from the philosophy of arithmetic. +A similar method, as I have shown elsewhere, can be applied to classes +themselves, which need not be supposed to have any metaphysical +reality, but can be regarded as symbolically constructed fictions. + +The method by which the construction proceeds is closely analogous in +these and all similar cases. Given a set of propositions nominally +dealing with the supposed inferred entities, we observe the properties +which are required of the supposed entities in order to make these +propositions true. By dint of a little logical ingenuity, we then +construct some logical function of less hypothetical entities which +has the requisite properties. This constructed function we substitute +for the supposed inferred entities, and thereby obtain a new and less +doubtful interpretation of the body of propositions in question. This +method, so fruitful in the philosophy of mathematics, will be found +equally applicable in the philosophy of physics, where, I do not +doubt, it would have been applied long ago but for the fact that all +who have studied this subject hitherto have been completely ignorant +of mathematical logic. I myself cannot claim originality in the +application of this method to physics, since I owe the suggestion and +the stimulus for its application entirely to my friend and +collaborator Dr. Whitehead, who is engaged in applying it to the more +mathematical portions of the region intermediate between sense-data +and the points, instants and particles of physics. + +A complete application of the method which substitutes constructions +for inferences would exhibit matter wholly in terms of sense-data, and +even, we may add, of the sense-data of a single person, since the +sense-data of others cannot be known without some element of +inference. This, however, must remain for the present an ideal, to be +approached as nearly as possible, but to be reached, if at all, only +after a long preliminary labour of which as yet we can only see the +very beginning. The inferences which are unavoidable can, however, be +subjected to certain guiding principles. In the first place they +should always be made perfectly explicit, and should be formulated in +the most general manner possible. In the second place the inferred +entities should, whenever this can be done, be similar to those whose +existence is given, rather than, like the Kantian _Ding an sich_, +something wholly remote from the data which nominally support the +inference. The inferred entities which I shall allow myself are of two +kinds: (_a_) the sense-data of other people, in favour of which there +is the evidence of testimony, resting ultimately upon the analogical +argument in favour of minds other than my own; (_b_) the "sensibilia" +which would appear from places where there happen to be no minds, and +which I suppose to be real although they are no one's data. Of these +two classes of inferred entities, the first will probably be allowed +to pass unchallenged. It would give me the greatest satisfaction to be +able to dispense with it, and thus establish physics upon a +solipsistic basis; but those--and I fear they are the majority--in +whom the human affections are stronger than the desire for logical +economy, will, no doubt, not share my desire to render solipsism +scientifically satisfactory. The second class of inferred entities +raises much more serious questions. It may be thought monstrous to +maintain that a thing can present any appearance at all in a place +where no sense organs and nervous structure exist through which it +could appear. I do not myself feel the monstrosity; nevertheless I +should regard these supposed appearances only in the light of a +hypothetical scaffolding, to be used while the edifice of physics is +being raised, though possibly capable of being removed as soon as the +edifice is completed. These "sensibilia" which are not data to anyone +are therefore to be taken rather as an illustrative hypothesis and as +an aid in preliminary statement than as a dogmatic part of the +philosophy of physics in its final form. + + +VII. PRIVATE SPACE AND THE SPACE OF PERSPECTIVES + +We have now to explain the ambiguity in the word "place," and how it +comes that two places of different sorts are associated with every +sense-datum, namely the place _at_ which it is and the place _from_ +which it is perceived. The theory to be advocated is closely analogous +to Leibniz's monadology, from which it differs chiefly in being less +smooth and tidy. + +The first fact to notice is that, so far as can be discovered, no +sensibile is ever a datum to two people at once. The things seen by +two different people are often closely similar, so similar that the +same _words_ can be used to denote them, without which communication +with others concerning sensible objects would be impossible. But, in +spite of this similarity, it would seem that some difference always +arises from difference in the point of view. Thus each person, so far +as his sense-data are concerned, lives in a private world. This +private world contains its own space, or rather spaces, for it would +seem that only experience teaches us to correlate the space of sight +with the space of touch and with the various other spaces of other +senses. This multiplicity of private spaces, however, though +interesting to the psychologist, is of no great importance in regard +to our present problem, since a merely solipsistic experience enables +us to correlate them into the one private space which embraces all our +own sense-data. The place _at_ which a sense-datum is, is a place in +private space. This place therefore is different from any place in the +private space of another percipient. For if we assume, as logical +economy demands, that all position is relative, a place is only +definable by the things in or around it, and therefore the same place +cannot occur in two private worlds which have no common constituent. +The question, therefore, of combining what we call different +appearances of the same thing in the same place does not arise, and +the fact that a given object appears to different spectators to have +different shapes and colours affords no argument against the physical +reality of all these shapes and colours. + +In addition to the private spaces belonging to the private worlds of +different percipients, there is, however, another space, in which one +whole private world counts as a point, or at least as a spatial unit. +This might be described as the space of points of view, since each +private world may be regarded as the appearance which the universe +presents from a certain point of view. I prefer, however, to speak of +it as the space of _perspectives_, in order to obviate the suggestion +that a private world is only real when someone views it. And for the +same reason, when I wish to speak of a private world without assuming +a percipient, I shall call it a "perspective." + +We have now to explain how the different perspectives are ordered in +one space. This is effected by means of the correlated "sensibilia" +which are regarded as the appearances, in different perspectives, of +one and the same thing. By moving, and by testimony, we discover that +two different perspectives, though they cannot both contain the same +"sensibilia," may nevertheless contain very similar ones; and the +spatial order of a certain group of "sensibilia" in a private space of +one perspective is found to be identical with, or very similar to, the +spatial order of the correlated "sensibilia" in the private space of +another perspective. In this way one "sensibile" in one perspective is +correlated with one "sensibile" in another. Such correlated +"sensibilia" will be called "appearances of one thing." In Leibniz's +monadology, since each monad mirrored the whole universe, there was in +each perspective a "sensibile" which was an appearance of each thing. +In our system of perspectives, we make no such assumption of +completeness. A given thing will have appearances in some +perspectives, but presumably not in certain others. The "thing" being +defined as the class of its appearances, if [kappa] is the class of +perspectives in which a certain thing [theta] appears, then [theta] is +a member of the multiplicative class of [kappa], [kappa] being a class +of mutually exclusive classes of "sensibilia." And similarly a +perspective is a member of the multiplicative class of the things +which appear in it. + +The arrangement of perspectives in a space is effected by means of the +differences between the appearances of a given thing in the various +perspectives. Suppose, say, that a certain penny appears in a number +of different perspectives; in some it looks larger and in some +smaller, in some it looks circular, in others it presents the +appearance of an ellipse of varying eccentricity. We may collect +together all those perspectives in which the appearance of the penny +is circular. These we will place on one straight line, ordering them +in a series by the variations in the apparent size of the penny. Those +perspectives in which the penny appears as a straight line of a +certain thickness will similarly be placed upon a plane (though in +this case there will be many different perspectives in which the penny +is of the same size; when one arrangement is completed these will form +a circle concentric with the penny), and ordered as before by the +apparent size of the penny. By such means, all those perspectives in +which the penny presents a visual appearance can be arranged in a +three-dimensional spatial order. Experience shows that the same +spatial order of perspectives would have resulted if, instead of the +penny, we had chosen any other thing which appeared in all the +perspectives in question, or any other method of utilising the +differences between the appearances of the same things in different +perspectives. It is this empirical fact which has made it possible to +construct the one all-embracing space of physics. + +The space whose construction has just been explained, and whose +elements are whole perspectives, will be called "perspective-space." + + +VIII. THE PLACING OF "THINGS" AND "SENSIBILIA" IN PERSPECTIVE SPACE + +The world which we have so far constructed is a world of six +dimensions, since it is a three-dimensional series of perspectives, +each of which is itself three-dimensional. We have now to explain the +correlation between the perspective space and the various private +spaces contained within the various perspectives severally. It is by +means of this correlation that the one three-dimensional space of +physics is constructed; and it is because of the unconscious +performance of this correlation that the distinction between +perspective space and the percipient's private space has been blurred, +with disastrous results for the philosophy of physics. Let us revert +to our penny: the perspectives in which the penny appears larger are +regarded as being nearer to the penny than those in which it appears +smaller, but as far as experience goes the apparent size of the penny +will not grow beyond a certain limit, namely, that where (as we say) +the penny is so near the eye that if it were any nearer it could not +be seen. By touch we may prolong the series until the penny touches +the eye, but no further. If we have been travelling along a line of +perspectives in the previously defined sense, we may, however, by +imagining the penny removed, prolong the line of perspectives by +means, say, of another penny; and the same may be done with any other +line of perspectives defined by means of the penny. All these lines +meet in a certain place, that is, in a certain perspective. This +perspective will be defined as "the place where the penny is." + +It is now evident in what sense two places in constructed physical +space are associated with a given "sensibile." There is first the +place which is the perspective of which the "sensibile" is a member. +This is the place _from_ which the "sensibile" appears. Secondly there +is the place where the thing is of which the "sensibile" is a member, +in other words an appearance; this is the place _at_ which the +"sensibile" appears. The "sensibile" which is a member of one +perspective is correlated with another perspective, namely, that which +is the place where the thing is of which the "sensibile" is an +appearance. To the psychologist the "place from which" is the more +interesting, and the "sensibile" accordingly appears to him subjective +and where the percipient is. To the physicist the "place at which" is +the more interesting, and the "sensibile" accordingly appears to him +physical and external. The causes, limits and partial justification of +each of these two apparently incompatible views are evident from the +above duplicity of places associated with a given "sensibile." + +We have seen that we can assign to a physical thing a place in the +perspective space. In this way different parts of our body acquire +positions in perspective space, and therefore there is a meaning +(whether true or false need not much concern us) in saying that the +perspective to which our sense-data belong is inside our head. Since +our mind is correlated with the perspective to which our sense-data +belong, we may regard this perspective as being the position of our +mind in perspective space. If, therefore, this perspective is, in the +above defined sense, inside our head, there is a good meaning for the +statement that the mind is in the head. We can now say of the various +appearances of a given thing that some of them are nearer to the thing +than others; those are nearer which belong to perspectives that are +nearer to "the place where the thing is." We can thus find a meaning, +true or false, for the statement that more is to be learnt about a +thing by examining it close to than by viewing it from a distance. We +can also find a meaning for the phrase "the things which intervene +between the subject and a thing of which an appearance is a datum to +him." One reason often alleged for the subjectivity of sense-data is +that the appearance of a thing may change when we find it hard to +suppose that the thing itself has changed--for example, when the +change is due to our shutting our eyes, or to our screwing them up so +as to make the thing look double. If the thing is defined as the class +of its appearances (which is the definition adopted above), there is +of course necessarily _some_ change in the thing whenever any one of +its appearances changes. Nevertheless there is a very important +distinction between two different ways in which the appearances may +change. If after looking at a thing I shut my eyes, the appearance of +my eyes changes in every perspective in which there is such an +appearance, whereas most of the appearances of the thing will remain +unchanged. We may say, as a matter of definition, that a thing changes +when, however near to the thing an appearance of it may be, there are +changes in appearances as near as, or still nearer to, the thing. On +the other hand we shall say that the change is in some other thing if +all appearances of the thing which are at not more than a certain +distance from the thing remain unchanged, while only comparatively +distant appearances of the thing are altered. From this consideration +we are naturally led to the consideration of _matter_, which must be +our next topic. + + +IX. THE DEFINITION OF MATTER + +We defined the "physical thing" as the class of its appearances, but +this can hardly be taken as a definition of matter. We want to be able +to express the fact that the appearance of a thing in a given +perspective is causally affected by the matter between the thing and +the perspective. We have found a meaning for "between a thing and a +perspective." But we want matter to be something other than the whole +class of appearances of a thing, in order to state the influence of +matter on appearances. + +We commonly assume that the information we get about a thing is more +accurate when the thing is nearer. Far off, we see it is a man; then +we see it is Jones; then we see he is smiling. Complete accuracy would +only be attainable as a limit: if the appearances of Jones as we +approach him tend towards a limit, that limit may be taken to be what +Jones really is. It is obvious that from the point of view of physics +the appearances of a thing close to "count" more than the appearances +far off. We may therefore set up the following tentative definition: + +The _matter_ of a given thing is the limit of its appearances as their +distance from the thing diminishes. + +It seems probable that there is something in this definition, but it +is not quite satisfactory, because empirically there is no such limit +to be obtained from sense-data. The definition will have to be eked +out by constructions and definitions. But probably it suggests the +right direction in which to look. + +We are now in a position to understand in outline the reverse journey +from matter to sense-data which is performed by physics. The +appearance of a thing in a given perspective is a function of the +matter composing the thing and of the intervening matter. The +appearance of a thing is altered by intervening smoke or mist, by blue +spectacles or by alterations in the sense-organs or nerves of the +percipient (which also must be reckoned as part of the intervening +medium). The nearer we approach to the thing, the less its appearance +is affected by the intervening matter. As we travel further and +further from the thing, its appearances diverge more and more from +their initial character; and the causal laws of their divergence are +to be stated in terms of the matter which lies between them and the +thing. Since the appearances at very small distances are less affected +by causes other than the thing itself, we come to think that the limit +towards which these appearances tend as the distance diminishes is +what the thing "really is," as opposed to what it merely seems to be. +This, together with its necessity for the statement of causal laws, +seems to be the source of the entirely erroneous feeling that matter +is more "real" than sense-data. + +Consider for example the infinite divisibility of matter. In looking +at a given thing and approaching it, one sense-datum will become +several, and each of these will again divide. Thus _one_ appearance +may represent _many_ things, and to this process there seems no end. +Hence in the limit, when we approach indefinitely near to the thing +there will be an indefinite number of units of matter corresponding to +what, at a finite distance, is only one appearance. This is how +infinite divisibility arises. + +The whole causal efficacy of a thing resides in its matter. This is in +some sense an empirical fact, but it would be hard to state it +precisely, because "causal efficacy" is difficult to define. + +What can be known empirically about the matter of a thing is only +approximate, because we cannot get to know the appearances of the +thing from very small distances, and cannot accurately infer the limit +of these appearances. But it _is_ inferred _approximately_ by means of +the appearances we can observe. It then turns out that these +appearances can be exhibited by physics as a function of the matter +in our immediate neighbourhood; e.g. the visual appearance of a +distant object is a function of the light-waves that reach the eyes. +This leads to confusions of thought, but offers no real difficulty. + +One appearance, of a visible object for example, is not sufficient to +determine its other simultaneous appearances, although it goes a +certain distance towards determining them. The determination of the +hidden structure of a thing, so far as it is possible at all, can only +be effected by means of elaborate dynamical inferences. + + +X. TIME[30] + +It seems that the one all-embracing time is a construction, like the +one all-embracing space. Physics itself has become conscious of this +fact through the discussions connected with relativity. + +Between two perspectives which both belong to one person's experience, +there will be a direct time-relation of before and after. This +suggests a way of dividing history in the same sort of way as it is +divided by different experiences, but without introducing experience +or anything mental: we may define a "biography" as everything that is +(directly) earlier or later than, or simultaneous with, a given +"sensibile." This will give a series of perspectives, which _might_ +all form parts of one person's experience, though it is not necessary +that all or any of them should actually do so. By this means, the +history of the world is divided into a number of mutually exclusive +biographies. + +We have now to correlate the times in the different biographies. The +natural thing would be to say that the appearances of a given +(momentary) thing in two different perspectives belonging to different +biographies are to be taken as simultaneous; but this is not +convenient. Suppose _A_ shouts to _B_, and _B_ replies as soon as he +hears _A's_ shout. Then between _A's_ hearing of his own shout and his +hearing of _B's_ there is an interval; thus if we made _A's_ and _B's_ +hearing of the same shout exactly simultaneous with each other, we +should have events exactly simultaneous with a given event but not +with each other. To obviate this, we assume a "velocity of sound." +That is, we assume that the time when _B_ hears _A's_ shout is +half-way between the time when _A_ hears his own shout and the time +when he hears _B's_. In this way the correlation is effected. + +What has been said about sound applies of course equally to light. The +general principle is that the appearances, in different perspectives, +which are to be grouped together as constituting what a certain thing +is at a certain moment, are not to be all regarded as being at that +moment. On the contrary they spread outward from the thing with +various velocities according to the nature of the appearances. Since +no _direct_ means exist of correlating the time in one biography with +the time in another, this temporal grouping of the appearances +belonging to a given thing at a given moment is in part conventional. +Its motive is partly to secure the verification of such maxims as that +events which are exactly simultaneous with the same event are exactly +simultaneous with one another, partly to secure convenience in the +formulation of causal laws. + + +XI. THE PERSISTENCE OF THINGS AND MATTER + +Apart from any of the fluctuating hypotheses of physics, three main +problems arise in connecting the world of physics with the world of +sense, namely: + + 1. the construction of a single space; + 2. the construction of a single time; + 3. the construction of permanent things or matter. + +We have already considered the first and second of these problems; it +remains to consider the third. + +We have seen how correlated appearances in different perspectives are +combined to form one "thing" at one moment in the all-embracing time +of physics. We have now to consider how appearances at different times +are combined as belonging to one "thing," and how we arrive at the +persistent "matter" of physics. The assumption of permanent substance, +which technically underlies the procedure of physics, cannot of course +be regarded as metaphysically legitimate: just as the one thing +simultaneously seen by many people is a construction, so the one thing +seen at different times by the same or different people must be a +construction, being in fact nothing but a certain grouping of certain +"sensibilia." + +We have seen that the momentary state of a "thing" is an assemblage of +"sensibilia," in different perspectives, not all simultaneous in the +one constructed time, but spreading out from "the place where the +thing is" with velocities depending upon the nature of the +"sensibilia." The time _at_ which the "thing" is in this state is the +lower limit of the times at which these appearances occur. We have now +to consider what leads us to speak of another set of appearances as +belonging to the same "thing" at a different time. + +For this purpose, we may, at least to begin with, confine ourselves +within a single biography. If we can always say when two "sensibilia" +in a given biography are appearances of one thing, then, since we have +seen how to connect "sensibilia" in different biographies as +appearances of the same momentary state of a thing, we shall have all +that is necessary for the complete construction of the history of a +thing. + +It is to be observed, to begin with, that the identity of a thing for +common sense is not always correlated with the identity of matter for +physics. A human body is one persisting thing for common sense, but +for physics its matter is constantly changing. We may say, broadly, +that the common-sense conception is based upon continuity in +appearances at the ordinary distances of sense-data, while the +physical conception is based upon the continuity of appearances at +very small distances from the thing. It is probable that the +common-sense conception is not capable of complete precision. Let us +therefore concentrate our attention upon the conception of the +persistence of matter in physics. + +The first characteristic of two appearances of the same piece of +matter at different times is _continuity_. The two appearances must be +connected by a series of intermediaries, which, if time and space form +compact series, must themselves form a compact series. The colour of +the leaves is different in autumn from what it is in summer; but we +believe that the change occurs gradually, and that, if the colours are +different at two given times, there are intermediate times at which +the colours are intermediate between those at the given times. + +But there are two considerations that are important as regards +continuity. + +First, it is largely hypothetical. We do not observe any one thing +continuously, and it is merely a hypothesis to assume that, while we +are not observing it, it passes through conditions intermediate +between those in which it is perceived. During uninterrupted +observation, it is true, continuity is nearly verified; but even here, +when motions are very rapid, as in the case of explosions, the +continuity is not actually capable of direct verification. Thus we can +only say that the sense-data are found to _permit_ a hypothetical +complement of "sensibilia" such as will preserve continuity, and that +therefore there _may_ be such a complement. Since, however, we have +already made such use of hypothetical "sensibilia," we will let this +point pass, and admit such "sensibilia" as are required to preserve +continuity. + +Secondly, continuity is not a sufficient criterion of material +identity. It is true that in many cases, such as rocks, mountains, +tables, chairs, etc., where the appearances change slowly, continuity +is sufficient, but in other cases, such as the parts of an +approximately homogeneous fluid, it fails us utterly. We can travel by +sensibly continuous gradations from any one drop of the sea at any one +time to any other drop at any other time. We infer the motions of +sea-water from the effects of the current, but they cannot be inferred +from direct sensible observation together with the assumption of +continuity. + +The characteristic required in addition to continuity is conformity +with the laws of dynamics. Starting from what common sense regards as +persistent things, and making only such modifications as from time to +time seem reasonable, we arrive at assemblages of "sensibilia" which +are found to obey certain simple laws, namely those of dynamics. By +regarding "sensibilia" at different times as belonging to the same +piece of matter, we are able to define _motion_, which presupposes the +assumption or construction of something persisting throughout the +time of the motion. The motions which are regarded as occurring, +during a period in which all the "sensibilia" and the times of their +appearance are given, will be different according to the manner in +which we combine "sensibilia" at different times as belonging to the +same piece of matter. Thus even when the whole history of the world is +given in every particular, the question what motions take place is +still to a certain extent arbitrary even after the assumption of +continuity. Experience shows that it is possible to determine motions +in such a way as to satisfy the laws of dynamics, and that this +determination, roughly and on the whole, is fairly in agreement with +the common-sense opinions about persistent things. This determination, +therefore, is adopted, and leads to a criterion by which we can +determine, sometimes practically, sometimes only theoretically, +whether two appearances at different times are to be regarded as +belonging to the same piece of matter. The persistence of all matter +throughout all time can, I imagine, be secured by definition. + +To recommend this conclusion, we must consider what it is that is +proved by the empirical success of physics. What is proved is that its +hypotheses, though unverifiable where they go beyond sense-data, are +at no point in contradiction with sense-data, but, on the contrary, +are ideally such as to render all sense-data calculable when a +sufficient collection of "sensibilia" is given. Now physics has found +it empirically possible to collect sense-data into series, each series +being regarded as belonging to one "thing," and behaving, with regard +to the laws of physics, in a way in which series not belonging to one +thing would in general not behave. If it is to be unambiguous whether +two appearances belong to the same thing or not, there must be only +one way of grouping appearances so that the resulting things obey the +laws of physics. It would be very difficult to prove that this is the +case, but for our present purposes we may let this point pass, and +assume that there is only one way. Thus we may lay down the following +definition: _Physical things are those series of appearances whose +matter obeys the laws of physics_. That such series exist is an +empirical fact, which constitutes the verifiability of physics. + + +XII. ILLUSIONS, HALLUCINATIONS, AND DREAMS + +It remains to ask how, in our system, we are to find a place for +sense-data which apparently fail to have the usual connection with the +world of physics. Such sense-data are of various kinds, requiring +somewhat different treatment. But all are of the sort that would be +called "unreal," and therefore, before embarking upon the discussion, +certain logical remarks must be made upon the conceptions of reality +and unreality. + +Mr. A. Wolf[31] says: + + "The conception of mind as a system of transparent activities is, + I think, also untenable because of its failure to account for the + very possibility of dreams and hallucinations. It seems impossible + to realise how a bare, transparent activity can be directed to + what is not there, to apprehend what is not given." + +This statement is one which, probably, most people would endorse. But +it is open to two objections. First it is difficult to see how an +activity, however un-"transparent," can be directed towards a nothing: +a term of a relation cannot be a mere nonentity. Secondly, no reason +is given, and I am convinced that none can be given, for the assertion +that dream-objects are not "there" and not "given." Let us take the +second point first. + +(1) The belief that dream-objects are not given comes, I think, from +failure to distinguish, as regards waking life, between the +sense-datum and the corresponding "thing." In dreams, there is no such +corresponding "thing" as the dreamer supposes; if, therefore, the +"thing" were given in waking life, as e.g. Meinong maintains,[32] then +there would be a difference in respect of givenness between dreams and +waking life. But if, as we have maintained, what is given is never the +thing, but merely one of the "sensibilia" which compose the thing, +then what we apprehend in a dream is just as much given as what we +apprehend in waking life. + +Exactly the same argument applies as to the dream-objects being +"there." They have their position in the private space of the +perspective of the dreamer; where they fail is in their correlation +with other private spaces and therefore with perspective space. But in +the only sense in which "there" can be a datum, they are "there" just +as truly as any of the sense-data of waking life. + +(2) The conception of "illusion" or "unreality," and the correlative +conception of "reality," are generally used in a way which embodies +profound logical confusions. Words that go in pairs, such as "real" +and "unreal," "existent" and "non-existent," "valid" and "invalid," +etc., are all derived from the one fundamental pair, "true" and +"false." Now "true" and "false" are applicable only--except in +derivative significations--to _propositions_. Thus wherever the above +pairs can be significantly applied, we must be dealing either with +propositions or with such incomplete phrases as only acquire meaning +when put into a context which, with them, forms a proposition. Thus +such pairs of words can be applied to _descriptions_,[33] but not to +proper names: in other words, they have no application whatever to +data, but only to entities or non-entities described in terms of data. + +Let us illustrate by the terms "existence" and "non-existence." Given +any datum _x_, it is meaningless either to assert or to deny that _x_ +"exists." We might be tempted to say: "Of course _x_ exists, for +otherwise it could not be a datum." But such a statement is really +meaningless, although it is significant and true to say "My present +sense-datum exists," and it may also be true that "_x_ is my present +sense-datum." The inference from these two propositions to "_x_ +exists" is one which seems irresistible to people unaccustomed to +logic; yet the apparent proposition inferred is not merely false, but +strictly meaningless. To say "My present sense-datum exists" is to say +(roughly): "There is an object of which 'my present sense-datum' is a +description." But we cannot say: "There is an object of which '_x_' is +a description," because '_x_' is (in the case we are supposing) a +name, not a description. Dr. Whitehead and I have explained this point +fully elsewhere (_loc. cit._) with the help of symbols, without which +it is hard to understand; I shall not therefore here repeat the +demonstration of the above propositions, but shall proceed with their +application to our present problem. + +The fact that "existence" is only applicable to descriptions is +concealed by the use of what are grammatically proper names in a way +which really transforms them into descriptions. It is, for example, a +legitimate question whether Homer existed; but here "Homer" means +"the author of the Homeric poems," and is a description. Similarly we +may ask whether God exists; but then "God" means "the Supreme Being" +or "the _ens realissimum_" or whatever other description we may +prefer. If "God" were a proper name, God would have to be a datum; and +then no question could arise as to His existence. The distinction +between existence and other predicates, which Kant obscurely felt, is +brought to light by the theory of descriptions, and is seen to remove +"existence" altogether from the fundamental notions of metaphysics. + +What has been said about "existence" applies equally to "reality," +which may, in fact, be taken as synonymous with "existence." +Concerning the immediate objects in illusions, hallucinations, and +dreams, it is meaningless to ask whether they "exist" or are "real." +There they are, and that ends the matter. But we may legitimately +inquire as to the existence or reality of "things" or other +"sensibilia" inferred from such objects. It is the unreality of these +"things" and other "sensibilia," together with a failure to notice +that they are not data, which has led to the view that the objects of +dreams are unreal. + +We may now apply these considerations in detail to the stock arguments +against realism, though what is to be said will be mainly a repetition +of what others have said before. + +(1) We have first the variety of normal appearances, supposed to be +incompatible. This is the case of the different shapes and colours +which a given thing presents to different spectators. Locke's water +which seems both hot and cold belongs to this class of cases. Our +system of different perspectives fully accounts for these cases, and +shows that they afford no argument against realism. + +(2) We have cases where the correlation between different senses is +unusual. The bent stick in water belongs here. People say it looks +bent but is straight: this only means that it is straight to the +touch, though bent to sight. There is no "illusion," but only a false +inference, if we think that the stick would feel bent to the touch. +The stick would look just as bent in a photograph, and, as Mr. +Gladstone used to say, "the photograph cannot lie."[34] The case of +seeing double also belongs here, though in this case the cause of the +unusual correlation is physiological, and would therefore not operate +in a photograph. It is a mistake to ask whether the "thing" is +duplicated when we see it double. The "thing" is a whole system of +"sensibilia," and it is only those visual "sensibilia" which are data +to the percipient that are duplicated. The phenomenon has a purely +physiological explanation; indeed, in view of our having two eyes, it +is in less need of explanation than the single visual sense-datum +which we normally obtain from the things on which we focus. + +(3) We come now to cases like dreams, which may, at the moment of +dreaming, contain nothing to arouse suspicion, but are condemned on the +ground of their supposed incompatibility with earlier and later data. Of +course it often happens that dream-objects fail to behave in the +accustomed manner: heavy objects fly, solid objects melt, babies turn +into pigs or undergo even greater changes. But none of these unusual +occurrences _need_ happen in a dream, and it is not on account of such +occurrences that dream-objects are called "unreal." It is their lack of +continuity with the dreamer's past and future that makes him, when he +wakes, condemn them; and it is their lack of correlation with other +private worlds that makes others condemn them. Omitting the latter +ground, our reason for condemning them is that the "things" which we +infer from them cannot be combined according to the laws of physics with +the "things" inferred from waking sense-data. This might be used to +condemn the "things" inferred from the data of dreams. Dream-data are no +doubt appearances of "things," but not of such "things" as the dreamer +supposes. I have no wish to combat psychological theories of dreams, +such as those of the psycho-analysts. But there certainly are cases +where (whatever psychological causes may contribute) the presence of +physical causes also is very evident. For instance, a door banging may +produce a dream of a naval engagement, with images of battleships and +sea and smoke. The whole dream will be an appearance of the door +banging, but owing to the peculiar condition of the body (especially the +brain) during sleep, this appearance is not that expected to be produced +by a door banging, and thus the dreamer is led to entertain false +beliefs. But his sense-data are still physical, and are such as a +completed physics would include and calculate. + +(4) The last class of illusions are those which cannot be discovered +within one person's experience, except through the discovery of +discrepancies with the experiences of others. Dreams might conceivably +belong to this class, if they were jointed sufficiently neatly into +waking life; but the chief instances are recurrent sensory +hallucinations of the kind that lead to insanity. What makes the +patient, in such cases, become what others call insane is the fact +that, within his own experience, there is nothing to show that the +hallucinatory sense-data do not have the usual kind of connection with +"sensibilia" in other perspectives. Of course he may learn this +through testimony, but he probably finds it simpler to suppose that +the testimony is untrue and that he is being wilfully deceived. There +is, so far as I can see, no theoretical criterion by which the patient +can decide, in such a case, between the two equally satisfactory +hypotheses of his madness and of his friends' mendacity. + +From the above instances it would appear that abnormal sense-data, of +the kind which we regard as deceptive, have intrinsically just the +same status as any others, but differ as regards their correlations or +causal connections with other "sensibilia" and with "things." Since +the usual correlations and connections become part of our unreflective +expectations, and even seem, except to the psychologist, to form part +of our data, it comes to be thought, mistakenly, that in such cases +the data are unreal, whereas they are merely the causes of false +inferences. The fact that correlations and connections of unusual +kinds occur adds to the difficulty of inferring things from sense and +of expressing physics in terms of sense-data. But the unusualness +would seem to be always physically or physiologically explicable, and +therefore raises only a complication, not a philosophical objection. + +I conclude, therefore, that no valid objection exists to the view +which regards sense-data as part of the actual substance of the +physical world, and that, on the other hand, this view is the only one +which accounts for the empirical verifiability of physics. In the +present paper, I have given only a rough preliminary sketch. In +particular, the part played by _time_ in the construction of the +physical world is, I think, more fundamental than would appear from +the above account. I should hope that, with further elaboration, the +part played by unperceived "sensibilia" could be indefinitely +diminished, probably by invoking the history of a "thing" to eke out +the inferences derivable from its momentary appearance. + +FOOTNOTES: + +[29] _Proc. Arist. Soc._, 1909-1910, pp. 191-218. + +[30] On this subject, compare _A Theory of Time and Space_, by Mr. +A.A. Robb (Camb. Univ. Press), which first suggested to me the views +advocated here, though I have, for present purposes, omitted what is +most interesting and novel in his theory. Mr. Robb has given a sketch +of his theory in a pamphlet with the same title (Heffer and Sons, +Cambridge, 1913). + +[31] "Natural Realism and Present Tendencies in Philosophy," _Proc. +Arist. Soc._, 1908-1909, p. 165. + +[32] _Die Erfahrungsgrundlagen unseres Wissens_, p. 28. + +[33] Cf. _Principia Mathematica_, Vol. I, * 14, and Introduction, +Chap. III. For the definition of _existence_, cf. * 14. 02. + +[34] Cf. Edwin B. Holt, _The Place of Illusory Experience in a +Realistic World._ "The New Realism," p. 303, both on this point and as +regards _seeing double_. + + + + +IX + +ON THE NOTION OF CAUSE + + +In the following paper I wish, first, to maintain that the word +"cause" is so inextricably bound up with misleading associations as to +make its complete extrusion from the philosophical vocabulary +desirable; secondly, to inquire what principle, if any, is employed in +science in place of the supposed "law of causality" which philosophers +imagine to be employed; thirdly, to exhibit certain confusions, +especially in regard to teleology and determinism, which appear to me +to be connected with erroneous notions as to causality. + +All philosophers, of every school, imagine that causation is one of +the fundamental axioms or postulates of science, yet, oddly enough, in +advanced sciences such as gravitational astronomy, the word "cause" +never occurs. Dr. James Ward, in his _Naturalism and Agnosticism_, +makes this a ground of complaint against physics: the business of +those who wish to ascertain the ultimate truth about the world, he +apparently thinks, should be the discovery of causes, yet physics +never even seeks them. To me it seems that philosophy ought not to +assume such legislative functions, and that the reason why physics has +ceased to look for causes is that, in fact, there are no such things. +The law of causality, I believe, like much that passes muster among +philosophers, is a relic of a bygone age, surviving, like the +monarchy, only because it is erroneously supposed to do no harm. In +order to find out what philosophers commonly understand by "cause," I +consulted Baldwin's _Dictionary_, and was rewarded beyond my +expectations, for I found the following three mutually incompatible +definitions:-- + + "CAUSALITY. (1) The necessary connection of events in the + time-series.... + + "CAUSE (notion of). Whatever may be included in the thought or + perception of a process as taking place in consequence of + another process.... + + "CAUSE AND EFFECT. (1) Cause and effect ... are correlative terms + denoting any two distinguishable things, phases, or aspects of + reality, which are so related to each other that whenever the + first ceases to exist the second comes into existence + immediately after, and whenever the second comes into existence + the first has ceased to exist immediately before." + +Let us consider these three definitions in turn. The first, obviously, +is unintelligible without a definition of "necessary." Under this +head, Baldwin's _Dictionary_ gives the following:-- + + "NECESSARY. That is necessary which not only is true, but would + be true under all circumstances. Something more than brute + compulsion is, therefore, involved in the conception; there is + a general law under which the thing takes place." + +The notion of cause is so intimately connected with that of necessity +that it will be no digression to linger over the above definition, +with a view to discovering, if possible, _some_ meaning of which it is +capable; for, as it stands, it is very far from having any definite +signification. + +The first point to notice is that, if any meaning is to be given to +the phrase "would be true under all circumstances," the subject of it +must be a propositional function, not a proposition.[35] A +proposition is simply true or false, and that ends the matter: there +can be no question of "circumstances." "Charles I's head was cut off" +is just as true in summer as in winter, on Sundays as on Mondays. Thus +when it is worth saying that something "would be true under all +circumstances," the something in question must be a propositional +function, i.e. an expression containing a variable, and becoming a +proposition when a value is assigned to the variable; the varying +"circumstances" alluded to are then the different values of which the +variable is capable. Thus if "necessary" means "what is true under all +circumstances," then "if _x_ is a man, _x_ is mortal" is necessary, +because it is true for any possible value of _x_. Thus we should be +led to the following definition:-- + + "NECESSARY is a predicate of a propositional function, meaning + that it is true for all possible values of its argument or + arguments." + +Unfortunately, however, the definition in Baldwin's _Dictionary_ says +that what is necessary is not only "true under all circumstances" but +is also "true." Now these two are incompatible. Only propositions can +be "true," and only propositional functions can be "true under all +circumstances." Hence the definition as it stands is nonsense. What is +meant seems to be this: "A proposition is necessary when it is a value +of a propositional function which is true under all circumstances, +i.e. for all values of its argument or arguments." But if we adopt +this definition, the same proposition will be necessary or contingent +according as we choose one or other of its terms as the argument to +our propositional function. For example, "if Socrates is a man, +Socrates is mortal," is necessary if Socrates is chosen as argument, +but not if _man_ or _mortal_ is chosen. Again, "if Socrates is a man, +Plato is mortal," will be necessary if either Socrates or _man_ is +chosen as argument, but not if Plato or _mortal_ is chosen. However, +this difficulty can be overcome by specifying the constituent which is +to be regarded as argument, and we thus arrive at the following +definition: + +"A proposition is _necessary_ with respect to a given constituent if +it remains true when that constituent is altered in any way compatible +with the proposition remaining significant." + +We may now apply this definition to the definition of causality quoted +above. It is obvious that the argument must be the time at which the +earlier event occurs. Thus an instance of causality will be such as: +"If the event [Math: e_{1}] occurs at the time [Math: t_{1}], it will +be followed by the event [Math: e_{2}]." This proposition is intended +to be necessary with respect to [Math: t_{1}], i.e. to remain true +however [Math: t_{1}] may be varied. Causality, as a universal law, +will then be the following: "Given any event [Math: t_{1}], there is +an event [Math: e_{2}] such that, whenever [Math: t_{1}] occurs, +[Math: e_{2}] occurs later." But before this can be considered +precise, we must specify how much later [Math: e_{2}] is to occur. +Thus the principle becomes:-- + +"Given any event [Math: e_{1}], there is an event [Math: e_{2}] and a +time-interval [tau] such that, whenever [Math: e_{1}] occurs, [Math: +e_{2}] follows after an interval [tau]." + +I am not concerned as yet to consider whether this law is true or +false. For the present, I am merely concerned to discover what the law +of causality is supposed to be. I pass, therefore, to the other +definitions quoted above. + +The second definition need not detain us long, for two reasons. First, +because it is psychological: not the "thought or perception" of a +process, but the process itself, must be what concerns us in +considering causality. Secondly, because it is circular: in speaking +of a process as "taking place in consequence of" another process, it +introduces the very notion of cause which was to be defined. + +The third definition is by far the most precise; indeed as regards +clearness it leaves nothing to be desired. But a great difficulty is +caused by the temporal contiguity of cause and effect which the +definition asserts. No two instants are contiguous, since the +time-series is compact; hence either the cause or the effect or both +must, if the definition is correct, endure for a finite time; indeed, +by the wording of the definition it is plain that both are assumed to +endure for a finite time. But then we are faced with a dilemma: if the +cause is a process involving change within itself, we shall require +(if causality is universal) causal relations between its earlier and +later parts; moreover, it would seem that only the later parts can be +relevant to the effect, since the earlier parts are not contiguous to +the effect, and therefore (by the definition) cannot influence the +effect. Thus we shall be led to diminish the duration of the cause +without limit, and however much we may diminish it, there will still +remain an earlier part which might be altered without altering the +effect, so that the true cause, as defined, will not have been +reached, for it will be observed that the definition excludes +plurality of causes. If, on the other hand, the cause is purely +static, involving no change within itself, then, in the first place, +no such cause is to be found in nature, and in the second place, it +seems strange--too strange to be accepted, in spite of bare logical +possibility--that the cause, after existing placidly for some time, +should suddenly explode into the effect, when it might just as well +have done so at any earlier time, or have gone on unchanged without +producing its effect. This dilemma, therefore, is fatal to the view +that cause and effect can be contiguous in time; if there are causes +and effects, they must be separated by a finite time-interval [tau], +as was assumed in the above interpretation of the first definition. + +What is essentially the same statement of the law of causality as the +one elicited above from the first of Baldwin's definitions is given by +other philosophers. Thus John Stuart Mill says:-- + +"The Law of Causation, the recognition of which is the main pillar of +inductive science, is but the familiar truth, that invariability of +succession is found by observation to obtain between every fact in +nature and some other fact which has preceded it."[36] + +And Bergson, who has rightly perceived that the law as stated by +philosophers is worthless, nevertheless continues to suppose that it +is used in science. Thus he says:-- + +"Now, it is argued, this law [the law of causality] means that every +phenomenon is determined by its conditions, or, in other words, that +the same causes produce the same effects."[37] + +And again:-- + +"We perceive physical phenomena, and these phenomena obey laws. This +means: (1) That phenomena _a_, _b_, _c_, _d_, previously perceived, +can occur again in the same shape; (2) that a certain phenomenon P, +which appeared after the conditions _a_, _b_, _c_, _d_, and after +these conditions only, will not fail to recur as soon as the same +conditions are again present."[38] + +A great part of Bergson's attack on science rests on the assumption +that it employs this principle. In fact, it employs no such principle, +but philosophers--even Bergson--are too apt to take their views on +science from each other, not from science. As to what the principle +is, there is a fair consensus among philosophers of different schools. +There are, however, a number of difficulties which at once arise. I +omit the question of plurality of causes for the present, since other +graver questions have to be considered. Two of these, which are forced +on our attention by the above statement of the law, are the +following:-- + + (1) What is meant by an "event"? + + (2) How long may the time-interval be between cause and effect? + +(1) An "event," in the statement of the law, is obviously intended to +be something that is likely to recur since otherwise the law becomes +trivial. It follows that an "event" is not a particular, but some +universal of which there may be many instances. It follows also that +an "event" must be something short of the whole state of the universe, +since it is highly improbable that this will recur. What is meant by +an "event" is something like striking a match, or dropping a penny +into the slot of an automatic machine. If such an event is to recur, +it must not be defined too narrowly: we must not state with what +degree of force the match is to be struck, nor what is to be the +temperature of the penny. For if such considerations were relevant, +our "event" would occur at most once, and the law would cease to give +information. An "event," then, is a universal defined sufficiently +widely to admit of many particular occurrences in time being instances +of it. + +(2) The next question concerns the time-interval. Philosophers, no +doubt, think of cause and effect as contiguous in time, but this, for +reasons already given, is impossible. Hence, since there are no +infinitesimal time-intervals, there must be some finite lapse of time +[tau] between cause and effect. This, however, at once raises +insuperable difficulties. However short we make the interval [tau], +something may happen during this interval which prevents the expected +result. I put my penny in the slot, but before I can draw out my +ticket there is an earthquake which upsets the machine and my +calculations. In order to be sure of the expected effect, we must know +that there is nothing in the environment to interfere with it. But +this means that the supposed cause is not, by itself, adequate to +insure the effect. And as soon as we include the environment, the +probability of repetition is diminished, until at last, when the whole +environment is included, the probability of repetition becomes almost +_nil_. + +In spite of these difficulties, it must, of course, be admitted that +many fairly dependable regularities of sequence occur in daily life. +It is these regularities that have suggested the supposed law of +causality; where they are found to fail, it is thought that a better +formulation could have been found which would have never failed. I am +far from denying that there may be such sequences which in fact never +do fail. It may be that there will never be an exception to the rule +that when a stone of more than a certain mass, moving with more than a +certain velocity, comes in contact with a pane of glass of less than +a certain thickness, the glass breaks. I also do not deny that the +observation of such regularities, even when they are not without +exceptions, is useful in the infancy of a science: the observation +that unsupported bodies in air usually fall was a stage on the way to +the law of gravitation. What I deny is that science assumes the +existence of invariable uniformities of sequence of this kind, or that +it aims at discovering them. All such uniformities, as we saw, depend +upon a certain vagueness in the definition of the "events." That +bodies fall is a vague qualitative statement; science wishes to know +how fast they fall. This depends upon the shape of the bodies and the +density of the air. It is true that there is more nearly uniformity +when they fall in a vacuum; so far as Galileo could observe, the +uniformity is then complete. But later it appeared that even there the +latitude made a difference, and the altitude. Theoretically, the +position of the sun and moon must make a difference. In short, every +advance in a science takes us farther away from the crude uniformities +which are first observed, into greater differentiation of antecedent +and consequent, and into a continually wider circle of antecedents +recognised as relevant. + +The principle "same cause, same effect," which philosophers imagine to +be vital to science, is therefore utterly otiose. As soon as the +antecedents have been given sufficiently fully to enable the +consequent to be calculated with some exactitude, the antecedents have +become so complicated that it is very unlikely they will ever recur. +Hence, if this were the principle involved, science would remain +utterly sterile. + +The importance of these considerations lies partly in the fact that +they lead to a more correct account of scientific procedure, partly in +the fact that they remove the analogy with human volition which makes +the conception of cause such a fruitful source of fallacies. The +latter point will become clearer by the help of some illustrations. +For this purpose I shall consider a few maxims which have played a +great part in the history of philosophy. + +(1) "Cause and effect must more or less resemble each other." This +principle was prominent in the philosophy of occasionalism, and is +still by no means extinct. It is still often thought, for example, +that mind could not have grown up in a universe which previously +contained nothing mental, and one ground for this belief is that +matter is too dissimilar from mind to have been able to cause it. Or, +more particularly, what are termed the nobler parts of our nature are +supposed to be inexplicable, unless the universe always contained +something at least equally noble which could cause them. All such +views seem to depend upon assuming some unduly simplified law of +causality; for, in any legitimate sense of "cause" and "effect," +science seems to show that they are usually very widely dissimilar, +the "cause" being, in fact, two states of the whole universe, and the +"effect" some particular event. + +(2) "Cause is analogous to volition, since there must be an +intelligible _nexus_ between cause and effect." This maxim is, I +think, often unconsciously in the imaginations of philosophers who +would reject it when explicitly stated. It is probably operative in +the view we have just been considering, that mind could not have +resulted from a purely material world. I do not profess to know what +is meant by "intelligible"; it seems to mean "familiar to +imagination." Nothing is less "intelligible," in any other sense, than +the connection between an act of will and its fulfilment. But +obviously the sort of nexus desired between cause and effect is such +as could only hold between the "events" which the supposed law of +causality contemplates; the laws which replace causality in such a +science as physics leave no room for any two events between which a +nexus could be sought. + +(3) "The cause _compels_ the effect in some sense in which the effect +does not compel the cause." This belief seems largely operative in the +dislike of determinism; but, as a matter of fact, it is connected with +our second maxim, and falls as soon as that is abandoned. We may +define "compulsion" as follows: "Any set of circumstances is said to +compel A when A desires to do something which the circumstances +prevent, or to abstain from something which the circumstances cause." +This presupposes that some meaning has been found for the word +"cause"--a point to which I shall return later. What I want to make +clear at present is that compulsion is a very complex notion, +involving thwarted desire. So long as a person does what he wishes to +do, there is no compulsion, however much his wishes may be calculable +by the help of earlier events. And where desire does not come in, +there can be no question of compulsion. Hence it is, in general, +misleading to regard the cause as compelling the effect. + +A vaguer form of the same maxim substitutes the word "determine" for +the word "compel"; we are told that the cause _determines_ the effect +in a sense in which the effect does not _determine_ the cause. It is +not quite clear what is meant by "determining"; the only precise +sense, so far as I know, is that of a function or one-many relation. +If we admit plurality of causes, but not of effects, that is, if we +suppose that, given the cause, the effect must be such and such, but, +given the effect, the cause may have been one of many alternatives, +then we may say that the cause determines the effect, but not the +effect the cause. Plurality of causes, however, results only from +conceiving the effect vaguely and narrowly and the cause precisely and +widely. Many antecedents may "cause" a man's death, because his death +is vague and narrow. But if we adopt the opposite course, taking as +the "cause" the drinking of a dose of arsenic, and as the "effect" the +whole state of the world five minutes later, we shall have plurality +of effects instead of plurality of causes. Thus the supposed lack of +symmetry between "cause" and "effect" is illusory. + +(4) "A cause cannot operate when it has ceased to exist, because what +has ceased to exist is nothing." This is a common maxim, and a still +more common unexpressed prejudice. It has, I fancy, a good deal to do +with the attractiveness of Bergson's "_durée_": since the past has +effects now, it must still exist in some sense. The mistake in this +maxim consists in the supposition that causes "operate" at all. A +volition "operates" when what it wills takes place; but nothing can +operate except a volition. The belief that causes "operate" results +from assimilating them, consciously or unconsciously, to volitions. We +have already seen that, if there are causes at all, they must be +separated by a finite interval of time from their effects, and thus +cause their effects after they have ceased to exist. + +It may be objected to the above definition of a volition "operating" +that it only operates when it "causes" what it wills, not when it +merely happens to be followed by what it wills. This certainly +represents the usual view of what is meant by a volition "operating," +but as it involves the very view of causation which we are engaged in +combating, it is not open to us as a definition. We may say that a +volition "operates" when there is some law in virtue of which a +similar volition in rather similar circumstances will usually be +followed by what it wills. But this is a vague conception, and +introduces ideas which we have not yet considered. What is chiefly +important to notice is that the usual notion of "operating" is not +open to us if we reject, as I contend that we should, the usual notion +of causation. + +(5) "A cause cannot operate except where it is." This maxim is very +widespread; it was urged against Newton, and has remained a source of +prejudice against "action at a distance." In philosophy it has led to +a denial of transient action, and thence to monism or Leibnizian +monadism. Like the analogous maxim concerning temporal contiguity, it +rests upon the assumption that causes "operate," i.e. that they are in +some obscure way analogous to volitions. And, as in the case of +temporal contiguity, the inferences drawn from this maxim are wholly +groundless. + +I return now to the question, What law or laws can be found to take +the place of the supposed law of causality? + +First, without passing beyond such uniformities of sequence as are +contemplated by the traditional law, we may admit that, if any such +sequence has been observed in a great many cases, and has never been +found to fail, there is an inductive probability that it will be found +to hold in future cases. If stones have hitherto been found to break +windows, it is probable that they will continue to do so. This, of +course, assumes the inductive principle, of which the truth may +reasonably be questioned; but as this principle is not our present +concern, I shall in this discussion treat it as indubitable. We may +then say, in the case of any such frequently observed sequence, that +the earlier event is the _cause_ and the later event the _effect_. + +Several considerations, however, make such special sequences very +different from the traditional relation of cause and effect. In the +first place, the sequence, in any hitherto unobserved instance, is no +more than probable, whereas the relation of cause and effect was +supposed to be necessary. I do not mean by this merely that we are not +sure of having discovered a true case of cause and effect; I mean +that, even when we have a case of cause and effect in our present +sense, all that is meant is that on grounds of observation, it is +probable that when one occurs the other will also occur. Thus in our +present sense, A may be the cause of B even if there actually are +cases where B does not follow A. Striking a match will be the cause of +its igniting, in spite of the fact that some matches are damp and fail +to ignite. + +In the second place, it will not be assumed that _every_ event has +some antecedent which is its cause in this sense; we shall only +believe in causal sequences where we find them, without any +presumption that they always are to be found. + +In the third place, _any_ case of sufficiently frequent sequence will +be causal in our present sense; for example, we shall not refuse to +say that night is the cause of day. Our repugnance to saying this +arises from the ease with which we can imagine the sequence to fail, +but owing to the fact that cause and effect must be separated by a +finite interval of time, _any_ such sequence _might_ fail through the +interposition of other circumstances in the interval. Mill, discussing +this instance of night and day, says:-- + +"It is necessary to our using the word cause, that we should believe +not only that the antecedent always _has_ been followed by the +consequent, but that as long as the present constitution of things +endures, it always _will_ be so."[39] + +In this sense, we shall have to give up the hope of finding causal +laws such as Mill contemplated; any causal sequence which we have +observed may at any moment be falsified without a falsification of any +laws of the kind that the more advanced sciences aim at establishing. + +In the fourth place, such laws of probable sequence, though useful in +daily life and in the infancy of a science, tend to be displaced by +quite different laws as soon as a science is successful. The law of +gravitation will illustrate what occurs in any advanced science. In +the motions of mutually gravitating bodies, there is nothing that can +be called a cause, and nothing that can be called an effect; there is +merely a formula. Certain differential equations can be found, which +hold at every instant for every particle of the system, and which, +given the configuration and velocities at one instant, or the +configurations at two instants, render the configuration at any other +earlier or later instant theoretically calculable. That is to say, the +configuration at any instant is a function of that instant and the +configurations at two given instants. This statement holds throughout +physics, and not only in the special case of gravitation. But there is +nothing that could be properly called "cause" and nothing that could +be properly called "effect" in such a system. + +No doubt the reason why the old "law of causality" has so long +continued to pervade the books of philosophers is simply that the idea +of a function is unfamiliar to most of them, and therefore they seek +an unduly simplified statement. There is no question of repetitions of +the "same" cause producing the "same" effect; it is not in any +sameness of causes and effects that the constancy of scientific law +consists, but in sameness of relations. And even "sameness of +relations" is too simple a phrase; "sameness of differential +equations" is the only correct phrase. It is impossible to state this +accurately in non-mathematical language; the nearest approach would be +as follows: "There is a constant relation between the state of the +universe at any instant and the rate of change in the rate at which +any part of the universe is changing at that instant, and this +relation is many-one, i.e. such that the rate of change in the rate of +change is determinate when the state of the universe is given." If the +"law of causality" is to be something actually discoverable in the +practice of science, the above proposition has a better right to the +name than any "law of causality" to be found in the books of +philosophers. + +In regard to the above principle, several observations must be made-- + +(1) No one can pretend that the above principle is _a priori_ or +self-evident or a "necessity of thought." Nor is it, in any sense, a +premiss of science: it is an empirical generalisation from a number of +laws which are themselves empirical generalisations. + +(2) The law makes no difference between past and future: the future +"determines" the past in exactly the same sense in which the past +"determines" the future. The word "determine," here, has a purely +logical significance: a certain number of variables "determine" +another variable if that other variable is a function of them. + +(3) The law will not be empirically verifiable unless the course of +events within some sufficiently small volume will be approximately +the same in any two states of the universe which only differ in regard +to what is at a considerable distance from the small volume in +question. For example, motions of planets in the solar system must be +approximately the same however the fixed stars may be distributed, +provided that all the fixed stars are very much farther from the sun +than the planets are. If gravitation varied directly as the distance, +so that the most remote stars made the most difference to the motions +of the planets, the world might be just as regular and just as much +subject to mathematical laws as it is at present, but we could never +discover the fact. + +(4) Although the old "law of causality" is not assumed by science, +something which we may call the "uniformity of nature" is assumed, or +rather is accepted on inductive grounds. The uniformity of nature does +not assert the trivial principle "same cause, same effect," but the +principle of the permanence of laws. That is to say, when a law +exhibiting, e.g. an acceleration as a function of the configuration +has been found to hold throughout the observable past, it is expected +that it will continue to hold in the future, or that, if it does not +itself hold, there is some other law, agreeing with the supposed law +as regards the past, which will hold for the future. The ground of +this principle is simply the inductive ground that it has been found +to be true in very many instances; hence the principle cannot be +considered certain, but only probable to a degree which cannot be +accurately estimated. + +The uniformity of nature, in the above sense, although it is assumed +in the practice of science, must not, in its generality, be regarded +as a kind of major premiss, without which all scientific reasoning +would be in error. The assumption that _all_ laws of nature are +permanent has, of course, less probability than the assumption that +this or that particular law is permanent; and the assumption that a +particular law is permanent for all time has less probability than the +assumption that it will be valid up to such and such a date. Science, +in any given case, will assume what the case requires, but no more. In +constructing the _Nautical Almanac_ for 1915 it will assume that the +law of gravitation will remain true up to the end of that year; but it +will make no assumption as to 1916 until it comes to the next volume +of the almanac. This procedure is, of course, dictated by the fact +that the uniformity of nature is not known _a priori_, but is an +empirical generalisation, like "all men are mortal." In all such +cases, it is better to argue immediately from the given particular +instances to the new instance, than to argue by way of a major +premiss; the conclusion is only probable in either case, but acquires +a higher probability by the former method than by the latter. + +In all science we have to distinguish two sorts of laws: first, those +that are empirically verifiable but probably only approximate; +secondly, those that are not verifiable, but may be exact. The law of +gravitation, for example, in its applications to the solar system, is +only empirically verifiable when it is assumed that matter outside the +solar system may be ignored for such purposes; we believe this to be +only approximately true, but we cannot empirically verify the law of +universal gravitation which we believe to be exact. This point is very +important in connection with what we may call "relatively isolated +systems." These may be defined as follows:-- + +A system relatively isolated during a given period is one which, +within some assignable margin of error, will behave in the same way +throughout that period, however the rest of the universe may be +constituted. + +A system may be called "practically isolated" during a given period +if, although there _might_ be states of the rest of the universe which +would produce more than the assigned margin of error, there is reason +to believe that such states do not in fact occur. + +Strictly speaking, we ought to specify the respect in which the system +is relatively isolated. For example, the earth is relatively isolated +as regards falling bodies, but not as regards tides; it is +_practically_ isolated as regards economic phenomena, although, if +Jevons' sunspot theory of commercial crises had been true, it would +not have been even practically isolated in this respect. + +It will be observed that we cannot prove in advance that a system is +isolated. This will be inferred from the observed fact that +approximate uniformities can be stated for this system alone. If the +complete laws for the whole universe were known, the isolation of a +system could be deduced from them; assuming, for example, the law of +universal gravitation, the practical isolation of the solar system in +this respect can be deduced by the help of the fact that there is very +little matter in its neighbourhood. But it should be observed that +isolated systems are only important as providing a possibility of +_discovering_ scientific laws; they have no theoretical importance in +the finished structure of a science. + +The case where one event A is said to "cause" another event B, which +philosophers take as fundamental, is really only the most simplified +instance of a practically isolated system. It may happen that, as a +result of general scientific laws, whenever A occurs throughout a +certain period, it is followed by B; in that case, A and B form a +system which is practically isolated throughout that period. It is, +however, to be regarded as a piece of good fortune if this occurs; it +will always be due to special circumstances, and would not have been +true if the rest of the universe had been different though subject to +the same laws. + +The essential function which causality has been supposed to perform is +the possibility of inferring the future from the past, or, more +generally, events at any time from events at certain assigned times. +Any system in which such inference is possible may be called a +"deterministic" system. We may define a deterministic system as +follows:-- + + A system is said to be "deterministic" when, given certain data, + [Math: e_{1}, e_{2}, ..., e_{n}, at times t_{1}, t_{2}, ..., + t_{n}] respectively, concerning this system, if [Math: E_{t}] is + the state of the system at any time _t_, there is a functional + relation of the form + +[Math: E_{t} = f (e_{1}, t_{1}, e_{2}, t_{2}, ..., e_{n}, t_{n}, t)]. (A) + + The system will be "deterministic throughout a given period" if + _t_, in the above formula, may be any time within that period, + though outside that period the formula may be no longer true. If + the universe, as a whole, is such a system, determinism is true of + the universe; if not, not. A system which is part of a + deterministic system I shall call "determined"; one which is not + part of any such system I shall call "capricious." + +The events [Math: e_{1}, e_{2}, ..., e_{n}] I shall call "determinants" +of the system. It is to be observed that a system which has one set of +determinants will in general have many. In the case of the motions of +the planets, for example, the configurations of the solar system at any +two given times will be determinants. + +We may take another illustration from the hypothesis of +psycho-physical parallelism. Let us assume, for the purposes of this +illustration, that to a given state of brain a given state of mind +always corresponds, and vice versa, i.e. that there is a one-one +relation between them, so that each is a function of the other. We may +also assume, what is practically certain, that to a given state of a +certain brain a given state of the whole material universe +corresponds, since it is highly improbable that a given brain is ever +twice in exactly the same state. Hence there will be a one-one +relation between the state of a given person's mind and the state of +the whole material universe. It follows that, if _n_ states of the +material universe are determinants of the material universe, then _n_ +states of a given man's mind are determinants of the whole material +and mental universe--assuming, that is to say, that psycho-physical +parallelism is true. + +The above illustration is important in connection with a certain +confusion which seems to have beset those who have philosophised on +the relation of mind and matter. It is often thought that, if the +state of the mind is determinate when the state of the brain is given, +and if the material world forms a deterministic system, then mind is +"subject" to matter in some sense in which matter is not "subject" to +mind. But if the state of the brain is also determinate when the state +of the mind is given, it must be exactly as true to regard matter as +subject to mind as it would be to regard mind as subject to matter. We +could, theoretically, work out the history of mind without ever +mentioning matter, and then, at the end, deduce that matter must +meanwhile have gone through the corresponding history. It is true that +if the relation of brain to mind were many-one, not one-one, there +would be a one-sided dependence of mind on brain, while conversely, if +the relation were one-many, as Bergson supposes, there would be a +one-aided dependence of brain on mind. But the dependence involved is, +in any case, only logical; it does not mean that we shall be +compelled to do things we desire not to do, which is what people +instinctively imagine it to mean. + +As another illustration we may take the case of mechanism and +teleology. A system may be defined as "mechanical" when it has a set +of determinants that are purely material, such as the positions of +certain pieces of matter at certain times. It is an open question +whether the world of mind and matter, as we know it, is a mechanical +system or not; let us suppose, for the sake of argument, that it is a +mechanical system. This supposition--so I contend--throws no light +whatever on the question whether the universe is or is not a +"teleological" system. It is difficult to define accurately what is +meant by a "teleological" system, but the argument is not much +affected by the particular definition we adopt. Broadly, a +teleological system is one in which purposes are realised, i.e. in +which certain desires--those that are deeper or nobler or more +fundamental or more universal or what not--are followed by their +realisation. Now the fact--if it be a fact--that the universe is +mechanical has no bearing whatever on the question whether it is +teleological in the above sense. There might be a mechanical system in +which all wishes were realised, and there might be one in which all +wishes were thwarted. The question whether, or how far, our actual +world is teleological, cannot, therefore, be settled by proving that +it is mechanical, and the desire that it should be teleological is no +ground for wishing it to be not mechanical. + +There is, in all these questions, a very great difficulty in avoiding +confusion between what we can infer and what is in fact determined. +Let us consider, for a moment, the various senses in which the future +may be "determined." There is one sense--and a very important one--in +which it is determined quite independently of scientific laws, namely, +the sense that it will be what it will be. We all regard the past as +determined simply by the fact that it has happened; but for the +accident that memory works backward and not forward, we should regard +the future as equally determined by the fact that it will happen. +"But," we are told, "you cannot alter the past, while you can to some +extent alter the future." This view seems to me to rest upon just +those errors in regard to causation which it has been my object to +remove. You cannot make the past other than it was--true, but this is +a mere application of the law of contradiction. If you already know +what the past was, obviously it is useless to wish it different. But +also you cannot make the future other than it will be; this again is +an application of the law of contradiction. And if you happen to know +the future--e.g. in the case of a forthcoming eclipse--it is just as +useless to wish it different as to wish the past different. "But," it +will be rejoined, "our wishes can _cause_ the future, sometimes, to be +different from what it would be if they did not exist, and they can +have no such effect upon the past." This, again, is a mere tautology. +An effect being _defined_ as something subsequent to its cause, +obviously we can have no _effect_ upon the past. But that does not +mean that the past would not have been different if our present wishes +had been different. Obviously, our present wishes are conditioned by +the past, and therefore could not have been different unless the past +had been different; therefore, if our present wishes were different, +the past would be different. Of course, the past cannot be different +from what it was, but no more can our present wishes be different from +what they are; this again is merely the law of contradiction. The +facts seem to be merely (1) that wishing generally depends upon +ignorance, and is therefore commoner in regard to the future than in +regard to the past; (2) that where a wish concerns the future, it and +its realisation very often form a "practically independent system," +i.e. many wishes regarding the future are realised. But there seems no +doubt that the main difference in our feelings arises from the +accidental fact that the past but not the future can be known by +memory. + +Although the sense of "determined" in which the future is determined +by the mere fact that it will be what it will be is sufficient (at +least so it seems to me) to refute some opponents of determinism, +notably M. Bergson and the pragmatists, yet it is not what most people +have in mind when they speak of the future as determined. What they +have in mind is a formula by means of which the future can be +exhibited, and at least theoretically calculated, as a function of the +past. But at this point we meet with a great difficulty, which besets +what has been said above about deterministic systems, as well as what +is said by others. + +If formulæ of any degree of complexity, however great, are admitted, +it would seem that any system, whose state at a given moment is a +function of certain measurable quantities, must be a deterministic +system. Let us consider, in illustration, a single material particle, +whose co-ordinates at time _t_ are [Math: x_{t}, y_{t}, z_{t}]. Then, +however, the particle moves, there must be, theoretically, functions +[Math: f_{1}, f_{2}, f_{3}], such that + +[Math: x_{t} = f_{t}(t), y_{t} = f_{2}(t), z_{t} = f_{3}(t).] + +It follows that, theoretically, the whole state of the material +universe at time _t_ must be capable of being exhibited as a function +of _t_. Hence our universe will be deterministic in the sense defined +above. But if this be true, no information is conveyed about the +universe in stating that it is deterministic. It is true that the +formulæ involved may be of strictly infinite complexity, and therefore +not practically capable of being written down or apprehended. But +except from the point of view of our knowledge, this might seem to be +a detail: in itself, if the above considerations are sound, the +material universe _must_ be deterministic, _must_ be subject to laws. + +This, however, is plainly not what was intended. The difference +between this view and the view intended may be seen as follows. Given +some formula which fits the facts hitherto--say the law of +gravitation--there will be an infinite number of other formulæ, not +empirically distinguishable from it in the past, but diverging from it +more and more in the future. Hence, even assuming that there are +persistent laws, we shall have no reason for assuming that the law of +the inverse square will hold in future; it may be some other hitherto +indistinguishable law that will hold. We cannot say that _every_ law +which has held hitherto must hold in the future, because past facts +which obey one law will also obey others, hitherto indistinguishable +but diverging in future. Hence there must, at every moment, be laws +hitherto unbroken which are now broken for the first time. What +science does, in fact, is to select the _simplest_ formula that will +fit the facts. But this, quite obviously, is merely a methodological +precept, not a law of Nature. If the simplest formula ceases, after a +time, to be applicable, the simplest formula that remains applicable +is selected, and science has no sense that an axiom has been +falsified. We are thus left with the brute fact that, in many +departments of science, quite simple laws have hitherto been found to +hold. This fact cannot be regarded as having any _a priori_ ground, +nor can it be used to support inductively the opinion that the same +laws will continue; for at every moment laws hitherto true are being +falsified, though in the advanced sciences these laws are less simple +than those that have remained true. Moreover it would be fallacious to +argue inductively from the state of the advanced sciences to the +future state of the others, for it may well be that the advanced +sciences are advanced simply because, hitherto, their subject-matter +has obeyed simple and easily ascertainable laws, while the +subject-matter of other sciences has not done so. + +The difficulty we have been considering seems to be met partly, if not +wholly, by the principle that the _time_ must not enter explicitly +into our formulæ. All mechanical laws exhibit acceleration as a +function of configuration, not of configuration and time jointly; and +this principle of the irrelevance of the time may be extended to all +scientific laws. In fact we might interpret the "uniformity of nature" +as meaning just this, that no scientific law involves the time as an +argument, unless, of course, it is given in an integrated form, in +which case _lapse_ of time, though not absolute time, may appear in +our formulæ. Whether this consideration suffices to overcome our +difficulty completely, I do not know; but in any case it does much to +diminish it. + +It will serve to illustrate what has been said if we apply it to the +question of free will. + +(1) Determinism in regard to the will is the doctrine that our +volitions belong to some deterministic system, i.e. are "determined" +in the sense defined above. Whether this doctrine is true or false, is +a mere question of fact; no _a priori_ considerations (if our previous +discussions have been correct) can exist on either side. On the one +hand, there is no _a priori_ category of causality, but merely certain +observed uniformities. As a matter of fact, there are observed +uniformities in regard to volitions; thus there is some empirical +evidence that volitions are determined. But it would be very rash to +maintain that the evidence is overwhelming, and it is quite possible +that some volitions, as well as some other things, are not determined, +except in the sense in which we found that everything must be +determined. + +(2) But, on the other hand, the subjective sense of freedom, sometimes +alleged against determinism, has no bearing on the question whatever. +The view that it has a bearing rests upon the belief that causes +compel their effects, or that nature enforces obedience to its laws as +governments do. These are mere anthropomorphic superstitions, due to +assimilation of causes with volitions and of natural laws with human +edicts. We feel that our will is not compelled, but that only means +that it is not other than we choose it to be. It is one of the +demerits of the traditional theory of causality that it has created an +artificial opposition between determinism and the freedom of which we +are introspectively conscious. + +(3) Besides the general question whether volitions are determined, +there is the further question whether they are _mechanically_ +determined, i.e. whether they are part of what was above defined as a +mechanical system. This is the question whether they form part of a +system with purely material determinants, i.e. whether there are laws +which, given certain material data, make all volitions functions of +those data. Here again, there is empirical evidence up to a point, but +it is not conclusive in regard to all volitions. It is important to +observe, however that even if volitions are part of a mechanical +system, this by no means implies any supremacy of matter over mind. It +may well be that the same system which is susceptible of material +determinants is also susceptible of mental determinants; thus a +mechanical system may be determined by sets of volitions, as well as +by sets of material facts. It would seem, therefore, that the reasons +which make people dislike the view that volitions are mechanically +determined are fallacious. + +(4) The notion of _necessity_, which is often associated with +determinism, is a confused notion not legitimately deducible from +determinism. Three meanings are commonly confounded when necessity is +spoken of:-- + +([alpha]) An _action_ is necessary when it will be performed +however much the agent may wish to do otherwise. Determinism does not +imply that actions are necessary in this sense. + +([beta]) A _propositional function_ is necessary when all its +values are true. This sense is not relevant to our present discussion. + +([gamma]) A _proposition_ is necessary with respect to a given +constituent when it is the value, with that constituent as argument, +of a necessary propositional function, in other words, when it remains +true however that constituent may be varied. In this sense, in a +deterministic system, the connection of a volition with its +determinants is necessary, if the time at which the determinants occur +be taken as the constituent to be varied, the time-interval between +the determinants and the volition being kept constant. But this sense +of necessity is purely logical, and has no emotional importance. + +We may now sum up our discussion of causality. We found first that the +law of causality, as usually stated by philosophers, is false, and is +not employed in science. We then considered the nature of scientific +laws, and found that, instead of stating that one event A is always +followed by another event B, they stated functional relations between +certain events at certain times, which we called determinants, and +other events at earlier or later times or at the same time. We were +unable to find any _a priori_ category involved: the existence of +scientific laws appeared as a purely empirical fact, not necessarily +universal, except in a trivial and scientifically useless form. We +found that a system with one set of determinants may very likely have +other sets of a quite different kind, that, for example, a +mechanically determined system may also be teleologically or +volitionally determined. Finally we considered the problem of free +will: here we found that the reasons for supposing volitions to be +determined are strong but not conclusive, and we decided that even if +volitions are mechanically determined, that is no reason for denying +freedom in the sense revealed by introspection, or for supposing that +mechanical events are not determined by volitions. The problem of free +will _versus_ determinism is therefore, if we were right, mainly +illusory, but in part not yet capable of being decisively solved. + +FOOTNOTES: + +[35] A propositional function is an expression containing a variable, +or undetermined constituent, and becoming a proposition as soon as a +definite value is assigned to the variable. Examples are: "A is A," +"_x_ is a number." The variable is called the _argument_ of the +function. + +[36] _Logic_, Bk. III, Chap. V, § 2. + +[37] _Time and Free Will_, p. 199. + +[38] _Time and Free Will._ p. 202. + +[39] _Loc. cit._, § 6 + + + + +X + +KNOWLEDGE BY ACQUAINTANCE AND KNOWLEDGE BY DESCRIPTION + + +The object of the following paper is to consider what it is that we +know in cases where we know propositions about "the so-and-so" without +knowing who or what the so-and-so is. For example, I know that the +candidate who gets most votes will be elected, though I do not know +who is the candidate who will get most votes. The problem I wish to +consider is: What do we know in these cases, where the subject is +merely described? I have considered this problem elsewhere[40] from a +purely logical point of view; but in what follows I wish to consider +the question in relation to theory of knowledge as well as in relation +to logic, and in view of the above-mentioned logical discussions, I +shall in this paper make the logical portion as brief as possible. + +In order to make clear the antithesis between "acquaintance" and +"description," I shall first of all try to explain what I mean by +"acquaintance." I say that I am _acquainted_ with an object when I +have a direct cognitive relation to that object, i.e. when I am +directly aware of the object itself. When I speak of a cognitive +relation here, I do not mean the sort of relation which constitutes +judgment, but the sort which constitutes presentation. In fact, I +think the relation of subject and object which I call acquaintance is +simply the converse of the relation of object and subject which +constitutes presentation. That is, to say that S has acquaintance with +O is essentially the same thing as to say that O is presented to S. +But the associations and natural extensions of the word _acquaintance_ +are different from those of the word _presentation_. To begin with, as +in most cognitive words, it is natural to say that I am acquainted +with an object even at moments when it is not actually before my mind, +provided it has been before my mind, and will be again whenever +occasion arises. This is the same sense in which I am said to know +that 2+2=4 even when I am thinking of something else. In the second +place, the word _acquaintance_ is designed to emphasise, more than the +word _presentation_, the relational character of the fact with which +we are concerned. There is, to my mind, a danger that, in speaking of +presentation, we may so emphasise the object as to lose sight of the +subject. The result of this is either to lead to the view that there +is no subject, whence we arrive at materialism; or to lead to the view +that what is presented is part of the subject, whence we arrive at +idealism, and should arrive at solipsism but for the most desperate +contortions. Now I wish to preserve the dualism of subject and object +in my terminology, because this dualism seems to me a fundamental fact +concerning cognition. Hence I prefer the word _acquaintance_ because +it emphasises the need of a subject which is acquainted. + +When we ask what are the kinds of objects with which we are +acquainted, the first and most obvious example is _sense-data_. When I +see a colour or hear a noise, I have direct acquaintance with the +colour or the noise. The sense-datum with which I am acquainted in +these cases is generally, if not always, complex. This is +particularly obvious in the case of sight. I do not mean, of course, +merely that the supposed physical object is complex, but that the +direct sensible object is complex and contains parts with spatial +relations. Whether it is possible to be aware of a complex without +being aware of its constituents is not an easy question, but on the +whole it would seem that there is no reason why it should not be +possible. This question arises in an acute form in connection with +self-consciousness, which we must now briefly consider. + +In introspection, we seem to be immediately aware of varying +complexes, consisting of objects in various cognitive and conative +relations to ourselves. When I see the sun, it often happens that I +am aware of my seeing the sun, in addition to being aware of the sun; +and when I desire food, it often happens that I am aware of my desire +for food. But it is hard to discover any state of mind in which I am +aware of myself alone, as opposed to a complex of which I am a +constituent. The question of the nature of self-consciousness is too +large and too slightly connected with our subject, to be argued at +length here. It is difficult, but probably not impossible, to account +for plain facts if we assume that we do not have acquaintance with +ourselves. It is plain that we are not only _acquainted_ with the +complex "Self-acquainted-with-A," but we also _know_ the proposition +"I am acquainted with A." Now here the complex has been analysed, and +if "I" does not stand for something which is a direct object of +acquaintance, we shall have to suppose that "I" is something known by +description. If we wished to maintain the view that there is no +acquaintance with Self, we might argue as follows: We are acquainted +with _acquaintance_, and we know that it is a relation. Also we are +acquainted with a complex in which we perceive that acquaintance is +the relating relation. Hence we know that this complex must have a +constituent which is that which is acquainted, i.e. must have a +subject-term as well as an object-term. This subject-term we define +as "I." Thus "I" means "the subject-term in awarenesses of which _I_ +am aware." But as a definition this cannot be regarded as a happy +effort. It would seem necessary, therefore, either to suppose that I +am acquainted with myself, and that "I," therefore, requires no +definition, being merely the proper name of a certain object, or to +find some other analysis of self-consciousness. Thus self-consciousness +cannot be regarded as throwing light on the question whether we can +know a complex without knowing its constituents. This question, +however, is not important for our present purposes, and I shall +therefore not discuss it further. + +The awarenesses we have considered so far have all been awarenesses of +particular existents, and might all in a large sense be called +sense-data. For, from the point of view of theory of knowledge, +introspective knowledge is exactly on a level with knowledge derived +from sight or hearing. But, in addition to awareness of the above kind +of objects, which may be called awareness of _particulars_; we have +also (though not quite in the same sense) what may be called awareness +of _universals_. Awareness of universals is called _conceiving_, and a +universal of which we are aware is called a _concept_. Not only are we +aware of particular yellows, but if we have seen a sufficient number +of yellows and have sufficient intelligence, we are aware of the +universal _yellow_; this universal is the subject in such judgments as +"yellow differs from blue" or "yellow resembles blue less than green +does." And the universal yellow is the predicate in such judgments as +"this is yellow," where "this" is a particular sense-datum. And +universal relations, too, are objects of awarenesses; up and down, +before and after, resemblance, desire, awareness itself, and so on, +would seem to be all of them objects of which we can be aware. + +In regard to relations, it might be urged that we are never aware of +the universal relation itself, but only of complexes in which it is a +constituent. For example, it may be said that we do not know directly +such a relation as _before_, though we understand such a proposition +as "this is before that," and may be directly aware of such a complex +as "this being before that." This view, however, is difficult to +reconcile with the fact that we often know propositions in which the +relation is the subject, or in which the relata are not definite given +objects, but "anything." For example, we know that if one thing is +before another, and the other before a third, then the first is before +the third; and here the things concerned are not definite things, but +"anything." It is hard to see how we could know such a fact about +"before" unless we were acquainted with "before," and not merely with +actual particular cases of one given object being before another given +object. And more directly: A judgment such as "this is before that," +where this judgment is derived from awareness of a complex, +constitutes an analysis, and we should not understand the analysis if +we were not acquainted with the meaning of the terms employed. Thus we +must suppose that we are acquainted with the meaning of "before," and +not merely with instances of it. + +There are thus at least two sorts of objects of which we are aware, +namely, particulars and universals. Among particulars I include all +existents, and all complexes of which one or more constituents are +existents, such as this-before-that, this-above-that, +the-yellowness-of-this. Among universals I include all objects of +which no particular is a constituent. Thus the disjunction +"universal-particular" includes all objects. We might also call it the +disjunction "abstract-concrete." It is not quite parallel with the +opposition "concept-percept," because things remembered or imagined +belong with particulars, but can hardly be called percepts. (On the +other hand, universals with which we are acquainted may be identified +with concepts.) + +It will be seen that among the objects with which we are acquainted +are not included physical objects (as opposed to sense-data), nor +other people's minds. These things are known to us by what I call +"knowledge by description," which we must now consider. + +By a "description" I mean any phrase of the form "a so-and-so" or "the +so-and-so." A phrase of the form "a so-and-so" I shall call an +"ambiguous" description; a phrase of the form "the so-and-so" (in the +singular) I shall call a "definite" description. Thus "a man" is an +ambiguous description, and "the man with the iron mask" is a definite +description. There are various problems connected with ambiguous +descriptions, but I pass them by, since they do not directly concern +the matter I wish to discuss. What I wish to discuss is the nature of +our knowledge concerning objects in cases where we know that there is +an object answering to a definite description, though we are not +_acquainted_ with any such object. This is a matter which is concerned +exclusively with _definite_ descriptions. I shall, therefore, in the +sequel, speak simply of "descriptions" when I mean "definite +descriptions." Thus a description will mean any phrase of the form +"the so-and-so" in the singular. + +I shall say that an object is "known by description" when we know that +it is "_the_ so-and-so," i.e. when we know that there is one object, +and no more, having a certain property; and it will generally be +implied that we do not have knowledge of the same object by +acquaintance. We know that the man with the iron mask existed, and +many propositions are known about him; but we do not know who he was. +We know that the candidate who gets most votes will be elected, and in +this case we are very likely also acquainted (in the only sense in +which one can be acquainted with some one else) with the man who is, +in fact, the candidate who will get most votes, but we do not know +which of the candidates he is, i.e. we do not know any proposition of +the form "A is the candidate who will get most votes" where A is one +of the candidates by name. We shall say that we have "_merely_ +descriptive knowledge" of the so-and-so when, although we know that +the so-and-so exists, and although we may possibly be acquainted with +the object which is, in fact, the so-and-so, yet we do not know any +proposition "_a_ is the so-and-so," where _a_ is something with which +we are acquainted. + +When we say "the so-and-so exists," we mean that there is just one +object which is the so-and-so. The proposition "_a_ is the so-and-so" +means that _a_ has the property so-and-so, and nothing else has. "Sir +Joseph Larmor is the Unionist candidate" means "Sir Joseph Larmor is a +Unionist candidate, and no one else is." "The Unionist candidate +exists" means "some one is a Unionist candidate, and no one else is." +Thus, when we are acquainted with an object which we know to be the +so-and-so, we know that the so-and-so exists but we may know that the +so-and-so exists when we are not acquainted with any object which we +know to be the so-and-so, and even when we are not acquainted with any +object which, in fact, is the so-and-so. + +Common words, even proper names, are usually really descriptions. That +is to say, the thought in the mind of a person using a proper name +correctly can generally only be expressed explicitly if we replace the +proper name by a description. Moreover, the description required to +express the thought will vary for different people, or for the same +person at different times. The only thing constant (so long as the +name is rightly used) is the object to which the name applies. But so +long as this remains constant, the particular description involved +usually makes no difference to the truth or falsehood of the +proposition in which the name appears. + +Let us take some illustrations. Suppose some statement made about +Bismarck. Assuming that there is such a thing as direct acquaintance +with oneself, Bismarck himself might have used his name directly to +designate the particular person with whom he was acquainted. In this +case, if he made a judgment about himself, he himself might be a +constituent of the judgment. Here the proper name has the direct use +which it always wishes to have, as simply standing for a certain +object, and not for a description of the object. But if a person who +knew Bismarck made a judgment about him, the case is different. What +this person was acquainted with were certain sense-data which he +connected (rightly, we will suppose) with Bismarck's body. His body as +a physical object, and still more his mind, were only known as the +body and the mind connected with these sense-data. That is, they were +known by description. It is, of course, very much a matter of chance +which characteristics of a man's appearance will come into a friend's +mind when he thinks of him; thus the description actually in the +friend's mind is accidental. The essential point is that he knows that +the various descriptions all apply to the same entity, in spite of +not being acquainted with the entity in question. + +When we, who did not know Bismarck, make a judgment about him, the +description in our minds will probably be some more or less vague mass +of historical knowledge--far more, in most cases, than is required to +identify him. But, for the sake of illustration, let us assume that we +think of him as "the first Chancellor of the German Empire." Here all +the words are abstract except "German." The word "German" will again +have different meanings for different people. To some it will recall +travels in Germany, to some the look of Germany on the map, and so on. +But if we are to obtain a description which we know to be applicable, +we shall be compelled, at some point, to bring in a reference to a +particular with which we are acquainted. Such reference is involved in +any mention of past, present, and future (as opposed to definite +dates), or of here and there, or of what others have told us. Thus it +would seem that, in some way or other, a description known to be +applicable to a particular must involve some reference to a particular +with which we are acquainted, if our knowledge about the thing +described is not to be merely what follows logically from the +description. For example, "the most long-lived of men" is a +description which must apply to some man, but we can make no judgments +concerning this man which involve knowledge about him beyond what the +description gives. If, however, we say, "the first Chancellor of the +German Empire was an astute diplomatist," we can only be assured of +the truth of our judgment in virtue of something with which we are +acquainted--usually a testimony heard or read. Considered +psychologically, apart from the information we convey to others, apart +from the fact about the actual Bismarck, which gives importance to +our judgment, the thought we really have contains the one or more +particulars involved, and otherwise consists wholly of concepts. All +names of places--London, England, Europe, the earth, the Solar +System--similarly involve, when used, descriptions which start from +some one or more particulars with which we are acquainted. I suspect +that even the Universe, as considered by metaphysics, involves such a +connection with particulars. In logic, on the contrary, where we are +concerned not merely with what does exist, but with whatever might or +could exist or be, no reference to actual particulars is involved. + +It would seem that, when we make a statement about something only +known by description, we often _intend_ to make our statement, not in +the form involving the description, but about the actual thing +described. That is to say, when we say anything about Bismarck, we +should like, if we could, to make the judgment which Bismarck alone +can make, namely, the judgment of which he himself is a constituent. +In this we are necessarily defeated, since the actual Bismarck is +unknown to us. But we know that there is an object B called Bismarck, +and that B was an astute diplomatist. We can thus _describe_ the +proposition we should like to affirm, namely, "B was an astute +diplomatist," where B is the object which was Bismarck. What enables +us to communicate in spite of the varying descriptions we employ is +that we know there is a true proposition concerning the actual +Bismarck, and that, however we may vary the description (so long as +the description is correct), the proposition described is still the +same. This proposition, which is described and is known to be true, is +what interests us; but we are not acquainted with the proposition +itself, and do not know _it_, though we know it is true. + +It will be seen that there are various stages in the removal from +acquaintance with particulars: there is Bismarck to people who knew +him, Bismarck to those who only know of him through history, the man +with the iron mask, the longest-lived of men. These are progressively +further removed from acquaintance with particulars, and there is a +similar hierarchy in the region of universals. Many universals, like +many particulars, are only known to us by description. But here, as in +the case of particulars, knowledge concerning what is known by +description is ultimately reducible to knowledge concerning what is +known by acquaintance. + +The fundamental epistemological principle in the analysis of +propositions containing descriptions is this: _Every proposition which +we can understand must be composed wholly of constituents with which +we are acquainted._ From what has been said already, it will be plain +why I advocate this principle, and how I propose to meet the case of +propositions which at first sight contravene it. Let us begin with the +reasons for supposing the principle true. + +The chief reason for supposing the principle true is that it seems +scarcely possible to believe that we can make a judgment or entertain +a supposition without knowing what it is that we are judging or +supposing about. If we make a judgment about (say) Julius Cæsar, it is +plain that the actual person who was Julius Cæsar is not a constituent +of the judgment. But before going further, it may be well to explain +what I mean when I say that this or that is a constituent of a +judgment, or of a proposition which we understand. To begin with +judgments: a judgment, as an occurrence, I take to be a relation of a +mind to several entities, namely, the entities which compose what is +judged. If, e.g. I judge that A loves B, the judgment as an event +consists in the existence, at a certain moment, of a specific +four-term relation, called _judging_, between me and A and love and B. +That is to say, at the time when I judge, there is a certain complex +whose terms are myself and A and love and B, and whose relating +relation is _judging_. My reasons for this view have been set forth +elsewhere,[41] and I shall not repeat them here. Assuming this view of +judgment, the constituents of the judgment are simply the constituents +of the complex which is the judgment. Thus, in the above case, the +constituents are myself and A and love and B and judging. But myself +and judging are constituents shared by all my judgments; thus the +_distinctive_ constituents of the particular judgment in question are +A and love and B. Coming now to what is meant by "understanding a +proposition," I should say that there is another relation possible +between me and A and love and B, which is called my _supposing_ that A +loves B.[42] When we can _suppose_ that A loves B, we "understand the +proposition" _A loves B_. Thus we often understand a proposition in +cases where we have not enough knowledge to make a judgment. +Supposing, like judging, is a many-term relation, of which a mind is +one term. The other terms of the relation are called the constituents +of the proposition supposed. Thus the principle which I enunciated may +be re-stated as follows: _Whenever a relation of supposing or judging +occurs, the terms to which the supposing or judging mind is related by +the relation of supposing or judging must be terms with which the mind +in question is acquainted._ This is merely to say that we cannot make +a judgment or a supposition without knowing what it is that we are +making our judgment or supposition about. It seems to me that the +truth of this principle is evident as soon as the principle is +understood; I shall, therefore, in what follows, assume the principle, +and use it as a guide in analysing judgments that contain +descriptions. + +Returning now to Julius Cæsar, I assume that it will be admitted that +he himself is not a constituent of any judgment which I can make. But +at this point it is necessary to examine the view that judgments are +composed of something called "ideas," and that it is the "idea" of +Julius Cæsar that is a constituent of my judgment. I believe the +plausibility of this view rests upon a failure to form a right theory +of descriptions. We may mean by my "idea" of Julius Cæsar the things +that I know about him, e.g. that he conquered Gaul, was assassinated +on the Ides of March, and is a plague to schoolboys. Now I am +admitting, and indeed contending, that in order to discover what is +actually in my mind when I judge about Julius Cæsar, we must +substitute for the proper name a description made up of some of the +things I know about him. (A description which will often serve to +express my thought is "the man whose name was _Julius Cæsar_." For +whatever else I may have forgotten about him, it is plain that when I +mention him I have not forgotten that that was his name.) But although +I think the theory that judgments consist of ideas may have been +suggested in some such way, yet I think the theory itself is +fundamentally mistaken. The view seems to be that there is some +mental existent which may be called the "idea" of something outside +the mind of the person who has the idea, and that, since judgment is a +mental event, its constituents must be constituents of the mind of the +person judging. But in this view ideas become a veil between us and +outside things--we never really, in knowledge, attain to the things we +are supposed to be knowing about, but only to the ideas of those +things. The relation of mind, idea, and object, on this view, is +utterly obscure, and, so far as I can see, nothing discoverable by +inspection warrants the intrusion of the idea between the mind and the +object. I suspect that the view is fostered by the dislike of +relations, and that it is felt the mind could not know objects unless +there were something "in" the mind which could be called the state of +knowing the object. Such a view, however, leads at once to a vicious +endless regress, since the relation of idea to object will have to be +explained by supposing that the idea itself has an idea of the object, +and so on _ad infinitum_. I therefore see no reason to believe that, +when we are acquainted with an object, there is in us something which +can be called the "idea" of the object. On the contrary, I hold that +acquaintance is wholly a relation, not demanding any such constituent +of the mind as is supposed by advocates of "ideas." This is, of +course, a large question, and one which would take us far from our +subject if it were adequately discussed. I therefore content myself +with the above indications, and with the corollary that, in judging, +the actual objects concerning which we judge, rather than any supposed +purely mental entities, are constituents of the complex which is the +judgment. + +When, therefore, I say that we must substitute for "Julius Cæsar" some +description of Julius Cæsar, in order to discover the meaning of a +judgment nominally about him, I am not saying that we must substitute +an idea. Suppose our description is "the man whose name was _Julius +Cæsar_." Let our judgment be "Julius Cæsar was assassinated." Then it +becomes "the man whose name was _Julius Cæsar_ was assassinated." Here +_Julius Cæsar_ is a noise or shape with which we are acquainted, and +all the other constituents of the judgment (neglecting the tense in +"was") are _concepts_ with which we are acquainted. Thus our judgment +is wholly reduced to constituents with which we are acquainted, but +Julius Cæsar himself has ceased to be a constituent of our judgment. +This, however, requires a proviso, to be further explained shortly, +namely that "the man whose name was _Julius Cæsar_" must not, as a +whole, be a constituent of our judgment, that is to say, this phrase +must not, as a whole, have a meaning which enters into the judgment. +Any right analysis of the judgment, therefore, must break up this +phrase, and not treat it as a subordinate complex which is part of the +judgment. The judgment "the man whose name was _Julius Cæsar_ was +assassinated" may be interpreted as meaning "one and only one man was +called _Julius Cæsar_, and that one was assassinated." Here it is +plain that there is no constituent corresponding to the phrase "the +man whose name was _Julius Cæsar_." Thus there is no reason to regard +this phrase as expressing a constituent of the judgment, and we have +seen that this phrase must be broken up if we are to be acquainted +with all the constituents of the judgment. This conclusion, which we +have reached from considerations concerned with the theory of +knowledge, is also forced upon us by logical considerations, which +must now be briefly reviewed. + +It is common to distinguish two aspects, _meaning_ and _denotation_, +such phrases as "the author of Waverley." The meaning will be a +certain complex, consisting (at least) of authorship and Waverley with +some relation; the denotation will be Scott. Similarly "featherless +bipeds" will have a complex meaning, containing as constituents the +presence of two feet and the absence of feathers, while its denotation +will be the class of men. Thus when we say "Scott is the author of +Waverley" or "men are the same as featherless bipeds," we are +asserting an identity of denotation, and this assertion is worth +making because of the diversity of meaning.[43] I believe that the +duality of meaning and denotation, though capable of a true +interpretation, is misleading if taken as fundamental. The denotation, +I believe, is not a constituent of the proposition, except in the case +of proper names, i.e. of words which do not assign a property to an +object, but merely and solely name it. And I should hold further that, +in this sense, there are only two words which are strictly proper +names of particulars, namely, "I" and "this."[44] + +One reason for not believing the denotation to be a constituent of the +proposition is that we may know the proposition even when we are not +acquainted with the denotation. The proposition "the author of +Waverley is a novelist" was known to people who did not know that "the +author of Waverley" denoted Scott. This reason has been already +sufficiently emphasised. + +A second reason is that propositions concerning "the so-and-so" are +possible even when "the so-and-so" has no denotation. Take, e.g. "the +golden mountain does not exist" or "the round square is +self-contradictory." If we are to preserve the duality of meaning and +denotation, we have to say, with Meinong, that there are such objects +as the golden mountain and the round square, although these objects do +not have being. We even have to admit that the existent round square +is existent, but does not exist.[45] Meinong does not regard this as a +contradiction, but I fail to see that it is not one. Indeed, it seems +to me evident that the judgment "there is no such object as the round +square" does not presuppose that there is such an object. If this is +admitted, however, we are led to the conclusion that, by parity of +form, no judgment concerning "the so-and-so" actually involves the +so-and-so as a constituent. + +Miss Jones[46] contends that there is no difficulty in admitting +contradictory predicates concerning such an object as "the present +King of France," on the ground that this object is in itself +contradictory. Now it might, of course, be argued that this object, +unlike the round square, is not self-contradictory, but merely +non-existent. This, however, would not go to the root of the matter. +The real objection to such an argument is that the law of +contradiction ought not to be stated in the traditional form "A is not +both B and not B," but in the form "no proposition is both true and +false." The traditional form only applies to certain propositions, +namely, to those which attribute a predicate to a subject. When the +law is stated of propositions, instead of being stated concerning +subjects and predicates, it is at once evident that propositions about +the present King of France or the round square can form no exception, +but are just as incapable of being both true and false as other +propositions. Miss Jones[47] argues that "Scott is the author of +Waverley" asserts identity of denotation between _Scott_ and _the +author of Waverley_. But there is some difficulty in choosing among +alternative meanings of this contention. In the first place, it should +be observed that _the author of Waverley_ is not a _mere_ name, like +_Scott_. _Scott_ is merely a noise or shape conventionally used to +designate a certain person; it gives us no information about that +person, and has nothing that can be called meaning as opposed to +denotation. (I neglect the fact, considered above, that even proper +names, as a rule, really stand for descriptions.) But _the author of +Waverley_ is not merely conventionally a name for Scott; the element +of mere convention belongs here to the separate words, _the_ and +_author_ and _of_ and _Waverley_. Given what these words stand for, +_the author of Waverley_ is no longer arbitrary. When it is said that +Scott is the author of Waverley, we are not stating that these are two +_names_ for one man, as we should be if we said "Scott is Sir Walter." +A man's name is what he is called, but however much Scott had been +called the author of Waverley, that would not have made him be the +author; it was necessary for him actually to write Waverley, which was +a fact having nothing to do with names. + +If, then, we are asserting identity of denotation, we must not mean by +_denotation_ the mere relation of a name to the thing named. In fact, +it would be nearer to the truth to say that the _meaning_ of "Scott" +is the _denotation_ of "the author of Waverley." The relation of +"Scott" to Scott is that "Scott" means Scott, just as the relation of +"author" to the concept which is so called is that "author" means this +concept. Thus if we distinguish meaning and denotation in "the author +of Waverley," we shall have to say that "Scott" has meaning but not +denotation. Also when we say "Scott is the author of Waverley," the +_meaning_ of "the author of Waverley" is relevant to our assertion. +For if the denotation alone were relevant, any other phrase with the +same denotation would give the same proposition. Thus "Scott is the +author of Marmion" would be the same proposition as "Scott is the +author of Waverley." But this is plainly not the case, since from the +first we learn that Scott wrote Marmion and from the second we learn +that he wrote Waverley, but the first tells us nothing about Waverley +and the second nothing about Marmion. Hence the meaning of "the author +of Waverley," as opposed to the denotation, is certainly relevant to +"Scott is the author of Waverley." + +We have thus agreed that "the author of Waverley" is not a mere name, +and that its meaning is relevant in propositions in which it occurs. +Thus if we are to say, as Miss Jones does, that "Scott is the author +of Waverley" asserts an identity of denotation, we must regard the +denotation of "the author of Waverley" as the denotation of what is +_meant_ by "the author of Waverley." Let us call the meaning of "the +author of Waverley" M. Thus M is what "the author of Waverley" means. +Then we are to suppose that "Scott is the author of Waverley" means +"Scott is the denotation of M." But here we are explaining our +proposition by another of the same form, and thus we have made no +progress towards a real explanation. "The denotation of M," like "the +author of Waverley," has both meaning and denotation, on the theory we +are examining. If we call its meaning M', our proposition becomes +"Scott is the denotation of M'." But this leads at once to an endless +regress. Thus the attempt to regard our proposition as asserting +identity of denotation breaks down, and it becomes imperative to find +some other analysis. When this analysis has been completed, we shall +be able to reinterpret the phrase "identity of denotation," which +remains obscure so long as it is taken as fundamental. + +The first point to observe is that, in any proposition about "the +author of Waverley," provided Scott is not explicitly mentioned, the +denotation itself, i.e. Scott, does not occur, but only the concept of +denotation, which will be represented by a variable. Suppose we say +"the author of Waverley was the author of Marmion," we are certainly +not saying that both were Scott--we may have forgotten that there was +such a person as Scott. We are saying that there is some man who was +the author of Waverley and the author of Marmion. That is to say, +there is some one who wrote Waverley and Marmion, and no one else +wrote them. Thus the identity is that of a variable, i.e. of an +indefinite subject, "some one." This is why we can understand +propositions about "the author of Waverley," without knowing who he +was. When we say "the author of Waverley was a poet," we mean "one and +only one man wrote Waverley, and he was a poet"; when we say "the +author of Waverley was Scott" we mean "one and only one man wrote +Waverley, and he was Scott." Here the identity is between a variable, +i.e. an indeterminate subject ("he"), and Scott; "the author of +Waverley" has been analysed away, and no longer appears as a +constituent of the proposition.[48] + +The reason why it is imperative to analyse away the phrase "the author +of Waverley" may be stated as follows. It is plain that when we say +"the author of Waverley is the author of Marmion," the _is_ expresses +identity. We have seen also that the common _denotation_, namely +Scott, is not a constituent of this proposition, while the _meanings_ +(if any) of "the author of Waverley" and "the author of Marmion" are +not identical. We have seen also that, in any sense in which the +meaning of a word is a constituent of a proposition in whose verbal +expression the word occurs, "Scott" means the actual man Scott, in the +same sense (so far as concerns our present discussion) in which +"author" means a certain universal. Thus, if "the author of Waverley" +were a subordinate complex in the above proposition, its _meaning_ +would have to be what was said to be identical with the _meaning_ of +"the author of Marmion." This is plainly not the case; and the only +escape is to say that "the author of Waverley" does not, by itself, +have a meaning, though phrases of which it is part do have a meaning. +That is, in a right analysis of the above proposition, "the author of +Waverley" must disappear. This is effected when the above proposition +is analysed as meaning: "Some one wrote Waverley and no one else did, +and that some one also wrote Marmion and no one else did." This may be +more simply expressed by saying that the propositional function "_x_ +wrote Waverley and Marmion, and no one else did" is capable of truth, +i.e. some value of _x_ makes it true, but no other value does. Thus +the true subject of our judgment is a propositional function, i.e. a +complex containing an undetermined constituent, and becoming a +proposition as soon as this constituent is determined. + +We may now define the denotation of a phrase. If we know that the +proposition "_a_ is the so-and-so" is true, i.e. that _a_ is so-and-so +and nothing else is, we call _a_ the denotation of the phrase "the +so-and-so." A very great many of the propositions we naturally make +about "the so-and-so" will remain true or remain false if we +substitute _a_ for "the so-and-so," where _a_ is the denotation of +"the so-and-so." Such propositions will also remain true or remain +false if we substitute for "the so-and-so" any other phrase having the +same denotation. Hence, as practical men, we become interested in the +denotation more than in the description, since the denotation decides +as to the truth or falsehood of so many statements in which the +description occurs. Moreover, as we saw earlier in considering the +relations of description and acquaintance, we often wish to reach the +denotation, and are only hindered by lack of acquaintance: in such +cases the description is merely the means we employ to get as near as +possible to the denotation. Hence it naturally comes to be supposed +that the denotation is part of the proposition in which the +description occurs. But we have seen, both on logical and on +epistemological grounds, that this is an error. The actual object (if +any) which is the denotation is not (unless it is explicitly +mentioned) a constituent of propositions in which descriptions occur; +and this is the reason why, in order to understand such propositions, +we need acquaintance with the constituents of the description, but do +not need acquaintance with its denotation. The first result of +analysis, when applied to propositions whose grammatical subject is +"the so-and-so," is to substitute a variable as subject; i.e. we +obtain a proposition of the form: "There is _something_ which alone is +so-and-so, and that _something_ is such-and-such." The further +analysis of propositions concerning "the so-and-so" is thus merged in +the problem of the nature of the variable, i.e. of the meanings of +_some_, _any_, and _all_. This is a difficult problem, concerning +which I do not intend to say anything at present. + +To sum up our whole discussion. We began by distinguishing two sorts +of knowledge of objects, namely, knowledge by _acquaintance_ and +knowledge by _description_. Of these it is only the former that brings +the object itself before the mind. We have acquaintance with +sense-data, with many universals, and possibly with ourselves, but not +with physical objects or other minds. We have _descriptive_ knowledge +of an object when we know that it is _the_ object having some property +or properties with which we are acquainted; that is to say, when we +know that the property or properties in question belong to one object +and no more, we are said to have knowledge of that one object by +description, whether or not we are acquainted with the object. Our +knowledge of physical objects and of other minds is only knowledge by +description, the descriptions involved being usually such as involve +sense-data. All propositions intelligible to us, whether or not they +primarily concern things only known to us by description, are composed +wholly of constituents with which we are acquainted, for a constituent +with which we are not acquainted is unintelligible to us. A judgment, +we found, is not composed of mental constituents called "ideas," but +consists of an occurrence whose constituents are a mind[49] and +certain objects, particulars or universals. (One at least must be a +universal.) When a judgment is rightly analysed, the objects which are +constituents of it must all be objects with which the mind which is a +constituent of it is acquainted. This conclusion forces us to analyse +descriptive phrases occurring in propositions, and to say that the +objects denoted by such phrases are not constituents of judgments in +which such phrases occur (unless these objects are explicitly +mentioned). This leads us to the view (recommended also on purely +logical grounds) that when we say "the author of Marmion was the +author of Waverley," Scott himself is not a constituent of our +judgment, and that the judgment cannot be explained by saying that it +affirms identity of denotation with diversity of meaning. It also, +plainly, does not assert identity of meaning. Such judgments, +therefore, can only be analysed by breaking up the descriptive +phrases, introducing a variable, and making propositional functions +the ultimate subjects. In fact, "the so-and-so is such-and-such" will +mean that "_x_ is so-and-so and nothing else is, and _x_ is +such-and-such" is capable of truth. The analysis of such judgments +involves many fresh problems, but the discussion of these problems is +not undertaken in the present paper. + +FOOTNOTES: + +[40] See references later. + +[41] _Philosophical Essays_, "The Nature of Truth." I have been +persuaded by Mr. Wittgenstein that this theory is somewhat unduly +simple, but the modification which I believe it to require does not +affect the above argument [1917]. + +[42] Cf. Meinong, _Ueber Annahmen_, _passim_. I formerly supposed, +contrary to Meinong's view, that the relationship of supposing might +be merely that of presentation. In this view I now think I was +mistaken, and Meinong is right. But my present view depends upon the +theory that both in judgment and in assumption there is no single +Objective, but the several constituents of the judgment or assumption +are in a many-term relation to the mind. + +[43] This view has been recently advocated by Miss E.E.C. Jones. "A +New Law of Thought and its Implications," _Mind_, January, 1911. + +[44] I should now exclude "I" from proper names in the strict sense, +and retain only "this" [1917]. + +[45] Meinong, _Ueber Annahmen_, 2nd ed., Leipzig, 1910, p. 141. + +[46] _Mind_, July, 1910, p. 380. + +[47] _Mind_, July, 1910, p. 379. + +[48] The theory which I am advocating is set forth fully, with the +logical grounds in its favour, in _Principia Mathematica_, Vol. I. +Introduction, Chap. III; also, less fully, in _Mind_, October, 1905. + +[49] I use this phrase merely to denote the something psychological +which enters into judgment, without intending to prejudge the question +as to what this something is. + + + + +INDEX + + +Achilles and the tortoise, 80 ff, 89 ff + +Acquaintance, the relation of, 209 ff + +Alexander, 125 + +American Realists, the, 134 + +Aristotle, 42, 76, 97 + + +Bacon, 41 + +Bergson, 14 ff, 22, 105, 128, 185 ff, 203 + +Berkeley, 97, 132 + +Blake, 1 + +Bosanquet, 99 + +Broad, 89 _n_ + + +Calculus, the, 82 + +Cantor, Georg, 64, 81 ff, 85, 91 + +Carlyle, 50, 82 + +Cause, the conception of, 135 _n_, 180 ff + +Christianity and renunciation, 51 + +Chuang Tz[)u], 106 + +Construction of permanent things and matter, 169 ff + +Constructions, logical, 155 ff + + +Darwin, 15, 23, 43 + +Dedekind, 64, 81 ff, 85 + +Descartes, 97, 126 + +Descriptions, 175, 214 ff + + +Education, 37 ff + +Euclid, 62, 92, 94 + +Evolutionism, 23 ff, 28 + + +Fano, 93 + +Faraday, 34 + +Free will, 205 ff + +Frege, 78 _n_ + + +Galileo, 42 + +Gladstone, 177 + +Good and evil, 26 ff + + +Hegel, 8, 10, 18, 85, 97, 105 ff + +Heine, 113 + +Heraclitus, 1 ff, 10 + +Hertz, 34 + +Holt, 177 _n_ + +Hume, 1, 97 + + +Infinite, the mathematical, 84 ff + + +James, William, 100 + +Jones, Miss E.E.C., 224 _n_, 225 + +Judgment, 219 ff + + +Kant, 85, 96, 97, 99, 118 ff + +Knowledge by acquaintance, 209 ff; + by description, 214 ff + + +Laplace, 23 + +Leibniz, 76, 79, 82 ff, 97, 126, 144, 160 + +Locke, 97 + +Logic, the laws of, 68 ff + + +Macaulay and Taylor's theorem, 95 + +Malthus, 43 + +Mathematics, 58 ff; + and the Metaphysicians, 74 ff; + and logic, 75 ff; + and the infinitesimal, 82 ff + +Matter, the nature of, 125 ff; + definition of, 164 ff + +Maxwell, 34 + +Meaning and denotation, 223 ff + +Meinong, 174, 220 _n_, 225 + +Militarism, 50 + +Mill, 185, 193 ff + +Mysticism and logic, 1 ff + + +Necessity, the notion of, 207 ff + +Nietzsche, 22, 50 + +Nunn, 125, 137 _n_, 153 + + +Parmenides, 7 ff, 18, 21 + +Particulars, awareness of, 210 ff + +Peano, 78 ff, 93 ff + +Perspectives, 139 ff; + the space of, 158 ff + +Philosophy and logic, 111 + +Physics, sense-data and, 145 ff + +Pierce, 76 _n_ + +Plato, 1 ff, 10, 30, 60, 97 + +Pragmatism, 22, 105 + + +Realism and the analytic method, 120 ff + +Reason and intuition, 12 ff + +Relatives, the logic of, 76 + +Robb, 167 _n_ + + +Santayana, 20 + +Sense-data, 147, 210 ff; + and physics, 145 ff + +Sensibilia, 148 ff + +Space, 138 ff; + private, 158 ff; + the logical problem, 114 ff; + the problem in physics, 115 ff; + the epistemological problem, 118 ff + +Systems, deterministic, 199; + practically isolated, 198; + relatively isolated, 197; + mechanical, 201 + + +Time, 10, 21 ff, 141 ff, 167 ff + +Tristram Shandy, the paradox of, 90 ff + + +Unity and Plurality, 18 ff + +Universals, awareness of, 212 ff + + +Ward, 180 + +Weierstrass, 80, 82, 95 + +Whitehead, 117, 157, 175 + +Wolf, 173 + + +Zeno the Eleatic, 64, 80, 84, 89 ff + + + + +[Illustration] + +GEORGE ALLEN & UNWIN LTD + +_London: 40 Museum Street, W.C. 1_ + +_Auckland: 24 Wyndham Street_ +_Bombay: 15 Graham Road, Ballard Estate, Bombay 1_ +_Calcutta: 17 Chittaranjan Avenue, Calcutta 13_ +_Cape Town: 109 Long Street_ +_Karachi: Metherson's Estate, Wood Street, Karachi 2_ +_New Delhi: 13-14 Ajmeri Gate Extension, New Delhi 1_ +_São Paulo: Avenida 9 de Julho 1138--Ap. 51_ +_Singapore, South East Asia and Far East, 36c, Prinsep Street_ +_Sydney, N.S.W.: Bradbury House, 55 York Street_ +_Toronto: 91 Wellington Street West_ + + * * * * * + + +-----------------------------------------------------------+ + | Typographical errors corrected in text: | + | | + | Page 111: charateristic replaced with characteristic | + | Page 71: inexpugnable replaced with inexpungable | + | | + +-----------------------------------------------------------+ + + * * * * * + + + + + + +End of the Project Gutenberg EBook of Mysticism and Logic and Other Essays, by +Bertrand Russell + +*** END OF THIS PROJECT GUTENBERG EBOOK MYSTICISM AND LOGIC *** + +***** This file should be named 25447-8.txt or 25447-8.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/2/5/4/4/25447/ + +Produced by Jeannie Howse, Adrian Mastronardi and the +Online Distributed Proofreading Team at http://www.pgdp.net +(This file was produced from images generously made +available by The Internet Archive/Canadian Libraries) + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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