,transcriber's ,notes3 ,every effort has been made to replicate this text as faithfully as possible1 including inconsistencies in spelling and hyphenation4 ,some corrections of spelling and punctuation have been made4 ,they are listed at the end of the text4 ,italic text has been marked with underscores4 ,greek text has been transliterated and marked with tildes4 ,,our ,,knowledge ,,of ,,the ,,external ,,world ,,as ,a ,,field ,,for ,,scientific ,,method ,,in ,,philosophy ,,by ,,the ,,same ,,author ,,introduction ,,to ,,mathematical ,,philosophy ,second ,edition4 ,demy #hvo1 #abs4 #f;d4 net4 ,,the ,,analysis ,,of ,,mind ,demy #hvo1 #afs4 net4 ,,principles ,,of ,,social ,,reconstruction ,seventh ,impression4 ,cr4 #hvo1 #es4 net2 ,limp1 #cs4 #f;d4 net4 ,,roads ,,to ,,freedom3 ,,socialism1 ,,anarchism ,,and ,,syndicalism ,fourth ,impression4 ,cr4 #hvo1 #es4 net2 ,limp1 #cs4 #f;d4 net4 ,,the ,,practice ,,and ,,theory ,,of ,,bolshevism ,second ,impression4 ,cr4 #hvo1 #fs4 net4 ,,our ,,knowledge ,,of ,,the ,,external ,,world ,,as ,a ,,field ,,for ,,scientific ,,method ,,in ,,philosophy ,,by ,,bertrand ,,russell1 ,f4,r4,s ,,london3 ,,george ,,allen ,,unwin ,,ltd ,,ruskin ,,house1 #dj ,,museum ,,street1 ,w4,c4 #a ,first published in #aiad by ,the ,open ,court ,publishing ,company ,reissued by ,george ,allen ,unwin ,ltd4 #aibb ,,preface ,the following lectures#a are an attempt to show1 by means of examples1 the nature1 capacity1 and limitations of the logical-analytic method in philosophy4 ,this method1 of which the first complete example is to be found in the writings of ,frege1 has gradually1 in the course of actual research1 increasingly forced itself upon me as something perfectly definite1 capable of embodiment in maxims1 and adequate1 in all branches of philosophy1 to yield whatever objective scientific knowledge it is possible to obtain4 ,most of the methods hitherto practised have professed to lead to more ambitious results than any that logical analysis can claim to reach1 but unfortunately these results have always been such as many competent philosophers considered inadmissible4 ,regarded merely as hypotheses and as aids to imagination1 the great systems of the past serve a very useful purpose1 and are abundantly worthy of study4 ,but something different is required if philosophy is to become a science1 and to aim at results independent of the tastes and temperament of the philosopher who advocates them4 ,in what follows1 ,i have endeavoured to show1 however imperfectly1 the way by which ,i believe that this desideratum is to be found4 #a ,delivered as ,lowell ,lectures in ,boston1 in ,march and ,april #aiad4 ,the central problem by which ,i have sought to illustrate method is the problem of the relation between the crude data of sense and the space1 time1 and matter of mathematical physics4 ,i have been made aware of the importance of this problem by my friend and collaborator ,dr ,whitehead1 to whom are due almost all the differences between the views advocated here and those suggested in ,the ,problems of ,philosophy4#b ,i owe to him the definition of points1 the suggestion for the treatment of instants and things1 and the whole conception of the world of physics as a construction rather than an inference4 ,what is said on these topics here is1 in fact1 a rough preliminary account of the more precise results which he is giving in the fourth volume of our ,principia ,mathematica4#c ,it will be seen that if his way of dealing with these topics is capable of being successfully carried through1 a wholly new light is thrown on the time-honoured controversies of realists and idealists1 and a method is obtained of solving all that is soluble in their problem4 #b ,london and ,new ,york1 #aiab 7,home ,university ,library74 #c ,the first volume was published at ,cambridge in #aiaj1 the second in #aiab1 and the third in #aiac4 ,the speculations of the past as to the reality or unreality of the world of physics were baffled1 at the outset1 by the absence of any satisfactory theory of the mathematical infinite4 ,this difficulty has been removed by the work of ,georg ,cantor4 ,but the positive and detailed solution of the problem by means of mathematical constructions based upon sensible objects as data has only been rendered possible by the growth of mathematical logic1 without which it is practically impossible to manipulate ideas of the requisite abstractness and complexity4 ,this aspect1 which is somewhat obscured in a merely popular outline such as is contained in the following lectures1 will become plain as soon as ,dr ,whitehead's work is published4 ,in pure logic1 which1 however1 will be very briefly discussed in these lectures1 ,i have had the benefit of vitally important discoveries1 not yet published1 by my friend ,mr ,ludwig ,wittgenstein4 ,since my purpose was to illustrate method1 ,i have included much that is tentative and incomplete1 for it is not by the study of finished structures alone that the manner of construction can be learnt4 ,except in regard to such matters as ,cantor's theory of infinity1 no finality is claimed for the theories suggested2 but ,i believe that where they are found to require modification1 this will be discovered by substantially the same method as that which at present makes them appear probable1 and it is on this ground that ,i ask the reader to be tolerant of their incompleteness4 ,cambridge1 ,june #aiad4 ,,contents ,,lecture ,,page ,i4 ,current ,tendencies #c ,,ii4 ,logic as the ,essence of ,philosophy #cc ,,iii4 ,on our ,knowledge of the ,external ,world #fc ,,iv4 ,the ,world of ,physics and the ,world of ,sense #aja ,v4 ,the ,theory of ,continuity #abi ,,vi4 ,the ,problem of ,infinity considered ,historically #aee ,,vii4 ,the ,positive ,theory of ,infinity #ahe ,,viii4 ,on the ,notion of ,cause1 with ,applications to the ,free-will ,problem #baa ,index #bdc ,,lecture ,i ,,current ,,tendencies ,philosophy1 from the earliest times1 has made greater claims1 and achieved fewer results1 than any other branch of learning4 ,ever since ,thales said that all is water1 philosophers have been ready with glib assertions about the sum-total of things2 and equally glib denials have come from other philosophers ever since ,thales was contradicted by ,anaximander4 ,i believe that the time has now arrived when this unsatisfactory state of things can be brought to an end4 ,in the following course of lectures ,i shall try1 chiefly by taking certain special problems as examples1 to indicate wherein the claims of philosophers have been excessive1 and why their achievements have not been greater4 ,the problems and the method of philosophy have1 ,i believe1 been misconceived by all schools1 many of its traditional problems being insoluble with our means of knowledge1 while other more neglected but not less important problems can1 by a more patient and more adequate method1 be solved with all the precision and certainty to which the most advanced sciences have attained4 ,among present-day philosophies1 we may distinguish three principal types1 often combined in varying proportions by a single philosopher1 but in essence and tendency distinct4 ,the first of these1 which ,i shall call the classical tradition1 descends in the main from ,kant and ,hegel2 it represents the attempt to adapt to present needs the methods and results of the great constructive philosophers from ,plato downwards4 ,the second type1 which may be called evolutionism1 derived its predominance from ,darwin1 and must be reckoned as having had ,herbert ,spencer for its first philosophical representative2 but in recent times it has become1 chiefly through ,william ,james and ,m4 ,bergson1 far bolder and far more searching in its innovations than it was in the hands of ,herbert ,spencer4 ,the third type1 which may be called logical atomism for want of a better name1 has gradually crept into philosophy through the critical scrutiny of mathematics4 ,this type of philosophy1 which is the one that ,i wish to advocate1 has not as yet many whole-hearted adherents1 but the new realism which owes its inception to ,harvard is very largely impregnated with its spirit4 ,it represents1 ,i believe1 the same kind of advance as was introduced into physics by ,galileo3 the substitution of piecemeal1 detailed1 and verifiable results for large untested generalities recommended only by a certain appeal to imagination4 ,but before we can understand the changes advocated by this new philosophy1 we must briefly examine and criticise the other two types with which it has to contend4 ,a4 ,the ,classical ,tradition ,twenty years ago1 the classical tradition1 having vanquished the opposing tradition of the ,english empiricists1 held almost unquestioned sway in all ,anglo-,saxon universities4 ,at the present day1 though it is losing ground1 many of the most prominent teachers still adhere to it4 ,in academic ,france1 in spite of ,m4 ,bergson1 it is far stronger than all its opponents combined2 and in ,germany it has many vigorous advocates4 ,nevertheless1 it represents on the whole a decaying force1 and it has failed to adapt itself to the temper of the age4 ,its advocates are1 in the main1 those whose extra-philosophical knowledge is literary1 rather than those who have felt the inspiration of science4 ,there are1 apart from reasoned arguments1 certain general intellectual forces against it--the same general forces which are breaking down the other great syntheses of the past1 and making our age one of bewildered groping where our ancestors walked in the clear daylight of unquestioning certainty4 ,the original impulse out of which the classical tradition developed was the naive faith of the ,greek philosophers in the omnipotence of reasoning4 ,the discovery of geometry had intoxicated them1 and its a priori deductive method appeared capable of universal application4 ,they would prove1 for instance1 that all reality is one1 that there is no such thing as change1 that the world of sense is a world of mere illusion2 and the strangeness of their results gave them no qualms because they believed in the correctness of their reasoning4 ,thus it came to be thought that by mere thinking the most surprising and important truths concerning the whole of reality could be established with a certainty which no contrary observations could shake4 ,as the vital impulse of the early philosophers died away1 its place was taken by authority and tradition1 reinforced1 in the ,middle ,ages and almost to our own day1 by systematic theology4 ,modern philosophy1 from ,descartes onwards1 though not bound by authority like that of the ,middle ,ages1 still accepted more or less uncritically the ,aristotelian logic4 ,moreover1 it still believed1 except in ,great ,britain1 that a priori reasoning could reveal otherwise undiscoverable secrets about the universe1 and could prove reality to be quite different from what1 to direct observation1 it appears to be4 ,it is this belief1 rather than any particular tenets resulting from it1 that ,i regard as the distinguishing characteristic of the classical tradition1 and as hitherto the main obstacle to a scientific attitude in philosophy4 ,the nature of the philosophy embodied in the classical tradition may be made clearer by taking a particular exponent as an illustration4 ,for this purpose1 let us consider for a moment the doctrines of ,mr ,bradley1 who is probably the most distinguished living representative of this school4 ,mr ,bradley's ,appearance and ,reality is a book consisting of two parts1 the first called ,appearance1 the second ,reality4 ,the first part examines and condemns almost all that makes up our everyday world3 things and qualities1 relations1 space and time1 change1 causation1 activity1 the self4 ,all these1 though in some sense facts which qualify reality1 are not real as they appear4 ,what is real is one single1 indivisible1 timeless whole1 called the ,absolute1 which is in some sense spiritual1 but does not consist of souls1 or of thought and will as we know them4 ,and all this is established by abstract logical reasoning professing to find self-contradictions in the categories condemned as mere appearance1 and to leave no tenable alternative to the kind of ,absolute which is finally affirmed to be real4 ,one brief example may suffice to illustrate ,mr ,bradley's method4 ,the world appears to be full of many things with various relations to each other--right and left1 before and after1 father and son1 and so on4 ,but relations1 according to ,mr ,bradley1 are found on examination to be self-contradictory and therefore impossible4 ,he first argues that1 if there are relations1 there must be qualities between which they hold4 ,this part of his argument need not detain us4 ,he then proceeds3 ,but how the relation can stand to the qualities is1 on the other side1 unintelligible4 ,if it is nothing to the qualities1 then they are not related at all2 and1 if so1 as we saw1 they have ceased to be qualities1 and their relation is a nonentity4 ,but if it is to be something to them1 then clearly we shall require a new connecting relation4 ,for the relation hardly can be the mere adjective of one or both of its terms2 or1 at least1 as such it seems indefensible4 ,and1 being something itself1 if it does not itself bear a relation to the terms1 in what intelligible way will it succeed in being anything to them8 ,but here again we are hurried off into the eddy of a hopeless process1 since we are forced to go on finding new relations without end4 ,the links are united by a link1 and this bond of union is a link which also has two ends2 and these require each a fresh link to connect them with the old4 ,the problem is to find how the relation can stand to its qualities1 and this problem is insoluble4#d #d ,appearance and ,reality1 pp4 #cb-cc4 ,i do not propose to examine this argument in detail1 or to show the exact points where1 in my opinion1 it is fallacious4 ,i have quoted it only as an example of method4 ,most people will admit1 ,i think1 that it is calculated to produce bewilderment rather than conviction1 because there is more likelihood of error in a very subtle1 abstract1 and difficult argument than in so patent a fact as the interrelatedness of the things in the world4 ,to the early ,greeks1 to whom geometry was practically the only known science1 it was possible to follow reasoning with assent even when it led to the strangest conclusions4 ,but to us1 with our methods of experiment and observation1 our knowledge of the long history of a priori errors refuted by empirical science1 it has become natural to suspect a fallacy in any deduction of which the conclusion appears to contradict patent facts4 ,it is easy to carry such suspicion too far1 and it is very desirable1 if possible1 actually to discover the exact nature of the error when it exists4 ,but there is no doubt that what we may call the empirical outlook has become part of most educated people's habit of mind2 and it is this1 rather than any definite argument1 that has diminished the hold of the classical tradition upon students of philosophy and the instructed public generally4 ,the function of logic in philosophy1 as ,i shall try to show at a later stage1 is all-important2 but ,i do not think its function is that which it has in the classical tradition4 ,in that tradition1 logic becomes constructive through negation4 ,where a number of alternatives seem1 at first sight1 to be equally possible1 logic is made to condemn all of them except one1 and that one is then pronounced to be realised in the actual world4 ,thus the world is constructed by means of logic1 with little or no appeal to concrete experience4 ,the true function of logic is1 in my opinion1 exactly the opposite of this4 ,as applied to matters of experience1 it is analytic rather than constructive2 taken a priori1 it shows the possibility of hitherto unsuspected alternatives more often than the impossibility of alternatives which seemed prima facie possible4 ,thus1 while it liberates imagination as to what the world may be1 it refuses to legislate as to what the world is4 ,this change1 which has been brought about by an internal revolution in logic1 has swept away the ambitious constructions of traditional metaphysics1 even for those whose faith in logic is greatest2 while to the many who regard logic as a chimera the paradoxical systems to which it has given rise do not seem worthy even of refutation4 ,thus on all sides these systems have ceased to attract1 and even the philosophical world tends more and more to pass them by4 ,one or two of the favourite doctrines of the school we are considering may be mentioned to illustrate the nature of its claims4 ,the universe1 it tells us1 is an organic unity1 like an animal or a perfect work of art4 ,by this it means1 roughly speaking1 that all the different parts fit together and co-operate1 and are what they are because of their place in the whole4 ,this belief is sometimes advanced dogmatically1 while at other times it is defended by certain logical arguments4 ,if it is true1 every part of the universe is a microcosm1 a miniature reflection of the whole4 ,if we knew ourselves thoroughly1 according to this doctrine1 we should know everything4 ,common sense would naturally object that there are people--say in ,china--with whom our relations are so indirect and trivial that we cannot infer anything important as to them from any fact about ourselves4 ,if there are living beings in ,mars or in more distant parts of the universe1 the same argument becomes even stronger4 ,but further1 perhaps the whole contents of the space and time in which we live form only one of many universes1 each seeming to itself complete4 ,and thus the conception of the necessary unity of all that is resolves itself into the poverty of imagination1 and a freer logic emancipates us from the strait-waistcoated benevolent institution which idealism palms off as the totality of being4 ,another very important doctrine held by most1 though not all1 of the school we are examining is the doctrine that all reality is what is called mental or spiritual1 or that1 at any rate1 all reality is dependent for its existence upon what is mental4 ,this view is often particularised into the form which states that the relation of knower and known is fundamental1 and that nothing can exist unless it either knows or is known4 ,here again the same legislative function is ascribed to a priori argumentation3 it is thought that there are contradictions in an unknown reality4 ,again1 if ,i am not mistaken1 the argument is fallacious1 and a better logic will show that no limits can be set to the extent and nature of the unknown4 ,and when ,i speak of the unknown1 ,i do not mean merely what we personally do not know1 but what is not known to any mind4 ,here as elsewhere1 while the older logic shut out possibilities and imprisoned imagination within the walls of the familiar1 the newer logic shows rather what may happen1 and refuses to decide as to what must happen4 ,the classical tradition in philosophy is the last surviving child of two very diverse parents3 the ,greek belief in reason1 and the mediaeval belief in the tidiness of the universe4 ,to the schoolmen1 who lived amid wars1 massacres1 and pestilences1 nothing appeared so delightful as safety and order4 ,in their idealising dreams1 it was safety and order that they sought3 the universe of ,thomas ,aquinas or ,dante is as small and neat as a ,dutch interior4 ,to us1 to whom safety has become monotony1 to whom the primeval savageries of nature are so remote as to become a mere pleasing condiment to our ordered routine1 the world of dreams is very different from what it was amid the wars of ,guelf and ,ghibelline4 ,hence ,william ,james's protest against what he calls the block universe of the classical tradition2 hence ,nietzsche's worship of force2 hence the verbal bloodthirstiness of many quiet literary men4 ,the barbaric substratum of human nature1 unsatisfied in action1 finds an outlet in imagination4 ,in philosophy1 as elsewhere1 this tendency is visible2 and it is this1 rather than formal argument1 that has thrust aside the classical tradition for a philosophy which fancies itself more virile and more vital4 ,b4 ,evolutionism ,evolutionism1 in one form or another1 is the prevailing creed of our time4 ,it dominates our politics1 our literature1 and not least our philosophy4 ,nietzsche1 pragmatism1 ,bergson1 are phases in its philosophic development1 and their popularity far beyond the circles of professional philosophers shows its consonance with the spirit of the age4 ,it believes itself firmly based on science1 a liberator of hopes1 an inspirer of an invigorating faith in human power1 a sure antidote to the ratiocinative authority of the ,greeks and the dogmatic authority of mediaeval systems4 ,against so fashionable and so agreeable a creed it may seem useless to raise a protest2 and with much of its spirit every modern man must be in sympathy4 ,but ,i think that1 in the intoxication of a quick success1 much that is important and vital to a true understanding of the universe has been forgotten4 ,something of ,hellenism must be combined with the new spirit before it can emerge from the ardour of youth into the wisdom of manhood4 ,and it is time to remember that biology is neither the only science1 nor yet the model to which all other sciences must adapt themselves4 ,evolutionism1 as ,i shall try to show1 is not a truly scientific philosophy1 either in its method or in the problems which it considers4 ,the true scientific philosophy is something more arduous and more aloof1 appealing to less mundane hopes1 and requiring a severer discipline for its successful practice4 ,darwin's ,origin of ,species persuaded the world that the difference between different species of animals and plants is not the fixed1 immutable difference that it appears to be4 ,the doctrine of natural kinds1 which had rendered classification easy and definite1 which was enshrined in the ,aristotelian tradition1 and protected by its supposed necessity for orthodox dogma1 was suddenly swept away for ever out of the biological world4 ,the difference between man and the lower animals1 which to our human conceit appears enormous1 was shown to be a gradual achievement1 involving intermediate beings who could not with certainty be placed either within or without the human family4 ,the sun and planets had already been shown by ,laplace to be very probably derived from a primitive more or less undifferentiated nebula4 ,thus the old fixed landmarks became wavering and indistinct1 and all sharp outlines were blurred4 ,things and species lost their boundaries1 and none could say where they began or where they ended4 ,but if human conceit was staggered for a moment by its kinship with the ape1 it soon found a way to reassert itself1 and that way is the philosophy of evolution4 ,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 known4 ,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 ideal slowly embodying itself in the actual4 ,but such a view1 though it might satisfy ,spencer and those whom we may call ,hegelian evolutionists1 could not be accepted as adequate by the more whole-hearted votaries of change4 ,an ideal to which the world continuously approaches is1 to these minds1 too dead and static to be inspiring4 ,not only the aspirations1 but the ideal too1 must change and develop with the course of evolution2 there must be no fixed goal1 but a continual fashioning of fresh needs by the impulse which is life and which alone gives unity to the process4 ,ever since the seventeenth century1 those whom ,william ,james described as the tender-minded have been engaged in a desperate struggle with the mechanical view of the course of nature which physical science seems to impose4 ,a great part of the attractiveness of the classical tradition was due to the partial escape from mechanism which it provided4 ,but now1 with the influence of biology1 the tender-minded believe that a more radical escape is possible1 sweeping aside not merely the laws of physics1 but the whole apparently immutable apparatus of logic1 with its fixed concepts1 its general principles1 and its reasonings which seem able to compel even the most unwilling assent4 ,the older kind of teleology1 therefore1 which regarded the ,end as a fixed goal1 already partially visible1 towards which we were gradually approaching1 is rejected by ,m4 ,bergson as not allowing enough for the absolute dominion of change4 ,after explaining why he does not accept mechanism1 he proceeds3#e ,but radical finalism is quite as unacceptable1 and for the same reason4 ,the doctrine of teleology1 in its extreme form1 as we find it in ,leibniz for example1 implies that things and beings merely realise a programme previously arranged4 ,but if there is nothing unforeseen1 no invention or creation in the universe1 time is useless again4 ,as in the mechanistic hypothesis1 here again it is supposed that all is given4 ,finalism thus understood is only inverted mechanism4 ,it springs from the same postulate1 with this sole difference1 that in the movement of our finite intellects along successive things1 whose successiveness is reduced to a mere appearance1 it holds in front of us the light with which it claims to guide us1 instead of putting it behind4 ,it substitutes the attraction of the future for the impulsion of the past4 ,but succession remains none the less a mere appearance1 as indeed does movement itself4 ,in the doctrine of ,leibniz1 time is reduced to a confused perception1 relative to the human standpoint1 a perception which would vanish1 like a rising mist1 for a mind seated at the centre of things4 ,yet finalism is not1 like mechanism1 a doctrine with fixed rigid outlines4 ,it admits of as many inflections as we like4 ,the mechanistic philosophy is to be taken or left3 it must be left if the least grain of dust1 by straying from the path foreseen by mechanics1 should show the slightest trace of spontaneity4 ,the doctrine of final causes1 on the contrary1 will never be definitively refuted4 ,if one form of it be put aside1 it will take another4 ,its principle1 which is essentially psychological1 is very flexible4 ,it is so extensible1 and thereby so comprehensive1 that one accepts something of it as soon as one rejects pure mechanism4 ,the theory we shall put forward in this book will therefore necessarily partake of finalism to a certain extent4 #e ,creative ,evolution1 ,english translation1 p4 #da4 ,m4 ,bergson's form of finalism depends upon his conception of life4 ,life1 in his philosophy1 is a continuous stream1 in which all divisions are artificial and unreal4 ,separate things1 beginnings and endings1 are mere convenient fictions3 there is only smooth1 unbroken transition4 ,the beliefs of to-day may count as true to-day1 if they carry us along the stream2 but to-morrow they will be false1 and must be replaced by new beliefs to meet the new situation4 ,all our thinking consists of convenient fictions1 imaginary congealings of the stream3 reality flows on in spite of all our fictions1 and though it can be lived1 it cannot be conceived in thought4 ,somehow1 without explicit statement1 the assurance is slipped in that the future1 though we cannot foresee it1 will be better than the past or the present3 the reader is like the child who expects a sweet because it has been told to open its mouth and shut its eyes4 ,logic1 mathematics1 physics disappear in this philosophy1 because they are too static2 what is real is an impulse and movement towards a goal which1 like the rainbow1 recedes as we advance1 and makes every place different when we reach it from what it appeared to be at a distance4 ,now ,i do not propose at present to enter upon a technical examination of this philosophy4 ,at present ,i wish to make only two criticisms of it--first1 that its truth does not follow from what science has rendered probable concerning the facts of evolution1 and secondly1 that the motives and interests which inspire it are so exclusively practical1 and the problems with which it deals are so special1 that it can hardly be regarded as really touching any of the questions that to my mind constitute genuine philosophy4 7#a7 ,what biology has rendered probable is that the diverse species arose by adaptation from a less differentiated ancestry4 ,this fact is in itself exceedingly interesting1 but it is not the kind of fact from which philosophical consequences follow4 ,philosophy is general1 and takes an impartial interest in all that exists4 ,the changes suffered by minute portions of matter on the earth's surface are very important to us as active sentient beings2 but to us as philosophers they have no greater interest than other changes in portions of matter elsewhere4 ,and if the changes on the earth's surface during the last few millions of years appear to our present ethical notions to be in the nature of a progress1 that gives no ground for believing that progress is a general law of the universe4 ,except under the influence of desire1 no one would admit for a moment so crude a generalisation from such a tiny selection of facts4 ,what does result1 not specially from biology1 but from all the sciences which deal with what exists1 is that we cannot understand the world unless we can understand change and continuity4 ,this is even more evident in physics than it is in biology4 ,but the analysis of change and continuity is not a problem upon which either physics or biology throws any light3 it is a problem of a new kind1 belonging to a different kind of study4 ,the question whether evolutionism offers a true or a false answer to this problem is not1 therefore1 a question to be solved by appeals to particular facts1 such as biology and physics reveal4 ,in assuming dogmatically a certain answer to this question1 evolutionism ceases to be scientific1 yet it is only in touching on this question that evolutionism reaches the subject-matter of philosophy4 ,evolutionism thus consists of two parts3 one not philosophical1 but only a hasty generalisation of the kind which the special sciences might hereafter confirm or confute2 the other not scientific1 but a mere unsupported dogma1 belonging to philosophy by its subject-matter1 but in no way deducible from the facts upon which evolution relies4 7#b7 ,the predominant interest of evolutionism is in the question of human destiny1 or at least of the destiny of ,life4 ,it is more interested in morality and happiness than in knowledge for its own sake4 ,it must be admitted that the same may be said of many other philosophies1 and that a desire for the kind of knowledge which philosophy really can give is very rare4 ,but if philosophy is to become scientific--and it is our object to discover how this can be achieved--it is necessary first and foremost that philosophers should acquire the disinterested intellectual curiosity which characterises the genuine man of science4 ,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 limits4 ,it is impossible to say how much the limits may be enlarged with the progress of science4 ,but what is evident is that any proposition about the future belongs by its subject-matter to some particular science1 and is to be ascertained1 if at all1 by the methods of that science4 ,philosophy is not a short cut to the same kind of results as those of the other sciences3 if it is to be a genuine study1 it must have a province of its own1 and aim at results which the other sciences can neither prove nor disprove4 ,the consideration that philosophy1 if there is such a study1 must consist of propositions which could not occur in the other sciences1 is one which has very far-reaching consequences4 ,all the questions which have what is called a human interest--such1 for example1 as the question of a future life--belong1 at least in theory1 to special sciences1 and are capable1 at least in theory1 of being decided by empirical evidence4 ,philosophers have too often1 in the past1 permitted themselves to pronounce on empirical questions1 and found themselves1 as a result1 in disastrous conflict with well-attested facts4 ,we must1 therefore1 renounce the hope that philosophy can promise satisfaction to our mundane desires4 ,what it can do1 when it is purified from all practical taint1 is to help us to understand the general aspects of the world and the logical analysis of familiar but complex things4 ,through this achievement1 by the suggestion of fruitful hypotheses1 it may be indirectly useful in other sciences1 notably mathematics1 physics1 and psychology4 ,but a genuinely scientific philosophy cannot hope to appeal to any except those who have the wish to understand1 to escape from intellectual bewilderment4 ,it offers1 in its own domain1 the kind of satisfaction which the other sciences offer4 ,but it does not offer1 or attempt to offer1 a solution of the problem of human destiny1 or of the destiny of the universe4 ,evolutionism1 if what has been said is true1 is to be regarded as a hasty generalisation from certain rather special facts1 accompanied by a dogmatic rejection of all attempts at analysis1 and inspired by interests which are practical rather than theoretical4 ,in spite1 therefore1 of its appeal to detailed results in various sciences1 it cannot be regarded as any more genuinely scientific than the classical tradition which it has replaced4 ,how philosophy is to be rendered scientific1 and what is the true subject-matter of philosophy1 ,i shall try to show first by examples of certain achieved results1 and then more generally4 ,we will begin with the problem of the physical conceptions of space and time and matter1 which1 as we have seen1 are challenged by the contentions of the evolutionists4 ,that these conceptions stand in need of reconstruction will be admitted1 and is indeed increasingly urged by physicists themselves4 ,it will also be admitted that the reconstruction must take more account of change and the universal flux than is done in the older mechanics with its fundamental conception of an indestructible matter4 ,but ,i do not think the reconstruction required is on ,bergsonian lines1 nor do ,i think that his rejection of logic can be anything but harmful4 ,i shall not1 however1 adopt the method of explicit controversy1 but rather the method of independent inquiry1 starting from what1 in a pre-philosophic stage1 appear to be facts1 and keeping always as close to these initial data as the requirements of consistency will permit4 ,although explicit controversy is almost always fruitless in philosophy1 owing to the fact that no two philosophers ever understand one another1 yet it seems necessary to say something at the outset in justification of the scientific as against the mystical attitude4 ,metaphysics1 from the first1 has been developed by the union or the conflict of these two attitudes4 ,among the earliest ,greek philosophers1 the ,ionians were more scientific and the ,sicilians more mystical4#f ,but among the latter1 ,pythagoras1 for example1 was in himself a curious mixture of the two tendencies3 the scientific attitude led him to his proposition on right-angled triangles1 while his mystic insight showed him that it is wicked to eat beans4 ,naturally enough1 his followers divided into two sects1 the lovers of right-angled triangles and the abhorrers of beans2 but the former sect died out1 leaving1 however1 a haunting flavour of mysticism over much ,greek mathematical speculation1 and in particular over ,plato's views on mathematics4 ,plato1 of course1 embodies both the scientific and the mystical attitudes in a higher form than his predecessors1 but the mystical attitude is distinctly the stronger of the two1 and secures ultimate victory whenever the conflict is sharp4 ,plato1 moreover1 adopted from the ,eleatics the device of using logic to defeat common sense1 and thus to leave the field clear for mysticism--a device still employed in our own day by the adherents of the classical tradition4 #f ,cf4 ,burnet1 ,early ,greek ,philosophy1 pp4 #he ff4 ,the logic used in defence of mysticism seems to me faulty as logic1 and in a later lecture ,i shall criticise it on this ground4 ,but the more thorough-going mystics do not employ logic1 which they despise3 they appeal instead directly to the immediate deliverance of their insight4 ,now1 although fully developed mysticism is rare in the ,west1 some tincture of it colours the thoughts of many people1 particularly as regards matters on which they have strong convictions not based on evidence4 ,in all who seek passionately for the fugitive and difficult goods1 the conviction is almost irresistible that there is in the world something deeper1 more significant1 than the multiplicity of little facts chronicled and classified by science4 ,behind the veil of these mundane things1 they feel1 something quite different obscurely shimmers1 shining forth clearly in the great moments of illumination1 which alone give anything worthy to be called real knowledge of truth4 ,to seek such moments1 therefore1 is to them the way of wisdom1 rather than1 like the man of science1 to observe coolly1 to analyse without emotion1 and to accept without question the equal reality of the trivial and the important4 ,of the reality or unreality of the mystic's world ,i know nothing4 ,i have no wish to deny it1 nor even to declare that the insight which reveals it is not a genuine insight4 ,what ,i do wish to maintain--and it is here that the scientific attitude becomes imperative--is that insight1 untested and unsupported1 is an insufficient guarantee of truth1 in spite of the fact that much of the most important truth is first suggested by its means4 ,it is common to speak of an opposition between instinct and reason2 in the eighteenth century1 the opposition was drawn in favour of reason1 but under the influence of ,rousseau and the romantic movement instinct was given the preference1 first by those who rebelled against artificial forms of government and thought1 and then1 as the purely rationalistic defence of traditional theology became increasingly difficult1 by all who felt in science a menace to creeds which they associated with a spiritual outlook on life and the world4 ,bergson1 under the name of intuition1 has raised instinct to the position of sole arbiter of metaphysical truth4 ,but in fact the opposition of instinct and reason is mainly illusory4 ,instinct1 intuition1 or insight is what first leads to the beliefs which subsequent reason confirms or confutes2 but the confirmation1 where it is possible1 consists1 in the last analysis1 of agreement with other beliefs no less instinctive4 ,reason is a harmonising1 controlling force rather than a creative one4 ,even in the most purely logical realms1 it is insight that first arrives at what is new4 ,where instinct and reason do sometimes conflict is in regard to single beliefs1 held instinctively1 and held with such determination that no degree of inconsistency with other beliefs leads to their abandonment4 ,instinct1 like all human faculties1 is liable to error4 ,those in whom reason is weak are often unwilling to admit this as regards themselves1 though all admit it in regard to others4 ,where instinct is least liable to error is in practical matters as to which right judgment is a help to survival2 friendship and hostility in others1 for instance1 are often felt with extraordinary discrimination through very careful disguises4 ,but even in such matters a wrong impression may be given by reserve or flattery2 and in matters less directly practical1 such as philosophy deals with1 very strong instinctive beliefs may be wholly mistaken1 as we may come to know through their perceived inconsistency with other equally strong beliefs4 ,it is such considerations that necessitate the harmonising mediation of reason1 which tests our beliefs by their mutual compatibility1 and examines1 in doubtful cases1 the possible sources of error on the one side and on the other4 ,in this there is no opposition to instinct as a whole1 but only to blind reliance upon some one interesting aspect of instinct to the exclusion of other more commonplace but not less trustworthy aspects4 ,it is such onesidedness1 not instinct itself1 that reason aims at correcting4 ,these more or less trite maxims may be illustrated by application to ,bergson's advocacy of intuition as against intellect4 ,there are1 he says1 two profoundly different ways of knowing a thing4 ,the first implies that we move round the object2 the second that we enter into it4 ,the first depends on the point of view at which we are placed and on the symbols by which we express ourselves4 ,the second neither depends on a point of view nor relies on any symbol4 ,the first kind of knowledge may be said to stop at the relative2 the second1 in those cases where it is possible1 to attain the absolute4#g ,the second of these1 which is intuition1 is1 he says1 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 7p4 #f74 ,in illustration1 he mentions self-knowledge3 there is one reality1 at least1 which we all seize from within1 by intuition and not by simple analysis4 ,it is our own personality in its flowing through time--our self which endures 7p4 #h74 ,the rest of ,bergson's philosophy consists in reporting1 through the imperfect medium of words1 the knowledge gained by intuition1 and the consequent complete condemnation of all the pretended knowledge derived from science and common sense4 #g ,introduction to ,metaphysics1 p4 #a4 ,this procedure1 since it takes sides in a conflict of instinctive beliefs1 stands in need of justification by proving the greater trustworthiness of the beliefs on one side than of those on the other4 ,bergson attempts this justification in two ways--first1 by explaining that intellect is a purely practical faculty designed to secure biological success2 secondly1 by mentioning remarkable feats of instinct in animals1 and by pointing out characteristics of the world which1 though intuition can apprehend them1 are baffling to intellect as he interprets it4 ,of ,bergson's theory that intellect is a purely practical faculty developed in the struggle for survival1 and not a source of true beliefs1 we may say1 first1 that it is only through intellect that we know of the struggle for survival and of the biological ancestry of man3 if the intellect is misleading1 the whole of this merely inferred history is presumably untrue4 ,if1 on the other hand1 we agree with ,m4 ,bergson in thinking that evolution took place as ,darwin believed1 then it is not only intellect1 but all our faculties1 that have been developed under the stress of practical utility4 ,intuition is seen at its best where it is directly useful--for example1 in regard to other people's characters and dispositions4 ,bergson apparently holds that capacity for this kind of knowledge is less explicable by the struggle for existence than1 for example1 capacity for pure mathematics4 ,yet the savage deceived by false friendship is likely to pay for his mistake with his life2 whereas even in the most civilised societies men are not put to death for mathematical incompetence4 ,all the most striking of his instances of intuition in animals have a very direct survival value4 ,the fact is1 of course1 that both intuition and intellect have been developed because they are useful1 and that1 speaking broadly1 they are useful when they give truth and become harmful when they give falsehood4 ,intellect1 in civilised man1 like artistic capacity1 has occasionally been developed beyond the point where it is useful to the individual2 intuition1 on the other hand1 seems on the whole to diminish as civilisation increases4 ,speaking broadly1 it is greater in children than in adults1 in the uneducated than in the educated4 ,probably in dogs it exceeds anything to be found in human beings4 ,but those who find in these facts a recommendation of intuition ought to return to running wild in the woods1 dyeing themselves with woad and living on hips and haws4 ,let us next examine whether intuition possesses any such infallibility as ,bergson claims for it4 ,the best instance of it1 according to him1 is our acquaintance with ourselves2 yet self-knowledge is proverbially rare and difficult4 ,most men1 for example1 have in their nature meannesses1 vanities1 and envies of which they are quite unconscious1 though even their best friends can perceive them without any difficulty4 ,it is true that intuition has a convincingness which is lacking to intellect3 while it is present1 it is almost impossible to doubt its truth4 ,but if it should appear1 on examination1 to be at least as fallible as intellect1 its greater subjective certainty becomes a demerit1 making it only the more irresistibly deceptive4 ,apart from self-knowledge1 one of the most notable examples of intuition is the knowledge people believe themselves to possess of those with whom they are in love3 the wall between different personalities seems to become transparent1 and people think they see into another soul as into their own4 ,yet deception in such cases is constantly practised with success2 and even where there is no intentional deception1 experience gradually proves1 as a rule1 that the supposed insight was illusory1 and that the slower1 more groping methods of the intellect are in the long run more reliable4 ,bergson maintains that intellect can only deal with things in so far as they resemble what has been experienced in the past1 while intuition has the power of apprehending the uniqueness and novelty that always belong to each fresh moment4 ,that there is something unique and new at every moment1 is certainly true2 it is also true that this cannot be fully expressed by means of intellectual concepts4 ,only direct acquaintance can give knowledge of what is unique and new4 ,but direct acquaintance of this kind is given fully in sensation1 and does not require1 so far as ,i can see1 any special faculty of intuition for its apprehension4 ,it is neither intellect nor intuition1 but sensation1 that supplies new data2 but when the data are new in any remarkable manner1 intellect is much more capable of dealing with them than intuition would be4 ,the hen with a brood of ducklings no doubt has intuitions which seem to place her inside them1 and not merely to know them analytically2 but when the ducklings take to the water1 the whole apparent intuition is seen to be illusory1 and the hen is left helpless on the shore4 ,intuition1 in fact1 is an aspect and development of instinct1 and1 like all instinct1 is admirable in those customary surroundings which have moulded the habits of the animal in question1 but totally incompetent as soon as the surroundings are changed in a way which demands some non-habitual mode of action4 ,the theoretical understanding of the world1 which is the aim of philosophy1 is not a matter of great practical importance to animals1 or to savages1 or even to most civilised men4 ,it is hardly to be supposed1 therefore1 that the rapid1 rough and ready methods of instinct or intuition will find in this field a favourable ground for their application4 ,it is the older kinds of activity1 which bring out our kinship with remote generations of animal and semi-human ancestors1 that show intuition at its best4 ,in such matters as self-preservation and love1 intuition will act sometimes 7though not always7 with a swiftness and precision which are astonishing to the critical intellect4 ,but philosophy is not one of the pursuits which illustrate our affinity with the past3 it is a highly refined1 highly civilised pursuit1 demanding1 for its success1 a certain liberation from the life of instinct1 and even1 at times1 a certain aloofness from all mundane hopes and fears4 ,it is not in philosophy1 therefore1 that we can hope to see intuition at its best4 ,on the contrary1 since the true objects of philosophy1 and the habits of thought demanded for their apprehension1 are strange1 unusual1 and remote1 it is here1 more almost than anywhere else1 that intellect proves superior to intuition1 and that quick unanalysed convictions are least deserving of uncritical acceptance4 ,before embarking upon the somewhat difficult and abstract discussions which lie before us1 it will be well to take a survey of the hopes we may retain and the hopes we must abandon4 ,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 which1 so far as ,i can see1 philosophy can do anything whatever to satisfy4 ,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 worlds3 it is not a sufficiently abstract difference to come within the province of philosophy4 ,love and hate1 for example1 are ethical opposites1 but to philosophy they are closely analogous attitudes towards objects4 ,the general form and structure of those attitudes towards objects which constitute mental phenomena is a problem for philosophy2 but the difference between love and hate is not a difference of form or structure1 and therefore belongs rather to the special science of psychology than to philosophy4 ,thus the ethical interests which have often inspired philosophers must remain in the background3 some kind of ethical interest may inspire the whole study1 but none must obtrude in the detail or be expected in the special results which are sought4 ,if this view seems at first sight disappointing1 we may remind ourselves that a similar change has been found necessary in all the other sciences4 ,the physicist or chemist is not now required to prove the ethical importance of his ions or atoms2 the biologist is not expected to prove the utility of the plants or animals which he dissects4 ,in pre-scientific ages this was not the case4 ,astronomy1 for example1 was studied because men believed in astrology3 it was thought that the movements of the planets had the most direct and important bearing upon the lives of human beings4 ,presumably1 when this belief decayed and the disinterested study of astronomy began1 many who had found astrology absorbingly interesting decided that astronomy had too little human interest to be worthy of study4 ,physics1 as it appears in ,plato's ,timaeus for example1 is full of ethical notions3 it is an essential part of its purpose to show that the earth is worthy of admiration4 ,the modern physicist1 on the contrary1 though he has no wish to deny that the earth is admirable1 is not concerned1 as physicist1 with its ethical attributes3 he is merely concerned to find out facts1 not to consider whether they are good or bad4 ,in psychology1 the scientific attitude is even more recent and more difficult than in the physical sciences3 it is natural to consider that human nature is either good or bad1 and to suppose that the difference between good and bad1 so all-important in practice1 must be important in theory also4 ,it is only during the last century that an ethically neutral science of psychology has grown up2 and here too ethical neutrality has been essential to scientific success4 ,in philosophy1 hitherto1 ethical neutrality has been seldom sought and hardly ever achieved4 ,men have remembered their wishes1 and have judged philosophies in relation to their wishes4 ,driven from the particular sciences1 the belief that the notions of good and evil must afford a key to the understanding of the world has sought a refuge in philosophy4 ,but even from this last refuge1 if philosophy is not to remain a set of pleasing dreams1 this belief must be driven forth4 ,it is a commonplace that happiness is not best achieved by those who seek it directly2 and it would seem that the same is true of the good4 ,in thought1 at any rate1 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 desires4 ,the immense extension of our knowledge of facts in recent times has had1 as it had in the ,renaissance1 two effects upon the general intellectual outlook4 ,on the one hand1 it has made men distrustful of the truth of wide1 ambitious systems3 theories come and go swiftly1 each serving1 for a moment1 to classify known facts and promote the search for new ones1 but each in turn proving inadequate to deal with the new facts when they have been found4 ,even those who invent the theories do not1 in science1 regard them as anything but a temporary makeshift4 ,the ideal of an all-embracing synthesis1 such as the ,middle ,ages believed themselves to have attained1 recedes further and further beyond the limits of what seems feasible4 ,in such a world1 as in the world of ,montaigne1 nothing seems worth while except the discovery of more and more facts1 each in turn the deathblow to some cherished theory2 the ordering intellect grows weary1 and becomes slovenly through despair4 ,on the other hand1 the new facts have brought new powers2 man's physical control over natural forces has been increasing with unexampled rapidity1 and promises to increase in the future beyond all easily assignable limits4 ,thus alongside of despair as regards ultimate theory there is an immense optimism as regards practice3 what man can do seems almost boundless4 ,the old fixed limits of human power1 such as death1 or the dependence of the race on an equilibrium of cosmic forces1 are forgotten1 and no hard facts are allowed to break in upon the dream of omnipotence4 ,no philosophy is tolerated which sets bounds to man's capacity of gratifying his wishes2 and thus the very despair of theory is invoked to silence every whisper of doubt as regards the possibilities of practical achievement4 ,in the welcoming of new fact1 and in the suspicion of dogmatism as regards the universe at large1 the modern spirit should1 ,i think1 be accepted as wholly an advance4 ,but both in its practical pretensions and in its theoretical despair it seems to me to go too far4 ,most of what is greatest in man is called forth in response to the thwarting of his hopes by immutable natural obstacles2 by the pretence of omnipotence1 he becomes trivial and a little absurd4 ,and on the theoretical side1 ultimate metaphysical truth1 though less all-embracing and harder of attainment than it appeared to some philosophers in the past1 can1 ,i believe1 be discovered by those who are willing to combine the hopefulness1 patience1 and open-mindedness of science with something of the ,greek feeling for beauty in the abstract world of logic and for the ultimate intrinsic value in the contemplation of truth4 ,the philosophy1 therefore1 which is to be genuinely inspired by the scientific spirit1 must deal with somewhat dry and abstract matters1 and must not hope to find an answer to the practical problems of life4 ,to those who wish to understand much of what has in the past been most difficult and obscure in the constitution of the universe1 it has great rewards to offer--triumphs as noteworthy as those of ,newton and ,darwin1 and as important in the long run1 for the moulding of our mental habits4 ,and it brings with it--as a new and powerful method of investigation always does--a sense of power and a hope of progress more reliable and better grounded than any that rests on hasty and fallacious generalisation as to the nature of the universe at large4 ,many hopes which inspired philosophers in the past it cannot claim to fulfil2 but other hopes1 more purely intellectual1 it can satisfy more fully than former ages could have deemed possible for human minds4 ,,lecture ,,ii ,,logic ,,as ,,the ,,essence ,,of ,,philosophy ,the topics we discussed in our first lecture1 and the topics we shall discuss later1 all reduce themselves1 in so far as they are genuinely philosophical1 to problems of logic4 ,this is not due to any accident1 but to the fact that every philosophical problem1 when it is subjected to the necessary analysis and purification1 is found either to be not really philosophical at all1 or else to be1 in the sense in which we are using the word1 logical4 ,but as the word logic is never used in the same sense by two different philosophers1 some explanation of what ,i mean by the word is indispensable at the outset4 ,logic1 in the ,middle ,ages1 and down to the present day in teaching1 meant no more than a scholastic collection of technical terms and rules of syllogistic inference4 ,aristotle had spoken1 and it was the part of humbler men merely to repeat the lesson after him4 ,the trivial nonsense embodied in this tradition is still set in examinations1 and defended by eminent authorities as an excellent propaedeutic1 i4e4 a training in those habits of solemn humbug which are so great a help in later life4 ,but it is not this that ,i mean to praise in saying that all philosophy is logic4 ,ever since the beginning of the seventeenth century1 all vigorous minds that have concerned themselves with inference have abandoned the mediaeval tradition1 and in one way or other have widened the scope of logic4 ,the first extension was the introduction of the inductive method by ,bacon and ,galileo--by the former in a theoretical and largely mistaken form1 by the latter in actual use in establishing the foundations of modern physics and astronomy4 ,this is probably the only extension of the old logic which has become familiar to the general educated public4 ,but induction1 important as it is when regarded as a method of investigation1 does not seem to remain when its work is done3 in the final form of a perfected science1 it would seem that everything ought to be deductive4 ,if induction remains at all1 which is a difficult question1 it will remain merely as one of the principles according to which deductions are effected4 ,thus the ultimate result of the introduction of the inductive method seems not the creation of a new kind of non-deductive reasoning1 but rather the widening of the scope of deduction by pointing out a way of deducing which is certainly not syllogistic1 and does not fit into the mediaeval scheme4 ,the question of the scope and validity of induction is of great difficulty1 and of great importance to our knowledge4 ,take such a question as1 ,will the sun rise to-morrow8 ,our first instinctive feeling is that we have abundant reason for saying that it will1 because it has risen on so many previous mornings4 ,now1 ,i do not myself know whether this does afford a ground or not1 but ,i am willing to suppose that it does4 ,the question which then arises is3 ,what is the principle of inference by which we pass from past sunrises to future ones8 ,the answer given by ,mill is that the inference depends upon the law of causation4 ,let us suppose this to be true2 then what is the reason for believing in the law of causation8 ,there are broadly three possible answers3 7#a7 that it is itself known a priori2 7#b7 that it is a postulate2 7#c7 that it is an empirical generalisation from past instances in which it has been found to hold4 ,the theory that causation is known a priori cannot be definitely refuted1 but it can be rendered very unplausible by the mere process of formulating the law exactly1 and thereby showing that it is immensely more complicated and less obvious than is generally supposed4 ,the theory that causation is a postulate1 i4e4 that it is something which we choose to assert although we know that it is very likely false1 is also incapable of refutation2 but it is plainly also incapable of justifying any use of the law in inference4 ,we are thus brought to the theory that the law is an empirical generalisation1 which is the view held by ,mill4 ,but if so1 how are empirical generalisations to be justified8 ,the evidence in their favour cannot be empirical1 since we wish to argue from what has been observed to what has not been observed1 which can only be done by means of some known relation of the observed and the unobserved2 but the unobserved1 by definition1 is not known empirically1 and therefore its relation to the observed1 if known at all1 must be known independently of empirical evidence4 ,let us see what ,mill says on this subject4 ,according to ,mill1 the law of causation is proved by an admittedly fallible process called induction by simple enumeration4 ,this process1 he says1 consists in ascribing the nature of general truths to all propositions which are true in every instance that we happen to know of4#h ,as regards its fallibility1 he asserts that the precariousness of the method of simple enumeration is in an inverse ratio to the largeness of the generalisation4 ,the process is delusive and insufficient1 exactly in proportion as the subject-matter of the observation is special and limited in extent4 ,as the sphere widens1 this unscientific method becomes less and less liable to mislead2 and the most universal class of truths1 the law of causation for instance1 and the principles of number and of geometry1 are duly and satisfactorily proved by that method alone1 nor are they susceptible of any other proof4#i #h ,logic1 book iii41 chapter iii41 ,sec4 #b4 #i ,book iii41 chapter xxi41 ,sec4 #c4 ,in the above statement1 there are two obvious lacunae3 7#a7 ,how is the method of simple enumeration itself justified8 7#b7 ,what logical principle1 if any1 covers the same ground as this method1 without being liable to its failures8 ,let us take the second question first4 ,a method of proof which1 when used as directed1 gives sometimes truth and sometimes falsehood--as the method of simple enumeration does--is obviously not a valid method1 for validity demands invariable truth4 ,thus1 if simple enumeration is to be rendered valid1 it must not be stated as ,mill states it4 ,we shall have to say1 at most1 that the data render the result probable4 ,causation holds1 we shall say1 in every instance we have been able to test2 therefore it probably holds in untested instances4 ,there are terrible difficulties in the notion of probability1 but we may ignore them at present4 ,we thus have what at least may be a logical principle1 since it is without exception4 ,if a proposition is true in every instance that we happen to know of1 and if the instances are very numerous1 then1 we shall say1 it becomes very probable1 on the data1 that it will be true in any further instance4 ,this is not refuted by the fact that what we declare to be probable does not always happen1 for an event may be probable on the data and yet not occur4 ,it is1 however1 obviously capable of further analysis1 and of more exact statement4 ,we shall have to say something like this3 that every instance of a proposition#aj being true increases the probability of its being true in a fresh instance1 and that a sufficient number of favourable instances will1 in the absence of instances to the contrary1 make the probability of the truth of a fresh instance approach indefinitely near to certainty4 ,some such principle as this is required if the method of simple enumeration is to be valid4 #aj ,or rather a propositional function4 ,but this brings us to our other question1 namely1 how is our principle known to be true8 ,obviously1 since it is required to justify induction1 it cannot be proved by induction2 since it goes beyond the empirical data1 it cannot be proved by them alone2 since it is required to justify all inferences from empirical data to what goes beyond them1 it cannot itself be even rendered in any degree probable by such data4 ,hence1 if it is known1 it is not known by experience1 but independently of experience4 ,i do not say that any such principle is known3 ,i only say that it is required to justify the inferences from experience which empiricists allow1 and that it cannot itself be justified empirically4#aa #aa ,the subject of causality and induction will be discussed again in ,lecture ,,viii4 ,a similar conclusion can be proved by similar arguments concerning any other logical principle4 ,thus logical knowledge is not derivable from experience alone1 and the empiricist's philosophy can therefore not be accepted in its entirety1 in spite of its excellence in many matters which lie outside logic4 ,hegel and his followers widened the scope of logic in quite a different way--a way which ,i believe to be fallacious1 but which requires discussion if only to show how their conception of logic differs from the conception which ,i wish to advocate4 ,in their writings1 logic is practically identical with metaphysics4 ,in broad outline1 the way this came about is as follows4 ,hegel believed that1 by means of a priori reasoning1 it could be shown that the world must have various important and interesting characteristics1 since any world without these characteristics would be impossible and self-contradictory4 ,thus what he calls logic is an investigation of the nature of the universe1 in so far as this can be inferred merely from the principle that the universe must be logically self-consistent4 ,i do not myself believe that from this principle alone anything of importance can be inferred as regards the existing universe4 ,but1 however that may be1 ,i should not regard ,hegel's reasoning1 even if it were valid1 as properly belonging to logic3 it would rather be an application of logic to the actual world4 ,logic itself would be concerned rather with such questions as what self-consistency is1 which ,hegel1 so far as ,i know1 does not discuss4 ,and though he criticises the traditional logic1 and professes to replace it by an improved logic of his own1 there is some sense in which the traditional logic1 with all its faults1 is uncritically and unconsciously assumed throughout his reasoning4 ,it is not in the direction advocated by him1 it seems to me1 that the reform of logic is to be sought1 but by a more fundamental1 more patient1 and less ambitious investigation into the presuppositions which his system shares with those of most other philosophers4 ,the way in which1 as it seems to me1 ,hegel's system assumes the ordinary logic which it subsequently criticises1 is exemplified by the general conception of categories with which he operates throughout4 ,this conception is1 ,i think1 essentially a product of logical confusion1 but it seems in some way to stand for the conception of qualities of ,reality as a whole4 ,mr ,bradley has worked out a theory according to which1 in all judgment1 we are ascribing a predicate to ,reality as a whole2 and this theory is derived from ,hegel4 ,now the traditional logic holds that every proposition ascribes a predicate to a subject1 and from this it easily follows that there can be only one subject1 the ,absolute1 for if there were two1 the proposition that there were two would not ascribe a predicate to either4 ,thus ,hegel's doctrine1 that philosophical propositions must be of the form1 the ,absolute is such-and-such1 depends upon the traditional belief in the universality of the subject-predicate form4 ,this belief1 being traditional1 scarcely self-conscious1 and not supposed to be important1 operates underground1 and is assumed in arguments which1 like the refutation of relations1 appear at first sight such as to establish its truth4 ,this is the most important respect in which ,hegel uncritically assumes the traditional logic4 ,other less important respects--though important enough to be the source of such essentially ,hegelian conceptions as the concrete universal and the union of identity in difference--will be found where he explicitly deals with formal logic4#ab #ab ,see the translation by ,h4 ,s4 ,macran1 ,hegel's ,doctrine of ,formal ,logic1 ,oxford1 #aiab4 ,hegel's argument in this portion of his ,logic depends throughout upon confusing the is of predication1 as in ,socrates is mortal1 with the is of identity1 as in ,socrates is the philosopher who drank the hemlock4 ,owing to this confusion1 he thinks that ,socrates and mortal must be identical4 ,seeing that they are different1 he does not infer1 as others would1 that there is a mistake somewhere1 but that they exhibit identity in difference4 ,again1 ,socrates is particular1 mortal is universal4 ,therefore1 he says1 since ,socrates is mortal1 it follows that the particular is the universal--taking the is to be throughout expressive of identity4 ,but to say the particular is the universal is self-contradictory4 ,again ,hegel does not suspect a mistake but proceeds to synthesise particular and universal in the individual1 or concrete universal4 ,this is an example of how1 for want of care at the start1 vast and imposing systems of philosophy are built upon stupid and trivial confusions1 which1 but for the almost incredible fact that they are unintentional1 one would be tempted to characterise as puns4 ,there is quite another direction in which a large technical development of logic has taken place3 ,i mean the direction of what is called logistic or mathematical logic4 ,this kind of logic is mathematical in two different senses3 it is itself a branch of mathematics1 and it is the logic which is specially applicable to other more traditional branches of mathematics4 ,historically1 it began as merely a branch of mathematics3 its special applicability to other branches is a more recent development4 ,in both respects1 it is the fulfilment of a hope which ,leibniz cherished throughout his life1 and pursued with all the ardour of his amazing intellectual energy4 ,much of his work on this subject has been published recently1 since his discoveries have been remade by others2 but none was published by him1 because his results persisted in contradicting certain points in the traditional doctrine of the syllogism4 ,we now know that on these points the traditional doctrine is wrong1 but respect for ,aristotle prevented ,leibniz from realising that this was possible4#ac #ac ,cf4 ,couturat1 ,la ,logique de ,leibniz1 pp4 #cfa1 #chf4 ,the modern development of mathematical logic dates from ,boole's ,laws of ,thought 7#ahed74 ,but in him and his successors1 before ,peano and ,frege1 the only thing really achieved1 apart from certain details1 was the invention of a mathematical symbolism for deducing consequences from the premisses which the newer methods shared with those of ,aristotle4 ,this subject has considerable interest as an independent branch of mathematics1 but it has very little to do with real logic4 ,the first serious advance in real logic since the time of the ,greeks was made independently by ,peano and ,frege--both mathematicians4 ,they both arrived at their logical results by an analysis of mathematics4 ,traditional logic regarded the two propositions1 ,socrates is mortal and ,all men are mortal1 as being of the same form2#ad ,peano and ,frege showed that they are utterly different in form4 ,the philosophical importance of logic may be illustrated by the fact that this confusion--which is still committed by most writers--obscured not only the whole study of the forms of judgment and inference1 but also the relations of things to their qualities1 of concrete existence to abstract concepts1 and of the world of sense to the world of ,platonic ideas4 ,peano and ,frege1 who pointed out the error1 did so for technical reasons1 and applied their logic mainly to technical developments2 but the philosophical importance of the advance which they made is impossible to exaggerate4 #ad ,it was often recognised that there was some difference between them1 but it was not recognised that the difference is fundamental1 and of very great importance4 ,mathematical logic1 even in its most modern form1 is not directly of philosophical importance except in its beginnings4 ,after the beginnings1 it belongs rather to mathematics than to philosophy4 ,of its beginnings1 which are the only part of it that can properly be called philosophical logic1 ,i shall speak shortly4 ,but even the later developments1 though not directly philosophical1 will be found of great indirect use in philosophising4 ,they enable us to deal easily with more abstract conceptions than merely verbal reasoning can enumerate2 they suggest fruitful hypotheses which otherwise could hardly be thought of2 and they enable us to see quickly what is the smallest store of materials with which a given logical or scientific edifice can be constructed4 ,not only ,frege's theory of number1 which we shall deal with in ,lecture ,,vii41 but the whole theory of physical concepts which will be outlined in our next two lectures1 is inspired by mathematical logic1 and could never have been imagined without it4 ,in both these cases1 and in many others1 we shall appeal to a certain principle called the principle of abstraction4 ,this principle1 which might equally well be called the principle which dispenses with abstraction1 and is one which clears away incredible accumulations of metaphysical lumber1 was directly suggested by mathematical logic1 and could hardly have been proved or practically used without its help4 ,the principle will be explained in our fourth lecture1 but its use may be briefly indicated in advance4 ,when a group of objects have that kind of similarity which we are inclined to attribute to possession of a common quality1 the principle in question shows that membership of the group will serve all the purposes of the supposed common quality1 and that therefore1 unless some common quality is actually known1 the group or class of similar objects may be used to replace the common quality1 which need not be assumed to exist4 ,in this and other ways1 the indirect uses of even the later parts of mathematical logic are very great2 but it is now time to turn our attention to its philosophical foundations4 ,in every proposition and in every inference there is1 besides the particular subject-matter concerned1 a certain form1 a way in which the constituents of the proposition or inference are put together4 ,if ,i say1 ,socrates is mortal1 ,jones is angry1 ,the sun is hot1 there is something in common in these three cases1 something indicated by the word is4 ,what is in common is the form of the proposition1 not an actual constituent4 ,if ,i say a number of things about ,socrates--that he was an ,athenian1 that he married ,xantippe1 that he drank the hemlock--there is a common constituent1 namely ,socrates1 in all the propositions ,i enunciate1 but they have diverse forms4 ,if1 on the other hand1 ,i take any one of these propositions and replace its constituents1 one at a time1 by other constituents1 the form remains constant1 but no constituent remains4 ,take 7say7 the series of propositions1 ,socrates drank the hemlock1 ,coleridge drank the hemlock1 ,coleridge drank opium1 ,coleridge ate opium4 ,the form remains unchanged throughout this series1 but all the constituents are altered4 ,thus form is not another constituent1 but is the way the constituents are put together4 ,it is forms1 in this sense1 that are the proper object of philosophical logic4 ,it is obvious that the knowledge of logical forms is something quite different from knowledge of existing things4 ,the form of ,socrates drank the hemlock is not an existing thing like ,socrates or the hemlock1 nor does it even have that close relation to existing things that drinking has4 ,it is something altogether more abstract and remote4 ,we might understand all the separate words of a sentence without understanding the sentence3 if a sentence is long and complicated1 this is apt to happen4 ,in such a case we have knowledge of the constituents1 but not of the form4 ,we may also have knowledge of the form without having knowledge of the constituents4 ,if ,i say1 ,rorarius drank the hemlock1 those among you who have never heard of ,rorarius 7supposing there are any7 will understand the form1 without having knowledge of all the constituents4 ,in order to understand a sentence1 it is necessary to have knowledge both of the constituents and of the particular instance of the form4 ,it is in this way that a sentence conveys information1 since it tells us that certain known objects are related according to a certain known form4 ,thus some kind of knowledge of logical forms1 though with most people it is not explicit1 is involved in all understanding of discourse4 ,it is the business of philosophical logic to extract this knowledge from its concrete integuments1 and to render it explicit and pure4 ,in all inference1 form alone is essential3 the particular subject-matter is irrelevant except as securing the truth of the premisses4 ,this is one reason for the great importance of logical form4 ,when ,i say1 ,socrates was a man1 all men are mortal1 therefore ,socrates was mortal1 the connection of premisses and conclusion does not in any way depend upon its being ,socrates and man and mortality that ,i am mentioning4 ,the general form of the inference may be expressed in some such words as1 ,if a thing has a certain property1 and whatever has this property has a certain other property1 then the thing in question also has that other property4 ,here no particular things or properties are mentioned3 the proposition is absolutely general4 ,all inferences1 when stated fully1 are instances of propositions having this kind of generality4 ,if they seem to depend upon the subject-matter otherwise than as regards the truth of the premisses1 that is because the premisses have not been all explicitly stated4 ,in logic1 it is a waste of time to deal with inferences concerning particular cases3 we deal throughout with completely general and purely formal implications1 leaving it to other sciences to discover when the hypotheses are verified and when they are not4 ,but the forms of propositions giving rise to inferences are not the simplest forms3 they are always hypothetical1 stating that if one proposition is true1 then so is another4 ,before considering inference1 therefore1 logic must consider those simpler forms which inference presupposes4 ,here the traditional logic failed completely3 it believed that there was only one form of simple proposition 7i4e4 of proposition not stating a relation between two or more other propositions71 namely1 the form which ascribes a predicate to a subject4 ,this is the appropriate form in assigning the qualities of a given thing--we may say this thing is round1 and red1 and so on4 ,grammar favours this form1 but philosophically it is so far from universal that it is not even very common4 ,if we say this thing is bigger than that1 we are not assigning a mere quality of this1 but a relation of this and that4 ,we might express the same fact by saying that thing is smaller than this1 where grammatically the subject is changed4 ,thus propositions stating that two things have a certain relation have a different form from subject-predicate propositions1 and the failure to perceive this difference or to allow for it has been the source of many errors in traditional metaphysics4 ,the belief or unconscious conviction that all propositions are of the subject-predicate form--in other words1 that every fact consists in some thing having some quality--has rendered most philosophers incapable of giving any account of the world of science and daily life4 ,if they had been honestly anxious to give such an account1 they would probably have discovered their error very quickly2 but most of them were less anxious to understand the world of science and daily life1 than to convict it of unreality in the interests of a super-sensible real world4 ,belief in the unreality of the world of sense arises with irresistible force in certain moods--moods which1 ,i imagine1 have some simple physiological basis1 but are none the less powerfully persuasive4 ,the conviction born of these moods is the source of most mysticism and of most metaphysics4 ,when the emotional intensity of such a mood subsides1 a man who is in the habit of reasoning will search for logical reasons in favour of the belief which he finds in himself4 ,but since the belief already exists1 he will be very hospitable to any reason that suggests itself4 ,the paradoxes apparently proved by his logic are really the paradoxes of mysticism1 and are the goal which he feels his logic must reach if it is to be in accordance with insight4 ,it is in this way that logic has been pursued by those of the great philosophers who were mystics--notably ,plato1 ,spinoza1 and ,hegel4 ,but since they usually took for granted the supposed insight of the mystic emotion1 their logical doctrines were presented with a certain dryness1 and were believed by their disciples to be quite independent of the sudden illumination from which they sprang4 ,nevertheless their origin clung to them1 and they remained--to borrow a useful word from ,mr ,santayana--malicious in regard to the world of science and common sense4 ,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 belief4 ,the logic of mysticism shows1 as is natural1 the defects which are inherent in anything malicious4 ,while the mystic mood is dominant1 the need of logic is not felt2 as the mood fades1 the impulse to logic reasserts itself1 but with a desire to retain the vanishing insight1 or at least to prove that it was insight1 and that what seems to contradict it is illusion4 ,the logic which thus arises is not quite disinterested or candid1 and is inspired by a certain hatred of the daily world to which it is to be applied4 ,such an attitude naturally does not tend to the best results4 ,everyone knows that to read an author simply in order to refute him is not the way to understand him2 and to read the book of ,nature with a conviction that it is all illusion is just as unlikely to lead to understanding4 ,if our logic is to find the common world intelligible1 it must not be hostile1 but must be inspired by a genuine acceptance such as is not usually to be found among metaphysicians4 ,traditional logic1 since it holds that all propositions have the subject-predicate form1 is unable to admit the reality of relations3 all relations1 it maintains1 must be reduced to properties of the apparently related terms4 ,there are many ways of refuting this opinion2 one of the easiest is derived from the consideration of what are called asymmetrical relations4 ,in order to explain this1 ,i will first explain two independent ways of classifying relations4 ,some relations1 when they hold between ,a and ,b1 also hold between ,b and ,a4 ,such1 for example1 is the relation brother or sister4 ,if ,a is a brother or sister of ,b1 then ,b is a brother or sister of ,a4 ,such again is any kind of similarity1 say similarity of colour4 ,any kind of dissimilarity is also of this kind3 if the colour of ,a is unlike the colour of ,b1 then the colour of ,b is unlike the colour of ,a4 ,relations of this sort are called symmetrical4 ,thus a relation is symmetrical if1 whenever it holds between ,a and ,b1 it also holds between ,b and ,a4 ,all relations that are not symmetrical are called non-symmetrical4 ,thus brother is non-symmetrical1 because1 if ,a is a brother of ,b1 it may happen that ,b is a sister of ,a4 ,a relation is called asymmetrical when1 if it holds between ,a and ,b1 it never holds between ,b and ,a4 ,thus husband1 father1 grandfather1 etc41 are asymmetrical relations4 ,so are before1 after1 greater1 above1 to the right of1 etc4 ,all the relations that give rise to series are of this kind4 ,classification into symmetrical1 asymmetrical1 and merely non-symmetrical relations is the first of the two classifications we had to consider4 ,the second is into transitive1 intransitive1 and merely non-transitive relations1 which are defined as follows4 ,a relation is said to be transitive1 if1 whenever it holds between ,a and ,b and also between ,b and ,c1 it holds between ,a and ,c4 ,thus before1 after1 greater1 above are transitive4 ,all relations giving rise to series are transitive1 but so are many others4 ,the transitive relations just mentioned were asymmetrical1 but many transitive relations are symmetrical--for instance1 equality in any respect1 exact identity of colour1 being equally numerous 7as applied to collections71 and so on4 ,a relation is said to be non-transitive whenever it is not transitive4 ,thus brother is non-transitive1 because a brother of one's brother may be oneself4 ,all kinds of dissimilarity are non-transitive4 ,a relation is said to be intransitive when1 if ,a has the relation to ,b1 and ,b to ,c1 ,a never has it to ,c4 ,thus father is intransitive4 ,so is such a relation as one inch taller or one year later4 ,let us now1 in the light of this classification1 return to the question whether all relations can be reduced to predications4 ,in the case of symmetrical relations--i4e4 relations which1 if they hold between ,a and ,b1 also hold between ,b and ,a--some kind of plausibility can be given to this doctrine4 ,a symmetrical relation which is transitive1 such as equality1 can be regarded as expressing possession of some common property1 while one which is not transitive1 such as inequality1 can be regarded as expressing possession of different properties4 ,but when we come to asymmetrical relations1 such as before and after1 greater and less1 etc41 the attempt to reduce them to properties becomes obviously impossible4 ,when1 for example1 two things are merely known to be unequal1 without our knowing which is greater1 we may say that the inequality results from their having different magnitudes1 because inequality is a symmetrical relation2 but to say that when one thing is greater than another1 and not merely unequal to it1 that means that they have different magnitudes1 is formally incapable of explaining the facts4 ,for if the other thing had been greater than the one1 the magnitudes would also have been different1 though the fact to be explained would not have been the same4 ,thus mere difference of magnitude is not all that is involved1 since1 if it were1 there would be no difference between one thing being greater than another1 and the other being greater than the one4 ,we shall have to say that the one magnitude is greater than the other1 and thus we shall have failed to get rid of the relation greater4 ,in short1 both possession of the same property and possession of different properties are symmetrical relations1 and therefore cannot account for the existence of asymmetrical relations4 ,asymmetrical relations are involved in all series--in space and time1 greater and less1 whole and part1 and many others of the most important characteristics of the actual world4 ,all these aspects1 therefore1 the logic which reduces everything to subjects and predicates is compelled to condemn as error and mere appearance4 ,to those whose logic is not malicious1 such a wholesale condemnation appears impossible4 ,and in fact there is no reason except prejudice1 so far as ,i can discover1 for denying the reality of relations4 ,when once their reality is admitted1 all logical grounds for supposing the world of sense to be illusory disappear4 ,if this is to be supposed1 it must be frankly and simply on the ground of mystic insight unsupported by argument4 ,it is impossible to argue against what professes to be insight1 so long as it does not argue in its own favour4 ,as logicians1 therefore1 we may admit the possibility of the mystic's world1 while yet1 so long as we do not have his insight1 we must continue to study the everyday world with which we are familiar4 ,but when he contends that our world is impossible1 then our logic is ready to repel his attack4 ,and the first step in creating the logic which is to perform this service is the recognition of the reality of relations4 ,relations which have two terms are only one kind of relations4 ,a relation may have three terms1 or four1 or any number4 ,relations of two terms1 being the simplest1 have received more attention than the others1 and have generally been alone considered by philosophers1 both those who accepted and those who denied the reality of relations4 ,but other relations have their importance1 and are indispensable in the solution of certain problems4 ,jealousy1 for example1 is a relation between three people4 ,professor ,royce mentions the relation giving3 when ,a gives ,b to ,c1 that is a relation of three terms4#ae ,when a man says to his wife3 ,my dear1 ,i wish you could induce ,angelina to accept ,edwin1 his wish constitutes a relation between four people1 himself1 his wife1 ,angelina1 and ,edwin4 ,thus such relations are by no means recondite or rare4 ,but in order to explain exactly how they differ from relations of two terms1 we must embark upon a classification of the logical forms of facts1 which is the first business of logic1 and the business in which the traditional logic has been most deficient4 #ae ,encyclopaedia of the ,philosophical ,sciences1 vol4 i4 p4 #ig4 ,the existing world consists of many things with many qualities and relations4 ,a complete description of the existing world would require not only a catalogue of the things1 but also a mention of all their qualities and relations4 ,we should have to know not only this1 that1 and the other thing1 but also which was red1 which yellow1 which was earlier than which1 which was between which two others1 and so on4 ,when ,i speak of a fact1 ,i do not mean one of the simple things in the world2 ,i mean that a certain thing has a certain quality1 or that certain things have a certain relation4 ,thus1 for example1 ,i should not call ,napoleon a fact1 but ,i should call it a fact that he was ambitious1 or that he married ,josephine4 ,now a fact1 in this sense1 is never simple1 but always has two or more constituents4 ,when it simply assigns a quality to a thing1 it has only two constituents1 the thing and the quality4 ,when it consists of a relation between two things1 it has three constituents1 the things and the relation4 ,when it consists of a relation between three things1 it has four constituents1 and so on4 ,the constituents of facts1 in the sense in which we are using the word fact1 are not other facts1 but are things and qualities or relations4 ,when we say that there are relations of more than two terms1 we mean that there are single facts consisting of a single relation and more than two things4 ,i do not mean that one relation of two terms may hold between ,a and ,b1 and also between ,a and ,c1 as1 for example1 a man is the son of his father and also the son of his mother4 ,this constitutes two distinct facts3 if we choose to treat it as one fact1 it is a fact which has facts for its constituents4 ,but the facts ,i am speaking of have no facts among their constituents1 but only things and relations4 ,for example1 when ,a is jealous of ,b on account of ,c1 there is only one fact1 involving three people2 there are not two instances of jealousy1 but only one4 ,it is in such cases that ,i speak of a relation of three terms1 where the simplest possible fact in which the relation occurs is one involving three things in addition to the relation4 ,and the same applies to relations of four terms or five or any other number4 ,all such relations must be admitted in our inventory of the logical forms of facts3 two facts involving the same number of things have the same form1 and two which involve different numbers of things have different forms4 ,given any fact1 there is an assertion which expresses the fact4 ,the fact itself is objective1 and independent of our thought or opinion about it2 but the assertion is something which involves thought1 and may be either true or false4 ,an assertion may be positive or negative3 we may assert that ,charles ,i4 was executed1 or that he did not die in his bed4 ,a negative assertion may be said to be a denial4 ,given a form of words which must be either true or false1 such as ,charles ,i4 died in his bed1 we may either assert or deny this form of words3 in the one case we have a positive assertion1 in the other a negative one4 ,a form of words which must be either true or false ,i shall call a proposition4 ,thus a proposition is the same as what may be significantly asserted or denied4 ,a proposition which expresses what we have called a fact1 i4e4 which1 when asserted1 asserts that a certain thing has a certain quality1 or that certain things have a certain relation1 will be called an atomic proposition1 because1 as we shall see immediately1 there are other propositions into which atomic propositions enter in a way analogous to that in which atoms enter into molecules4 ,atomic propositions1 although1 like facts1 they may have any one of an infinite number of forms1 are only one kind of propositions4 ,all other kinds are more complicated4 ,in order to preserve the parallelism in language as regards facts and propositions1 we shall give the name atomic facts to the facts we have hitherto been considering4 ,thus atomic facts are what determine whether atomic propositions are to be asserted or denied4 ,whether an atomic proposition1 such as this is red1 or this is before that1 is to be asserted or denied can only be known empirically4 ,perhaps one atomic fact may sometimes be capable of being inferred from another1 though this seems very doubtful2 but in any case it cannot be inferred from premisses no one of which is an atomic fact4 ,it follows that1 if atomic facts are to be known at all1 some at least must be known without inference4 ,the atomic facts which we come to know in this way are the facts of sense-perception2 at any rate1 the facts of sense-perception are those which we most obviously and certainly come to know in this way4 ,if we knew all atomic facts1 and also knew that there were none except those we knew1 we should1 theoretically1 be able to infer all truths of whatever form4#af ,thus logic would then supply us with the whole of the apparatus required4 ,but in the first acquisition of knowledge concerning atomic facts1 logic is useless4 ,in pure logic1 no atomic fact is ever mentioned3 we confine ourselves wholly to forms1 without asking ourselves what objects can fill the forms4 ,thus pure logic is independent of atomic facts2 but conversely1 they are1 in a sense1 independent of logic4 ,pure logic and atomic facts are the two poles1 the wholly a priori and the wholly empirical4 ,but between the two lies a vast intermediate region1 which we must now briefly explore4 #af ,this perhaps requires modification in order to include such facts as beliefs and wishes1 since such facts apparently contain propositions as components4 ,such facts1 though not strictly atomic1 must be supposed included if the statement in the text is to be true4 ,molecular propositions are such as contain conjunctions--if1 or1 and1 unless1 etc4--and such words are the marks of a molecular proposition4 ,consider such an assertion as1 ,if it rains1 ,i shall bring my umbrella4 ,this assertion is just as capable of truth or falsehood as the assertion of an atomic proposition1 but it is obvious that either the corresponding fact1 or the nature of the correspondence with fact1 must be quite different from what it is in the case of an atomic proposition4 ,whether it rains1 and whether ,i bring my umbrella1 are each severally matters of atomic fact1 ascertainable by observation4 ,but the connection of the two involved in saying that if the one happens1 then the other will happen1 is something radically different from either of the two separately4 ,it does not require for its truth that it should actually rain1 or that ,i should actually bring my umbrella2 even if the weather is cloudless1 it may still be true that ,i should have brought my umbrella if the weather had been different4 ,thus we have here a connection of two propositions1 which does not depend upon whether they are to be asserted or denied1 but only upon the second being inferable from the first4 ,such propositions1 therefore1 have a form which is different from that of any atomic proposition4 ,such propositions are important to logic1 because all inference depends upon them4 ,if ,i have told you that if it rains ,i shall bring my umbrella1 and if you see that there is a steady downpour1 you can infer that ,i shall bring my umbrella4 ,there can be no inference except where propositions are connected in some such way1 so that from the truth or falsehood of the one something follows as to the truth or falsehood of the other4 ,it seems to be the case that we can sometimes know molecular propositions1 as in the above instance of the umbrella1 when we do not know whether the component atomic propositions are true or false4 ,the practical utility of inference rests upon this fact4 ,the next kind of propositions we have to consider are general propositions1 such as all men are mortal1 all equilateral triangles are equiangular4 ,and with these belong propositions in which the word some occurs1 such as some men are philosophers or some philosophers are not wise4 ,these are the denials of general propositions1 namely 7in the above instances71 of all men are non-philosophers and all philosophers are wise4 ,we will call propositions containing the word some negative general propositions1 and those containing the word all positive general propositions4 ,these propositions1 it will be seen1 begin to have the appearance of the propositions in logical text-books4 ,but their peculiarity and complexity are not known to the text-books1 and the problems which they raise are only discussed in the most superficial manner4 ,when we were discussing atomic facts1 we saw that we should be able1 theoretically1 to infer all other truths by logic if we knew all atomic facts and also knew that there were no other atomic facts besides those we knew4 ,the knowledge that there are no other atomic facts is positive general knowledge2 it is the knowledge that all atomic facts are known to me1 or at least all atomic facts are in this collection--however the collection may be given4 ,it is easy to see that general propositions1 such as all men are mortal1 cannot be known by inference from atomic facts alone4 ,if we could know each individual man1 and know that he was mortal1 that would not enable us to know that all men are mortal1 unless we knew that those were all the men there are1 which is a general proposition4 ,if we knew every other existing thing throughout the universe1 and knew that each separate thing was not an immortal man1 that would not give us our result unless we knew that we had explored the whole universe1 i4e4 unless we knew all things belong to this collection of things ,i have examined4 ,thus general truths cannot be inferred from particular truths alone1 but must1 if they are to be known1 be either self-evident1 or inferred from premisses of which at least one is a general truth4 ,but all empirical evidence is of particular truths4 ,hence1 if there is any knowledge of general truths at all1 there must be some knowledge of general truths which is independent of empirical evidence1 i4e4 does not depend upon the data of sense4 ,the above conclusion1 of which we had an instance in the case of the inductive principle1 is important1 since it affords a refutation of the older empiricists4 ,they believed that all our knowledge is derived from the senses and dependent upon them4 ,we see that1 if this view is to be maintained1 we must refuse to admit that we know any general propositions4 ,it is perfectly possible logically that this should be the case1 but it does not appear to be so in fact1 and indeed no one would dream of maintaining such a view except a theorist at the last extremity4 ,we must therefore admit that there is general knowledge not derived from sense1 and that some of this knowledge is not obtained by inference but is primitive4 ,such general knowledge is to be found in logic4 ,whether there is any such knowledge not derived from logic1 ,i do not know2 but in logic1 at any rate1 we have such knowledge4 ,it will be remembered that we excluded from pure logic such propositions as1 ,socrates is a man1 all men are mortal1 therefore ,socrates is mortal1 because ,socrates and man and mortal are empirical terms1 only to be understood through particular experience4 ,the corresponding proposition in pure logic is3 ,if anything has a certain property1 and whatever has this property has a certain other property1 then the thing in question has the other property4 ,this proposition is absolutely general3 it applies to all things and all properties4 ,and it is quite self-evident4 ,thus in such propositions of pure logic we have the self-evident general propositions of which we were in search4 ,a proposition such as1 ,if ,socrates is a man1 and all men are mortal1 then ,socrates is mortal1 is true in virtue of its form alone4 ,its truth1 in this hypothetical form1 does not depend upon whether ,socrates actually is a man1 nor upon whether in fact all men are mortal2 thus it is equally true when we substitute other terms for ,socrates and man and mortal4 ,the general truth of which it is an instance is purely formal1 and belongs to logic4 ,since it does not mention any particular thing1 or even any particular quality or relation1 it is wholly independent of the accidental facts of the existent world1 and can be known1 theoretically1 without any experience of particular things or their qualities and relations4 ,logic1 we may say1 consists of two parts4 ,the first part investigates what propositions are and what forms they may have2 this part enumerates the different kinds of atomic propositions1 of molecular propositions1 of general propositions1 and so on4 ,the second part consists of certain supremely general propositions1 which assert the truth of all propositions of certain forms4 ,this second part merges into pure mathematics1 whose propositions all turn out1 on analysis1 to be such general formal truths4 ,the first part1 which merely enumerates forms1 is the more difficult1 and philosophically the more important2 and it is the recent progress in this first part1 more than anything else1 that has rendered a truly scientific discussion of many philosophical problems possible4 ,the problem of the nature of judgment or belief may be taken as an example of a problem whose solution depends upon an adequate inventory of logical forms4 ,we have already seen how the supposed universality of the subject-predicate form made it impossible to give a right analysis of serial order1 and therefore made space and time unintelligible4 ,but in this case it was only necessary to admit relations of two terms4 ,the case of judgment demands the admission of more complicated forms4 ,if all judgments were true1 we might suppose that a judgment consisted in apprehension of a fact1 and that the apprehension was a relation of a mind to the fact4 ,from poverty in the logical inventory1 this view has often been held4 ,but it leads to absolutely insoluble difficulties in the case of error4 ,suppose ,i believe that ,charles ,i4 died in his bed4 ,there is no objective fact ,charles ,i4's death in his bed to which ,i can have a relation of apprehension4 ,charles ,i4 and death and his bed are objective1 but they are not1 except in my thought1 put together as my false belief supposes4 ,it is therefore necessary1 in analysing a belief1 to look for some other logical form than a two-term relation4 ,failure to realise this necessity has1 in my opinion1 vitiated almost everything that has hitherto been written on the theory of knowledge1 making the problem of error insoluble and the difference between belief and perception inexplicable4 ,modern logic1 as ,i hope is now evident1 has the effect of enlarging our abstract imagination1 and providing an infinite number of possible hypotheses to be applied in the analysis of any complex fact4 ,in this respect it is the exact opposite of the logic practised by the classical tradition4 ,in that logic1 hypotheses which seem prima facie possible are professedly proved impossible1 and it is decreed in advance that reality must have a certain special character4 ,in modern logic1 on the contrary1 while the prima facie hypotheses as a rule remain admissible1 others1 which only logic would have suggested1 are added to our stock1 and are very often found to be indispensable if a right analysis of the facts is to be obtained4 ,the old logic put thought in fetters1 while the new logic gives it wings4 ,it has1 in my opinion1 introduced the same kind of advance into philosophy as ,galileo introduced into physics1 making it possible at last to see what kinds of problems may be capable of solution1 and what kinds must be abandoned as beyond human powers4 ,and where a solution appears possible1 the new logic provides a method which enables us to obtain results that do not merely embody personal idiosyncrasies1 but must command the assent of all who are competent to form an opinion4 ,,lecture ,,iii ,,on ,,our ,,knowledge ,,of ,,the ,,external ,,world ,philosophy may be approached by many roads1 but one of the oldest and most travelled is the road which leads through doubt as to the reality of the world of sense4 ,in ,indian mysticism1 in ,greek and modern monistic philosophy from ,parmenides onward1 in ,berkeley1 in modern physics1 we find sensible appearance criticised and condemned for a bewildering variety of motives4 ,the mystic condemns it on the ground of immediate knowledge of a more real and significant world behind the veil2 ,parmenides and ,plato condemn it because its continual flux is thought inconsistent with the unchanging nature of the abstract entities revealed by logical analysis2 ,berkeley brings several weapons1 but his chief is the subjectivity of sense-data1 their dependence upon the organisation and point of view of the spectator2 while modern physics1 on the basis of sensible evidence itself1 maintains a mad dance of electrons which has1 superficially at least1 very little resemblance to the immediate objects of sight or touch4 ,every one of these lines of attack raises vital and interesting problems4 ,the mystic1 so long as he merely reports a positive revelation1 cannot be refuted2 but when he denies reality to objects of sense1 he may be questioned as to what he means by reality1 and may be asked how their unreality follows from the supposed reality of his super-sensible world4 ,in answering these questions1 he is led to a logic which merges into that of ,parmenides and ,plato and the idealist tradition4 ,the logic of the idealist tradition has gradually grown very complex and very abstruse1 as may be seen from the ,bradleian sample considered in our first lecture4 ,if we attempted to deal fully with this logic1 we should not have time to reach any other aspect of our subject2 we will therefore1 while acknowledging that it deserves a long discussion1 pass by its central doctrines with only such occasional criticism as may serve to exemplify other topics1 and concentrate our attention on such matters as its objections to the continuity of motion and the infinity of space and time--objections which have been fully answered by modern mathematicians in a manner constituting an abiding triumph for the method of logical analysis in philosophy4 ,these objections and the modern answers to them will occupy our fifth1 sixth1 and seventh lectures4 ,berkeley's attack1 as reinforced by the physiology of the sense-organs and nerves and brain1 is very powerful4 ,i think it must be admitted as probable that the immediate objects of sense depend for their existence upon physiological conditions in ourselves1 and that1 for example1 the coloured surfaces which we see cease to exist when we shut our eyes4 ,but it would be a mistake to infer that they are dependent upon mind1 not real while we see them1 or not the sole basis for our knowledge of the external world4 ,this line of argument will be developed in the present lecture4 ,the discrepancy between the world of physics and the world of sense1 which we shall consider in our fourth lecture1 will be found to be more apparent than real1 and it will be shown that whatever there is reason to believe in physics can probably be interpreted in terms of sense4 ,the instrument of discovery throughout is modern logic1 a very different science from the logic of the text-books and also from the logic of idealism4 ,our second lecture has given a short account of modern logic and of its points of divergence from the various traditional kinds of logic4 ,in our last lecture1 after a discussion of causality and free will1 we shall try to reach a general account of the logical-analytic method of scientific philosophy1 and a tentative estimate of the hopes of philosophical progress which it allows us to entertain4 ,in this lecture1 ,i wish to apply the logical-analytic method to one of the oldest problems of philosophy1 namely1 the problem of our knowledge of the external world4 ,what ,i have to say on this problem does not amount to an answer of a definite and dogmatic kind2 it amounts only to an analysis and statement of the questions involved1 with an indication of the directions in which evidence may be sought4 ,but although not yet a definite solution1 what can be said at present seems to me to throw a completely new light on the problem1 and to be indispensable1 not only in seeking the answer1 but also in the preliminary question as to what parts of our problem may possibly have an ascertainable answer4 ,in every philosophical problem1 our investigation starts from what may be called data1 by which ,i mean matters of common knowledge1 vague1 complex1 inexact1 as common knowledge always is1 but yet somehow commanding our assent as on the whole and in some interpretation pretty certainly true4 ,in the case of our present problem1 the common knowledge involved is of various kinds4 ,there is first our acquaintance with particular objects of daily life--furniture1 houses1 towns1 other people1 and so on4 ,then there is the extension of such particular knowledge to particular things outside our personal experience1 through history and geography1 newspapers1 etc4 ,and lastly1 there is the systematisation of all this knowledge of particulars by means of physical science1 which derives immense persuasive force from its astonishing power of foretelling the future4 ,we are quite willing to admit that there may be errors of detail in this knowledge1 but we believe them to be discoverable and corrigible by the methods which have given rise to our beliefs1 and we do not1 as practical men1 entertain for a moment the hypothesis that the whole edifice may be built on insecure foundations4 ,in the main1 therefore1 and without absolute dogmatism as to this or that special portion1 we may accept this mass of common knowledge as affording data for our philosophical analysis4 ,it may be said--and this is an objection which must be met at the outset--that it is the duty of the philosopher to call in question the admittedly fallible beliefs of daily life1 and to replace them by something more solid and irrefragable4 ,in a sense this is true1 and in a sense it is effected in the course of analysis4 ,but in another sense1 and a very important one1 it is quite impossible4 ,while admitting that doubt is possible with regard to all our common knowledge1 we must nevertheless accept that knowledge in the main if philosophy is to be possible at all4 ,there is not any superfine brand of knowledge1 obtainable by the philosopher1 which can give us a standpoint from which to criticise the whole of the knowledge of daily life4 ,the most that can be done is to examine and purify our common knowledge by an internal scrutiny1 assuming the canons by which it has been obtained1 and applying them with more care and with more precision4 ,philosophy cannot boast of having achieved such a degree of certainty that it can have authority to condemn the facts of experience and the laws of science4 ,the philosophic scrutiny1 therefore1 though sceptical in regard to every detail1 is not sceptical as regards the whole4 ,that is to say1 its criticism of details will only be based upon their relation to other details1 not upon some external criterion which can be applied to all the details equally4 ,the reason for this abstention from a universal criticism is not any dogmatic confidence1 but its exact opposite2 it is not that common knowledge must be true1 but that we possess no radically different kind of knowledge derived from some other source4 ,universal scepticism1 though logically irrefutable1 is practically barren2 it can only1 therefore1 give a certain flavour of hesitancy to our beliefs1 and cannot be used to substitute other beliefs for them4 ,although data can only be criticised by other data1 not by an outside standard1 yet we may distinguish different grades of certainty in the different kinds of common knowledge which we enumerated just now4 ,what does not go beyond our own personal sensible acquaintance must be for us the most certain3 the evidence of the senses is proverbially the least open to question4 ,what depends on testimony1 like the facts of history and geography which are learnt from books1 has varying degrees of certainty according to the nature and extent of the testimony4 ,doubts as to the existence of ,napoleon can only be maintained for a joke1 whereas the historicity of ,agamemnon is a legitimate subject of debate4 ,in science1 again1 we find all grades of certainty short of the highest4 ,the law of gravitation1 at least as an approximate truth1 has acquired by this time the same kind of certainty as the existence of ,napoleon1 whereas the latest speculations concerning the constitution of matter would be universally acknowledged to have as yet only a rather slight probability in their favour4 ,these varying degrees of certainty attaching to different data may be regarded as themselves forming part of our data2 they1 along with the other data1 lie within the vague1 complex1 inexact body of knowledge which it is the business of the philosopher to analyse4 ,the first thing that appears when we begin to analyse our common knowledge is that some of it is derivative1 while some is primitive2 that is to say1 there is some that we only believe because of something else from which it has been inferred in some sense1 though not necessarily in a strict logical sense1 while other parts are believed on their own account1 without the support of any outside evidence4 ,it is obvious that the senses give knowledge of the latter kind3 the immediate facts perceived by sight or touch or hearing do not need to be proved by argument1 but are completely self-evident4 ,psychologists1 however1 have made us aware that what is actually given in sense is much less than most people would naturally suppose1 and that much of what at first sight seems to be given is really inferred4 ,this applies especially in regard to our space-perceptions4 ,for instance1 we instinctively infer the real size and shape of a visible object from its apparent size and shape1 according to its distance and our point of view4 ,when we hear a person speaking1 our actual sensations usually miss a great deal of what he says1 and we supply its place by unconscious inference2 in a foreign language1 where this process is more difficult1 we find ourselves apparently grown deaf1 requiring1 for example1 to be much nearer the stage at a theatre than would be necessary in our own country4 ,thus the first step in the analysis of data1 namely1 the discovery of what is really given in sense1 is full of difficulty4 ,we will1 however1 not linger on this point2 so long as its existence is realised1 the exact outcome does not make any very great difference in our main problem4 ,the next step in our analysis must be the consideration of how the derivative parts of our common knowledge arise4 ,here we become involved in a somewhat puzzling entanglement of logic and psychology4 ,psychologically1 a belief may be called derivative whenever it is caused by one or more other beliefs1 or by some fact of sense which is not simply what the belief asserts4 ,derivative beliefs in this sense constantly arise without any process of logical inference1 merely by association of ideas or some equally extra-logical process4 ,from the expression of a man's face we judge as to what he is feeling3 we say we see that he is angry1 when in fact we only see a frown4 ,we do not judge as to his state of mind by any logical process3 the judgment grows up1 often without our being able to say what physical mark of emotion we actually saw4 ,in such a case1 the knowledge is derivative psychologically2 but logically it is in a sense primitive1 since it is not the result of any logical deduction4 ,there may or may not be a possible deduction leading to the same result1 but whether there is or not1 we certainly do not employ it4 ,if we call a belief logically primitive when it is not actually arrived at by a logical inference1 then innumerable beliefs are logically primitive which psychologically are derivative4 ,the separation of these two kinds of primitiveness is vitally important to our present discussion4 ,when we reflect upon the beliefs which are logically but not psychologically primitive1 we find that1 unless they can on reflection be deduced by a logical process from beliefs which are also psychologically primitive1 our confidence in their truth tends to diminish the more we think about them4 ,we naturally believe1 for example1 that tables and chairs1 trees and mountains1 are still there when we turn our backs upon them4 ,i do not wish for a moment to maintain that this is certainly not the case1 but ,i do maintain that the question whether it is the case is not to be settled off-hand on any supposed ground of obviousness4 ,the belief that they persist is1 in all men except a few philosophers1 logically primitive1 but it is not psychologically primitive2 psychologically1 it arises only through our having seen those tables and chairs1 trees and mountains4 ,as soon as the question is seriously raised whether1 because we have seen them1 we have a right to suppose that they are there still1 we feel that some kind of argument must be produced1 and that if none is forthcoming1 our belief can be no more than a pious opinion4 ,we do not feel this as regards the immediate objects of sense3 there they are1 and as far as their momentary existence is concerned1 no further argument is required4 ,there is accordingly more need of justifying our psychologically derivative beliefs than of justifying those that are primitive4 ,we are thus led to a somewhat vague distinction between what we may call hard data and soft data4 ,this distinction is a matter of degree1 and must not be pressed2 but if not taken too seriously it may help to make the situation clear4 ,i mean by hard data those which resist the solvent influence of critical reflection1 and by soft data those which1 under the operation of this process1 become to our minds more or less doubtful4 ,the hardest of hard data are of two sorts3 the particular facts of sense1 and the general truths of logic4 ,the more we reflect upon these1 the more we realise exactly what they are1 and exactly what a doubt concerning them really means1 the more luminously certain do they become4 ,verbal doubt concerning even these is possible1 but verbal doubt may occur when what is nominally being doubted is not really in our thoughts1 and only words are actually present to our minds4 ,real doubt1 in these two cases1 would1 ,i think1 be pathological4 ,at any rate1 to me they seem quite certain1 and ,i shall assume that you agree with me in this4 ,without this assumption1 we are in danger of falling into that universal scepticism which1 as we saw1 is as barren as it is irrefutable4 ,if we are to continue philosophising1 we must make our bow to the sceptical hypothesis1 and1 while admitting the elegant terseness of its philosophy1 proceed to the consideration of other hypotheses which1 though perhaps not certain1 have at least as good a right to our respect as the hypothesis of the sceptic4 ,applying our distinction of hard and soft data to psychologically derivative but logically primitive beliefs1 we shall find that most1 if not all1 are to be classed as soft data4 ,they may be found1 on reflection1 to be capable of logical proof1 and they then again become believed1 but no longer as data4 ,as data1 though entitled to a certain limited respect1 they cannot be placed on a level with the facts of sense or the laws of logic4 ,the kind of respect which they deserve seems to me such as to warrant us in hoping1 though not too confidently1 that the hard data may prove them to be at least probable4 ,also1 if the hard data are found to throw no light whatever upon their truth or falsehood1 we are justified1 ,i think1 in giving rather more weight to the hypothesis of their truth than to the hypothesis of their falsehood4 ,for the present1 however1 let us confine ourselves to the hard data1 with a view to discovering what sort of world can be constructed by their means alone4 ,our data now are primarily the facts of sense 7i4e4 of our own sense-data7 and the laws of logic4 ,but even the severest scrutiny will allow some additions to this slender stock4 ,some facts of memory--especially of recent memory--seem to have the highest degree of certainty4 ,some introspective facts are as certain as any facts of sense4 ,and facts of sense themselves must1 for our present purposes1 be interpreted with a certain latitude4 ,spatial and temporal relations must sometimes be included1 for example in the case of a swift motion falling wholly within the specious present4 ,and some facts of comparison1 such as the likeness or unlikeness of two shades of colour1 are certainly to be included among hard data4 ,also we must remember that the distinction of hard and soft data is psychological and subjective1 so that1 if there are other minds than our own--which at our present stage must be held doubtful--the catalogue of hard data may be different for them from what it is for us4 ,certain common beliefs are undoubtedly excluded from hard data4 ,such is the belief which led us to introduce the distinction1 namely1 that sensible objects in general persist when we are not perceiving them4 ,such also is the belief in other people's minds3 this belief is obviously derivative from our perception of their bodies1 and is felt to demand logical justification as soon as we become aware of its derivativeness4 ,belief in what is reported by the testimony of others1 including all that we learn from books1 is of course involved in the doubt as to whether other people have minds at all4 ,thus the world from which our reconstruction is to begin is very fragmentary4 ,the best we can say for it is that it is slightly more extensive than the world at which ,descartes arrived by a similar process1 since that world contained nothing except himself and his thoughts4 ,we are now in a position to understand and state the problem of our knowledge of the external world1 and to remove various misunderstandings which have obscured the meaning of the problem4 ,the problem really is3 ,can the existence of anything other than our own hard data be inferred from the existence of those data8 ,but before considering this problem1 let us briefly consider what the problem is not4 ,when we speak of the external world in this discussion1 we must not mean spatially external1 unless space is interpreted in a peculiar and recondite manner4 ,the immediate objects of sight1 the coloured surfaces which make up the visible world1 are spatially external in the natural meaning of this phrase4 ,we feel them to be there as opposed to here2 without making any assumption of an existence other than hard data1 we can more or less estimate the distance of a coloured surface4 ,it seems probable that distances1 provided they are not too great1 are actually given more or less roughly in sight2 but whether this is the case or not1 ordinary distances can certainly be estimated approximately by means of the data of sense alone4 ,the immediately given world is spatial1 and is further not wholly contained within our own bodies4 ,thus our knowledge of what is external in this sense is not open to doubt4 ,another form in which the question is often put is3 ,can we know of the existence of any reality which is independent of ourselves8 ,this form of the question suffers from the ambiguity of the two words independent and self4 ,to take the ,self first3 the question as to what is to be reckoned part of the ,self and what is not1 is a very difficult one4 ,among many other things which we may mean by the ,self1 two may be selected as specially important1 namely1 7#a7 the bare subject which thinks and is aware of objects1 7#b7 the whole assemblage of things that would necessarily cease to exist if our lives came to an end4 ,the bare subject1 if it exists at all1 is an inference1 and is not part of the data2 therefore this meaning of ,self may be ignored in our present inquiry4 ,the second meaning is difficult to make precise1 since we hardly know what things depend upon our lives for their existence4 ,and in this form1 the definition of ,self introduces the word depend1 which raises the same questions as are raised by the word independent4 ,let us therefore take up the word independent1 and return to the ,self later4 ,when we say that one thing is independent of another1 we may mean either that it is logically possible for the one to exist without the other1 or that there is no causal relation between the two such that the one only occurs as the effect of the other4 ,the only way1 so far as ,i know1 in which one thing can be logically dependent upon another is when the other is part of the one4 ,the existence of a book1 for example1 is logically dependent upon that of its pages3 without the pages there would be no book4 ,thus in this sense the question1 ,can we know of the existence of any reality which is independent of ourselves8 reduces to the question1 ,can we know of the existence of any reality of which our ,self is not part8 ,in this form1 the question brings us back to the problem of defining the ,self2 but ,i think1 however the ,self may be defined1 even when it is taken as the bare subject1 it cannot be supposed to be part of the immediate object of sense2 thus in this form of the question we must admit that we can know of the existence of realities independent of ourselves4 ,the question of causal dependence is much more difficult4 ,to know that one kind of thing is causally independent of another1 we must know that it actually occurs without the other4 ,now it is fairly obvious that1 whatever legitimate meaning we give to the ,self1 our thoughts and feelings are causally dependent upon ourselves1 i4e4 do not occur when there is no ,self for them to belong to4 ,but in the case of objects of sense this is not obvious2 indeed1 as we saw1 the common-sense view is that such objects persist in the absence of any percipient4 ,if this is the case1 then they are causally independent of ourselves2 if not1 not4 ,thus in this form the question reduces to the question whether we can know that objects of sense1 or any other objects not our own thoughts and feelings1 exist at times when we are not perceiving them4 ,this form1 in which the difficult word independent no longer occurs1 is the form in which we stated the problem a minute ago4 ,our question in the above form raises two distinct problems1 which it is important to keep separate4 ,first1 can we know that objects of sense1 or very similar objects1 exist at times when we are not perceiving them8 ,secondly1 if this cannot be known1 can we know that other objects1 inferable from objects of sense but not necessarily resembling them1 exist either when we are perceiving the objects of sense or at any other time8 ,this latter problem arises in philosophy as the problem of the thing in itself1 and in science as the problem of matter as assumed in physics4 ,we will consider this latter problem first4 ,owing to the fact that we feel passive in sensation1 we naturally suppose that our sensations have outside causes4 ,now it is necessary here first of all to distinguish between 7#a7 our sensation1 which is a mental event consisting in our being aware of a sensible object1 and 7#b7 the sensible object of which we are aware in sensation4 ,when ,i speak of the sensible object1 it must be understood that ,i do not mean such a thing as a table1 which is both visible and tangible1 can be seen by many people at once1 and is more or less permanent4 ,what ,i mean is just that patch of colour which is momentarily seen when we look at the table1 or just that particular hardness which is felt when we press it1 or just that particular sound which is heard when we rap it4 ,each of these ,i call a sensible object1 and our awareness of it ,i call a sensation4 ,now our sense of passivity1 if it really afforded any argument1 would only tend to show that the sensation has an outside cause2 this cause we should naturally seek in the sensible object4 ,thus there is no good reason1 so far1 for supposing that sensible objects must have outside causes4 ,but both the thing-in-itself of philosophy and the matter of physics present themselves as outside causes of the sensible object as much as of the sensation4 ,what are the grounds for this common opinion8 ,in each case1 ,i think1 the opinion has resulted from the combination of a belief that something which can persist independently of our consciousness makes itself known in sensation1 with the fact that our sensations often change in ways which seem to depend upon us rather than upon anything which would be supposed to persist independently of us4 ,at first1 we believe unreflectingly that everything is as it seems to be1 and that1 if we shut our eyes1 the objects we had been seeing remain as they were though we no longer see them4 ,but there are arguments against this view1 which have generally been thought conclusive4 ,it is extraordinarily difficult to see just what the arguments prove2 but if we are to make any progress with the problem of the external world1 we must try to make up our minds as to these arguments4 ,a table viewed from one place presents a different appearance from that which it presents from another place4 ,this is the language of common sense1 but this language already assumes that there is a real table of which we see the appearances4 ,let us try to state what is known in terms of sensible objects alone1 without any element of hypothesis4 ,we find that as we walk round the table1 we perceive a series of gradually changing visible objects4 ,but in speaking of walking round the table1 we have still retained the hypothesis that there is a single table connected with all the appearances4 ,what we ought to say is that1 while we have those muscular and other sensations which make us say we are walking1 our visual sensations change in a continuous way1 so that1 for example1 a striking patch of colour is not suddenly replaced by something wholly different1 but is replaced by an insensible gradation of slightly different colours with slightly different shapes4 ,this is what we really know by experience1 when we have freed our minds from the assumption of permanent things with changing appearances4 ,what is really known is a correlation of muscular and other bodily sensations with changes in visual sensations4 ,but walking round the table is not the only way of altering its appearance4 ,we can shut one eye1 or put on blue spectacles1 or look through a microscope4 ,all these operations1 in various ways1 alter the visual appearance which we call that of the table4 ,more distant objects will also alter their appearance if 7as we say7 the state of the atmosphere changes--if there is fog or rain or sunshine4 ,physiological changes also alter the appearances of things4 ,if we assume the world of common sense1 all these changes1 including those attributed to physiological causes1 are changes in the intervening medium4 ,it is not quite so easy as in the former case to reduce this set of facts to a form in which nothing is assumed beyond sensible objects4 ,anything intervening between ourselves and what we see must be invisible3 our view in every direction is bounded by the nearest visible object4 ,it might be objected that a dirty pane of glass1 for example1 is visible although we can see things through it4 ,but in this case we really see a spotted patchwork3 the dirtier specks in the glass are visible1 while the cleaner parts are invisible and allow us to see what is beyond4 ,thus the discovery that the intervening medium affects the appearances of things cannot be made by means of the sense of sight alone4 ,let us take the case of the blue spectacles1 which is the simplest1 but may serve as a type for the others4 ,the frame of the spectacles is of course visible1 but the blue glass1 if it is clean1 is not visible4 ,the blueness1 which we say is in the glass1 appears as being in the objects seen through the glass4 ,the glass itself is known by means of the sense of touch4 ,in order to know that it is between us and the objects seen through it1 we must know how to correlate the space of touch with the space of sight4 ,this correlation itself1 when stated in terms of the data of sense alone1 is by no means a simple matter4 ,but it presents no difficulties of principle1 and may therefore be supposed accomplished4 ,when it has been accomplished1 it becomes possible to attach a meaning to the statement that the blue glass1 which we can touch1 is between us and the object seen1 as we say1 through it4 ,but we have still not reduced our statement completely to what is actually given in sense4 ,we have fallen into the assumption that the object of which we are conscious when we touch the blue spectacles still exists after we have ceased to touch them4 ,so long as we are touching them1 nothing except our finger can be seen through the part touched1 which is the only part where we immediately know that there is something4 ,if we are to account for the blue appearance of objects other than the spectacles1 when seen through them1 it might seem as if we must assume that the spectacles still exist when we are not touching them2 and if this assumption really is necessary1 our main problem is answered3 we have means of knowing of the present existence of objects not given in sense1 though of the same kind as objects formerly given in sense4 ,it may be questioned1 however1 whether this assumption is actually unavoidable1 though it is unquestionably the most natural one to make4 ,we may say that the object of which we become aware when we touch the spectacles continues to have effects afterwards1 though perhaps it no longer exists4 ,in this view1 the supposed continued existence of sensible objects after they have ceased to be sensible will be a fallacious inference from the fact that they still have effects4 ,it is often supposed that nothing which has ceased to exist can continue to have effects1 but this is a mere prejudice1 due to a wrong conception of causality4 ,we cannot1 therefore1 dismiss our present hypothesis on the ground of a priori impossibility1 but must examine further whether it can really account for the facts4 ,it may be said that our hypothesis is useless in the case when the blue glass is never touched at all4 ,how1 in that case1 are we to account for the blue appearance of objects8 ,and more generally1 what are we to make of the hypothetical sensations of touch which we associate with untouched visible objects1 which we know would be verified if we chose1 though in fact we do not verify them8 ,must not these be attributed to permanent possession1 by the objects1 of the properties which touch would reveal8 ,let us consider the more general question first4 ,experience has taught us that where we see certain kinds of coloured surfaces we can1 by touch1 obtain certain expected sensations of hardness or softness1 tactile shape1 and so on4 ,this leads us to believe that what is seen is usually tangible1 and that it has1 whether we touch it or not1 the hardness or softness which we should expect to feel if we touched it4 ,but the mere fact that we are able to infer what our tactile sensations would be shows that it is not logically necessary to assume tactile qualities before they are felt4 ,all that is really known is that the visual appearance in question1 together with touch1 will lead to certain sensations1 which can necessarily be determined in terms of the visual appearance1 since otherwise they could not be inferred from it4 ,we can now give a statement of the experienced facts concerning the blue spectacles1 which will supply an interpretation of common-sense beliefs without assuming anything beyond the existence of sensible objects at the times when they are sensible4 ,by experience of the correlation of touch and sight sensations1 we become able to associate a certain place in touch-space with a certain corresponding place in sight-space4 ,sometimes1 namely in the case of transparent things1 we find that there is a tangible object in a touch-place without there being any visible object in the corresponding sight-place4 ,but in such a case as that of the blue spectacles1 we find that whatever object is visible beyond the empty sight-place in the same line of sight has a different colour from what it has when there is no tangible object in the intervening touch-place2 and as we move the tangible object in touch-space1 the blue patch moves in sight-space4 ,if now we find a blue patch moving in this way in sight-space1 when we have no sensible experience of an intervening tangible object1 we nevertheless infer that1 if we put our hand at a certain place in touch-space1 we should experience a certain touch-sensation4 ,if we are to avoid non-sensible objects1 this must be taken as the whole of our meaning when we say that the blue spectacles are in a certain place1 though we have not touched them1 and have only seen other things rendered blue by their interposition4 ,i think it may be laid down quite generally that1 in so far as physics or common sense is verifiable1 it must be capable of interpretation in terms of actual sense-data alone4 ,the reason for this is simple4 ,verification consists always in the occurrence of an expected sense-datum4 ,astronomers tell us there will be an eclipse of the moon3 we look at the moon1 and find the earth's shadow biting into it1 that is to say1 we see an appearance quite different from that of the usual full moon4 ,now if an expected sense-datum constitutes a verification1 what was asserted must have been about sense-data2 or1 at any rate1 if part of what was asserted was not about sense-data1 then only the other part has been verified4 ,there is in fact a certain regularity or conformity to law about the occurrence of sense-data1 but the sense-data that occur at one time are often causally connected with those that occur at quite other times1 and not1 or at least not very closely1 with those that occur at neighbouring times4 ,if ,i look at the moon and immediately afterwards hear a train coming1 there is no very close causal connection between my two sense-data2 but if ,i look at the moon on two nights a week apart1 there is a very close causal connection between the two sense-data4 ,the simplest1 or at least the easiest1 statement of the connection is obtained by imagining a real moon which goes on whether ,i look at it or not1 providing a series of possible sense-data of which only those are actual which belong to moments when ,i choose to look at the moon4 ,but the degree of verification obtainable in this way is very small4 ,it must be remembered that1 at our present level of doubt1 we are not at liberty to accept testimony4 ,when we hear certain noises1 which are those we should utter if we wished to express a certain thought1 we assume that that thought1 or one very like it1 has been in another mind1 and has given rise to the expression which we hear4 ,if at the same time we see a body resembling our own1 moving its lips as we move ours when we speak1 we cannot resist the belief that it is alive1 and that the feelings inside it continue when we are not looking at it4 ,when we see our friend drop a weight upon his toe1 and hear him say--what we should say in similar circumstances1 the phenomena can no doubt be explained without assuming that he is anything but a series of shapes and noises seen and heard by us1 but practically no man is so infected with philosophy as not to be quite certain that his friend has felt the same kind of pain as he himself would feel4 ,we will consider the legitimacy of this belief presently2 for the moment1 ,i only wish to point out that it needs the same kind of justification as our belief that the moon exists when we do not see it1 and that1 without it1 testimony heard or read is reduced to noises and shapes1 and cannot be regarded as evidence of the facts which it reports4 ,the verification of physics which is possible at our present level is1 therefore1 only that degree of verification which is possible by one man's unaided observations1 which will not carry us very far towards the establishment of a whole science4 ,before proceeding further1 let us summarise the argument so far as it has gone4 ,the problem is3 ,can the existence of anything other than our own hard data be inferred from these data8 ,it is a mistake to state the problem in the form3 ,can we know of the existence of anything other than ourselves and our states8 or3 ,can we know of the existence of anything independent of ourselves8 because of the extreme difficulty of defining self and independent precisely4 ,the felt passivity of sensation is irrelevant1 since1 even if it proved anything1 it could only prove that sensations are caused by sensible objects4 ,the natural naive belief is that things seen persist1 when unseen1 exactly or approximately as they appeared when seen2 but this belief tends to be dispelled by the fact that what common sense regards as the appearance of one object changes with what common sense regards as changes in the point of view and in the intervening medium1 including in the latter our own sense-organs and nerves and brain4 ,this fact1 as just stated1 assumes1 however1 the common-sense world of stable objects which it professes to call in question2 hence1 before we can discover its precise bearing on our problem1 we must find a way of stating it which does not involve any of the assumptions which it is designed to render doubtful4 ,what we then find1 as the bare outcome of experience1 is that gradual changes in certain sense-data are correlated with gradual changes in certain others1 or 7in the case of bodily motions7 with the other sense-data themselves4 ,the assumption that sensible objects persist after they have ceased to be sensible--for example1 that the hardness of a visible body1 which has been discovered by touch1 continues when the body is no longer touched--may be replaced by the statement that the effects of sensible objects persist1 i4e4 that what happens now can only be accounted for1 in many cases1 by taking account of what happened at an earlier time4 ,everything that one man1 by his own personal experience1 can verify in the account of the world given by common sense and physics1 will be explicable by some such means1 since verification consists merely in the occurrence of an expected sense-datum4 ,but what depends upon testimony1 whether heard or read1 cannot be explained in this way1 since testimony depends upon the existence of minds other than our own1 and thus requires a knowledge of something not given in sense4 ,but before examining the question of our knowledge of other minds1 let us return to the question of the thing-in-itself1 namely1 to the theory that what exists at times when we are not perceiving a given sensible object is something quite unlike that object1 something which1 together with us and our sense-organs1 causes our sensations1 but is never itself given in sensation4 ,the thing-in-itself1 when we start from common-sense assumptions1 is a fairly natural outcome of the difficulties due to the changing appearances of what is supposed to be one object4 ,it is supposed that the table 7for example7 causes our sense-data of sight and touch1 but must1 since these are altered by the point of view and the intervening medium1 be quite different from the sense-data to which it gives rise4 ,there is1 in this theory1 a tendency to a confusion from which it derives some of its plausibility1 namely1 the confusion between a sensation as a psychical occurrence and its object4 ,a patch of colour1 even if it only exists when it is seen1 is still something quite different from the seeing of it3 the seeing of it is mental1 but the patch of colour is not4 ,this confusion1 however1 can be avoided without our necessarily abandoning the theory we are examining4 ,the objection to it1 ,i think1 lies in its failure to realise the radical nature of the reconstruction demanded by the difficulties to which it points4 ,we cannot speak legitimately of changes in the point of view and the intervening medium until we have already constructed some world more stable than that of momentary sensation4 ,our discussion of the blue spectacles and the walk round the table has1 ,i hope1 made this clear4 ,but what remains far from clear is the nature of the reconstruction required4 ,although we cannot rest content with the above theory1 in the terms in which it is stated1 we must nevertheless treat it with a certain respect1 for it is in outline the theory upon which physical science and physiology are built1 and it must1 therefore1 be susceptible of a true interpretation4 ,let us see how this is to be done4 ,the first thing to realise is that there are no such things as illusions of sense4 ,objects of sense1 even when they occur in dreams1 are the most indubitably real objects known to us4 ,what1 then1 makes us call them unreal in dreams8 ,merely the unusual nature of their connection with other objects of sense4 ,i dream that ,i am in ,america1 but ,i wake up and find myself in ,england without those intervening days on the ,atlantic which1 alas6 are inseparably connected with a real visit to ,america4 ,objects of sense are called real when they have the kind of connection with other objects of sense which experience has led us to regard as normal2 when they fail in this1 they are called illusions4 ,but what is illusory is only the inferences to which they give rise2 in themselves1 they are every bit as real as the objects of waking life4 ,and conversely1 the sensible objects of waking life must not be expected to have any more intrinsic reality than those of dreams4 ,dreams and waking life1 in our first efforts at construction1 must be treated with equal respect2 it is only by some reality not merely sensible that dreams can be condemned4 ,accepting the indubitable momentary reality of objects of sense1 the next thing to notice is the confusion underlying objections derived from their changeableness4 ,as we walk round the table1 its aspect changes2 but it is thought impossible to maintain either that the table changes1 or that its various aspects can all really exist in the same place4 ,if we press one eyeball1 we shall see two tables2 but it is thought preposterous to maintain that there are really two tables4 ,such arguments1 however1 seem to involve the assumption that there can be something more real than objects of sense4 ,if we see two tables1 then there are two visual tables4 ,it is perfectly true that1 at the same moment1 we may discover by touch that there is only one tactile table4 ,this makes us declare the two visual tables an illusion1 because usually one visual object corresponds to one tactile object4 ,but all that we are warranted in saying is that1 in this case1 the manner of correlation of touch and sight is unusual4 ,again1 when the aspect of the table changes as we walk round it1 and we are told there cannot be so many different aspects in the same place1 the answer is simple3 what does the critic of the table mean by the same place8 ,the use of such a phrase presupposes that all our difficulties have been solved2 as yet1 we have no right to speak of a place except with reference to one given set of momentary sense-data4 ,when all are changed by a bodily movement1 no place remains the same as it was4 ,thus the difficulty1 if it exists1 has at least not been rightly stated4 ,we will now make a new start1 adopting a different method4 ,instead of inquiring what is the minimum of assumption by which we can explain the world of sense1 we will1 in order to have a model hypothesis as a help for the imagination1 construct one possible 7not necessary7 explanation of the facts4 ,it may perhaps then be possible to pare away what is superfluous in our hypothesis1 leaving a residue which may be regarded as the abstract answer to our problem4 ,let us imagine that each mind looks out upon the world1 as in ,leibniz's monadology1 from a point of view peculiar to itself2 and for the sake of simplicity let us confine ourselves to the sense of sight1 ignoring minds which are devoid of this sense4 ,each mind sees at each moment an immensely complex three-dimensional world2 but there is absolutely nothing which is seen by two minds simultaneously4 ,when we say that two people see the same thing1 we always find that1 owing to difference of point of view1 there are differences1 however slight1 between their immediate sensible objects4 7,i am here assuming the validity of testimony1 but as we are only constructing a possible theory1 that is a legitimate assumption47 ,the three-dimensional world seen by one mind therefore contains no place in common with that seen by another1 for places can only be constituted by the things in or around them4 ,hence we may suppose1 in spite of the differences between the different worlds1 that each exists entire exactly as it is perceived1 and might be exactly as it is even if it were not perceived4 ,we may further suppose that there are an infinite number of such worlds which are in fact unperceived4 ,if two men are sitting in a room1 two somewhat similar worlds are perceived by them2 if a third man enters and sits between them1 a third world1 intermediate between the two previous worlds1 begins to be perceived4 ,it is true that we cannot reasonably suppose just this world to have existed before1 because it is conditioned by the sense-organs1 nerves1 and brain of the newly arrived man2 but we can reasonably suppose that some aspect of the universe existed from that point of view1 though no one was perceiving it4 ,the system consisting of all views of the universe perceived and unperceived1 ,i shall call the system of perspectives2 ,i shall confine the expression private worlds to such views of the universe as are actually perceived4 ,thus a private world is a perceived perspective2 but there may be any number of unperceived perspectives4 ,two men are sometimes found to perceive very similar perspectives1 so similar that they can use the same words to describe them4 ,they say they see the same table1 because the differences between the two tables they see are slight and not practically important4 ,thus it is possible1 sometimes1 to establish a correlation by similarity between a great many of the things of one perspective1 and a great many of the things of another4 ,in case the similarity is very great1 we say the points of view of the two perspectives are near together in space2 but this space in which they are near together is totally different from the spaces inside the two perspectives4 ,it is a relation between the perspectives1 and is not in either of them2 no one can perceive it1 and if it is to be known it can be only by inference4 ,between two perceived perspectives which are similar1 we can imagine a whole series of other perspectives1 some at least unperceived1 and such that between any two1 however similar1 there are others still more similar4 ,in this way the space which consists of relations between perspectives can be rendered continuous1 and 7if we choose7 three-dimensional4 ,we can now define the momentary common-sense thing1 as opposed to its momentary appearances4 ,by the similarity of neighbouring perspectives1 many objects in the one can be correlated with objects in the other1 namely1 with the similar objects4 ,given an object in one perspective1 form the system of all the objects correlated with it in all the perspectives2 that system may be identified with the momentary common-sense thing4 ,thus an aspect of a thing is a member of the system of aspects which is the thing at that moment4 7,the correlation of the times of different perspectives raises certain complications1 of the kind considered in the theory of relativity2 but we may ignore these at present47 ,all the aspects of a thing are real1 whereas the thing is a mere logical construction4 ,it has1 however1 the merit of being neutral as between different points of view1 and of being visible to more than one person1 in the only sense in which it can ever be visible1 namely1 in the sense that each sees one of its aspects4 ,it will be observed that1 while each perspective contains its own space1 there is only one space in which the perspectives themselves are the elements4 ,there are as many private spaces as there are perspectives2 there are therefore at least as many as there are percipients1 and there may be any number of others which have a merely material existence and are not seen by anyone4 ,but there is only one perspective-space1 whose elements are single perspectives1 each with its own private space4 ,we have now to explain how the private space of a single perspective is correlated with part of the one all-embracing perspective space4 ,perspective space is the system of points of view of private spaces 7perspectives71 or1 since points of view have not been defined1 we may say it is the system of the private spaces themselves4 ,these private spaces will each count as one point1 or at any rate as one element1 in perspective space4 ,they are ordered by means of their similarities4 ,suppose1 for example1 that we start from one which contains the appearance of a circular disc1 such as would be called a penny1 and suppose this appearance1 in the perspective in question1 is circular1 not elliptic4 ,we can then form a whole series of perspectives containing a graduated series of circular aspects of varying sizes3 for this purpose we only have to move 7as we say7 towards the penny or away from it4 ,the perspectives in which the penny looks circular will be said to lie on a straight line in perspective space1 and their order on this line will be that of the sizes of the circular aspects4 ,moreover--though this statement must be noticed and subsequently examined--the perspectives in which the penny looks big will be said to be nearer to the penny than those in which it looks small4 ,it is to be remarked also that any other thing than our penny might have been chosen to define the relations of our perspectives in perspective space1 and that experience shows that the same spatial order of perspectives would have resulted4 ,in order to explain the correlation of private spaces with perspective space1 we have first to explain what is meant by the place 7in perspective space7 where a thing is4 ,for this purpose1 let us again consider the penny which appears in many perspectives4 ,we formed a straight line of perspectives in which the penny looked circular1 and we agreed that those in which it looked larger were to be considered as nearer to the penny4 ,we can form another straight line of perspectives in which the penny is seen end-on and looks like a straight line of a certain thickness4 ,these two lines will meet in a certain place in perspective space1 i4e4 in a certain perspective1 which may be defined as the place 7in perspective space7 where the penny is4 ,it is true that1 in order to prolong our lines until they reach this place1 we shall have to make use of other things besides the penny1 because1 so far as experience goes1 the penny ceases to present any appearance after we have come so near to it that it touches the eye4 ,but this raises no real difficulty1 because the spatial order of perspectives is found empirically to be independent of the particular things chosen for defining the order4 ,we can1 for example1 remove our penny and prolong each of our two straight lines up to their intersection by placing other pennies further off in such a way that the aspects of the one are circular where those of our original penny were circular1 and the aspects of the other are straight where those of our original penny were straight4 ,there will then be just one perspective in which one of the new pennies looks circular and the other straight4 ,this will be1 by definition1 the place where the original penny was in perspective space4 ,the above is1 of course1 only a first rough sketch of the way in which our definition is to be reached4 ,it neglects the size of the penny1 and it assumes that we can remove the penny without being disturbed by any simultaneous changes in the positions of other things4 ,but it is plain that such niceties cannot affect the principle1 and can only introduce complications in its application4 ,having now defined the perspective which is the place where a given thing is1 we can understand what is meant by saying that the perspectives in which a thing looks large are nearer to the thing than those in which it looks small3 they are1 in fact1 nearer to the perspective which is the place where the thing is4 ,we can now also explain the correlation between a private space and parts of perspective space4 ,if there is an aspect of a given thing in a certain private space1 then we correlate the place where this aspect is in the private space with the place where the thing is in perspective space4 ,we may define here as the place1 in perspective space1 which is occupied by our private world4 ,thus we can now understand what is meant by speaking of a thing as near to or far from here4 ,a thing is near to here if the place where it is is near to my private world4 ,we can also understand what is meant by saying that our private world is inside our head2 for our private world is a place in perspective space1 and may be part of the place where our head is4 ,it will be observed that two places in perspective space are associated with every aspect of a thing3 namely1 the place where the thing is1 and the place which is the perspective of which the aspect in question forms part4 ,every aspect of a thing is a member of two different classes of aspects1 namely3 7#a7 the various aspects of the thing1 of which at most one appears in any given perspective2 7#b7 the perspective of which the given aspect is a member1 i4e4 that in which the thing has the given aspect4 ,the physicist naturally classifies aspects in the first way1 the psychologist in the second4 ,the two places associated with a single aspect correspond to the two ways of classifying it4 ,we may distinguish the two places as that at which1 and that from which1 the aspect appears4 ,the place at which is the place of the thing to which the aspect belongs2 the place from which is the place of the perspective to which the aspect belongs4 ,let us now endeavour to state the fact that the aspect which a thing presents at a given place is affected by the intervening medium4 ,the aspects of a thing in different perspectives are to be conceived as spreading outwards from the place where the thing is1 and undergoing various changes as they get further away from this place4 ,the laws according to which they change cannot be stated if we only take account of the aspects that are near the thing1 but require that we should also take account of the things that are at the places from which these aspects appear4 ,this empirical fact can1 therefore1 be interpreted in terms of our construction4 ,we have now constructed a largely hypothetical picture of the world1 which contains and places the experienced facts1 including those derived from testimony4 ,the world we have constructed can1 with a certain amount of trouble1 be used to interpret the crude facts of sense1 the facts of physics1 and the facts of physiology4 ,it is therefore a world which may be actual4 ,it fits the facts1 and there is no empirical evidence against it2 it also is free from logical impossibilities4 ,but have we any good reason to suppose that it is real8 ,this brings us back to our original problem1 as to the grounds for believing in the existence of anything outside my private world4 ,what we have derived from our hypothetical construction is that there are no grounds against the truth of this belief1 but we have not derived any positive grounds in its favour4 ,we will resume this inquiry by taking up again the question of testimony and the evidence for the existence of other minds4 ,it must be conceded to begin with that the argument in favour of the existence of other people's minds cannot be conclusive4 ,a phantasm of our dreams will appear to have a mind--a mind to be annoying1 as a rule4 ,it will give unexpected answers1 refuse to conform to our desires1 and show all those other signs of intelligence to which we are accustomed in the acquaintances of our waking hours4 ,and yet1 when we are awake1 we do not believe that the phantasm was1 like the appearances of people in waking life1 representative of a private world to which we have no direct access4 ,if we are to believe this of the people we meet when we are awake1 it must be on some ground short of demonstration1 since it is obviously possible that what we call waking life may be only an unusually persistent and recurrent nightmare4 ,it may be that our imagination brings forth all that other people seem to say to us1 all that we read in books1 all the daily1 weekly1 monthly1 and quarterly journals that distract our thoughts1 all the advertisements of soap and all the speeches of politicians4 ,this may be true1 since it cannot be shown to be false1 yet no one can really believe it4 ,is there any logical ground for regarding this possibility as improbable8 ,or is there nothing beyond habit and prejudice8 ,the minds of other people are among our data1 in the very wide sense in which we used the word at first4 ,that is to say1 when we first begin to reflect1 we find ourselves already believing in them1 not because of any argument1 but because the belief is natural to us4 ,it is1 however1 a psychologically derivative belief1 since it results from observation of people's bodies2 and along with other such beliefs1 it does not belong to the hardest of hard data1 but becomes1 under the influence of philosophic reflection1 just sufficiently questionable to make us desire some argument connecting it with the facts of sense4 ,the obvious argument is1 of course1 derived from analogy4 ,other people's bodies behave as ours do when we have certain thoughts and feelings2 hence1 by analogy1 it is natural to suppose that such behaviour is connected with thoughts and feelings like our own4 ,someone says1 ,look out6 and we find we are on the point of being killed by a motor-car2 we therefore attribute the words we heard to the person in question having seen the motor-car first1 in which case there are existing things of which we are not directly conscious4 ,but this whole scene1 with our inference1 may occur in a dream1 in which case the inference is generally considered to be mistaken4 ,is there anything to make the argument from analogy more cogent when we are 7as we think7 awake8 ,the analogy in waking life is only to be preferred to that in dreams on the ground of its greater extent and consistency4 ,if a man were to dream every night about a set of people whom he never met by day1 who had consistent characters and grew older with the lapse of years1 he might1 like the man in ,calderon's play1 find it difficult to decide which was the dream-world and which was the so-called real world4 ,it is only the failure of our dreams to form a consistent whole either with each other or with waking life that makes us condemn them4 ,certain uniformities are observed in waking life1 while dreams seem quite erratic4 ,the natural hypothesis would be that demons and the spirits of the dead visit us while we sleep2 but the modern mind1 as a rule1 refuses to entertain this view1 though it is hard to see what could be said against it4 ,on the other hand1 the mystic1 in moments of illumination1 seems to awaken from a sleep which has filled all his mundane life3 the whole world of sense becomes phantasmal1 and he sees1 with the clarity and convincingness that belongs to our morning realisation after dreams1 a world utterly different from that of our daily cares and troubles4 ,who shall condemn him8 ,who shall justify him8 ,or who shall justify the seeming solidity of the common objects among which we suppose ourselves to live8 ,the hypothesis that other people have minds must1 ,i think1 be allowed to be not susceptible of any very strong support from the analogical argument4 ,at the same time1 it is a hypothesis which systematises a vast body of facts and never leads to any consequences which there is reason to think false4 ,there is therefore nothing to be said against its truth1 and good reason to use it as a working hypothesis4 ,when once it is admitted1 it enables us to extend our knowledge of the sensible world by testimony1 and thus leads to the system of private worlds which we assumed in our hypothetical construction4 ,in actual fact1 whatever we may try to think as philosophers1 we cannot help believing in the minds of other people1 so that the question whether our belief is justified has a merely speculative interest4 ,and if it is justified1 then there is no further difficulty of principle in that vast extension of our knowledge1 beyond our own private data1 which we find in science and common sense4 ,this somewhat meagre conclusion must not be regarded as the whole outcome of our long discussion4 ,the problem of the connection of sense with objective reality has commonly been dealt with from a standpoint which did not carry initial doubt so far as we have carried it2 most writers1 consciously or unconsciously1 have assumed that the testimony of others is to be admitted1 and therefore 7at least by implication7 that others have minds4 ,their difficulties have arisen after this admission1 from the differences in the appearance which one physical object presents to two people at the same time1 or to one person at two times between which it cannot be supposed to have changed4 ,such difficulties have made people doubtful how far objective reality could be known by sense at all1 and have made them suppose that there were positive arguments against the view that it can be so known4 ,our hypothetical construction meets these arguments1 and shows that the account of the world given by common sense and physical science can be interpreted in a way which is logically unobjectionable1 and finds a place for all the data1 both hard and soft4 ,it is this hypothetical construction1 with its reconciliation of psychology and physics1 which is the chief outcome of our discussion4 ,probably the construction is only in part necessary as an initial assumption1 and can be obtained from more slender materials by the logical methods of which we shall have an example in the definitions of points1 instants1 and particles2 but ,i do not yet know to what lengths this diminution in our initial assumptions can be carried4 ,,lecture ,,iv ,,the ,,world ,,of ,,physics ,,and ,,the ,,world ,,of ,,sense ,among the objections to the reality of objects of sense1 there is one which is derived from the apparent difference between matter as it appears in physics and things as they appear in sensation4 ,men of science1 for the most part1 are willing to condemn immediate data as merely subjective1 while yet maintaining the truth of the physics inferred from those data4 ,but such an attitude1 though it may be capable of justification1 obviously stands in need of it2 and the only justification possible must be one which exhibits matter as a logical construction from sense-data--unless1 indeed1 there were some wholly a priori principle by which unknown entities could be inferred from such as are known4 ,it is therefore necessary to find some way of bridging the gulf between the world of physics and the world of sense1 and it is this problem which will occupy us in the present lecture4 ,physicists appear to be unconscious of the gulf1 while psychologists1 who are conscious of it1 have not the mathematical knowledge required for spanning it4 ,the problem is difficult1 and ,i do not know its solution in detail4 ,all that ,i can hope to do is to make the problem felt1 and to indicate the kind of methods by which a solution is to be sought4 ,let us begin by a brief description of the two contrasted worlds4 ,we will take first the world of physics1 for1 though the other world is given while the physical world is inferred1 to us now the world of physics is the more familiar1 the world of pure sense having become strange and difficult to rediscover4 ,physics started from the common-sense belief in fairly permanent and fairly rigid bodies--tables and chairs1 stones1 mountains1 the earth and moon and sun4 ,this common-sense belief1 it should be noticed1 is a piece of audacious metaphysical theorising2 objects are not continually present to sensation1 and it may be doubted whether they are there when they are not seen or felt4 ,this problem1 which has been acute since the time of ,berkeley1 is ignored by common sense1 and has therefore hitherto been ignored by physicists4 ,we have thus here a first departure from the immediate data of sensation1 though it is a departure merely by way of extension1 and was probably made by our savage ancestors in some very remote prehistoric epoch4 ,but tables and chairs1 stones and mountains1 are not quite permanent or quite rigid4 ,tables and chairs lose their legs1 stones are split by frost1 and mountains are cleft by earthquakes and eruptions4 ,then there are other things1 which seem material1 and yet present almost no permanence or rigidity4 ,breath1 smoke1 clouds1 are examples of such things--so1 in a lesser degree1 are ice and snow2 and rivers and seas1 though fairly permanent1 are not in any degree rigid4 ,breath1 smoke1 clouds1 and generally things that can be seen but not touched1 were thought to be hardly real2 to this day the usual mark of a ghost is that it can be seen but not touched4 ,such objects were peculiar in the fact that they seemed to disappear completely1 not merely to be transformed into something else4 ,ice and snow1 when they disappear1 are replaced by water2 and it required no great theoretical effort to invent the hypothesis that the water was the same thing as the ice and snow1 but in a new form4 ,solid bodies1 when they break1 break into parts which are practically the same in shape and size as they were before4 ,a stone can be hammered into a powder1 but the powder consists of grains which retain the character they had before the pounding4 ,thus the ideal of absolutely rigid and absolutely permanent bodies1 which early physicists pursued throughout the changing appearances1 seemed attainable by supposing ordinary bodies to be composed of a vast number of tiny atoms4 ,this billiard-ball view of matter dominated the imagination of physicists until quite modern times1 until1 in fact1 it was replaced by the electromagnetic theory1 which in its turn is developing into a new atomism4 ,apart from the special form of the atomic theory which was invented for the needs of chemistry1 some kind of atomism dominated the whole of traditional dynamics1 and was implied in every statement of its laws and axioms4 ,the pictorial accounts which physicists give of the material world as they conceive it undergo violent changes under the influence of modifications in theory which are much slighter than the layman might suppose from the alterations of the description4 ,certain features1 however1 have remained fairly stable4 ,it is always assumed that there is something indestructible which is capable of motion in space2 what is indestructible is always very small1 but does not always occupy a mere point in space4 ,there is supposed to be one all-embracing space in which the motion takes place1 and until lately we might have assumed one all-embracing time also4 ,but the principle of relativity has given prominence to the conception of local time1 and has somewhat diminished men's confidence in the one even-flowing stream of time4 ,without dogmatising as to the ultimate outcome of the principle of relativity1 however1 we may safely say1 ,i think1 that it does not destroy the possibility of correlating different local times1 and does not therefore have such far-reaching philosophical consequences as is sometimes supposed4 ,in fact1 in spite of difficulties as to measurement1 the one all-embracing time still1 ,i think1 underlies all that physics has to say about motion4 ,we thus have still in physics1 as we had in ,newton's time1 a set of indestructible entities which may be called particles1 moving relatively to each other in a single space and a single time4 ,the world of immediate data is quite different from this4 ,nothing is permanent2 even the things that we think are fairly permanent1 such as mountains1 only become data when we see them1 and are not immediately given as existing at other moments4 ,so far from one all-embracing space being given1 there are several spaces for each person1 according to the different senses which give relations that may be called spatial4 ,experience teaches us to obtain one space from these by correlation1 and experience1 together with instinctive theorising1 teaches us to correlate our spaces with those which we believe to exist in the sensible worlds of other people4 ,the construction of a single time offers less difficulty so long as we confine ourselves to one person's private world1 but the correlation of one private time with another is a matter of great difficulty4 ,thus1 apart from any of the fluctuating hypotheses of physics1 three main problems arise in connecting the world of physics with the world of sense1 namely 7#a7 the construction of permanent things1 7#b7 the construction of a single space1 and 7#c7 the construction of a single time4 ,we will consider these three problems in succession4 7#a7 ,the belief in indestructible things very early took the form of atomism4 ,the underlying motive in atomism was not1 ,i think1 any empirical success in interpreting phenomena1 but rather an instinctive belief that beneath all the changes of the sensible world there must be something permanent and unchanging4 ,this belief was1 no doubt1 fostered and nourished by its practical successes1 culminating in the conservation of mass2 but it was not produced by these successes4 ,on the contrary1 they were produced by it4 ,philosophical writers on physics sometimes speak as though the conservation of something or other were essential to the possibility of science1 but this1 ,i believe1 is an entirely erroneous opinion4 ,if the a priori belief in permanence had not existed1 the same laws which are now formulated in terms of this belief might just as well have been formulated without it4 ,why should we suppose that1 when ice melts1 the water which replaces it is the same thing in a new form8 ,merely because this supposition enables us to state the phenomena in a way which is consonant with our prejudices4 ,what we really know is that1 under certain conditions of temperature1 the appearance we call ice is replaced by the appearance we call water4 ,we can give laws according to which the one appearance will be succeeded by the other1 but there is no reason except prejudice for regarding both as appearances of the same substance4 ,one task1 if what has just been said is correct1 which confronts us in trying to connect the world of sense with the world of physics1 is the task of reconstructing the conception of matter without the a priori beliefs which historically gave rise to it4 ,in spite of the revolutionary results of modern physics1 the empirical successes of the conception of matter show that there must be some legitimate conception which fulfils roughly the same functions4 ,the time has hardly come when we can state precisely what this legitimate conception is1 but we can see in a general way what it must be like4 ,for this purpose1 it is only necessary to take our ordinary common-sense statements and reword them without the assumption of permanent substance4 ,we say1 for example1 that things change gradually--sometimes very quickly1 but not without passing through a continuous series of intermediate states4 ,what this means is that1 given any sensible appearance1 there will usually be1 if we watch1 a continuous series of appearances connected with the given one1 leading on by imperceptible gradations to the new appearances which common-sense regards as those of the same thing4 ,thus a thing may be defined as a certain series of appearances1 connected with each other by continuity and by certain causal laws4 ,in the case of slowly changing things1 this is easily seen4 ,consider1 say1 a wall-paper which fades in the course of years4 ,it is an effort not to conceive of it as one thing whose colour is slightly different at one time from what it is at another4 ,but what do we really know about it8 ,we know that under suitable circumstances--i4e4 when we are1 as is said1 in the room--we perceive certain colours in a certain pattern3 not always precisely the same colours1 but sufficiently similar to feel familiar4 ,if we can state the laws according to which the colour varies1 we can state all that is empirically verifiable2 the assumption that there is a constant entity1 the wall-paper1 which has these various colours at various times1 is a piece of gratuitous metaphysics4 ,we may1 if we like1 define the wall-paper as the series of its aspects4 ,these are collected together by the same motives which led us to regard the wall-paper as one thing1 namely a combination of sensible continuity and causal connection4 ,more generally1 a thing will be defined as a certain series of aspects1 namely those which would commonly be said to be of the thing4 ,to say that a certain aspect is an aspect of a certain thing will merely mean that it is one of those which1 taken serially1 are the thing4 ,everything will then proceed as before3 whatever was verifiable is unchanged1 but our language is so interpreted as to avoid an unnecessary metaphysical assumption of permanence4 ,the above extrusion of permanent things affords an example of the maxim which inspires all scientific philosophising1 namely ,occam's razor3 ,entities are not to be multiplied without necessity4 ,in other words1 in dealing with any subject-matter1 find out what entities are undeniably involved1 and state everything in terms of these entities4 ,very often the resulting statement is more complicated and difficult than one which1 like common sense and most philosophy1 assumes hypothetical entities whose existence there is no good reason to believe in4 ,we find it easier to imagine a wall-paper with changing colours than to think merely of the series of colours2 but it is a mistake to suppose that what is easy and natural in thought is what is most free from unwarrantable assumptions1 as the case of things very aptly illustrates4 ,the above summary account of the genesis of things1 though it may be correct in outline1 has omitted some serious difficulties which it is necessary briefly to consider4 ,starting from a world of helter-skelter sense-data1 we wish to collect them into series1 each of which can be regarded as consisting of the successive appearances of one thing4 ,there is1 to begin with1 some conflict between what common sense regards as one thing1 and what physics regards an unchanging collection of particles4 ,to common sense1 a human body is one thing1 but to science the matter composing it is continually changing4 ,this conflict1 however1 is not very serious1 and may1 for our rough preliminary purpose1 be largely ignored4 ,the problem is3 by what principles shall we select certain data from the chaos1 and call them all appearances of the same thing8 ,a rough and approximate answer to this question is not very difficult4 ,there are certain fairly stable collections of appearances1 such as landscapes1 the furniture of rooms1 the faces of acquaintances4 ,in these cases1 we have little hesitation in regarding them on successive occasions as appearances of one thing or collection of things4 ,but1 as the ,comedy of ,errors illustrates1 we may be led astray if we judge by mere resemblance4 ,this shows that something more is involved1 for two different things may have any degree of likeness up to exact similarity4 ,another insufficient criterion of one thing is continuity4 ,as we have already seen1 if we watch what we regard as one changing thing1 we usually find its changes to be continuous so far as our senses can perceive4 ,we are thus led to assume that1 if we see two finitely different appearances at two different times1 and if we have reason to regard them as belonging to the same thing1 then there was a continuous series of intermediate states of that thing during the time when we were not observing it4 ,and so it comes to be thought that continuity of change is necessary and sufficient to constitute one thing4 ,but in fact it is neither4 ,it is not necessary1 because the unobserved states1 in the case where our attention has not been concentrated on the thing throughout1 are purely hypothetical1 and cannot possibly be our ground for supposing the earlier and later appearances to belong to the same thing2 on the contrary1 it is because we suppose this that we assume intermediate unobserved states4 ,continuity is also not sufficient1 since we can1 for example1 pass by sensibly continuous gradations from any one drop of the sea to any other drop4 ,the utmost we can say is that discontinuity during uninterrupted observation is as a rule a mark of difference between things1 though even this cannot be said in such cases as sudden explosions4 ,the assumption of continuity is1 however1 successfully made in physics4 ,this proves something1 though not anything of very obvious utility to our present problem3 it proves that nothing in the known world is inconsistent with the hypothesis that all changes are really continuous1 though from too great rapidity or from our lack of observation they may not always appear continuous4 ,in this hypothetical sense1 continuity may be allowed to be a necessary condition if two appearances are to be classed as appearances of the same thing4 ,but it is not a sufficient condition1 as appears from the instance of the drops in the sea4 ,thus something more must be sought before we can give even the roughest definition of a thing4 ,what is wanted further seems to be something in the nature of fulfilment of causal laws4 ,this statement1 as it stands1 is very vague1 but we will endeavour to give it precision4 ,when ,i speak of causal laws1 ,i mean any laws which connect events at different times1 or even1 as a limiting case1 events at the same time provided the connection is not logically demonstrable4 ,in this very general sense1 the laws of dynamics are causal laws1 and so are the laws correlating the simultaneous appearances of one thing to different senses4 ,the question is3 ,how do such laws help in the definition of a thing8 ,to answer this question1 we must consider what it is that is proved by the empirical success of physics4 ,what is proved is that its hypotheses1 though unverifiable where they go beyond sense-data1 are at no point in contradiction with sense-data1 but1 on the contrary1 are ideally such as to render all sense-data calculable from a sufficient collection of data all belonging to a given period of time4 ,now physics has found it empirically possible to collect sense-data into series1 each series being regarded as belonging to one thing1 and behaving1 with regard to the laws of physics1 in a way in which series not belonging to one thing would in general not behave4 ,if it is to be unambiguous whether two appearances belong to the same thing or not1 there must be only one way of grouping appearances so that the resulting things obey the laws of physics4 ,it would be very difficult to prove that this is the case1 but for our present purposes we may let this point pass1 and assume that there is only one way4 ,we must include in our definition of a thing those of its aspects1 if any1 which are not observed4 ,thus we may lay down the following definition3 ,things are those series of aspects which obey the laws of physics4 ,that such series exist is an empirical fact1 which constitutes the verifiability of physics4 ,it may still be objected that the matter of physics is something other than series of sense-data4 ,sense-data1 it may be said1 belong to psychology and are1 at any rate in some sense1 subjective1 whereas physics is quite independent of psychological considerations1 and does not assume that its matter only exists when it is perceived4 ,to this objection there are two answers1 both of some importance4 7a7 ,we have been considering1 in the above account1 the question of the verifiability of physics4 ,now verifiability is by no means the same thing as truth2 it is1 in fact1 something far more subjective and psychological4 ,for a proposition to be verifiable1 it is not enough that it should be true1 but it must also be such as we can discover to be true4 ,thus verifiability depends upon our capacity for acquiring knowledge1 and not only upon the objective truth4 ,in physics1 as ordinarily set forth1 there is much that is unverifiable3 there are hypotheses as to 7,a7 how things would appear to a spectator in a place where1 as it happens1 there is no spectator2 7,b7 how things would appear at times when1 in fact1 they are not appearing to anyone2 7,c7 things which never appear at all4 ,all these are introduced to simplify the statement of the causal laws1 but none of them form an integral part of what is known to be true in physics4 ,this brings us to our second answer4 7b7 ,if physics is to consist wholly of propositions known to be true1 or at least capable of being proved or disproved1 the three kinds of hypothetical entities we have just enumerated must all be capable of being exhibited as logical functions of sense-data4 ,in order to show how this might possibly be done1 let us recall the hypothetical ,leibnizian universe of ,lecture ,,iii4 ,in that universe1 we had a number of perspectives1 two of which never had any entity in common1 but often contained entities which could be sufficiently correlated to be regarded as belonging to the same thing4 ,we will call one of these an actual private world when there is an actual spectator to which it appears1 and ideal when it is merely constructed on principles of continuity4 ,a physical thing consists1 at each instant1 of the whole set of its aspects at that instant1 in all the different worlds2 thus a momentary state of a thing is a whole set of aspects4 ,an ideal appearance will be an aspect merely calculated1 but not actually perceived by any spectator4 ,an ideal state of a thing will be a state at a moment when all its appearances are ideal4 ,an ideal thing will be one whose states at all times are ideal4 ,ideal appearances1 states1 and things1 since they are calculated1 must be functions of actual appearances1 states1 and things2 in fact1 ultimately1 they must be functions of actual appearances4 ,thus it is unnecessary1 for the enunciation of the laws of physics1 to assign any reality to ideal elements3 it is enough to accept them as logical constructions1 provided we have means of knowing how to determine when they become actual4 ,this1 in fact1 we have with some degree of approximation2 the starry heaven1 for instance1 becomes actual whenever we choose to look at it4 ,it is open to us to believe that the ideal elements exist1 and there can be no reason for disbelieving this2 but unless in virtue of some a priori law we cannot know it1 for empirical knowledge is confined to what we actually observe4 7#b7 ,the three main conceptions of physics are space1 time1 and matter4 ,some of the problems raised by the conception of matter have been indicated in the above discussion of things4 ,but space and time also raise difficult problems of much the same kind1 namely1 difficulties in reducing the haphazard untidy world of immediate sensation to the smooth orderly world of geometry and kinematics4 ,let us begin with the consideration of space4 ,people who have never read any psychology seldom realise how much mental labour has gone into the construction of the one all-embracing space into which all sensible objects are supposed to fit4 ,kant1 who was unusually ignorant of psychology1 described space as an infinite given whole1 whereas a moment's psychological reflection shows that a space which is infinite is not given1 while a space which can be called given is not infinite4 ,what the nature of given space really is1 is a difficult question1 upon which psychologists are by no means agreed4 ,but some general remarks may be made1 which will suffice to show the problems1 without taking sides on any psychological issue still in debate4 ,the first thing to notice is that different senses have different spaces4 ,the space of sight is quite different from the space of touch3 it is only by experience in infancy that we learn to correlate them4 ,in later life1 when we see an object within reach1 we know how to touch it1 and more or less what it will feel like2 if we touch an object with our eyes shut1 we know where we should have to look for it1 and more or less what it would look like4 ,but this knowledge is derived from early experience of the correlation of certain kinds of touch-sensations with certain kinds of sight-sensations4 ,the one space into which both kinds of sensations fit is an intellectual construction1 not a datum4 ,and besides touch and sight1 there are other kinds of sensation which give other1 though less important spaces3 these also have to be fitted into the one space by means of experienced correlations4 ,and as in the case of things1 so here3 the one all-embracing space1 though convenient as a way of speaking1 need not be supposed really to exist4 ,all that experience makes certain is the several spaces of the several senses1 correlated by empirically discovered laws4 ,the one space may turn out to be valid as a logical construction1 compounded of the several spaces1 but there is no good reason to assume its independent metaphysical reality4 ,another respect in which the spaces of immediate experience differ from the space of geometry and physics is in regard to points4 ,the space of geometry and physics consists of an infinite number of points1 but no one has ever seen or touched a point4 ,if there are points in a sensible space1 they must be an inference4 ,it is not easy to see any way in which1 as independent entities1 they could be validly inferred from the data2 thus here again1 we shall have1 if possible1 to find some logical construction1 some complex assemblage of immediately given objects1 which will have the geometrical properties required of points4 ,it is customary to think of points as simple and infinitely small1 but geometry in no way demands that we should think of them in this way4 ,all that is necessary for geometry is that they should have mutual relations possessing certain enumerated abstract properties1 and it may be that an assemblage of data of sensation will serve this purpose4 ,exactly how this is to be done1 ,i do not yet know1 but it seems fairly certain that it can be done4 ,the following illustrative method1 simplified so as to be easily manipulated1 has been invented by ,dr ,whitehead for the purpose of showing how points might be manufactured from sense-data4 ,we have first of all to observe that there are no infinitesimal sense-data3 any surface we can see1 for example1 must be of some finite extent4 ,but what at first appears as one undivided whole is often found1 under the influence of attention1 to split up into parts contained within the whole4 ,thus one spatial object may be contained within another1 and entirely enclosed by the other4 ,this relation of enclosure1 by the help of some very natural hypotheses1 will enable us to define a point as a certain class of spatial objects1 namely all those 7as it will turn out in the end7 which would naturally be said to contain the point4 ,in order to obtain a definition of a point in this way1 we proceed as follows3 ,given any set of volumes or surfaces1 they will not in general converge into one point4 ,but if they get smaller and smaller1 while of any two of the set there is always one that encloses the other1 then we begin to have the kind of conditions which would enable us to treat them as having a point for their limit4 ,the hypotheses required for the relation of enclosure are that 7#a7 it must be transitive2 7#b7 of two different spatial objects1 it is impossible for each to enclose the other1 but a single spatial object always encloses itself2 7#c7 any set of spatial objects such that there is at least one spatial object enclosed by them all has a lower limit or minimum1 i4e4 an object enclosed by all of them and enclosing all objects which are enclosed by all of them2 7#d7 to prevent trivial exceptions1 we must add that there are to be instances of enclosure1 i4e4 there are really to be objects of which one encloses the other4 ,when an enclosure-relation has these properties1 we will call it a point-producer4 ,given any relation of enclosure1 we will call a set of objects an enclosure-series if1 of any two of them1 one is contained in the other4 ,we require a condition which shall secure that an enclosure-series converges to a point1 and this is obtained as follows3 ,let our enclosure-series be such that1 given any other enclosure-series of which there are members enclosed in any arbitrarily chosen member of our first series1 then there are members of our first series enclosed in any arbitrarily chosen member of our second series4 ,in this case1 our first enclosure-series may be called a punctual enclosure-series4 ,then a point is all the objects which enclose members of a given punctual enclosure-series4 ,in order to ensure infinite divisibility1 we require one further property to be added to those defining point-producers1 namely that any object which encloses itself also encloses an object other than itself4 ,the points generated by point-producers with this property will be found to be such as geometry requires4 7#c7 ,the question of time1 so long as we confine ourselves to one private world1 is rather less complicated than that of space1 and we can see pretty clearly how it might be dealt with by such methods as we have been considering4 ,events of which we are conscious do not last merely for a mathematical instant1 but always for some finite time1 however short4 ,even if there be a physical world such as the mathematical theory of motion supposes1 impressions on our sense-organs produce sensations which are not merely and strictly instantaneous1 and therefore the objects of sense of which we are immediately conscious are not strictly instantaneous4 ,instants1 therefore1 are not among the data of experience1 and1 if legitimate1 must be either inferred or constructed4 ,it is difficult to see how they can be validly inferred2 thus we are left with the alternative that they must be constructed4 ,how is this to be done8 ,immediate experience provides us with two time-relations among events3 they may be simultaneous1 or one may be earlier and the other later4 ,these two are both part of the crude data2 it is not the case that only the events are given1 and their time-order is added by our subjective activity4 ,the time-order1 within certain limits1 is as much given as the events4 ,in any story of adventure you will find such passages as the following3 ,with a cynical smile he pointed the revolver at the breast of the dauntless youth4 ',at the word three ,i shall fire1' he said4 ,the words one and two had already been spoken with a cool and deliberate distinctness4 ,the word three was forming on his lips4 ,at this moment a blinding flash of lightning rent the air4 ,here we have simultaneity--not due1 as ,kant would have us believe1 to the subjective mental apparatus of the dauntless youth1 but given as objectively as the revolver and the lightning4 ,and it is equally given in immediate experience that the words one and two come earlier than the flash4 ,these time-relations hold between events which are not strictly instantaneous4 ,thus one event may begin sooner than another1 and therefore be before it1 but may continue after the other has begun1 and therefore be also simultaneous with it4 ,if it persists after the other is over1 it will also be later than the other4 ,earlier1 simultaneous1 and later1 are not inconsistent with each other when we are concerned with events which last for a finite time1 however short2 they only become inconsistent when we are dealing with something instantaneous4 ,it is to be observed that we cannot give what may be called absolute dates1 but only dates determined by events4 ,we cannot point to a time itself1 but only to some event occurring at that time4 ,there is therefore no reason in experience to suppose that there are times as opposed to events3 the events1 ordered by the relations of simultaneity and succession1 are all that experience provides4 ,hence1 unless we are to introduce superfluous metaphysical entities1 we must1 in defining what mathematical physics can regard as an instant1 proceed by means of some construction which assumes nothing beyond events and their temporal relations4 ,if we wish to assign a date exactly by means of events1 how shall we proceed8 ,if we take any one event1 we cannot assign our date exactly1 because the event is not instantaneous1 that is to say1 it may be simultaneous with two events which are not simultaneous with each other4 ,in order to assign a date exactly1 we must be able1 theoretically1 to determine whether any given event is before1 at1 or after this date1 and we must know that any other date is either before or after this date1 but not simultaneous with it4 ,suppose1 now1 instead of taking one event ,a1 we take two events ,a and ,b1 and suppose ,a and ,b partly overlap1 but ,b ends before ,a ends4 ,then an event which is simultaneous with both ,a and ,b must exist during the time when ,a and ,b overlap2 thus we have come rather nearer to a precise date than when we considered ,a and ,b alone4 ,let ,c be an event which is simultaneous with both ,a and ,b1 but which ends before either ,a or ,b has ended4 ,then an event which is simultaneous with ,a and ,b and ,c must exist during the time when all three overlap1 which is a still shorter time4 ,proceeding in this way1 by taking more and more events1 a new event which is dated as simultaneous with all of them becomes gradually more and more accurately dated4 ,this suggests a way by which a completely accurate date can be defined4 ,a ,b ,c ,let us take a group of events of which any two overlap1 so that there is some time1 however short1 when they all exist4 ,if there is any other event which is simultaneous with all of these1 let us add it to the group2 let us go on until we have constructed a group such that no event outside the group is simultaneous with all of them1 but all the events inside the group are simultaneous with each other4 ,let us define this whole group as an instant of time4 ,it remains to show that it has the properties we expect of an instant4 ,what are the properties we expect of instants8 ,first1 they must form a series3 of any two1 one must be before the other1 and the other must be not before the one2 if one is before another1 and the other before a third1 the first must be before the third4 ,secondly1 every event must be at a certain number of instants2 two events are simultaneous if they are at the same instant1 and one is before the other if there is an instant1 at which the one is1 which is earlier than some instant at which the other is4 ,thirdly1 if we assume that there is always some change going on somewhere during the time when any given event persists1 the series of instants ought to be compact1 i4e4 given any two instants1 there ought to be other instants between them4 ,do instants1 as we have defined them1 have these properties8 ,we shall say that an event is at an instant when it is a member of the group by which the instant is constituted2 and we shall say that one instant is before another if the group which is the one instant contains an event which is earlier than1 but not simultaneous with1 some event in the group which is the other instant4 ,when one event is earlier than1 but not simultaneous with another1 we shall say that it wholly precedes the other4 ,now we know that of two events which are not simultaneous1 there must be one which wholly precedes the other1 and in that case the other cannot also wholly precede the one2 we also know that1 if one event wholly precedes another1 and the other wholly precedes a third1 then the first wholly precedes the third4 ,from these facts it is easy to deduce that the instants as we have defined them form a series4 ,we have next to show that every event is at at least one instant1 i4e4 that1 given any event1 there is at least one class1 such as we used in defining instants1 of which it is a member4 ,for this purpose1 consider all the events which are simultaneous with a given event1 and do not begin later1 i4e4 are not wholly after anything simultaneous with it4 ,we will call these the initial contemporaries of the given event4 ,it will be found that this class of events is the first instant at which the given event exists1 provided every event wholly after some contemporary of the given event is wholly after some initial contemporary of it4 ,finally1 the series of instants will be compact if1 given any two events of which one wholly precedes the other1 there are events wholly after the one and simultaneous with something wholly before the other4 ,whether this is the case or not1 is an empirical question2 but if it is not1 there is no reason to expect the time-series to be compact4#ag #ag ,the assumptions made concerning time-relations in the above are as follows3-- ,i4 ,in order to secure that instants form a series1 we assume3 7a7 ,no event wholly precedes itself4 7,an event is defined as whatever is simultaneous with something or other47 7b7 ,if one event wholly precedes another1 and the other wholly precedes a third1 then the first wholly precedes the third4 7c7 ,if one event wholly precedes another1 it is not simultaneous with it4 7d7 ,of two events which are not simultaneous1 one must wholly precede the other4 ,,ii4 ,in order to secure that the initial contemporaries of a given event should form an instant1 we assume3 7e7 ,an event wholly after some contemporary of a given event is wholly after some initial contemporary of the given event4 ,,iii4 ,in order to secure that the series of instants shall be compact1 we assume3 7f7 ,if one event wholly precedes another1 there is an event wholly after the one and simultaneous with something wholly before the other4 ,this assumption entails the consequence that if one event covers the whole of a stretch of time immediately preceding another event1 then it must have at least one instant in common with the other event2 i4e4 it is impossible for one event to cease just before another begins4 ,i do not know whether this should be regarded as inadmissible4 ,for a mathematico-logical treatment of the above topics1 cf4 ,n4 ,wilner1 ,a ,contribution to the ,theory of ,relative ,position1 ,proc4 ,camb4 ,phil4 ,soc41 xvii4 #e1 pp4 #dda-ddi4 ,thus our definition of instants secures all that mathematics requires1 without having to assume the existence of any disputable metaphysical entities4 ,instants may also be defined by means of the enclosure-relation1 exactly as was done in the case of points4 ,one object will be temporally enclosed by another when it is simultaneous with the other1 but not before or after it4 ,whatever encloses temporally or is enclosed temporally we shall call an event4 ,in order that the relation of temporal enclosure may be a point-producer1 we require 7#a7 that it should be transitive1 i4e4 that if one event encloses another1 and the other a third1 then the first encloses the third2 7#b7 that every event encloses itself1 but if one event encloses another different event1 then the other does not enclose the one2 7#c7 that given any set of events such that there is at least one event enclosed by all of them1 then there is an event enclosing all that they all enclose1 and itself enclosed by all of them2 7#d7 that there is at least one event4 ,to ensure infinite divisibility1 we require also that every event should enclose events other than itself4 ,assuming these characteristics1 temporal enclosure is an infinitely divisible point-producer4 ,we can now form an enclosure-series of events1 by choosing a group of events such that of any two there is one which encloses the other2 this will be a punctual enclosure-series if1 given any other enclosure-series such that every member of our first series encloses some member of our second1 then every member of our second series encloses some member of our first4 ,then an instant is the class of all events which enclose members of a given punctual enclosure-series4 ,the correlation of the times of different private worlds so as to produce the one all-embracing time of physics is a more difficult matter4 ,we saw1 in ,lecture ,,iii41 that different private worlds often contain correlated appearances1 such as common sense would regard as appearances of the same thing4 ,when two appearances in different worlds are so correlated as to belong to one momentary state of a thing1 it would be natural to regard them as simultaneous1 and as thus affording a simple means of correlating different private times4 ,but this can only be regarded as a first approximation4 ,what we call one sound will be heard sooner by people near the source of the sound than by people further from it1 and the same applies1 though in a less degree1 to light4 ,thus two correlated appearances in different worlds are not necessarily to be regarded as occurring at the same date in physical time1 though they will be parts of one momentary state of a thing4 ,the correlation of different private times is regulated by the desire to secure the simplest possible statement of the laws of physics1 and thus raises rather complicated technical problems2 but from the point of view of philosophical theory1 there is no very serious difficulty of principle involved4 ,the above brief outline must not be regarded as more than tentative and suggestive4 ,it is intended merely to show the kind of way in which1 given a world with the kind of properties that psychologists find in the world of sense1 it may be possible1 by means of purely logical constructions1 to make it amenable to mathematical treatment by defining series or classes of sense-data which can be called respectively particles1 points1 and instants4 ,if such constructions are possible1 then mathematical physics is applicable to the real world1 in spite of the fact that its particles1 points1 and instants are not to be found among actually existing entities4 ,the problem which the above considerations are intended to elucidate is one whose importance and even existence has been concealed by the unfortunate separation of different studies which prevails throughout the civilised world4 ,physicists1 ignorant and contemptuous of philosophy1 have been content to assume their particles1 points1 and instants in practice1 while conceding1 with ironical politeness1 that their concepts laid no claim to metaphysical validity4 ,metaphysicians1 obsessed by the idealistic opinion that only mind is real1 and the ,parmenidean belief that the real is unchanging1 repeated one after another the supposed contradictions in the notions of matter1 space1 and time1 and therefore naturally made no endeavour to invent a tenable theory of particles1 points1 and instants4 ,psychologists1 who have done invaluable work in bringing to light the chaotic nature of the crude materials supplied by unmanipulated sensation1 have been ignorant of mathematics and modern logic1 and have therefore been content to say that matter1 space1 and time are intellectual constructions1 without making any attempt to show in detail either how the intellect can construct them1 or what secures the practical validity which physics shows them to possess4 ,philosophers1 it is to be hoped1 will come to recognise that they cannot achieve any solid success in such problems without some slight knowledge of logic1 mathematics1 and physics2 meanwhile1 for want of students with the necessary equipment1 this vital problem remains unattempted and unknown4 ,there are1 it is true1 two authors1 both physicists1 who have done something1 though not much1 to bring about a recognition of the problem as one demanding study4 ,these two authors are ,poincare and ,mach1 ,poincare especially in his ,science and ,hypothesis1 ,mach especially in his ,analysis of ,sensations4 ,both of them1 however1 admirable as their work is1 seem to me to suffer from a general philosophical bias4 ,poincare is ,kantian1 while ,mach is ultra-empiricist2 with ,poincare almost all the mathematical part of physics is merely conventional1 while with ,mach the sensation as a mental event is identified with its object as a part of the physical world4 ,nevertheless1 both these authors1 and especially ,mach1 deserve mention as having made serious contributions to the consideration of our problem4 ,when a point or an instant is defined as a class of sensible qualities1 the first impression produced is likely to be one of wild and wilful paradox4 ,certain considerations apply here1 however1 which will again be relevant when we come to the definition of numbers4 ,there is a whole type of problems which can be solved by such definitions1 and almost always there will be at first an effect of paradox4 ,given a set of objects any two of which have a relation of the sort called symmetrical and transitive1 it is almost certain that we shall come to regard them as all having some common quality1 or as all having the same relation to some one object outside the set4 ,this kind of case is important1 and ,i shall therefore try to make it clear even at the cost of some repetition of previous definitions4 ,a relation is said to be symmetrical when1 if one term has this relation to another1 then the other also has it to the one4 ,thus brother or sister is a symmetrical relation3 if one person is a brother or a sister of another1 then the other is a brother or sister of the one4 ,simultaneity1 again1 is a symmetrical relation2 so is equality in size4 ,a relation is said to be transitive when1 if one term has this relation to another1 and the other to a third1 then the one has it to the third4 ,the symmetrical relations mentioned just now are also transitive--provided1 in the case of brother or sister1 we allow a person to be counted as his or her own brother or sister1 and provided1 in the case of simultaneity1 we mean complete simultaneity1 i4e4 beginning and ending together4 ,but many relations are transitive without being symmetrical--for instance1 such relations as greater1 earlier1 to the right of1 ancestor of1 in fact all such relations as give rise to series4 ,other relations are symmetrical without being transitive--for example1 difference in any respect4 ,if ,a is of a different age from ,b1 and ,b of a different age from ,c1 it does not follow that ,a is of a different age from ,c4 ,simultaneity1 again1 in the case of events which last for a finite time1 will not necessarily be transitive if it only means that the times of the two events overlap4 ,if ,a ends just after ,b has begun1 and ,b ends just after ,c has begun1 ,a and ,b will be simultaneous in this sense1 and so will ,b and ,c1 but ,a and ,c may well not be simultaneous4 ,all the relations which can naturally be represented as equality in any respect1 or as possession of a common property1 are transitive and symmetrical--this applies1 for example1 to such relations as being of the same height or weight or colour4 ,owing to the fact that possession of a common property gives rise to a transitive symmetrical relation1 we come to imagine that wherever such a relation occurs it must be due to a common property4 ,being equally numerous is a transitive symmetrical relation of two collections2 hence we imagine that both have a common property1 called their number4 ,existing at a given instant 7in the sense in which we defined an instant7 is a transitive symmetrical relation2 hence we come to think that there really is an instant which confers a common property on all the things existing at that instant4 ,being states of a given thing is a transitive symmetrical relation2 hence we come to imagine that there really is a thing1 other than the series of states1 which accounts for the transitive symmetrical relation4 ,in all such cases1 the class of terms that have the given transitive symmetrical relation to a given term will fulfil all the formal requisites of a common property of all the members of the class4 ,since there certainly is the class1 while any other common property may be illusory1 it is prudent1 in order to avoid needless assumptions1 to substitute the class for the common property which would be ordinarily assumed4 ,this is the reason for the definitions we have adopted1 and this is the source of the apparent paradoxes4 ,no harm is done if there are such common properties as language assumes1 since we do not deny them1 but merely abstain from asserting them4 ,but if there are not such common properties in any given case1 then our method has secured us against error4 ,in the absence of special knowledge1 therefore1 the method we have adopted is the only one which is safe1 and which avoids the risk of introducing fictitious metaphysical entities4 ,,lecture ,v ,,the ,,theory ,,of ,,continuity ,the theory of continuity1 with which we shall be occupied in the present lecture1 is1 in most of its refinements and developments1 a purely mathematical subject--very beautiful1 very important1 and very delightful1 but not1 strictly speaking1 a part of philosophy4 ,the logical basis of the theory alone belongs to philosophy1 and alone will occupy us to-night4 ,the way the problem of continuity enters into philosophy is1 broadly speaking1 the following3 ,space and time are treated by mathematicians as consisting of points and instants1 but they also have a property1 easier to feel than to define1 which is called continuity1 and is thought by many philosophers to be destroyed when they are resolved into points and instants4 ,zeno1 as we shall see1 proved that analysis into points and instants was impossible if we adhered to the view that the number of points or instants in a finite space or time must be finite4 ,later philosophers1 believing infinite number to be self-contradictory1 have found here an antinomy3 ,spaces and times could not consist of a finite number of points and instants1 for such reasons as ,zeno's2 they could not consist of an infinite number of points and instants1 because infinite numbers were supposed to be self-contradictory4 ,therefore spaces and times1 if real at all1 must not be regarded as composed of points and instants4 ,but even when points and instants1 as independent entities1 are discarded1 as they were by the theory advocated in our last lecture1 the problems of continuity1 as ,i shall try to show presently1 remain1 in a practically unchanged form4 ,let us therefore1 to begin with1 admit points and instants1 and consider the problems in connection with this simpler or at least more familiar hypothesis4 ,the argument against continuity1 in so far as it rests upon the supposed difficulties of infinite numbers1 has been disposed of by the positive theory of the infinite1 which will be considered in ,lecture ,,vii4 ,but there remains a feeling--of the kind that led ,zeno to the contention that the arrow in its flight is at rest--which suggests that points and instants1 even if they are infinitely numerous1 can only give a jerky motion1 a succession of different immobilities1 not the smooth transitions with which the senses have made us familiar4 ,this feeling is due1 ,i believe1 to a failure to realise imaginatively1 as well as abstractly1 the nature of continuous series as they appear in mathematics4 ,when a theory has been apprehended logically1 there is often a long and serious labour still required in order to feel it3 it is necessary to dwell upon it1 to thrust out from the mind1 one by one1 the misleading suggestions of false but more familiar theories1 to acquire the kind of intimacy which1 in the case of a foreign language1 would enable us to think and dream in it1 not merely to construct laborious sentences by the help of grammar and dictionary4 ,it is1 ,i believe1 the absence of this kind of intimacy which makes many philosophers regard the mathematical doctrine of continuity as an inadequate explanation of the continuity which we experience in the world of sense4 ,in the present lecture1 ,i shall first try to explain in outline what the mathematical theory of continuity is in its philosophically important essentials4 ,the application to actual space and time will not be in question to begin with4 ,i do not see any reason to suppose that the points and instants which mathematicians introduce in dealing with space and time are actual physically existing entities1 but ,i do see reason to suppose that the continuity of actual space and time may be more or less analogous to mathematical continuity4 ,the theory of mathematical continuity is an abstract logical theory1 not dependent for its validity upon any properties of actual space and time4 ,what is claimed for it is that1 when it is understood1 certain characteristics of space and time1 previously very hard to analyse1 are found not to present any logical difficulty4 ,what we know empirically about space and time is insufficient to enable us to decide between various mathematically possible alternatives1 but these alternatives are all fully intelligible and fully adequate to the observed facts4 ,for the present1 however1 it will be well to forget space and time and the continuity of sensible change1 in order to return to these topics equipped with the weapons provided by the abstract theory of continuity4 ,continuity1 in mathematics1 is a property only possible to a series of terms1 i4e4 to terms arranged in an order1 so that we can say of any two that one comes before the other4 ,numbers in order of magnitude1 the points on a line from left to right1 the moments of time from earlier to later1 are instances of series4 ,the notion of order1 which is here introduced1 is one which is not required in the theory of cardinal number4 ,it is possible to know that two classes have the same number of terms without knowing any order in which they are to be taken4 ,we have an instance of this in such a case as ,english husbands and ,english wives3 we can see that there must be the same number of husbands as of wives1 without having to arrange them in a series4 ,but continuity1 which we are now to consider1 is essentially a property of an order3 it does not belong to a set of terms in themselves1 but only to a set in a certain order4 ,a set of terms which can be arranged in one order can always also be arranged in other orders1 and a set of terms which can be arranged in a continuous order can always also be arranged in orders which are not continuous4 ,thus the essence of continuity must not be sought in the nature of the set of terms1 but in the nature of their arrangement in a series4 ,mathematicians have distinguished different degrees of continuity1 and have confined the word continuous1 for technical purposes1 to series having a certain high degree of continuity4 ,but for philosophical purposes1 all that is important in continuity is introduced by the lowest degree of continuity1 which is called compactness4 ,a series is called compact when no two terms are consecutive1 but between any two there are others4 ,one of the simplest examples of a compact series is the series of fractions in order of magnitude4 ,given any two fractions1 however near together1 there are other fractions greater than the one and smaller than the other1 and therefore no two fractions are consecutive4 ,there is no fraction1 for example1 which is next after #a_/b3 if we choose some fraction which is very little greater than #a_/b1 say #ea_/ajj we can find others1 such as #aja_/bjj1 which are nearer to #a_/b4 ,thus between any two fractions1 however little they differ1 there are an infinite number of other fractions4 ,mathematical space and time also have this property of compactness1 though whether actual space and time have it is a further question1 dependent upon empirical evidence1 and probably incapable of being answered with certainty4 ,in the case of abstract objects such as fractions1 it is perhaps not very difficult to realise the logical possibility of their forming a compact series4 ,the difficulties that might be felt are those of infinity1 for in a compact series the number of terms between any two given terms must be infinite4 ,but when these difficulties have been solved1 the mere compactness in itself offers no great obstacle to the imagination4 ,in more concrete cases1 however1 such as motion1 compactness becomes much more repugnant to our habits of thought4 ,it will therefore be desirable to consider explicitly the mathematical account of motion1 with a view to making its logical possibility felt4 ,the mathematical account of motion is perhaps artificially simplified when regarded as describing what actually occurs in the physical world2 but what actually occurs must be capable1 by a certain amount of logical manipulation1 of being brought within the scope of the mathematical account1 and must1 in its analysis1 raise just such problems as are raised in their simplest form by this account4 ,neglecting1 therefore1 for the present1 the question of its physical adequacy1 let us devote ourselves merely to considering its possibility as a formal statement of the nature of motion4 ,in order to simplify our problem as much as possible1 let us imagine a tiny speck of light moving along a scale4 ,what do we mean by saying that the motion is continuous8 ,it is not necessary for our purposes to consider the whole of what the mathematician means by this statement3 only part of what he means is philosophically important4 ,one part of what he means is that1 if we consider any two positions of the speck occupied at any two instants1 there will be other intermediate positions occupied at intermediate instants4 ,however near together we take the two positions1 the speck will not jump suddenly from the one to the other1 but will pass through an infinite number of other positions on the way4 ,every distance1 however small1 is traversed by passing through all the infinite series of positions between the two ends of the distance4 ,but at this point imagination suggests that we may describe the continuity of motion by saying that the speck always passes from one position at one instant to the next position at the next instant4 ,as soon as we say this or imagine it1 we fall into error1 because there is no next point or next instant4 ,if there were1 we should find ,zeno's paradoxes1 in some form1 unavoidable1 as will appear in our next lecture4 ,one simple paradox may serve as an illustration4 ,if our speck is in motion along the scale throughout the whole of a certain time1 it cannot be at the same point at two consecutive instants4 ,but it cannot1 from one instant to the next1 travel further than from one point to the next1 for if it did1 there would be no instant at which it was in the positions intermediate between that at the first instant and that at the next1 and we agreed that the continuity of motion excludes the possibility of such sudden jumps4 ,it follows that our speck must1 so long as it moves1 pass from one point at one instant to the next point at the next instant4 ,thus there will be just one perfectly definite velocity with which all motions must take place3 no motion can be faster than this1 and no motion can be slower4 ,since this conclusion is false1 we must reject the hypothesis upon which it is based1 namely that there are consecutive points and instants4#ah ,hence the continuity of motion must not be supposed to consist in a body's occupying consecutive positions at consecutive times4 #ah ,the above paradox is essentially the same as ,zeno's argument of the stadium which will be considered in our next lecture4 ,the difficulty to imagination lies chiefly1 ,i think1 in keeping out the suggestion of infinitesimal distances and times4 ,suppose we halve a given distance1 and then halve the half1 and so on1 we can continue the process as long as we please1 and the longer we continue it1 the smaller the resulting distance becomes4 ,this infinite divisibility seems1 at first sight1 to imply that there are infinitesimal distances1 i4e4 distances so small that any finite fraction of an inch would be greater4 ,this1 however1 is an error4 ,the continued bisection of our distance1 though it gives us continually smaller distances1 gives us always finite distances4 ,if our original distance was an inch1 we reach successively half an inch1 a quarter of an inch1 an eighth1 a sixteenth1 and so on2 but every one of this infinite series of diminishing distances is finite4 ,but1 it may be said1 in the end the distance will grow infinitesimal4 ,no1 because there is no end4 ,the process of bisection is one which can1 theoretically1 be carried on for ever1 without any last term being attained4 ,thus infinite divisibility of distances1 which must be admitted1 does not imply that there are distances so small that any finite distance would be larger4 ,it is easy1 in this kind of question1 to fall into an elementary logical blunder4 ,given any finite distance1 we can find a smaller distance2 this may be expressed in the ambiguous form there is a distance smaller than any finite distance4 ,but if this is then interpreted as meaning there is a distance such that1 whatever finite distance may be chosen1 the distance in question is smaller1 then the statement is false4 ,common language is ill adapted to expressing matters of this kind1 and philosophers who have been dependent on it have frequently been misled by it4 ,in a continuous motion1 then1 we shall say that at any given instant the moving body occupies a certain position1 and at other instants it occupies other positions2 the interval between any two instants and between any two positions is always finite1 but the continuity of the motion is shown in the fact that1 however near together we take the two positions and the two instants1 there are an infinite number of positions still nearer together1 which are occupied at instants that are also still nearer together4 ,the moving body never jumps from one position to another1 but always passes by a gradual transition through an infinite number of intermediaries4 ,at a given instant1 it is where it is1 like ,zeno's arrow2#ai but we cannot say that it is at rest at the instant1 since the instant does not last for a finite time1 and there is not a beginning and end of the instant with an interval between them4 ,rest consists in being in the same position at all the instants throughout a certain finite period1 however short2 it does not consist simply in a body's being where it is at a given instant4 ,this whole theory1 as is obvious1 depends upon the nature of compact series1 and demands1 for its full comprehension1 that compact series should have become familiar and easy to the imagination as well as to deliberate thought4 #ai ,see next lecture4 ,what is required may be expressed in mathematical language by saying that the position of a moving body must be a continuous function of the time4 ,to define accurately what this means1 we proceed as follows4 ,consider a particle which1 at the moment t1 is at the point ,p4 ,choose now any small portion ,p#a,p#b of the path of the particle1 this portion being one which contains ,p4 ,we say then that1 if the motion of the particle is continuous at the time t1 it must be possible to find two instants t#a1 t#b1 one earlier than t and one later1 such that throughout the whole time from t#a to t#b 7both included71 the particle lies between ,p#a and ,p#b4 ,and we say that this must still hold however small we make the portion ,p#a,p#b4 ,when this is the case1 we say that the motion is continuous at the time t2 and when the motion is continuous at all times1 we say that the motion as a whole is continuous4 ,it is obvious that if the particle were to jump suddenly from ,p to some other point ,q1 our definition would fail for all intervals ,p#a,p#b which were too small to include ,q4 ,thus our definition affords an analysis of the continuity of motion1 while admitting points and instants and denying infinitesimal distances in space or periods in time4 ,p#a ,p ,p#b ,q ------------------------ ,philosophers1 mostly in ignorance of the mathematician's analysis1 have adopted other and more heroic methods of dealing with the prima facie difficulties of continuous motion4 ,a typical and recent example of philosophic theories of motion is afforded by ,bergson1 whose views on this subject ,i have examined elsewhere4#bj #bj ,monist1 ,july #aiab1 pp4 #ccg-cda4 ,apart from definite arguments1 there are certain feelings1 rather than reasons1 which stand in the way of an acceptance of the mathematical account of motion4 ,to begin with1 if a body is moving at all fast1 we see its motion just as we see its colour4 ,a slow motion1 like that of the hour-hand of a watch1 is only known in the way which mathematics would lead us to expect1 namely by observing a change of position after a lapse of time2 but1 when we observe the motion of the second-hand1 we do not merely see first one position and then another--we see something as directly sensible as colour4 ,what is this something that we see1 and that we call visible motion8 ,whatever it is1 it is not the successive occupation of successive positions3 something beyond the mathematical theory of motion is required to account for it4 ,opponents of the mathematical theory emphasise this fact4 ,your theory1 they say1 may be very logical1 and might apply admirably to some other world2 but in this actual world1 actual motions are quite different from what your theory would declare them to be1 and require1 therefore1 some different philosophy from yours for their adequate explanation4 ,the objection thus raised is one which ,i have no wish to underrate1 but ,i believe it can be fully answered without departing from the methods and the outlook which have led to the mathematical theory of motion4 ,let us1 however1 first try to state the objection more fully4 ,if the mathematical theory is adequate1 nothing happens when a body moves except that it is in different places at different times4 ,but in this sense the hour-hand and the second-hand are equally in motion1 yet in the second-hand there is something perceptible to our senses which is absent in the hour-hand4 ,we can see1 at each moment1 that the second-hand is moving1 which is different from seeing it first in one place and then in another4 ,this seems to involve our seeing it simultaneously in a number of places1 although it must also involve our seeing that it is in some of these places earlier than in others4 ,if1 for example1 ,i move my hand quickly from left to right1 you seem to see the whole movement at once1 in spite of the fact that you know it begins at the left and ends at the right4 ,it is this kind of consideration1 ,i think1 which leads ,bergson and many others to regard a movement as really one indivisible whole1 not the series of separate states imagined by the mathematician4 ,to this objection there are three supplementary answers1 physiological1 psychological1 and logical4 ,we will consider them successively4 7#a7 ,the physiological answer merely shows that1 if the physical world is what the mathematician supposes1 its sensible appearance may nevertheless be expected to be what it is4 ,the aim of this answer is thus the modest one of showing that the mathematical account is not impossible as applied to the physical world2 it does not even attempt to show that this account is necessary1 or that an analogous account applies in psychology4 ,when any nerve is stimulated1 so as to cause a sensation1 the sensation does not cease instantaneously with the cessation of the stimulus1 but dies away in a short finite time4 ,a flash of lightning1 brief as it is to our sight1 is briefer still as a physical phenomenon3 we continue to see it for a few moments after the light-waves have ceased to strike the eye4 ,thus in the case of a physical motion1 if it is sufficiently swift1 we shall actually at one instant see the moving body throughout a finite portion of its course1 and not only at the exact spot where it is at that instant4 ,sensations1 however1 as they die away1 grow gradually fainter2 thus the sensation due to a stimulus which is recently past is not exactly like the sensation due to a present stimulus4 ,it follows from this that1 when we see a rapid motion1 we shall not only see a number of positions of the moving body simultaneously1 but we shall see them with different degrees of intensity--the present position most vividly1 and the others with diminishing vividness1 until sensation fades away into immediate memory4 ,this state of things accounts fully for the perception of motion4 ,a motion is perceived1 not merely inferred1 when it is sufficiently swift for many positions to be sensible at one time2 and the earlier and later parts of one perceived motion are distinguished by the less and greater vividness of the sensations4 ,this answer shows that physiology can account for our perception of motion4 ,but physiology1 in speaking of stimulus and sense-organs and a physical motion distinct from the immediate object of sense1 is assuming the truth of physics1 and is thus only capable of showing the physical account to be possible1 not of showing it to be necessary4 ,this consideration brings us to the psychological answer4 7#b7 ,the psychological answer to our difficulty about motion is part of a vast theory1 not yet worked out1 and only capable1 at present1 of being vaguely outlined4 ,we considered this theory in the third and fourth lectures2 for the present1 a mere sketch of its application to our present problem must suffice4 ,the world of physics1 which was assumed in the physiological answer1 is obviously inferred from what is given in sensation2 yet as soon as we seriously consider what is actually given in sensation1 we find it apparently very different from the world of physics4 ,the question is thus forced upon us3 ,is the inference from sense to physics a valid one8 ,i believe the answer to be affirmative1 for reasons which ,i suggested in the third and fourth lectures2 but the answer cannot be either short or easy4 ,it consists1 broadly speaking1 in showing that1 although the particles1 points1 and instants with which physics operates are not themselves given in experience1 and are very likely not actually existing things1 yet1 out of the materials provided in sensation1 it is possible to make logical constructions having the mathematical properties which physics assigns to particles1 points1 and instants4 ,if this can be done1 then all the propositions of physics can be translated1 by a sort of dictionary1 into propositions about the kinds of objects which are given in sensation4 ,applying these general considerations to the case of motion1 we find that1 even within the sphere of immediate sense-data1 it is necessary1 or at any rate more consonant with the facts than any other equally simple view1 to distinguish instantaneous states of objects1 and to regard such states as forming a compact series4 ,let us consider a body which is moving swiftly enough for its motion to be perceptible1 and long enough for its motion to be not wholly comprised in one sensation4 ,then1 in spite of the fact that we see a finite extent of the motion at one instant1 the extent which we see at one instant is different from that which we see at another4 ,thus we are brought back1 after all1 to a series of momentary views of the moving body1 and this series will be compact1 like the former physical series of points4 ,in fact1 though the terms of the series seem different1 the mathematical character of the series is unchanged1 and the whole mathematical theory of motion will apply to it verbatim4 ,when we are considering the actual data of sensation in this connection1 it is important to realise that two sense-data may be1 and must sometimes be1 really different when we cannot perceive any difference between them4 ,an old but conclusive reason for believing this was emphasised by ,poincare4#ba ,in all cases of sense-data capable of gradual change1 we may find one sense-datum indistinguishable from another1 and that other indistinguishable from a third1 while yet the first and third are quite easily distinguishable4 ,suppose1 for example1 a person with his eyes shut is holding a weight in his hand1 and someone noiselessly adds a small extra weight4 ,if the extra weight is small enough1 no difference will be perceived in the sensation4 ,after a time1 another small extra weight may be added1 and still no change will be perceived2 but if both extra weights had been added at once1 it may be that the change would be quite easily perceptible4 ,or1 again1 take shades of colour4 ,it would be easy to find three stuffs of such closely similar shades that no difference could be perceived between the first and second1 nor yet between the second and third1 while yet the first and third would be distinguishable4 ,in such a case1 the second shade cannot be the same as the first1 or it would be distinguishable from the third2 nor the same as the third1 or it would be distinguishable from the first4 ,it must1 therefore1 though indistinguishable from both1 be really intermediate between them4 #ba ,le continu mathematique1 ,revue de ,metaphysique et de ,morale1 vol4 i4 p4 #bi4 ,such considerations as the above show that1 although we cannot distinguish sense-data unless they differ by more than a certain amount1 it is perfectly reasonable to suppose that sense-data of a given kind1 such as weights or colours1 really form a compact series4 ,the objections which may be brought from a psychological point of view against the mathematical theory of motion are not1 therefore1 objections to this theory properly understood1 but only to a quite unnecessary assumption of simplicity in the momentary object of sense4 ,of the immediate object of sense1 in the case of a visible motion1 we may say that at each instant it is in all the positions which remain sensible at that instant2 but this set of positions changes continuously from moment to moment1 and is amenable to exactly the same mathematical treatment as if it were a mere point4 ,when we assert that some mathematical account of phenomena is correct1 all that we primarily assert is that something definable in terms of the crude phenomena satisfies our formulae2 and in this sense the mathematical theory of motion is applicable to the data of sensation as well as to the supposed particles of abstract physics4 ,there are a number of distinct questions which are apt to be confused when the mathematical continuum is said to be inadequate to the facts of sense4 ,we may state these1 in order of diminishing generality1 as follows3-- 7a7 ,are series possessing mathematical continuity logically possible8 7b7 ,assuming that they are possible logically1 are they not impossible as applied to actual sense-data1 because1 among actual sense-data1 there are no such fixed mutually external terms as are to be found1 e4g41 in the series of fractions8 7c7 ,does not the assumption of points and instants make the whole mathematical account fictitious8 7d7 ,finally1 assuming that all these objections have been answered1 is there1 in actual empirical fact1 any sufficient reason to believe the world of sense continuous8 ,let us consider these questions in succession4 7a7 ,the question of the logical possibility of the mathematical continuum turns partly on the elementary misunderstandings we considered at the beginning of the present lecture1 partly on the possibility of the mathematical infinite1 which will occupy our next two lectures1 and partly on the logical form of the answer to the ,bergsonian objection which we stated a few minutes ago4 ,i shall say no more on this topic at present1 since it is desirable first to complete the psychological answer4 7b7 ,the question whether sense-data are composed of mutually external units is not one which can be decided by empirical evidence4 ,it is often urged that1 as a matter of immediate experience1 the sensible flux is devoid of divisions1 and is falsified by the dissections of the intellect4 ,now ,i have no wish to argue that this view is contrary to immediate experience3 ,i wish only to maintain that it is essentially incapable of being proved by immediate experience4 ,as we saw1 there must be among sense-data differences so slight as to be imperceptible3 the fact that sense-data are immediately given does not mean that their differences also must be immediately given 7though they may be74 ,suppose1 for example1 a coloured surface on which the colour changes gradually--so gradually that the difference of colour in two very neighbouring portions is imperceptible1 while the difference between more widely separated portions is quite noticeable4 ,the effect produced1 in such a case1 will be precisely that of interpenetration1 of transition which is not a matter of discrete units4 ,and since it tends to be supposed that the colours1 being immediate data1 must appear different if they are different1 it seems easily to follow that interpenetration must be the ultimately right account4 ,but this does not follow4 ,it is unconsciously assumed1 as a premiss for a reductio ad absurdum of the analytic view1 that1 if ,a and ,b are immediate data1 and ,a differs from ,b1 then the fact that they differ must also be an immediate datum4 ,it is difficult to say how this assumption arose1 but ,i think it is to be connected with the confusion between acquaintance and knowledge about4 ,acquaintance1 which is what we derive from sense1 does not1 theoretically at least1 imply even the smallest knowledge about1 i4e4 it does not imply knowledge of any proposition concerning the object with which we are acquainted4 ,it is a mistake to speak as if acquaintance had degrees3 there is merely acquaintance and non-acquaintance4 ,when we speak of becoming better acquainted1 as for instance with a person1 what we must mean is1 becoming acquainted with more parts of a certain whole2 but the acquaintance with each part is either complete or nonexistent4 ,thus it is a mistake to say that if we were perfectly acquainted with an object we should know all about it4 ,knowledge about is knowledge of propositions1 which is not involved necessarily in acquaintance with the constituents of the propositions4 ,to know that two shades of colour are different is knowledge about them2 hence acquaintance with the two shades does not in any way necessitate the knowledge that they are different4 ,from what has just been said it follows that the nature of sense-data cannot be validly used to prove that they are not composed of mutually external units4 ,it may be admitted1 on the other hand1 that nothing in their empirical character specially necessitates the view that they are composed of mutually external units4 ,this view1 if it is held1 must be held on logical1 not on empirical1 grounds4 ,i believe that the logical grounds are adequate to the conclusion4 ,they rest1 at bottom1 upon the impossibility of explaining complexity without assuming constituents4 ,it is undeniable that the visual field1 for example1 is complex2 and so far as ,i can see1 there is always self-contradiction in the theories which1 while admitting this complexity1 attempt to deny that it results from a combination of mutually external units4 ,but to pursue this topic would lead us too far from our theme1 and ,i shall therefore say no more about it at present4 7c7 ,it is sometimes urged that the mathematical account of motion is rendered fictitious by its assumption of points and instants4 ,now there are here two different questions to be distinguished4 ,there is the question of absolute or relative space and time1 and there is the question whether what occupies space and time must be composed of elements which have no extension or duration4 ,and each of these questions in turn may take two forms1 namely3 7,a7 is the hypothesis consistent with the facts and with logic8 7,b7 is it necessitated by the facts or by logic8 ,i wish to answer1 in each case1 yes to the first form of the question1 and no to the second4 ,but in any case the mathematical account of motion will not be fictitious1 provided a right interpretation is given to the words point and instant4 ,a few words on each alternative will serve to make this clear4 ,formally1 mathematics adopts an absolute theory of space and time1 i4e4 it assumes that1 besides the things which are in space and time1 there are also entities1 called points and instants1 which are occupied by things4 ,this view1 however1 though advocated by ,newton1 has long been regarded by mathematicians as merely a convenient fiction4 ,there is1 so far as ,i can see1 no conceivable evidence either for or against it4 ,it is logically possible1 and it is consistent with the facts4 ,but the facts are also consistent with the denial of spatial and temporal entities over and above things with spatial and temporal relations4 ,hence1 in accordance with ,occam's razor1 we shall do well to abstain from either assuming or denying points and instants4 ,this means1 so far as practical working out is concerned1 that we adopt the relational theory2 for in practice the refusal to assume points and instants has the same effect as the denial of them4 ,but in strict theory the two are quite different1 since the denial introduces an element of unverifiable dogma which is wholly absent when we merely refrain from the assertion4 ,thus1 although we shall derive points and instants from things1 we shall leave the bare possibility open that they may also have an independent existence as simple entities4 ,we come now to the question whether the things in space and time are to be conceived as composed of elements without extension or duration1 i4e4 of elements which only occupy a point and an instant4 ,physics1 formally1 assumes in its differential equations that things consist of elements which occupy only a point at each instant1 but persist throughout time4 ,for reasons explained in ,lecture ,,iv41 the persistence of things through time is to be regarded as the formal result of a logical construction1 not as necessarily implying any actual persistence4 ,the same motives1 in fact1 which lead to the division of things into point-particles1 ought presumably to lead to their division into instant-particles1 so that the ultimate formal constituent of the matter in physics will be a point-instant-particle4 ,but such objects1 as well as the particles of physics1 are not data4 ,the same economy of hypothesis1 which dictates the practical adoption of a relative rather than an absolute space and time1 also dictates the practical adoption of material elements which have a finite extension and duration4 ,since1 as we saw in ,lecture ,,iv41 points and instants can be constructed as logical functions of such elements1 the mathematical account of motion1 in which a particle passes continuously through a continuous series of points1 can be interpreted in a form which assumes only elements which agree with our actual data in having a finite extension and duration4 ,thus1 so far as the use of points and instants is concerned1 the mathematical account of motion can be freed from the charge of employing fictions4 7d7 ,but we must now face the question3 ,is there1 in actual empirical fact1 any sufficient reason to believe the world of sense continuous8 ,the answer here must1 ,i think1 be in the negative4 ,we may say that the hypothesis of continuity is perfectly consistent with the facts and with logic1 and that it is technically simpler than any other tenable hypothesis4 ,but since our powers of discrimination among very similar sensible objects are not infinitely precise1 it is quite impossible to decide between different theories which only differ in regard to what is below the margin of discrimination4 ,if1 for example1 a coloured surface which we see consists of a finite number of very small surfaces1 and if a motion which we see consists1 like a cinematograph1 of a large finite number of successive positions1 there will be nothing empirically discoverable to show that objects of sense are not continuous4 ,in what is called experienced continuity1 such as is said to be given in sense1 there is a large negative element3 absence of perception of difference occurs in cases which are thought to give perception of absence of difference4 ,when1 for example1 we cannot distinguish a colour ,a from a colour ,b1 nor a colour ,b from a colour ,c1 but can distinguish ,a from ,c1 the indistinguishability is a purely negative fact1 namely1 that we do not perceive a difference4 ,even in regard to immediate data1 this is no reason for denying that there is a difference4 ,thus1 if we see a coloured surface whose colour changes gradually1 its sensible appearance if the change is continuous will be indistinguishable from what it would be if the change were by small finite jumps4 ,if this is true1 as it seems to be1 it follows that there can never be any empirical evidence to demonstrate that the sensible world is continuous1 and not a collection of a very large finite number of elements of which each differs from its neighbour in a finite though very small degree4 ,the continuity of space and time1 the infinite number of different shades in the spectrum1 and so on1 are all in the nature of unverifiable hypotheses--perfectly possible logically1 perfectly consistent with the known facts1 and simpler technically than any other tenable hypotheses1 but not the sole hypotheses which are logically and empirically adequate4 ,if a relational theory of instants is constructed1 in which an instant is defined as a group of events simultaneous with each other and not all simultaneous with any event outside the group1 then if our resulting series of instants is to be compact1 it must be possible1 if x wholly precedes y1 to find an event z1 simultaneous with part of x1 which wholly precedes some event which wholly precedes y4 ,now this requires that the number of events concerned should be infinite in any finite period of time4 ,if this is to be the case in the world of one man's sense-data1 and if each sense-datum is to have not less than a certain finite temporal extension1 it will be necessary to assume that we always have an infinite number of sense-data simultaneous with any given sense-datum4 ,applying similar considerations to space1 and assuming that sense-data are to have not less than a certain spatial extension1 it will be necessary to suppose that an infinite number of sense-data overlap spatially with any given sense-datum4 ,this hypothesis is possible1 if we suppose a single sense-datum1 e4g4 in sight1 to be a finite surface1 enclosing other surfaces which are also single sense-data4 ,but there are difficulties in such a hypothesis1 and ,i do not know whether these difficulties could be successfully met4 ,if they cannot1 we must do one of two things3 either declare that the world of one man's sense-data is not continuous1 or else refuse to admit that there is any lower limit to the duration and extension of a single sense-datum4 ,i do not know what is the right course to adopt as regards these alternatives4 ,the logical analysis we have been considering provides the apparatus for dealing with the various hypotheses1 and the empirical decision between them is a problem for the psychologist4 7#c7 ,we have now to consider the logical answer to the alleged difficulties of the mathematical theory of motion1 or rather to the positive theory which is urged on the other side4 ,the view urged explicitly by ,bergson1 and implied in the doctrines of many philosophers1 is1 that a motion is something indivisible1 not validly analysable into a series of states4 ,this is part of a much more general doctrine1 which holds that analysis always falsifies1 because the parts of a complex whole are different1 as combined in that whole1 from what they would otherwise be4 ,it is very difficult to state this doctrine in any form which has a precise meaning4 ,often arguments are used which have no bearing whatever upon the question4 ,it is urged1 for example1 that when a man becomes a father1 his nature is altered by the new relation in which he finds himself1 so that he is not strictly identical with the man who was previously not a father4 ,this may be true1 but it is a causal psychological fact1 not a logical fact4 ,the doctrine would require that a man who is a father cannot be strictly identical with a man who is a son1 because he is modified in one way by the relation of fatherhood and in another by that of sonship4 ,in fact1 we may give a precise statement of the doctrine we are combating in the form3 ,there can never be two facts concerning the same thing4 ,a fact concerning a thing always is or involves a relation to one or more entities2 thus two facts concerning the same thing would involve two relations of the same thing4 ,but the doctrine in question holds that a thing is so modified by its relations that it cannot be the same in one relation as in another4 ,hence1 if this doctrine is true1 there can never be more than one fact concerning any one thing4 ,i do not think the philosophers in question have realised that this is the precise statement of the view they advocate1 because in this form the view is so contrary to plain truth that its falsehood is evident as soon as it is stated4 ,the discussion of this question1 however1 involves so many logical subtleties1 and is so beset with difficulties1 that ,i shall not pursue it further at present4 ,when once the above general doctrine is rejected1 it is obvious that1 where there is change1 there must be a succession of states4 ,there cannot be change--and motion is only a particular case of change--unless there is something different at one time from what there is at some other time4 ,change1 therefore1 must involve relations and complexity1 and must demand analysis4 ,so long as our analysis has only gone as far as other smaller changes1 it is not complete2 if it is to be complete1 it must end with terms that are not changes1 but are related by a relation of earlier and later4 ,in the case of changes which appear continuous1 such as motions1 it seems to be impossible to find anything other than change so long as we deal with finite periods of time1 however short4 ,we are thus driven back1 by the logical necessities of the case1 to the conception of instants without duration1 or at any rate without any duration which even the most delicate instruments can reveal4 ,this conception1 though it can be made to seem difficult1 is really easier than any other that the facts allow4 ,it is a kind of logical framework into which any tenable theory must fit--not necessarily itself the statement of the crude facts1 but a form in which statements which are true of the crude facts can be made by a suitable interpretation4 ,the direct consideration of the crude facts of the physical world has been undertaken in earlier lectures2 in the present lecture1 we have only been concerned to show that nothing in the crude facts is inconsistent with the mathematical doctrine of continuity1 or demands a continuity of a radically different kind from that of mathematical motion4 ,,lecture ,,vi ,,the ,,problem ,,of ,,infinity ,,considered ,,historically ,it will be remembered that1 when we enumerated the grounds upon which the reality of the sensible world has been questioned1 one of those mentioned was the supposed impossibility of infinity and continuity4 ,in view of our earlier discussion of physics1 it would seem that no conclusive empirical evidence exists in favour of infinity or continuity in objects of sense or in matter4 ,nevertheless1 the explanation which assumes infinity and continuity remains incomparably easier and more natural1 from a scientific point of view1 than any other1 and since ,georg ,cantor has shown that the supposed contradictions are illusory1 there is no longer any reason to struggle after a finitist explanation of the world4 ,the supposed difficulties of continuity all have their source in the fact that a continuous series must have an infinite number of terms1 and are in fact difficulties concerning infinity4 ,hence1 in freeing the infinite from contradiction1 we are at the same time showing the logical possibility of continuity as assumed in science4 ,the kind of way in which infinity has been used to discredit the world of sense may be illustrated by ,kant's first two antinomies4 ,in the first1 the thesis states3 ,the world has a beginning in time1 and as regards space is enclosed within limits2 the antithesis states3 ,the world has no beginning and no limits in space1 but is infinite in respect of both time and space4 ,kant professes to prove both these propositions1 whereas1 if what we have said on modern logic has any truth1 it must be impossible to prove either4 ,in order1 however1 to rescue the world of sense1 it is enough to destroy the proof of one of the two4 ,for our present purpose1 it is the proof that the world is finite that interests us4 ,kant's argument as regards space here rests upon his argument as regards time4 ,we need therefore only examine the argument as regards time4 ,what he says is as follows3 ,for let us assume that the world has no beginning as regards time1 so that up to every given instant an eternity has elapsed1 and therefore an infinite series of successive states of the things in the world has passed by4 ,but the infinity of a series consists just in this1 that it can never be completed by successive synthesis4 ,therefore an infinite past world-series is impossible1 and accordingly a beginning of the world is a necessary condition of its existence2 which was the first thing to be proved4 ,many different criticisms might be passed on this argument1 but we will content ourselves with a bare minimum4 ,to begin with1 it is a mistake to define the infinity of a series as impossibility of completion by successive synthesis4 ,the notion of infinity1 as we shall see in the next lecture1 is primarily a property of classes1 and only derivatively applicable to series2 classes which are infinite are given all at once by the defining property of their members1 so that there is no question of completion or of successive synthesis4 ,and the word synthesis1 by suggesting the mental activity of synthesising1 introduces1 more or less surreptitiously1 that reference to mind by which all ,kant's philosophy was infected4 ,in the second place1 when ,kant says that an infinite series can never be completed by successive synthesis1 all that he has even conceivably a right to say is that it cannot be completed in a finite time4 ,thus what he really proves is1 at most1 that if the world had no beginning1 it must have already existed for an infinite time4 ,this1 however1 is a very poor conclusion1 by no means suitable for his purposes4 ,and with this result we might1 if we chose1 take leave of the first antinomy4 ,it is worth while1 however1 to consider how ,kant came to make such an elementary blunder4 ,what happened in his imagination was obviously something like this3 ,starting from the present and going backwards in time1 we have1 if the world had no beginning1 an infinite series of events4 ,as we see from the word synthesis1 he imagined a mind trying to grasp these successively1 in the reverse order to that in which they had occurred1 i4e4 going from the present backwards4 ,this series is obviously one which has no end4 ,but the series of events up to the present has an end1 since it ends with the present4 ,owing to the inveterate subjectivism of his mental habits1 he failed to notice that he had reversed the sense of the series by substituting backward synthesis for forward happening1 and thus he supposed that it was necessary to identify the mental series1 which had no end1 with the physical series1 which had an end but no beginning4 ,it was this mistake1 ,i think1 which1 operating unconsciously1 led him to attribute validity to a singularly flimsy piece of fallacious reasoning4 ,the second antinomy illustrates the dependence of the problem of continuity upon that of infinity4 ,the thesis states3 ,every complex substance in the world consists of simple parts1 and there exists everywhere nothing but the simple or what is composed of it4 ,the antithesis states3 ,no complex thing in the world consists of simple parts1 and everywhere in it there exists nothing simple4 ,here1 as before1 the proofs of both thesis and antithesis are open to criticism1 but for the purpose of vindicating physics and the world of sense it is enough to find a fallacy in one of the proofs4 ,we will choose for this purpose the proof of the antithesis1 which begins as follows3 ,assume that a complex thing 7as substance7 consists of simple parts4 ,since all external relation1 and therefore all composition out of substances1 is only possible in space1 the space occupied by a complex thing must consist of as many parts as the thing consists of4 ,now space does not consist of simple parts1 but of spaces4 ,the rest of his argument need not concern us1 for the nerve of the proof lies in the one statement3 ,space does not consist of simple parts1 but of spaces4 ,this is like ,bergson's objection to the absurd proposition that motion is made up of immobilities4 ,kant does not tell us why he holds that a space must consist of spaces rather than of simple parts4 ,geometry regards space as made up of points1 which are simple2 and although1 as we have seen1 this view is not scientifically or logically necessary1 it remains prima facie possible1 and its mere possibility is enough to vitiate ,kant's argument4 ,for1 if his proof of the thesis of the antinomy were valid1 and if the antithesis could only be avoided by assuming points1 then the antinomy itself would afford a conclusive reason in favour of points4 ,why1 then1 did ,kant think it impossible that space should be composed of points8 ,i think two considerations probably influenced him4 ,in the first place1 the essential thing about space is spatial order1 and mere points1 by themselves1 will not account for spatial order4 ,it is obvious that his argument assumes absolute space2 but it is spatial relations that are alone important1 and they cannot be reduced to points4 ,this ground for his view depends1 therefore1 upon his ignorance of the logical theory of order and his oscillations between absolute and relative space4 ,but there is also another ground for his opinion1 which is more relevant to our present topic4 ,this is the ground derived from infinite divisibility4 ,a space may be halved1 and then halved again1 and so on ad infinitum1 and at every stage of the process the parts are still spaces1 not points4 ,in order to reach points by such a method1 it would be necessary to come to the end of an unending process1 which is impossible4 ,but just as an infinite class can be given all at once by its defining concept1 though it cannot be reached by successive enumeration1 so an infinite set of points can be given all at once as making up a line or area or volume1 though they can never be reached by the process of successive division4 ,thus the infinite divisibility of space gives no ground for denying that space is composed of points4 ,kant does not give his grounds for this denial1 and we can therefore only conjecture what they were4 ,but the above two grounds1 which we have seen to be fallacious1 seem sufficient to account for his opinion1 and we may therefore conclude that the antithesis of the second antinomy is unproved4 ,the above illustration of ,kant's antinomies has only been introduced in order to show the relevance of the problem of infinity to the problem of the reality of objects of sense4 ,in the remainder of the present lecture1 ,i wish to state and explain the problem of infinity1 to show how it arose1 and to show the irrelevance of all the solutions proposed by philosophers4 ,in the following lecture1 ,i shall try to explain the true solution1 which has been discovered by the mathematicians1 but nevertheless belongs essentially to philosophy4 ,the solution is definitive1 in the sense that it entirely satisfies and convinces all who study it carefully4 ,for over two thousand years the human intellect was baffled by the problem2 its many failures and its ultimate success make this problem peculiarly apt for the illustration of method4 ,the problem appears to have first arisen in some such way as the following4#bb ,pythagoras and his followers1 who were interested1 like ,descartes1 in the application of number to geometry1 adopted in that science more arithmetical methods than those with which ,euclid has made us familiar4 ,they1 or their contemporaries the atomists1 believed1 apparently1 that space is composed of indivisible points1 while time is composed of indivisible instants4#bc ,this belief would not1 by itself1 have raised the difficulties which they encountered1 but it was presumably accompanied by another belief1 that the number of points in any finite area or of instants in any finite period must be finite4 ,i do not suppose that this latter belief was a conscious one1 because probably no other possibility had occurred to them4 ,but the belief nevertheless operated1 and very soon brought them into conflict with facts which they themselves discovered4 ,before explaining how this occurred1 however1 it is necessary to say one word in explanation of the phrase finite number4 ,the exact explanation is a matter for our next lecture2 for the present1 it must suffice to say that ,i mean #j and #a and #b and #c and so on1 for ever--in other words1 any number that can be obtained by successively adding ones4 ,this includes all the numbers that can be expressed by means of our ordinary numerals1 and since such numbers can be made greater and greater1 without ever reaching an unsurpassable maximum1 it is easy to suppose that there are no other numbers4 ,but this supposition1 natural as it is1 is mistaken4 #bb ,in what concerns the early ,greek philosophers1 my knowledge is largely derived from ,burnet's valuable work1 ,early ,greek ,philosophy 7#bnd ed41 ,london1 #aijh74 ,i have also been greatly assisted by ,mr ,d4 ,s4 ,robertson of ,trinity ,college1 who has supplied the deficiencies of my knowledge of ,greek1 and brought important references to my notice4 #bc ,cf4 ,aristotle1 ,metaphysics1 ,m4 #f1 #ajhj;b1 #ah sqq41 and #ajhc;b1 #h sqq4 ,whether the ,pythagoreans themselves believed space and time to be composed of indivisible points and instants is a debatable question4#bd ,it would seem that the distinction between space and matter had not yet been clearly made1 and that therefore1 when an atomistic view is expressed1 it is difficult to decide whether particles of matter or points of space are intended4 ,there is an interesting passage#be in ,aristotle's ,physics1#bf where he says3 ,the ,pythagoreans all maintained the existence of the void1 and said that it enters into the heaven itself from the boundless breath1 inasmuch as the heaven breathes in the void also2 and the void differentiates natures1 as if it were a sort of separation of consecutives1 and as if it were their differentiation2 and that this also is what is first in numbers1 for it is the void which differentiates them4 #bd ,there is some reason to think that the ,pythagoreans distinguished between discrete and continuous quantity4 ,g4 ,j4 ,allman1 in his ,greek ,geometry from ,thales to ,euclid1 says 7p4 #bc73 ,the ,pythagoreans made a fourfold division of mathematical science1 attributing one of its parts to the how many1 to poson1 and the other to the how much1 to pelikon2 and they assigned to each of these parts a twofold division4 ,for they said that discrete quantity1 or the how many1 either subsists by itself or must be considered with relation to some other2 but that continued quantity1 or the how much1 is either stable or in motion4 ,hence they affirmed that arithmetic contemplates that discrete quantity which subsists by itself1 but music that which is related to another2 and that geometry considers continued quantity so far as it is immovable2 but astronomy 7ten sphairiken7 contemplates continued quantity so far as it is of a self-motive nature4 7,proclus1 ed4 ,friedlein1 p4 #ce4 ,as to the distinction between to pelikon1 continuous1 and to poson1 discrete quantity1 see ,iambl41 in ,nicomachi ,geraseni ,arithmeticam introductionem1 ed4 ,tennulius1 p4 #adh47 ,cf4 p4 #dh4 #be ,referred to by ,burnet1 op4 cit41 p4 #abj4 #bf iv41 #f4 #bac;b1 #bb2 ,h4 ,ritter and ,l4 ,preller1 ,historia ,philosophiae ,graecae1 #hth ed41 ,gotha1 #ahih1 p4 #ge 7this work will be referred to in future as ,r4 ,p474 ,this seems to imply that they regarded matter as consisting of atoms with empty space in between4 ,but if so1 they must have thought space could be studied by only paying attention to the atoms1 for otherwise it would be hard to account for their arithmetical methods in geometry1 or for their statement that things are numbers4 ,the difficulty which beset the ,pythagoreans in their attempts to apply numbers arose through their discovery of incommensurables1 and this1 in turn1 arose as follows4 ,pythagoras1 as we all learnt in youth1 discovered the proposition that the sum of the squares on the sides of a right-angled triangle is equal to the square on the hypotenuse4 ,it is said that he sacrificed an ox when he discovered this theorem2 if so1 the ox was the first martyr to science4 ,but the theorem1 though it has remained his chief claim to immortality1 was soon found to have a consequence fatal to his whole philosophy4 ,consider the case of a right-angled triangle whose two sides are equal1 such a triangle as is formed by two sides of a square and a diagonal4 ,here1 in virtue of the theorem1 the square on the diagonal is double of the square on either of the sides4 ,but ,pythagoras or his early followers easily proved that the square of one whole number cannot be double of the square of another4#bg ,thus the length of the side and the length of the diagonal are incommensurable2 that is to say1 however small a unit of length you take1 if it is contained an exact number of times in the side1 it is not contained any exact number of times in the diagonal1 and vice versa4 #bg ,the ,pythagorean proof is roughly as follows4 ,if possible1 let the ratio of the diagonal to the side of a square be m_/n1 where m and n are whole numbers having no common factor4 ,then we must have m#b #bn#b4 ,now the square of an odd number is odd1 but m#b1 being equal to #bn#b1 is even4 ,hence m must be even4 ,but the square of an even number divides by #d1 therefore n#b1 which is half of m#b1 must be even4 ,therefore n must be even4 ,but1 since m is even1 and m and n have no common factor1 n must be odd4 ,thus n must be both odd and even1 which is impossible2 and therefore the diagonal and the side cannot have a rational ratio4 ,now this fact might have been assimilated by some philosophies without any great difficulty1 but to the philosophy of ,pythagoras it was absolutely fatal4 ,pythagoras held that number is the constitutive essence of all things1 yet no two numbers could express the ratio of the side of a square to the diagonal4 ,it would seem probable that we may expand his difficulty1 without departing from his thought1 by assuming that he regarded the length of a line as determined by the number of atoms contained in it--a line two inches long would contain twice as many atoms as a line one inch long1 and so on4 ,but if this were the truth1 then there must be a definite numerical ratio between any two finite lengths1 because it was supposed that the number of atoms in each1 however large1 must be finite4 ,here there was an insoluble contradiction4 ,the ,pythagoreans1 it is said1 resolved to keep the existence of incommensurables a profound secret1 revealed only to a few of the supreme heads of the sect2 and one of their number1 ,hippasos of ,metapontion1 is even said to have been shipwrecked at sea for impiously disclosing the terrible discovery to their enemies4 ,it must be remembered that ,pythagoras was the founder of a new religion as well as the teacher of a new science3 if the science came to be doubted1 the disciples might fall into sin1 and perhaps even eat beans1 which according to ,pythagoras is as bad as eating parents' bones4 ,the problem first raised by the discovery of incommensurables proved1 as time went on1 to be one of the most severe and at the same time most far-reaching problems that have confronted the human intellect in its endeavour to understand the world4 ,it showed at once that numerical measurement of lengths1 if it was to be made accurate1 must require an arithmetic more advanced and more difficult than any that the ancients possessed4 ,they therefore set to work to reconstruct geometry on a basis which did not assume the universal possibility of numerical measurement--a reconstruction which1 as may be seen in ,euclid1 they effected with extraordinary skill and with great logical acumen4 ,the moderns1 under the influence of ,cartesian geometry1 have reasserted the universal possibility of numerical measurement1 extending arithmetic1 partly for that purpose1 so as to include what are called irrational numbers1 which give the ratios of incommensurable lengths4 ,but although irrational numbers have long been used without a qualm1 it is only in quite recent years that logically satisfactory definitions of them have been given4 ,with these definitions1 the first and most obvious form of the difficulty which confronted the ,pythagoreans has been solved2 but other forms of the difficulty remain to be considered1 and it is these that introduce us to the problem of infinity in its pure form4 ,we saw that1 accepting the view that a length is composed of points1 the existence of incommensurables proves that every finite length must contain an infinite number of points4 ,in other words1 if we were to take away points one by one1 we should never have taken away all the points1 however long we continued the process4 ,the number of points1 therefore1 cannot be counted1 for counting is a process which enumerates things one by one4 ,the property of being unable to be counted is characteristic of infinite collections1 and is a source of many of their paradoxical qualities4 ,so paradoxical are these qualities that until our own day they were thought to constitute logical contradictions4 ,a long line of philosophers1 from ,zeno#bh to ,m4 ,bergson1 have based much of their metaphysics upon the supposed impossibility of infinite collections4 ,broadly speaking1 the difficulties were stated by ,zeno1 and nothing material was added until we reach ,bolzano's ,paradoxien des ,unendlichen1 a little work written in #ahdg-h1 and published posthumously in #ahea4 ,intervening attempts to deal with the problem are futile and negligible4 ,the definitive solution of the difficulties is due1 not to ,bolzano1 but to ,georg ,cantor1 whose work on this subject first appeared in #ahhb4 #bh ,in regard to ,zeno and the ,pythagoreans1 ,i have derived much valuable information and criticism from ,mr ,p4 ,e4 ,b4 ,jourdain4 ,in order to understand ,zeno1 and to realise how little modern orthodox metaphysics has added to the achievements of the ,greeks1 we must consider for a moment his master ,parmenides1 in whose interest the paradoxes were invented4#bi ,parmenides expounded his views in a poem divided into two parts1 called the way of truth and the way of opinion--like ,mr ,bradley's ,appearance and ,reality1 except that ,parmenides tells us first about reality and then about appearance4 ,the way of opinion1 in his philosophy1 is1 broadly speaking1 ,pythagoreanism2 it begins with a warning3 ,here ,i shall close my trustworthy speech and thought about the truth4 ,henceforward learn the opinions of mortals1 giving ear to the deceptive ordering of my words4 ,what has gone before has been revealed by a goddess1 who tells him what really is4 ,reality1 she says1 is uncreated1 indestructible1 unchanging1 indivisible2 it is immovable in the bonds of mighty chains1 without beginning and without end2 since coming into being and passing away have been driven afar1 and true belief has cast them away4 ,the fundamental principle of his inquiry is stated in a sentence which would not be out of place in ,hegel3#cj ,thou canst not know what is not--that is impossible--nor utter it2 for it is the same thing that can be thought and that can be4 ,and again3 ,it needs must be that what can be thought and spoken of is2 for it is possible for it to be1 and it is not possible for what is nothing to be4 ,the impossibility of change follows from this principle2 for what is past can be spoken of1 and therefore1 by the principle1 still is4 #bi ,so ,plato makes ,zeno say in the ,parmenides1 apropos of his philosophy as a whole2 and all internal and external evidence supports this view4 #cj ,with ,parmenides1 ,hegel says1 philosophising proper began4 ,werke 7edition of #ahdj71 vol4 xiii4 p4 #bgd4 ,the great conception of a reality behind the passing illusions of sense1 a reality one1 indivisible1 and unchanging1 was thus introduced into ,western philosophy by ,parmenides1 not1 it would seem1 for mystical or religious reasons1 but on the basis of a logical argument as to the impossibility of not-being4 ,all the great metaphysical systems--notably those of ,plato1 ,spinoza1 and ,hegel--are the outcome of this fundamental idea4 ,it is difficult to disentangle the truth and the error in this view4 ,the contention that time is unreal and that the world of sense is illusory must1 ,i think1 be regarded as based upon fallacious reasoning4 ,nevertheless1 there is some sense--easier to feel than to state--in which time is an unimportant and superficial characteristic of reality4 ,past and future must be acknowledged to be as real as the present1 and a certain emancipation from slavery to time is essential to philosophic thought4 ,the importance of time is rather practical than theoretical1 rather in relation to our desires than in relation to truth4 ,a truer image of the world1 ,i think1 is obtained by picturing things as entering into the stream of time from an eternal world outside1 than from a view which regards time as the devouring tyrant of all that is4 ,both in thought and in feeling1 to realise the unimportance of time is the gate of wisdom4 ,but unimportance is not unreality2 and therefore what we shall have to say about ,zeno's arguments in support of ,parmenides must be mainly critical4 ,the relation of ,zeno to ,parmenides is explained by ,plato#ca in the dialogue in which ,socrates1 as a young man1 learns logical acumen and philosophic disinterestedness from their dialectic4 ,i quote from ,jowett's translation3 ,i see1 ,parmenides1 said ,socrates1 that ,zeno is your second self in his writings too2 he puts what you say in another way1 and would fain deceive us into believing that he is telling us what is new4 ,for you1 in your poems1 say ,all is one1 and of this you adduce excellent proofs2 and he on the other hand says ,there is no ,many2 and on behalf of this he offers overwhelming evidence4 ,to deceive the world1 as you have done1 by saying the same thing in different ways1 one of you affirming the one1 and the other denying the many1 is a strain of art beyond the reach of most of us4 ,yes1 ,socrates1 said ,zeno4 ,but although you are as keen as a ,spartan hound in pursuing the track1 you do not quite apprehend the true motive of the composition1 which is not really such an ambitious work as you imagine2 for what you speak of was an accident2 ,i had no serious intention of deceiving the world4 ,the truth is1 that these writings of mine were meant to protect the arguments of ,parmenides against those who scoff at him and show the many ridiculous and contradictory results which they suppose to follow from the affirmation of the one4 ,my answer is an address to the partisans of the many1 whose attack ,i return with interest by retorting upon them that their hypothesis of the being of the many if carried out appears in a still more ridiculous light than the hypothesis of the being of the one4 #ca ,parmenides1 #abh ,a-,d4 ,zeno's four arguments against motion were intended to exhibit the contradictions that result from supposing that there is such a thing as change1 and thus to support the ,parmenidean doctrine that reality is unchanging4#cb ,unfortunately1 we only know his arguments through ,aristotle1#cc who stated them in order to refute them4 ,those philosophers in the present day who have had their doctrines stated by opponents will realise that a just or adequate presentation of ,zeno's position is hardly to be expected from ,aristotle2 but by some care in interpretation it seems possible to reconstruct the so-called sophisms which have been refuted by every tyro from that day to this4 #cb ,this interpretation is combated by ,milhaud1 ,les philosophes-geometres de la ,grece1 p4 #adj n41 but his reasons do not seem to me convincing4 ,all the interpretations in what follows are open to question1 but all have the support of reputable authorities4 #cc ,physics1 vi4 #i4 #bcif 7,r4,p4 #acf-aci74 ,zeno's arguments would seem to be ad hominem2 that is to say1 they seem to assume premisses granted by his opponents1 and to show that1 granting these premisses1 it is possible to deduce consequences which his opponents must deny4 ,in order to decide whether they are valid arguments or sophisms1 it is necessary to guess at the tacit premisses1 and to decide who was the homo at whom they were aimed4 ,some maintain that they were aimed at the ,pythagoreans1#cd while others have held that they were intended to refute the atomists4#ce ,m4 ,evellin1 on the contrary1 holds that they constitute a refutation of infinite divisibility1#cf while ,m4 ,g4 ,noel1 in the interests of ,hegel1 maintains that the first two arguments refute infinite divisibility1 while the next two refute indivisibles4#cg ,amid such a bewildering variety of interpretations1 we can at least not complain of any restrictions on our liberty of choice4 #cd ,cf4 ,gaston ,milhaud1 ,les philosophes-geometres de la ,grece1 p4 #adj n42 ,paul ,tannery1 ,pour l'histoire de la science hellene1 p4 #bdi2 ,burnet1 op4 cit41 p4 #cfb4 #ce ,cf4 ,r4 ,k4 ,gaye1 ,on ,aristotle1 ,physics1 ,z ix4 ,journal of ,philology1 vol4 xxxi41 esp4 p4 #aaa4 ,also ,moritz ,cantor1 ,vorlesungen ueber ,geschichte der ,mathematik1 #ast ed41 vol4 i41 #ahhj1 p4 #afh1 who1 however1 subsequently adopted ,paul ,tannery's opinion1 ,vorlesungen1 #crd ed4 7vol4 i4 p4 #bjj74 #cf ,le mouvement et les partisans des indivisibles1 ,revue de ,metaphysique et de ,morale1 vol4 i4 pp4 #chb-cie4 #cg ,le mouvement et les arguments de ,zenon d',elee1 ,revue de ,metaphysique et de ,morale1 vol4 i4 pp4 #ajg-abe4 ,the historical questions raised by the above-mentioned discussions are no doubt largely insoluble1 owing to the very scanty material from which our evidence is derived4 ,the points which seem fairly clear are the following3 7#a7 ,that1 in spite of ,,mm4 ,milhaud and ,paul ,tannery1 ,zeno is anxious to prove that motion is really impossible1 and that he desires to prove this because he follows ,parmenides in denying plurality2#ch 7#b7 that the third and fourth arguments proceed on the hypothesis of indivisibles1 a hypothesis which1 whether adopted by the ,pythagoreans or not1 was certainly much advocated1 as may be seen from the treatise ,on ,indivisible ,lines attributed to ,aristotle4 ,as regards the first two arguments1 they would seem to be valid on the hypothesis of indivisibles1 and also1 without this hypothesis1 to be such as would be valid if the traditional contradictions in infinite numbers were insoluble1 which they are not4 #ch ,cf4 ,m4 ,brochard1 ,les pretendus sophismes de ,zenon d',elee1 ,revue de ,metaphysique et de ,morale1 vol4 i4 pp4 #bji-bae4 ,we may conclude1 therefore1 that ,zeno's polemic is directed against the view that space and time consist of points and instants2 and that as against the view that a finite stretch of space or time consists of a finite number of points and instants1 his arguments are not sophisms1 but perfectly valid4 ,the conclusion which ,zeno wishes us to draw is that plurality is a delusion1 and spaces and times are really indivisible4 ,the other conclusion which is possible1 namely1 that the number of points and instants is infinite1 was not tenable so long as the infinite was infected with contradictions4 ,in a fragment which is not one of the four famous arguments against motion1 ,zeno says3 ,if things are a many1 they must be just as many as they are1 and neither more nor less4 ,now1 if they are as many as they are1 they will be finite in number4 ,if things are a many1 they will be infinite in number2 for there will always be other things between them1 and others again between these4 ,and so things are infinite in number4#ci #ci ,simplicius1 ,phys41 #adj1 #bh ,d 7,r4,p4 #acc72 ,burnet1 op4 cit41 pp4 #cfd-cfe4 ,this argument attempts to prove that1 if there are many things1 the number of them must be both finite and infinite1 which is impossible2 hence we are to conclude that there is only one thing4 ,but the weak point in the argument is the phrase3 ,if they are just as many as they are1 they will be finite in number4 ,this phrase is not very clear1 but it is plain that it assumes the impossibility of definite infinite numbers4 ,without this assumption1 which is now known to be false1 the arguments of ,zeno1 though they suffice 7on certain very reasonable assumptions7 to dispel the hypothesis of finite indivisibles1 do not suffice to prove that motion and change and plurality are impossible4 ,they are not1 however1 on any view1 mere foolish quibbles3 they are serious arguments1 raising difficulties which it has taken two thousand years to answer1 and which even now are fatal to the teachings of most philosophers4 ,the first of ,zeno's arguments is the argument of the race-course1 which is paraphrased by ,burnet as follows3#dj ,you cannot get to the end of a race-course4 ,you cannot traverse an infinite number of points in a finite time4 ,you must traverse the half of any given distance before you traverse the whole1 and the half of that again before you can traverse it4 ,this goes on ad infinitum1 so that there are an infinite number of points in any given space1 and you cannot touch an infinite number one by one in a finite time4#da #dj ,op4 cit41 p4 #cfg4 #da ,aristotle's words are3 ,the first is the one on the non-existence of motion on the ground that what is moved must always attain the middle point sooner than the end-point1 on which we gave our opinion in the earlier part of our discourse4 ,phys41 vi4 #i4 #ici;,b 7,r4,p4 #acf74 ,aristotle seems to refer to ,phys41 vi4 #b4 #bbc;,,ab ,r4,p4 #acf;,a3 ,all space is continuous1 for time and space are divided into the same and equal divisions4444 ,wherefore also ,zeno's argument is fallacious1 that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time4 ,for there are two senses in which the term 'infinite' is applied both to length and to time1 and in fact to all continuous things1 either in regard to divisibility1 or in regard to the ends4 ,now it is not possible to touch things infinite in regard to number in a finite time1 but it is possible to touch things infinite in regard to divisibility3 for time itself also is infinite in this sense4 ,so that in fact we go through an infinite1 space in an infinite time and not in a finite time1 and we touch infinite things with infinite things1 not with finite things4 ,philoponus1 a sixth-century commentator 7,r4,p4 #acf;,a1 ,exc4 ,paris ,philop4 in ,arist4 ,phys41 #hjc1 #b4 ,vit471 gives the following illustration3 ,for if a thing were moved the space of a cubit in one hour1 since in every space there are an infinite number of points1 the thing moved must needs touch all the points of the space3 it will then go through an infinite collection in a finite time1 which is impossible4 ,zeno appeals here1 in the first place1 to the fact that any distance1 however small1 can be halved4 ,from this it follows1 of course1 that there must be an infinite number of points in a line4 ,but1 ,aristotle represents him as arguing1 you cannot touch an infinite number of points one by one in a finite time4 ,the words one by one are important4 7#a7 ,if all the points touched are concerned1 then1 though you pass through them continuously1 you do not touch them one by one4 ,that is to say1 after touching one1 there is not another which you touch next3 no two points are next each other1 but between any two there are always an infinite number of others1 which cannot be enumerated one by one4 7#b7 ,if1 on the other hand1 only the successive middle points are concerned1 obtained by always halving what remains of the course1 then the points are reached one by one1 and1 though they are infinite in number1 they are in fact all reached in a finite time4 ,his argument to the contrary may be supposed to appeal to the view that a finite time must consist of a finite number of instants1 in which case what he says would be perfectly true on the assumption that the possibility of continued dichotomy is undeniable4 ,if1 on the other hand1 we suppose the argument directed against the partisans of infinite divisibility1 we must suppose it to proceed as follows3#db ,the points given by successive halving of the distances still to be traversed are infinite in number1 and are reached in succession1 each being reached a finite time later than its predecessor2 but the sum of an infinite number of finite times must be infinite1 and therefore the process will never be completed4 ,it is very possible that this is historically the right interpretation1 but in this form the argument is invalid4 ,if half the course takes half a minute1 and the next quarter takes a quarter of a minute1 and so on1 the whole course will take a minute4 ,the apparent force of the argument1 on this interpretation1 lies solely in the mistaken supposition that there cannot be anything beyond the whole of an infinite series1 which can be seen to be false by observing that #a is beyond the whole of the infinite series #a_/b1 #c_/d1 #g_/h1 #ae_/af1 444 #db ,cf4 ,mr ,c4 ,d4 ,broad1 ,note on ,achilles and the ,tortoise1 ,mind1 ,n4,s41 vol4 xxii4 pp4 #cah-i4 ,the second of ,zeno's arguments is the one concerning ,achilles and the tortoise1 which has achieved more notoriety than the others4 ,it is paraphrased by ,burnet as follows3#dc ,achilles will never overtake the tortoise4 ,he must first reach the place from which the tortoise started4 ,by that time the tortoise will have got some way ahead4 ,achilles must then make up that1 and again the tortoise will be ahead4 ,he is always coming nearer1 but he never makes up to it4#dd #dc ,op4 cit4 #dd ,aristotle's words are3 ,the second is the so-called ,achilles4 ,it consists in this1 that the slower will never be overtaken in its course by the quickest1 for the pursuer must always come first to the point from which the pursued has just departed1 so that the slower must necessarily be always still more or less in advance4 ,phys41 vi4 #i4 #bci;,b 7,r4,p4 #acg74 ,this argument is essentially the same as the previous one4 ,it shows that1 if ,achilles ever overtakes the tortoise1 it must be after an infinite number of instants have elapsed since he started4 ,this is in fact true2 but the view that an infinite number of instants make up an infinitely long time is not true1 and therefore the conclusion that ,achilles will never overtake the tortoise does not follow4 ,the third argument1#de that of the arrow1 is very interesting4 ,the text has been questioned4 ,burnet accepts the alterations of ,zeller1 and paraphrases thus3 ,the arrow in flight is at rest4 ,for1 if everything is at rest when it occupies a space equal to itself1 and what is in flight at any given moment always occupies a space equal to itself1 it cannot move4 #de ,phys41 vi4 #i4 #bci;,b 7,r4,p4 #ach74 ,but according to ,prantl1 the literal translation of the unemended text of ,aristotle's statement of the argument is as follows3 ,if everything1 when it is behaving in a uniform manner1 is continually either moving or at rest1 but what is moving is always in the now1 then the moving arrow is motionless4 ,this form of the argument brings out its force more clearly than ,burnet's paraphrase4 ,here1 if not in the first two arguments1 the view that a finite part of time consists of a finite series of successive instants seems to be assumed2 at any rate the plausibility of the argument seems to depend upon supposing that there are consecutive instants4 ,throughout an instant1 it is said1 a moving body is where it is3 it cannot move during the instant1 for that would require that the instant should have parts4 ,thus1 suppose we consider a period consisting of a thousand instants1 and suppose the arrow is in flight throughout this period4 ,at each of the thousand instants1 the arrow is where it is1 though at the next instant it is somewhere else4 ,it is never moving1 but in some miraculous way the change of position has to occur between the instants1 that is to say1 not at any time whatever4 ,this is what ,m4 ,bergson calls the cinematographic representation of reality4 ,the more the difficulty is meditated1 the more real it becomes4 ,the solution lies in the theory of continuous series3 we find it hard to avoid supposing that1 when the arrow is in flight1 there is a next position occupied at the next moment2 but in fact there is no next position and no next moment1 and when once this is imaginatively realised1 the difficulty is seen to disappear4 ,the fourth and last of ,zeno's arguments is#df the argument of the stadium4 #df ,phys41 vi4 #i4 #bci;,b 7,r4,p4 #aci74 ,the argument as stated by ,burnet is as follows3 ,first ,position4 ,second ,position4 ,a 4444 ,a 4444 ,b 4444 ,b 4444 ,c 4444 ,c 4444 ,half the time may be equal to double the time4 ,let us suppose three rows of bodies1 one of which 7,a7 is at rest while the other two 7,b1 ,c7 are moving with equal velocity in opposite directions4 ,by the time they are all in the same part of the course1 ,b will have passed twice as many of the bodies in ,c as in ,a4 ,therefore the time which it takes to pass ,c is twice as long as the time it takes to pass ,a4 ,but the time which ,b and ,c take to reach the position of ,a is the same4 ,therefore double the time is equal to the half4 ,gaye#dg devoted an interesting article to the interpretation of this argument4 ,his translation of ,aristotle's statement is as follows3 ,the fourth argument is that concerning the two rows of bodies1 each row being composed of an equal number of bodies of equal size1 passing each other on a race-course as they proceed with equal velocity in opposite directions1 the one row originally occupying the space between the goal and the middle point of the course1 and the other that between the middle point and the starting-post4 ,this1 he thinks1 involves the conclusion that half a given time is equal to double the time4 ,the fallacy of the reasoning lies in the assumption that a body occupies an equal time in passing with equal velocity a body that is in motion and a body of equal size that is at rest1 an assumption which is false4 ,for instance 7so runs the argument71 let ,a ,a 444 be the stationary bodies of equal size1 ,b ,b 444 the bodies1 equal in number and in size to ,a ,a 4441 originally occupying the half of the course from the starting-post to the middle of the ,a's1 and ,c ,c 444 those originally occupying the other half from the goal to the middle of the ,a's1 equal in number1 size1 and velocity1 to ,b ,b 444 ,then three consequences follow4 ,first1 as the ,b's and ,c's pass one another1 the first ,b reaches the last ,c at the same moment at which the first ,c reaches the last ,b4 ,secondly1 at this moment the first ,c has passed all the ,a's1 whereas the first ,b has passed only half the ,a's and has consequently occupied only half the time occupied by the first ,c1 since each of the two occupies an equal time in passing each ,a4 ,thirdly1 at the same moment all the ,b's have passed all the ,c's3 for the first ,c and the first ,b will simultaneously reach the opposite ends of the course1 since 7so says ,zeno7 the time occupied by the first ,c in passing each of the ,b's is equal to that occupied by it in passing each of the ,a's1 because an equal time is occupied by both the first ,b and the first ,c in passing all the ,a's4 ,this is the argument3 but it presupposes the aforesaid fallacious assumption4 #dg ,loc4 cit4 ,first ,position4 ,second ,position4 ,b ,b' ,b ,b ,b' ,b 4 4 4 4 4 4 ,a ,a' ,a ,a ,a' ,a 4 4 4 4 4 4 ,c ,c' ,c ,c ,c' ,c 4 4 4 4 4 4 ,this argument is not quite easy to follow1 and it is only valid as against the assumption that a finite time consists of a finite number of instants4 ,we may re-state it in different language4 ,let us suppose three drill-sergeants1 ,a1 ,a'1 and ,a1 standing in a row1 while the two files of soldiers march past them in opposite directions4 ,at the first moment which we consider1 the three men ,b1 ,b'1 ,b in one row1 and the three men ,c1 ,c'1 ,c in the other row1 are respectively opposite to ,a1 ,a'1 and ,a4 ,at the very next moment1 each row has moved on1 and now ,b and ,c are opposite ,a'4 ,thus ,b and ,c are opposite each other4 ,when1 then1 did ,b pass ,c'8 ,it must have been somewhere between the two moments which we supposed consecutive1 and therefore the two moments cannot really have been consecutive4 ,it follows that there must be other moments between any two given moments1 and therefore that there must be an infinite number of moments in any given interval of time4 ,the above difficulty1 that ,b must have passed ,c' at some time between two consecutive moments1 is a genuine one1 but is not precisely the difficulty raised by ,zeno4 ,what ,zeno professes to prove is that half of a given time is equal to double that time4 ,the most intelligible explanation of the argument known to me is that of ,gaye4#dh ,since1 however1 his explanation is not easy to set forth shortly1 ,i will re-state what seems to me to be the logical essence of ,zeno's contention4 ,if we suppose that time consists of a series of consecutive instants1 and that motion consists in passing through a series of consecutive points1 then the fastest possible motion is one which1 at each instant1 is at a point consecutive to that at which it was at the previous instant4 ,any slower motion must be one which has intervals of rest interspersed1 and any faster motion must wholly omit some points4 ,all this is evident from the fact that we cannot have more than one event for each instant4 ,but now1 in the case of our ,a's and ,b's and ,c's1 ,b is opposite a fresh ,a every instant1 and therefore the number of ,a's passed gives the number of instants since the beginning of the motion4 ,but during the motion ,b has passed twice as many ,c's1 and yet cannot have passed more than one each instant4 ,hence the number of instants since the motion began is twice the number of ,a's passed1 though we previously found it was equal to this number4 ,from this result1 ,zeno's conclusion follows4 #dh ,loc4 cit41 p4 #aje4 ,zeno's arguments1 in some form1 have afforded grounds for almost all the theories of space and time and infinity which have been constructed from his day to our own4 ,we have seen that all his arguments are valid 7with certain reasonable hypotheses7 on the assumption that finite spaces and times consist of a finite number of points and instants1 and that the third and fourth almost certainly in fact proceeded on this assumption1 while the first and second1 which were perhaps intended to refute the opposite assumption1 were in that case fallacious4 ,we may therefore escape from his paradoxes either by maintaining that1 though space and time do consist of points and instants1 the number of them in any finite interval is infinite2 or by denying that space and time consist of points and instants at all2 or lastly1 by denying the reality of space and time altogether4 ,it would seem that ,zeno himself1 as a supporter of ,parmenides1 drew the last of these three possible deductions1 at any rate in regard to time4 ,in this a very large number of philosophers have followed him4 ,many others1 like ,m4 ,bergson1 have preferred to deny that space and time consist of points and instants4 ,either of these solutions will meet the difficulties in the form in which ,zeno raised them4 ,but1 as we saw1 the difficulties can also be met if infinite numbers are admissible4 ,and on grounds which are independent of space and time1 infinite numbers1 and series in which no two terms are consecutive1 must in any case be admitted4 ,consider1 for example1 all the fractions less than #a1 arranged in order of magnitude4 ,between any two of them1 there are others1 for example1 the arithmetical mean of the two4 ,thus no two fractions are consecutive1 and the total number of them is infinite4 ,it will be found that much of what ,zeno says as regards the series of points on a line can be equally well applied to the series of fractions4 ,and we cannot deny that there are fractions1 so that two of the above ways of escape are closed to us4 ,it follows that1 if we are to solve the whole class of difficulties derivable from ,zeno's by analogy1 we must discover some tenable theory of infinite numbers4 ,what1 then1 are the difficulties which1 until the last thirty years1 led philosophers to the belief that infinite numbers are impossible8 ,the difficulties of infinity are of two kinds1 of which the first may be called sham1 while the others involve1 for their solution1 a certain amount of new and not altogether easy thinking4 ,the sham difficulties are those suggested by the etymology1 and those suggested by confusion of the mathematical infinite with what philosophers impertinently call the true infinite4 ,etymologically1 infinite should mean having no end4 ,but in fact some infinite series have ends1 some have not2 while some collections are infinite without being serial1 and can therefore not properly be regarded as either endless or having ends4 ,the series of instants from any earlier one to any later one 7both included7 is infinite1 but has two ends2 the series of instants from the beginning of time to the present moment has one end1 but is infinite4 ,kant1 in his first antinomy1 seems to hold that it is harder for the past to be infinite than for the future to be so1 on the ground that the past is now completed1 and that nothing infinite can be completed4 ,it is very difficult to see how he can have imagined that there was any sense in this remark2 but it seems most probable that he was thinking of the infinite as the unended4 ,it is odd that he did not see that the future too has one end at the present1 and is precisely on a level with the past4 ,his regarding the two as different in this respect illustrates just that kind of slavery to time which1 as we agreed in speaking of ,parmenides1 the true philosopher must learn to leave behind him4 ,the confusions introduced into the notions of philosophers by the so-called true infinite are curious4 ,they see that this notion is not the same as the mathematical infinite1 but they choose to believe that it is the notion which the mathematicians are vainly trying to reach4 ,they therefore inform the mathematicians1 kindly but firmly1 that they are mistaken in adhering to the false infinite1 since plainly the true infinite is something quite different4 ,the reply to this is that what they call the true infinite is a notion totally irrelevant to the problem of the mathematical infinite1 to which it has only a fanciful and verbal analogy4 ,so remote is it that ,i do not propose to confuse the issue by even mentioning what the true infinite is4 ,it is the false infinite that concerns us1 and we have to show that the epithet false is undeserved4 ,there are1 however1 certain genuine difficulties in understanding the infinite1 certain habits of mind derived from the consideration of finite numbers1 and easily extended to infinite numbers under the mistaken notion that they represent logical necessities4 ,for example1 every number that we are accustomed to1 except #j1 has another number immediately before it1 from which it results by adding #a2 but the first infinite number does not have this property4 ,the numbers before it form an infinite series1 containing all the ordinary finite numbers1 having no maximum1 no last finite number1 after which one little step would plunge us into the infinite4 ,if it is assumed that the first infinite number is reached by a succession of small steps1 it is easy to show that it is self-contradictory4 ,the first infinite number is1 in fact1 beyond the whole unending series of finite numbers4 ,but1 it will be said1 there cannot be anything beyond the whole of an unending series4 ,this1 we may point out1 is the very principle upon which ,zeno relies in the arguments of the race-course and the ,achilles4 ,take the race-course3 there is the moment when the runner still has half his distance to run1 then the moment when he still has a quarter1 then when he still has an eighth1 and so on in a strictly unending series4 ,beyond the whole of this series is the moment when he reaches the goal4 ,thus there certainly can be something beyond the whole of an unending series4 ,but it remains to show that this fact is only what might have been expected4 ,the difficulty1 like most of the vaguer difficulties besetting the mathematical infinite1 is derived1 ,i think1 from the more or less unconscious operation of the idea of counting4 ,if you set to work to count the terms in an infinite collection1 you will never have completed your task4 ,thus1 in the case of the runner1 if half1 three-quarters1 seven-eighths1 and so on of the course were marked1 and the runner was not allowed to pass any of the marks until the umpire said ,now1 then ,zeno's conclusion would be true in practice1 and he would never reach the goal4 ,but it is not essential to the existence of a collection1 or even to knowledge and reasoning concerning it1 that we should be able to pass its terms in review one by one4 ,this may be seen in the case of finite collections2 we can speak of mankind or the human race1 though many of the individuals in this collection are not personally known to us4 ,we can do this because we know of various characteristics which every individual has if he belongs to the collection1 and not if he does not4 ,and exactly the same happens in the case of infinite collections3 they may be known by their characteristics although their terms cannot be enumerated4 ,in this sense1 an unending series may nevertheless form a whole1 and there may be new terms beyond the whole of it4 ,some purely arithmetical peculiarities of infinite numbers have also caused perplexity4 ,for instance1 an infinite number is not increased by adding one to it1 or by doubling it4 ,such peculiarities have seemed to many to contradict logic1 but in fact they only contradict confirmed mental habits4 ,the whole difficulty of the subject lies in the necessity of thinking in an unfamiliar way1 and in realising that many properties which we have thought inherent in number are in fact peculiar to finite numbers4 ,if this is remembered1 the positive theory of infinity1 which will occupy the next lecture1 will not be found so difficult as it is to those who cling obstinately to the prejudices instilled by the arithmetic which is learnt in childhood4 ,,lecture ,,vii ,,the ,,positive ,,theory ,,of ,,infinity ,the positive theory of infinity1 and the general theory of number to which it has given rise1 are among the triumphs of scientific method in philosophy1 and are therefore specially suitable for illustrating the logical-analytic character of that method4 ,the work in this subject has been done by mathematicians1 and its results can be expressed in mathematical symbolism4 ,why1 then1 it may be said1 should the subject be regarded as philosophy rather than as mathematics8 ,this raises a difficult question1 partly concerned with the use of words1 but partly also of real importance in understanding the function of philosophy4 ,every subject-matter1 it would seem1 can give rise to philosophical investigations as well as to the appropriate science1 the difference between the two treatments being in the direction of movement and in the kind of truths which it is sought to establish4 ,in the special sciences1 when they have become fully developed1 the movement is forward and synthetic1 from the simpler to the more complex4 ,but in philosophy we follow the inverse direction3 from the complex and relatively concrete we proceed towards the simple and abstract by means of analysis1 seeking1 in the process1 to eliminate the particularity of the original subject-matter1 and to confine our attention entirely to the logical form of the facts concerned4 ,between philosophy and pure mathematics there is a certain affinity1 in the fact that both are general and a priori4 ,neither of them asserts propositions which1 like those of history and geography1 depend upon the actual concrete facts being just what they are4 ,we may illustrate this characteristic by means of ,leibniz's conception of many possible worlds1 of which one only is actual4 ,in all the many possible worlds1 philosophy and mathematics will be the same2 the differences will only be in respect of those particular facts which are chronicled by the descriptive sciences4 ,any quality1 therefore1 by which our actual world is distinguished from other abstractly possible worlds1 must be ignored by mathematics and philosophy alike4 ,mathematics and philosophy differ1 however1 in their manner of treating the general properties in which all possible worlds agree2 for while mathematics1 starting from comparatively simple propositions1 seeks to build up more and more complex results by deductive synthesis1 philosophy1 starting from data which are common knowledge1 seeks to purify and generalise them into the simplest statements of abstract form that can be obtained from them by logical analysis4 ,the difference between philosophy and mathematics may be illustrated by our present problem1 namely1 the nature of number4 ,both start from certain facts about numbers which are evident to inspection4 ,but mathematics uses these facts to deduce more and more complicated theorems1 while philosophy seeks1 by analysis1 to go behind these facts to others1 simpler1 more fundamental1 and inherently more fitted to form the premisses of the science of arithmetic4 ,the question1 ,what is a number8 is the pre-eminent philosophic question in this subject1 but it is one which the mathematician as such need not ask1 provided he knows enough of the properties of numbers to enable him to deduce his theorems4 ,we1 since our object is philosophical1 must grapple with the philosopher's question4 ,the answer to the question1 ,what is a number8 which we shall reach in this lecture1 will be found to give also1 by implication1 the answer to the difficulties of infinity which we considered in the previous lecture4 ,the question ,what is a number8 is one which1 until quite recent times1 was never considered in the kind of way that is capable of yielding a precise answer4 ,philosophers were content with some vague dictum such as1 ,number is unity in plurality4 ,a typical definition of the kind that contented philosophers is the following from ,sigwart's ,logic 7,sec4 #ff1 section #c73 ,every number is not merely a plurality1 but a plurality thought as held together and closed1 and to that extent as a unity4 ,now there is in such definitions a very elementary blunder1 of the same kind that would be committed if we said yellow is a flower because some flowers are yellow4 ,take1 for example1 the number #c4 ,a single collection of three things might conceivably be described as a plurality thought as held together and closed1 and to that extent as a unity2 but a collection of three things is not the number #c4 ,the number #c is something which all collections of three things have in common1 but is not itself a collection of three things4 ,the definition1 therefore1 apart from any other defects1 has failed to reach the necessary degree of abstraction3 the number #c is something more abstract than any collection of three things4 ,such vague philosophic definitions1 however1 remained inoperative because of their very vagueness4 ,what most men who thought about numbers really had in mind was that numbers are the result of counting4 ,on the consciousness of the law of counting1 says ,sigwart at the beginning of his discussion of number1 rests the possibility of spontaneously prolonging the series of numbers ad infinitum4 ,it is this view of number as generated by counting which has been the chief psychological obstacle to the understanding of infinite numbers4 ,counting1 because it is familiar1 is erroneously supposed to be simple1 whereas it is in fact a highly complex process1 which has no meaning unless the numbers reached in counting have some significance independent of the process by which they are reached4 ,and infinite numbers cannot be reached at all in this way4 ,the mistake is of the same kind as if cows were defined as what can be bought from a cattle-merchant4 ,to a person who knew several cattle-merchants1 but had never seen a cow1 this might seem an admirable definition4 ,but if in his travels he came across a herd of wild cows1 he would have to declare that they were not cows at all1 because no cattle-merchant could sell them4 ,so infinite numbers were declared not to be numbers at all1 because they could not be reached by counting4 ,it will be worth while to consider for a moment what counting actually is4 ,we count a set of objects when we let our attention pass from one to another1 until we have attended once to each1 saying the names of the numbers in order with each successive act of attention4 ,the last number named in this process is the number of the objects1 and therefore counting is a method of finding out what the number of the objects is4 ,but this operation is really a very complicated one1 and those who imagine that it is the logical source of number show themselves remarkably incapable of analysis4 ,in the first place1 when we say one1 two1 three 444 as we count1 we cannot be said to be discovering the number of the objects counted unless we attach some meaning to the words one1 two1 three1 444 ,a child may learn to know these words in order1 and to repeat them correctly like the letters of the alphabet1 without attaching any meaning to them4 ,such a child may count correctly from the point of view of a grown-up listener1 without having any idea of numbers at all4 ,the operation of counting1 in fact1 can only be intelligently performed by a person who already has some idea what the numbers are2 and from this it follows that counting does not give the logical basis of number4 ,again1 how do we know that the last number reached in the process of counting is the number of the objects counted8 ,this is just one of those facts that are too familiar for their significance to be realised2 but those who wish to be logicians must acquire the habit of dwelling upon such facts4 ,there are two propositions involved in this fact3 first1 that the number of numbers from #a up to any given number is that given number--for instance1 the number of numbers from #a to #ajj is a hundred2 secondly1 that if a set of numbers can be used as names of a set of objects1 each number occurring only once1 then the number of numbers used as names is the same as the number of objects4 ,the first of these propositions is capable of an easy arithmetical proof so long as finite numbers are concerned2 but with infinite numbers1 after the first1 it ceases to be true4 ,the second proposition remains true1 and is in fact1 as we shall see1 an immediate consequence of the definition of number4 ,but owing to the falsehood of the first proposition where infinite numbers are concerned1 counting1 even if it were practically possible1 would not be a valid method of discovering the number of terms in an infinite collection1 and would in fact give different results according to the manner in which it was carried out4 ,there are two respects in which the infinite numbers that are known differ from finite numbers3 first1 infinite numbers have1 while finite numbers have not1 a property which ,i shall call reflexiveness2 secondly1 finite numbers have1 while infinite numbers have not1 a property which ,i shall call inductiveness4 ,let us consider these two properties successively4 7#a7 ,reflexiveness4--,a number is said to be reflexive when it is not increased by adding #a to it4 ,it follows at once that any finite number can be added to a reflexive number without increasing it4 ,this property of infinite numbers was always thought1 until recently1 to be self-contradictory2 but through the work of ,georg ,cantor it has come to be recognised that1 though at first astonishing1 it is no more self-contradictory than the fact that people at the antipodes do not tumble off4 ,in virtue of this property1 given any infinite collection of objects1 any finite number of objects can be added or taken away without increasing or diminishing the number of the collection4 ,even an infinite number of objects may1 under certain conditions1 be added or taken away without altering the number4 ,this may be made clearer by the help of some examples4 ,imagine all the natural numbers #j1 #a1 #b1 #c1 444 to be written down in a row1 and immediately beneath them write down the numbers #a1 #b1 #c1 #d1 4441 so that #a is under #j1 #b is under #a1 and so on4 ,then every number in the top row has a number directly under it in the bottom row1 and no number occurs twice in either row4 ,it follows that the number of numbers in the two rows must be the same4 ,but all the numbers that occur in the bottom row also occur in the top row1 and one more1 namely #j2 thus the number of terms in the top row is obtained by adding one to the number of the bottom row4 ,so long1 therefore1 as it was supposed that a number must be increased by adding #a to it1 this state of things constituted a contradiction1 and led to the denial that there are infinite numbers4 #j1 #a1 #b1 #c1 444 n 444 #a1 #b1 #c1 #d1 444 n #a 444 ,the following example is even more surprising4 ,write the natural numbers #a1 #b1 #c1 #d1 444 in the top row1 and the even numbers #b1 #d1 #f1 #h1 444 in the bottom row1 so that under each number in the top row stands its double in the bottom row4 ,then1 as before1 the number of numbers in the two rows is the same1 yet the second row results from taking away all the odd numbers--an infinite collection--from the top row4 ,this example is given by ,leibniz to prove that there can be no infinite numbers4 ,he believed in infinite collections1 but1 since he thought that a number must always be increased when it is added to and diminished when it is subtracted from1 he maintained that infinite collections do not have numbers4 ,the number of all numbers1 he says1 implies a contradiction1 which ,i show thus3 ,to any number there is a corresponding number equal to its double4 ,therefore the number of all numbers is not greater than the number of even numbers1 i4e4 the whole is not greater than its part4#di ,in dealing with this argument1 we ought to substitute the number of all finite numbers for the number of all numbers2 we then obtain exactly the illustration given by our two rows1 one containing all the finite numbers1 the other only the even finite numbers4 ,it will be seen that ,leibniz regards it as self-contradictory to maintain that the whole is not greater than its part4 ,but the word greater is one which is capable of many meanings2 for our purpose1 we must substitute the less ambiguous phrase containing a greater number of terms4 ,in this sense1 it is not self-contradictory for whole and part to be equal2 it is the realisation of this fact which has made the modern theory of infinity possible4 #di ,phil4 ,werke1 ,gerhardt's edition1 vol4 i4 p4 #cch4 ,there is an interesting discussion of the reflexiveness of infinite wholes in the first of ,galileo's ,dialogues on ,motion4 ,i quote from a translation published in #agcj4ej ,the personages in the dialogue are ,salviati1 ,sagredo1 and ,simplicius1 and they reason as follows3 ,simp4 ,here already arises a ,doubt which ,i think is not to be resolv'd2 and that is this3 ,since 'tis plain that one ,line is given greater than another1 and since both contain infinite ,points1 we must surely necessarily infer1 that we have found in the same ,species something greater than ,infinite1 since the ,infinity of ,points of the greater ,line exceeds the ,infinity of ,points of the lesser4 ,but now1 to assign an ,infinite greater than an ,infinite1 is what ,i can't possibly conceive4 ,salv4 ,these are some of those ,difficulties which arise from ,discourses which our finite ,understanding makes about ,infinites1 by ascribing to them ,attributes which we give to ,things finite and terminate1 which ,i think most improper1 because those ,attributes of ,majority1 ,minority1 and ,equality1 agree not with ,infinities1 of which we can't say that one is greater than1 less than1 or equal to another4 ,for ,proof whereof ,i have something come into my ,head1 which 7that ,i may be the better understood7 ,i will propose by way of ,interrogatories to ,simplicius1 who started this ,difficulty4 ,to begin then3 ,i suppose you know which are square ,numbers1 and which not8 ,simp4 ,i know very well that a square ,number is that which arises from the ,multiplication of any ,number into itself2 thus #d and #i are square ,numbers1 that arising from #b1 and this from #c1 multiplied by themselves4 ,salv4 ,very well2 ,and you also know1 that as the ,products are call'd ,squares1 the ,factors are call'd ,roots3 ,and that the other ,numbers1 which proceed not from ,numbers multiplied into themselves1 are not ,squares4 ,whence taking in all ,numbers1 both ,squares and ,not ,squares1 if ,i should say1 that the ,not ,squares are more than the ,squares1 should ,i not be in the right8 ,simp4 ,most certainly4 ,salv4 ,if ,i go on with you then1 and ask you1 ,how many squar'd ,numbers there are8 you may truly answer1 ,that there are as many as are their proper ,roots1 since every ,square has its own ,root1 and every ,root its own ,square1 and since no ,square has more than one ,root1 nor any ,root more than one ,square4 ,simp4 ,very true4 ,salv4 ,but now1 if ,i should ask how many ,roots there are1 you can't deny but there are as many as there are ,numbers1 since there's no ,number but what's the ,root to some ,square4 ,and this being granted1 we may likewise affirm1 that there are as many square ,numbers1 as there are ,numbers2 for there are as many ,squares as there are ,roots1 and as many ,roots as ,numbers4 ,and yet in the ,beginning of this1 we said1 there were many more ,numbers than ,squares1 the greater ,part of ,numbers being not ,squares3 ,and tho' the ,number of ,squares decreases in a greater proportion1 as we go on to bigger ,numbers1 for count to an ,hundred you'll find #aj ,squares1 viz4 #a1 #d1 #i1 #af1 #be1 #cf1 #di1 #fd1 #ha1 #ajj1 which is the same as to say the #ajth ,part are ,squares2 in ,ten thousand only the #ajjth ,part are ,squares2 in a ,million only the #ajjjth3 ,and yet in an infinite ,number1 if we can but comprehend it1 we may say the ,squares are as many as all the ,numbers taken together4 ,sagr4 ,what must be determin'd then in this ,case8 ,salv4 ,i see no other way1 but by saying that all ,numbers are infinite2 ,squares are ,infinite1 their ,roots ,infinite1 and that the ,number of ,squares is not less than the ,number of ,numbers1 nor this less than that3 and then by concluding that the ,attributes or ,terms of ,equality1 ,majority1 and ,minority1 have no ,place in ,infinites1 but are confin'd to terminate ,quantities4 #ej ,mathematical ,discourses concerning two new sciences relating to mechanics and local motion1 in four dialogues4 ,by ,galileo ,galilei1 ,chief ,philosopher and ,mathematician to the ,grand ,duke of ,tuscany4 ,done into ,english from the ,italian1 by ,tho4 ,weston1 late ,master1 and now published by ,john ,weston1 present ,master1 of the ,academy at ,greenwich4 ,see pp4 #df ff4 ,the way in which the problem is expounded in the above discussion is worthy of ,galileo1 but the solution suggested is not the right one4 ,it is actually the case that the number of square 7finite7 numbers is the same as the number of 7finite7 numbers4 ,the fact that1 so long as we confine ourselves to numbers less than some given finite number1 the proportion of squares tends towards zero as the given finite number increases1 does not contradict the fact that the number of all finite squares is the same as the number of all finite numbers4 ,this is only an instance of the fact1 now familiar to mathematicians1 that the limit of a function as the variable approaches a given point may not be the same as its value when the variable actually reaches the given point4 ,but although the infinite numbers which ,galileo discusses are equal1 ,cantor has shown that what ,simplicius could not conceive is true1 namely1 that there are an infinite number of different infinite numbers1 and that the conception of greater and less can be perfectly well applied to them4 ,the whole of ,simplicius's difficulty comes1 as is evident1 from his belief that1 if greater and less can be applied1 a part of an infinite collection must have fewer terms than the whole2 and when this is denied1 all contradictions disappear4 ,as regards greater and less lengths of lines1 which is the problem from which the above discussion starts1 that involves a meaning of greater and less which is not arithmetical4 ,the number of points is the same in a long line and in a short one1 being in fact the same as the number of points in all space4 ,the greater and less of metrical geometry involves the new metrical conception of congruence1 which cannot be developed out of arithmetical considerations alone4 ,but this question has not the fundamental importance which belongs to the arithmetical theory of infinity4 7#b7 ,non-inductiveness4--,the second property by which infinite numbers are distinguished from finite numbers is the property of non-inductiveness4 ,this will be best explained by defining the positive property of inductiveness which characterises the finite numbers1 and which is named after the method of proof known as mathematical induction4 ,let us first consider what is meant by calling a property hereditary in a given series4 ,take such a property as being named ,jones4 ,if a man is named ,jones1 so is his son2 we will therefore call the property of being called ,jones hereditary with respect to the relation of father and son4 ,if a man is called ,jones1 all his descendants in the direct male line are called ,jones2 this follows from the fact that the property is hereditary4 ,now1 instead of the relation of father and son1 consider the relation of a finite number to its immediate successor1 that is1 the relation which holds between #j and #a1 between #a and #b1 between #b and #c1 and so on4 ,if a property of numbers is hereditary with respect to this relation1 then if it belongs to 7say7 #ajj1 it must belong also to all finite numbers greater than #ajj2 for1 being hereditary1 it belongs to #aja because it belongs to #ajj1 and it belongs to #ajb because it belongs to #aja1 and so on--where the and so on will take us1 sooner or later1 to any finite number greater than #ajj4 ,thus1 for example1 the property of being greater than #ii is hereditary in the series of finite numbers2 and generally1 a property is hereditary in this series when1 given any number that possesses the property1 the next number must always also possess it4 ,it will be seen that a hereditary property1 though it must belong to all the finite numbers greater than a given number possessing the property1 need not belong to all the numbers less than this number4 ,for example1 the hereditary property of being greater than #ii belongs to #ajj and all greater numbers1 but not to any smaller number4 ,similarly1 the hereditary property of being called ,jones belongs to all the descendants 7in the direct male line7 of those who have this property1 but not to all their ancestors1 because we reach at last a first ,jones1 before whom the ancestors have no surname4 ,it is obvious1 however1 that any hereditary property possessed by ,adam must belong to all men2 and similarly any hereditary property possessed by #j must belong to all finite numbers4 ,this is the principle of what is called mathematical induction4 ,it frequently happens1 when we wish to prove that all finite numbers have some property1 that we have first to prove that #j has the property1 and then that the property is hereditary1 i4e4 that1 if it belongs to a given number1 then it belongs to the next number4 ,owing to the fact that such proofs are called inductive1 ,i shall call the properties to which they are applicable inductive properties4 ,thus an inductive property of numbers is one which is hereditary and belongs to #j4 ,taking any one of the natural numbers1 say #bi1 it is easy to see that it must have all inductive properties4 ,for since such properties belong to #j and are hereditary1 they belong to #a2 therefore1 since they are hereditary1 they belong to #b1 and so on2 by twenty-nine repetitions of such arguments we show that they belong to #bi4 ,we may define the inductive numbers as all those that possess all inductive properties2 they will be the same as what are called the natural numbers1 i4e4 the ordinary finite whole numbers4 ,to all such numbers1 proofs by mathematical induction can be validly applied4 ,they are those numbers1 we may loosely say1 which can be reached from #j by successive additions of #a2 in other words1 they are all the numbers that can be reached by counting4 ,but beyond all these numbers1 there are the infinite numbers1 and infinite numbers do not have all inductive properties4 ,such numbers1 therefore1 may be called non-inductive4 ,all those properties of numbers which are proved by an imaginary step-by-step process from one number to the next are liable to fail when we come to infinite numbers4 ,the first of the infinite numbers has no immediate predecessor1 because there is no greatest finite number2 thus no succession of steps from one number to the next will ever reach from a finite number to an infinite one1 and the step-by-step method of proof fails4 ,this is another reason for the supposed self-contradictions of infinite numbers4 ,many of the most familiar properties of numbers1 which custom had led people to regard as logically necessary1 are in fact only demonstrable by the step-by-step method1 and fail to be true of infinite numbers4 ,but so soon as we realise the necessity of proving such properties by mathematical induction1 and the strictly limited scope of this method of proof1 the supposed contradictions are seen to contradict1 not logic1 but only our prejudices and mental habits4 ,the property of being increased by the addition of #a--;i4e4 the property of non-reflexiveness--may serve to illustrate the limitations of mathematical induction4 ,it is easy to prove that #j is increased by the addition of #a1 and that1 if a given number is increased by the addition of #a1 so is the next number1 i4e4 the number obtained by the addition of #a4 ,it follows that each of the natural numbers is increased by the addition of #a4 ,this follows generally from the general argument1 and follows for each particular case by a sufficient number of applications of the argument4 ,we first prove that #j is not equal to #a2 then1 since the property of being increased by #a is hereditary1 it follows that #a is not equal to #b2 hence it follows that #b is not equal to #c2 if we wish to prove that #cj1jjj is not equal to #cj1jja1 we can do so by repeating this reasoning #cj1jjj times4 ,but we cannot prove in this way that all numbers are increased by the addition of #a2 we can only prove that this holds of the numbers attainable by successive additions of #a starting from #j4 ,the reflexive numbers1 which lie beyond all those attainable in this way1 are as a matter of fact not increased by the addition of #a4 ,the two properties of reflexiveness and non-inductiveness1 which we have considered as characteristics of infinite numbers1 have not so far been proved to be always found together4 ,it is known that all reflexive numbers are non-inductive1 but it is not known that all non-inductive numbers are reflexive4 ,fallacious proofs of this proposition have been published by many writers1 including myself1 but up to the present no valid proof has been discovered4 ,the infinite numbers actually known1 however1 are all reflexive as well as non-inductive2 thus1 in mathematical practice1 if not in theory1 the two properties are always associated4 ,for our purposes1 therefore1 it will be convenient to ignore the bare possibility that there may be non-inductive non-reflexive numbers1 since all known numbers are either inductive or reflexive4 ,when infinite numbers are first introduced to people1 they are apt to refuse the name of numbers to them1 because their behaviour is so different from that of finite numbers that it seems a wilful misuse of terms to call them numbers at all4 ,in order to meet this feeling1 we must now turn to the logical basis of arithmetic1 and consider the logical definition of numbers4 ,the logical definition of numbers1 though it seems an essential support to the theory of infinite numbers1 was in fact discovered independently and by a different man4 ,the theory of infinite numbers--that is to say1 the arithmetical as opposed to the logical part of the theory--was discovered by ,georg ,cantor1 and published by him in #ahhb-c4ea ,the definition of number was discovered about the same time by a man whose great genius has not received the recognition it deserves--,i mean ,gottlob ,frege of ,jena4 ,his first work1 ,begriffsschrift1 published in #ahgi1 contained the very important theory of hereditary properties in a series to which ,i alluded in connection with inductiveness4 ,his definition of number is contained in his second work1 published in #ahhd1 and entitled ,die ,grundlagen der ,arithmetik1 eine logisch-mathematische ,untersuchung ueber den ,begriff der ,zahl4#eb ,it is with this book that the logical theory of arithmetic begins1 and it will repay us to consider ,frege's analysis in some detail4 #ea ,in his ,grundlagen einer allgemeinen ,mannichfaltigkeitslehre and in articles in ,acta ,mathematica1 vol4 ii4 #eb ,the definition of number contained in this book1 and elaborated in the ,grundgesetze der ,arithmetik 7vol4 i41 #ahic2 vol4 ii41 #aijc71 was rediscovered by me in ignorance of ,frege's work4 ,i wish to state as emphatically as possible--what seems still often ignored--that his discovery antedated mine by eighteen years4 ,frege begins by noting the increased desire for logical strictness in mathematical demonstrations which distinguishes modern mathematicians from their predecessors1 and points out that this must lead to a critical investigation of the definition of number4 ,he proceeds to show the inadequacy of previous philosophical theories1 especially of the synthetic a priori theory of ,kant and the empirical theory of ,mill4 ,this brings him to the question3 ,what kind of object is it that number can properly be ascribed to8 ,he points out that physical things may be regarded as one or many3 for example1 if a tree has a thousand leaves1 they may be taken altogether as constituting its foliage1 which would count as one1 not as a thousand2 and one pair of boots is the same object as two boots4 ,it follows that physical things are not the subjects of which number is properly predicated2 for when we have discovered the proper subjects1 the number to be ascribed must be unambiguous4 ,this leads to a discussion of the very prevalent view that number is really something psychological and subjective1 a view which ,frege emphatically rejects4 ,number1 he says1 is as little an object of psychology or an outcome of psychical processes as the ,north ,sea4444 ,the botanist wishes to state something which is just as much a fact when he gives the number of petals in a flower as when he gives its colour4 ,the one depends as little as the other upon our caprice4 ,there is therefore a certain similarity between number and colour2 but this does not consist in the fact that both are sensibly perceptible in external things1 but in the fact that both are objective 7p4 #cd74 ,i distinguish the objective1 he continues1 from the palpable1 the spatial1 the actual4 ,the earth's axis1 the centre of mass of the solar system1 are objective1 but ,i should not call them actual1 like the earth itself 7p4 #ce74 ,he concludes that number is neither spatial and physical1 nor subjective1 but non-sensible and objective4 ,this conclusion is important1 since it applies to all the subject-matter of mathematics and logic4 ,most philosophers have thought that the physical and the mental between them exhausted the world of being4 ,some have argued that the objects of mathematics were obviously not subjective1 and therefore must be physical and empirical2 others have argued that they were obviously not physical1 and therefore must be subjective and mental4 ,both sides were right in what they denied1 and wrong in what they asserted2 ,frege has the merit of accepting both denials1 and finding a third assertion by recognising the world of logic1 which is neither mental nor physical4 ,the fact is1 as ,frege points out1 that no number1 not even #a1 is applicable to physical things1 but only to general terms or descriptions1 such as man1 satellite of the earth1 satellite of ,venus4 ,the general term man is applicable to a certain number of objects3 there are in the world so and so many men4 ,the unity which philosophers rightly feel to be necessary for the assertion of a number is the unity of the general term1 and it is the general term which is the proper subject of number4 ,and this applies equally when there is one object or none which falls under the general term4 ,satellite of the earth is a term only applicable to one object1 namely1 the moon4 ,but one is not a property of the moon itself1 which may equally well be regarded as many molecules3 it is a property of the general term earth's satellite4 ,similarly1 #j is a property of the general term satellite of ,venus1 because ,venus has no satellite4 ,here at last we have an intelligible theory of the number #j4 ,this was impossible if numbers applied to physical objects1 because obviously no physical object could have the number #j4 ,thus1 in seeking our definition of number we have arrived so far at the result that numbers are properties of general terms or general descriptions1 not of physical things or of mental occurrences4 ,instead of speaking of a general term1 such as man1 as the subject of which a number can be asserted1 we may1 without making any serious change1 take the subject as the class or collection of objects--i4e4 mankind in the above instance--to which the general term in question is applicable4 ,two general terms1 such as man and featherless biped1 which are applicable to the same collection of objects1 will obviously have the same number of instances2 thus the number depends upon the class1 not upon the selection of this or that general term to describe it1 provided several general terms can be found to describe the same class4 ,but some general term is always necessary in order to describe a class4 ,even when the terms are enumerated1 as this and that and the other1 the collection is constituted by the general property of being either this1 or that1 or the other1 and only so acquires the unity which enables us to speak of it as one collection4 ,and in the case of an infinite class1 enumeration is impossible1 so that description by a general characteristic common and peculiar to the members of the class is the only possible description4 ,here1 as we see1 the theory of number to which ,frege was led by purely logical considerations becomes of use in showing how infinite classes can be amenable to number in spite of being incapable of enumeration4 ,frege next asks the question3 ,when do two collections have the same number of terms8 ,in ordinary life1 we decide this question by counting2 but counting1 as we saw1 is impossible in the case of infinite collections1 and is not logically fundamental with finite collections4 ,we want1 therefore1 a different method of answering our question4 ,an illustration may help to make the method clear4 ,i do not know how many married men there are in ,england1 but ,i do know that the number is the same as the number of married women4 ,the reason ,i know this is that the relation of husband and wife relates one man to one woman and one woman to one man4 ,a relation of this sort is called a one-one relation4 ,the relation of father to son is called a one-many relation1 because a man can have only one father but may have many sons2 conversely1 the relation of son to father is called a many-one relation4 ,but the relation of husband to wife 7in ,christian countries7 is called one-one1 because a man cannot have more than one wife1 or a woman more than one husband4 ,now1 whenever there is a one-one relation between all the terms of one collection and all the terms of another severally1 as in the case of ,english husbands and ,english wives1 the number of terms in the one collection is the same as the number in the other2 but when there is not such a relation1 the number is different4 ,this is the answer to the question3 ,when do two collections have the same number of terms8 ,we can now at last answer the question3 ,what is meant by the number of terms in a given collection8 ,when there is a one-one relation between all the terms of one collection and all the terms of another severally1 we shall say that the two collections are similar4 ,we have just seen that two similar collections have the same number of terms4 ,this leads us to define the number of a given collection as the class of all collections that are similar to it2 that is to say1 we set up the following formal definition3 ,the number of terms in a given class is defined as meaning the class of all classes that are similar to the given class4 ,this definition1 as ,frege 7expressing it in slightly different terms7 showed1 yields the usual arithmetical properties of numbers4 ,it is applicable equally to finite and infinite numbers1 and it does not require the admission of some new and mysterious set of metaphysical entities4 ,it shows that it is not physical objects1 but classes or the general terms by which they are defined1 of which numbers can be asserted2 and it applies to #j and #a without any of the difficulties which other theories find in dealing with these two special cases4 ,the above definition is sure to produce1 at first sight1 a feeling of oddity1 which is liable to cause a certain dissatisfaction4 ,it defines the number #b1 for instance1 as the class of all couples1 and the number #c as the class of all triads4 ,this does not seem to be what we have hitherto been meaning when we spoke of #b and #c1 though it would be difficult to say what we had been meaning4 ,the answer to a feeling cannot be a logical argument1 but nevertheless the answer in this case is not without importance4 ,in the first place1 it will be found that when an idea which has grown familiar as an unanalysed whole is first resolved accurately into its component parts--which is what we do when we define it--there is almost always a feeling of unfamiliarity produced by the analysis1 which tends to cause a protest against the definition4 ,in the second place1 it may be admitted that the definition1 like all definitions1 is to a certain extent arbitrary4 ,in the case of the small finite numbers1 such as #b and #c1 it would be possible to frame definitions more nearly in accordance with our unanalysed feeling of what we mean2 but the method of such definitions would lack uniformity1 and would be found to fail sooner or later--at latest when we reached infinite numbers4 ,in the third place1 the real desideratum about such a definition as that of number is not that it should represent as nearly as possible the ideas of those who have not gone through the analysis required in order to reach a definition1 but that it should give us objects having the requisite properties4 ,numbers1 in fact1 must satisfy the formulae of arithmetic2 any indubitable set of objects fulfilling this requirement may be called numbers4 ,so far1 the simplest set known to fulfil this requirement is the set introduced by the above definition4 ,in comparison with this merit1 the question whether the objects to which the definition applies are like or unlike the vague ideas of numbers entertained by those who cannot give a definition1 is one of very little importance4 ,all the important requirements are fulfilled by the above definition1 and the sense of oddity which is at first unavoidable will be found to wear off very quickly with the growth of familiarity4 ,there is1 however1 a certain logical doctrine which may be thought to form an objection to the above definition of numbers as classes of classes--,i mean the doctrine that there are no such objects as classes at all4 ,it might be thought that this doctrine would make havoc of a theory which reduces numbers to classes1 and of the many other theories in which we have made use of classes4 ,this1 however1 would be a mistake3 none of these theories are any the worse for the doctrine that classes are fictions4 ,what the doctrine is1 and why it is not destructive1 ,i will try briefly to explain4 ,on account of certain rather complicated difficulties1 culminating in definite contradictions1 ,i was led to the view that nothing that can be said significantly about things1 i4e4 particulars1 can be said significantly 7i4e4 either truly or falsely7 about classes of things4 ,that is to say1 if1 in any sentence in which a thing is mentioned1 you substitute a class for the thing1 you no longer have a sentence that has any meaning3 the sentence is no longer either true or false1 but a meaningless collection of words4 ,appearances to the contrary can be dispelled by a moment's reflection4 ,for example1 in the sentence1 ,adam is fond of apples1 you may substitute mankind1 and say1 ,mankind is fond of apples4 ,but obviously you do not mean that there is one individual1 called mankind1 which munches apples3 you mean that the separate individuals who compose mankind are each severally fond of apples4 ,now1 if nothing that can be said significantly about a thing can be said significantly about a class of things1 it follows that classes of things cannot have the same kind of reality as things have2 for if they had1 a class could be substituted for a thing in a proposition predicating the kind of reality which would be common to both4 ,this view is really consonant to common sense4 ,in the third or fourth century ,b4,c4 there lived a ,chinese philosopher named ,hui ,tzu1 who maintained that a bay horse and a dun cow are three2 because taken separately they are two1 and taken together they are one3 two and one make three4#ec ,the author from whom ,i quote says that ,hui ,tzu was particularly fond of the quibbles which so delighted the sophists or unsound reasoners of ancient ,greece1 and this no doubt represents the judgment of common sense upon such arguments4 ,yet if collections of things were things1 his contention would be irrefragable4 ,it is only because the bay horse and the dun cow taken together are not a new thing that we can escape the conclusion that there are three things wherever there are two4 #ec ,giles1 ,the ,civilisation of ,china 7,home ,university ,library71 p4 #adg4 ,when it is admitted that classes are not things1 the question arises3 ,what do we mean by statements which are nominally about classes8 ,take such a statement as1 ,the class of people interested in mathematical logic is not very numerous4 ,obviously this reduces itself to1 ,not very many people are interested in mathematical logic4 ,for the sake of definiteness1 let us substitute some particular number1 say #c1 for very many4 ,then our statement is1 ,not three people are interested in mathematical logic4 ,this may be expressed in the form3 ,if x is interested in mathematical logic1 and also y is interested1 and also z is interested1 then x is identical with y1 or x is identical with z1 or y is identical with z4 ,here there is no longer any reference at all to a class4 ,in some such way1 all statements nominally about a class can be reduced to statements about what follows from the hypothesis of anything's having the defining property of the class4 ,all that is wanted1 therefore1 in order to render the verbal use of classes legitimate1 is a uniform method of interpreting propositions in which such a use occurs1 so as to obtain propositions in which there is no longer any such use4 ,the definition of such a method is a technical matter1 which ,dr ,whitehead and ,i have dealt with elsewhere1 and which we need not enter into on this occasion4#ed #ed ,cf4 ,principia ,mathematica1 ,sec4 #bj1 and ,introduction1 chapter iii4 ,if the theory that classes are merely symbolic is accepted1 it follows that numbers are not actual entities1 but that propositions in which numbers verbally occur have not really any constituents corresponding to numbers1 but only a certain logical form which is not a part of propositions having this form4 ,this is in fact the case with all the apparent objects of logic and mathematics4 ,such words as or1 not1 if1 there is1 identity1 greater1 plus1 nothing1 everything1 function1 and so on1 are not names of definite objects1 like ,john or ,jones1 but are words which require a context in order to have meaning4 ,all of them are formal1 that is to say1 their occurrence indicates a certain form of proposition1 not a certain constituent4 ,logical constants1 in short1 are not entities2 the words expressing them are not names1 and cannot significantly be made into logical subjects except when it is the words themselves1 as opposed to their meanings1 that are being discussed4#ee ,this fact has a very important bearing on all logic and philosophy1 since it shows how they differ from the special sciences4 ,but the questions raised are so large and so difficult that it is impossible to pursue them further on this occasion4 #ee ,in the above remarks ,i am making use of unpublished work by my friend ,ludwig ,wittgenstein4 ,,lecture ,,viii ,,on ,,the ,,notion ,,of ,,cause1 ,,with ,,applications ,,to ,,the ,,free-,,will ,,problem ,the nature of philosophic analysis1 as illustrated in our previous lectures1 can now be stated in general terms4 ,we start from a body of common knowledge1 which constitutes our data4 ,on examination1 the data are found to be complex1 rather vague1 and largely interdependent logically4 ,by analysis we reduce them to propositions which are as nearly as possible simple and precise1 and we arrange them in deductive chains1 in which a certain number of initial propositions form a logical guarantee for all the rest4 ,these initial propositions are premisses for the body of knowledge in question4 ,premisses are thus quite different from data--they are simpler1 more precise1 and less infected with logical redundancy4 ,if the work of analysis has been performed completely1 they will be wholly free from logical redundancy1 wholly precise1 and as simple as is logically compatible with their leading to the given body of knowledge4 ,the discovery of these premisses belongs to philosophy2 but the work of deducing the body of common knowledge from them belongs to mathematics1 if mathematics is interpreted in a somewhat liberal sense4 ,but besides the logical analysis of the common knowledge which forms our data1 there is the consideration of its degree of certainty4 ,when we have arrived at its premisses1 we may find that some of them seem open to doubt1 and we may find further that this doubt extends to those of our original data which depend upon these doubtful premisses4 ,in our third lecture1 for example1 we saw that the part of physics which depends upon testimony1 and thus upon the existence of other minds than our own1 does not seem so certain as the part which depends exclusively upon our own sense-data and the laws of logic4 ,similarly1 it used to be felt that the parts of geometry which depend upon the axiom of parallels have less certainty than the parts which are independent of this premiss4 ,we may say1 generally1 that what commonly passes as knowledge is not all equally certain1 and that1 when analysis into premisses has been effected1 the degree of certainty of any consequence of the premisses will depend upon that of the most doubtful premiss employed in proving this consequence4 ,thus analysis into premisses serves not only a logical purpose1 but also the purpose of facilitating an estimate as to the degree of certainty to be attached to this or that derivative belief4 ,in view of the fallibility of all human beliefs1 this service seems at least as important as the purely logical services rendered by philosophical analysis4 ,in the present lecture1 ,i wish to apply the analytic method to the notion of cause1 and to illustrate the discussion by applying it to the problem of free will4 ,for this purpose ,i shall inquire3 ,i41 what is meant by a causal law2 ,,ii41 what is the evidence that causal laws have held hitherto2 ,,iii41 what is the evidence that they will continue to hold in the future2 ,,iv41 how the causality which is used in science differs from that of common sense and traditional philosophy2 ,v41 what new light is thrown on the question of free will by our analysis of the notion of cause4 ,i4 ,by a causal law ,i mean any general proposition in virtue of which it is possible to infer the existence of one thing or event from the existence of another or of a number of others4 ,if you hear thunder without having seen lightning1 you infer that there nevertheless was a flash1 because of the general proposition1 ,all thunder is preceded by lightning4 ,when ,robinson ,crusoe sees a footprint1 he infers a human being1 and he might justify his inference by the general proposition1 ,all marks in the ground shaped like a human foot are subsequent to a human being's standing where the marks are4 ,when we see the sun set1 we expect that it will rise again the next day4 ,when we hear a man speaking1 we infer that he has certain thoughts4 ,all these inferences are due to causal laws4 ,a causal law1 we said1 allows us to infer the existence of one thing 7or event7 from the existence of one or more others4 ,the word thing here is to be understood as only applying to particulars1 i4e4 as excluding such logical objects as numbers or classes or abstract properties and relations1 and including sense-data1 with whatever is logically of the same type as sense-data4#ef ,in so far as a causal law is directly verifiable1 the thing inferred and the thing from which it is inferred must both be data1 though they need not both be data at the same time4 ,in fact1 a causal law which is being used to extend our knowledge of existence must be applied to what1 at the moment1 is not a datum2 it is in the possibility of such application that the practical utility of a causal law consists4 ,the important point1 for our present purpose1 however1 is that what is inferred is a thing1 a particular1 an object having the kind of reality that belongs to objects of sense1 not an abstract object such as virtue or the square root of two4 #ef ,thus we are not using thing here in the sense of a class of correlated aspects1 as we did in ,lecture ,,iii4 ,each aspect will count separately in stating causal laws4 ,but we cannot become acquainted with a particular except by its being actually given4 ,hence the particular inferred by a causal law must be only described with more or less exactness2 it cannot be named until the inference is verified4 ,moreover1 since the causal law is general1 and capable of applying to many cases1 the given particular from which we infer must allow the inference in virtue of some general characteristic1 not in virtue of its being just the particular that it is4 ,this is obvious in all our previous instances3 we infer the unperceived lightning from the thunder1 not in virtue of any peculiarity of the thunder1 but in virtue of its resemblance to other claps of thunder4 ,thus a causal law must state that the existence of a thing of a certain sort 7or of a number of things of a number of assigned sorts7 implies the existence of another thing having a relation to the first which remains invariable so long as the first is of the kind in question4 ,it is to be observed that what is constant in a causal law is not the object or objects given1 nor yet the object inferred1 both of which may vary within wide limits1 but the relation between what is given and what is inferred4 ,the principle1 same cause1 same effect1 which is sometimes said to be the principle of causality1 is much narrower in its scope than the principle which really occurs in science2 indeed1 if strictly interpreted1 it has no scope at all1 since the same cause never recurs exactly4 ,we shall return to this point at a later stage of the discussion4 ,the particular which is inferred may be uniquely determined by the causal law1 or may be only described in such general terms that many different particulars might satisfy the description4 ,this depends upon whether the constant relation affirmed by the causal law is one which only one term can have to the data1 or one which many terms may have4 ,if many terms may have the relation in question1 science will not be satisfied until it has found some more stringent law1 which will enable us to determine the inferred things uniquely4 ,since all known things are in time1 a causal law must take account of temporal relations4 ,it will be part of the causal law to state a relation of succession or coexistence between the thing given and the thing inferred4 ,when we hear thunder and infer that there was lightning1 the law states that the thing inferred is earlier than the thing given4 ,conversely1 when we see lightning and wait expectantly for the thunder1 the law states that the thing given is earlier than the thing inferred4 ,when we infer a man's thoughts from his words1 the law states that the two are 7at least approximately7 simultaneous4 ,if a causal law is to achieve the precision at which science aims1 it must not be content with a vague earlier or later1 but must state how much earlier or how much later4 ,that is to say1 the time-relation between the thing given and the thing inferred ought to be capable of exact statement2 and usually the inference to be drawn is different according to the length and direction of the interval4 ,a quarter of an hour ago this man was alive2 an hour hence he will be cold4 ,such a statement involves two causal laws1 one inferring from a datum something which existed a quarter of an hour ago1 the other inferring from the same datum something which will exist an hour hence4 ,often a causal law involves not one datum1 but many1 which need not be all simultaneous with each other1 though their time-relations must be given4 ,the general scheme of a causal law will be as follows3 ,whenever things occur in certain relations to each other 7among which their time-relations must be included71 then a thing having a fixed relation to these things will occur at a date fixed relatively to their dates4 ,the things given will not1 in practice1 be things that only exist for an instant1 for such things1 if there are any1 can never be data4 ,the things given will each occupy some finite time4 ,they may be not static things1 but processes1 especially motions4 ,we have considered in an earlier lecture the sense in which a motion may be a datum1 and need not now recur to this topic4 ,it is not essential to a causal law that the object inferred should be later than some or all of the data4 ,it may equally well be earlier or at the same time4 ,the only thing essential is that the law should be such as to enable us to infer the existence of an object which we can more or less accurately describe in terms of the data4 ,,ii4 ,i come now to our second question1 namely3 ,what is the nature of the evidence that causal laws have held hitherto1 at least in the observed portions of the past8 ,this question must not be confused with the further question3 ,does this evidence warrant us in assuming the truth of causal laws in the future and in unobserved portions of the past8 ,for the present1 ,i am only asking what are the grounds which lead to a belief in causal laws1 not whether these grounds are adequate to support the belief in universal causation4 ,the first step is the discovery of approximate unanalysed uniformities of sequence or coexistence4 ,after lightning comes thunder1 after a blow received comes pain1 after approaching a fire comes warmth2 again1 there are uniformities of coexistence1 for example between touch and sight1 between certain sensations in the throat and the sound of one's own voice1 and so on4 ,every such uniformity of sequence or coexistence1 after it has been experienced a certain number of times1 is followed by an expectation that it will be repeated on future occasions1 i4e4 that where one of the correlated events is found1 the other will be found also4 ,the connection of experienced past uniformity with expectation as to the future is just one of those uniformities of sequence which we have observed to be true hitherto4 ,this affords a psychological account of what may be called the animal belief in causation1 because it is something which can be observed in horses and dogs1 and is rather a habit of acting than a real belief4 ,so far1 we have merely repeated ,hume1 who carried the discussion of cause up to this point1 but did not1 apparently1 perceive how much remained to be said4 ,is there1 in fact1 any characteristic1 such as might be called causality or uniformity1 which is found to hold throughout the observed past8 ,and if so1 how is it to be stated8 ,the particular uniformities which we mentioned before1 such as lightning being followed by thunder1 are not found to be free from exceptions4 ,we sometimes see lightning without hearing thunder2 and although1 in such a case1 we suppose that thunder might have been heard if we had been nearer to the lightning1 that is a supposition based on theory1 and therefore incapable of being invoked to support the theory4 ,what does seem1 however1 to be shown by scientific experience is this3 that where an observed uniformity fails1 some wider uniformity can be found1 embracing more circumstances1 and subsuming both the successes and the failures of the previous uniformity4 ,unsupported bodies in air fall1 unless they are balloons or aeroplanes2 but the principles of mechanics give uniformities which apply to balloons and aeroplanes just as accurately as to bodies that fall4 ,there is much that is hypothetical and more or less artificial in the uniformities affirmed by mechanics1 because1 when they cannot otherwise be made applicable1 unobserved bodies are inferred in order to account for observed peculiarities4 ,still1 it is an empirical fact that it is possible to preserve the laws by assuming such bodies1 and that they never have to be assumed in circumstances in which they ought to be observable4 ,thus the empirical verification of mechanical laws may be admitted1 although we must also admit that it is less complete and triumphant than is sometimes supposed4 ,assuming now1 what must be admitted to be doubtful1 that the whole of the past has proceeded according to invariable laws1 what can we say as to the nature of these laws8 ,they will not be of the simple type which asserts that the same cause always produces the same effect4 ,we may take the law of gravitation as a sample of the kind of law that appears to be verified without exception4 ,in order to state this law in a form which observation can confirm1 we will confine it to the solar system4 ,it then states that the motions of planets and their satellites have at every instant an acceleration compounded of accelerations towards all the other bodies in the solar system1 proportional to the masses of those bodies and inversely proportional to the squares of their distances4 ,in virtue of this law1 given the state of the solar system throughout any finite time1 however short1 its state at all earlier and later times is determinate except in so far as other forces than gravitation or other bodies than those in the solar system have to be taken into consideration4 ,but other forces1 so far as science can discover1 appear to be equally regular1 and equally capable of being summed up in single causal laws4 ,if the mechanical account of matter were complete1 the whole physical history of the universe1 past and future1 could be inferred from a sufficient number of data concerning an assigned finite time1 however short4 ,in the mental world1 the evidence for the universality of causal laws is less complete than in the physical world4 ,psychology cannot boast of any triumph comparable to gravitational astronomy4 ,nevertheless1 the evidence is not very greatly less than in the physical world4 ,the crude and approximate causal laws from which science starts are just as easy to discover in the mental sphere as in the physical4 ,in the world of sense1 there are to begin with the correlations of sight and touch and so on1 and the facts which lead us to connect various kinds of sensations with eyes1 ears1 nose1 tongue1 etc4 ,then there are such facts as that our body moves in answer to our volitions4 ,exceptions exist1 but are capable of being explained as easily as the exceptions to the rule that unsupported bodies in air fall4 ,there is1 in fact1 just such a degree of evidence for causal laws in psychology as will warrant the psychologist in assuming them as a matter of course1 though not such a degree as will suffice to remove all doubt from the mind of a sceptical inquirer4 ,it should be observed that causal laws in which the given term is mental and the inferred term physical1 or vice versa1 are at least as easy to discover as causal laws in which both terms are mental4 ,it will be noticed that1 although we have spoken of causal laws1 we have not hitherto introduced the word cause4 ,at this stage1 it will be well to say a few words on legitimate and illegitimate uses of this word4 ,the word cause1 in the scientific account of the world1 belongs only to the early stages1 in which small preliminary1 approximate generalisations are being ascertained with a view to subsequent larger and more invariable laws4 ,we may say1 ,arsenic causes death1 so long as we are ignorant of the precise process by which the result is brought about4 ,but in a sufficiently advanced science1 the word cause will not occur in any statement of invariable laws4 ,there is1 however1 a somewhat rough and loose use of the word cause which may be preserved4 ,the approximate uniformities which lead to its pre-scientific employment may turn out to be true in all but very rare and exceptional circumstances1 perhaps in all circumstances that actually occur4 ,in such cases1 it is convenient to be able to speak of the antecedent event as the cause and the subsequent event as the effect4 ,in this sense1 provided it is realised that the sequence is not necessary and may have exceptions1 it is still possible to employ the words cause and effect4 ,it is in this sense1 and in this sense only1 that we shall intend the words when we speak of one particular event causing another particular event1 as we must sometimes do if we are to avoid intolerable circumlocution4 ,,iii4 ,we come now to our third question1 namely3 ,what reason can be given for believing that causal laws will hold in future1 or that they have held in unobserved portions of the past8 ,what we have said so far is that there have been hitherto certain observed causal laws1 and that all the empirical evidence we possess is compatible with the view that everything1 both mental and physical1 so far as our observation has extended1 has happened in accordance with causal laws4 ,the law of universal causation1 suggested by these facts1 may be enunciated as follows3 ,there are such invariable relations between different events at the same or different times that1 given the state of the whole universe throughout any finite time1 however short1 every previous and subsequent event can theoretically be determined as a function of the given events during that time4 ,have we any reason to believe this universal law8 ,or1 to ask a more modest question1 have we any reason to believe that a particular causal law1 such as the law of gravitation1 will continue to hold in the future8 ,among observed causal laws is this1 that observation of uniformities is followed by expectation of their recurrence4 ,a horse who has been driven always along a certain road expects to be driven along that road again2 a dog who is always fed at a certain hour expects food at that hour and not at any other4 ,such expectations1 as ,hume pointed out1 explain only too well the common-sense belief in uniformities of sequence1 but they afford absolutely no logical ground for beliefs as to the future1 not even for the belief that we shall continue to expect the continuation of experienced uniformities1 for that is precisely one of those causal laws for which a ground has to be sought4 ,if ,hume's account of causation is the last word1 we have not only no reason to suppose that the sun will rise to-morrow1 but no reason to suppose that five minutes hence we shall still expect it to rise to-morrow4 ,it may1 of course1 be said that all inferences as to the future are in fact invalid1 and ,i do not see how such a view could be disproved4 ,but1 while admitting the legitimacy of such a view1 we may nevertheless inquire3 ,if inferences as to the future are valid1 what principle must be involved in making them8 ,the principle involved is the principle of induction1 which1 if it is true1 must be an a priori logical law1 not capable of being proved or disproved by experience4 ,it is a difficult question how this principle ought to be formulated2 but if it is to warrant the inferences which we wish to make by its means1 it must lead to the following proposition3 ,if1 in a great number of instances1 a thing of a certain kind is associated in a certain way with a thing of a certain other kind1 it is probable that a thing of the one kind is always similarly associated with a thing of the other kind2 and as the number of instances increases1 the probability approaches indefinitely near to certainty4 ,it may well be questioned whether this proposition is true2 but if we admit it1 we can infer that any characteristic of the whole of the observed past is likely to apply to the future and to the unobserved past4 ,this proposition1 therefore1 if it is true1 will warrant the inference that causal laws probably hold at all times1 future as well as past2 but without this principle1 the observed cases of the truth of causal laws afford no presumption as to the unobserved cases1 and therefore the existence of a thing not directly observed can never be validly inferred4 ,it is thus the principle of induction1 rather than the law of causality1 which is at the bottom of all inferences as to the existence of things not immediately given4 ,with the principle of induction1 all that is wanted for such inferences can be proved2 without it1 all such inferences are invalid4 ,this principle has not received the attention which its great importance deserves4 ,those who were interested in deductive logic naturally enough ignored it1 while those who emphasised the scope of induction wished to maintain that all logic is empirical1 and therefore could not be expected to realise that induction itself1 their own darling1 required a logical principle which obviously could not be proved inductively1 and must therefore be a priori if it could be known at all4 ,the view that the law of causality itself is a priori cannot1 ,i think1 be maintained by anyone who realises what a complicated principle it is4 ,in the form which states that every event has a cause it looks simple2 but on examination1 cause is merged in causal law1 and the definition of a causal law is found to be far from simple4 ,there must necessarily be some a priori principle involved in inference from the existence of one thing to that of another1 if such inference is ever valid2 but it would appear from the above analysis that the principle in question is induction1 not causality4 ,whether inferences from past to future are valid depends wholly1 if our discussion has been sound1 upon the inductive principle3 if it is true1 such inferences are valid1 and if it is false1 they are invalid4 ,,iv4 ,i come now to the question how the conception of causal laws which we have arrived at is related to the traditional conception of cause as it occurs in philosophy and common sense4 ,historically1 the notion of cause has been bound up with that of human volition4 ,the typical cause would be the fiat of a king4 ,the cause is supposed to be active1 the effect passive4 ,from this it is easy to pass on to the suggestion that a true cause must contain some prevision of the effect2 hence the effect becomes the end at which the cause aims1 and teleology replaces causation in the explanation of nature4 ,but all such ideas1 as applied to physics1 are mere anthropomorphic superstitions4 ,it is as a reaction against these errors that ,mach and others have urged a purely descriptive view of physics3 physics1 they say1 does not aim at telling us why things happen1 but only how they happen4 ,and if the question why8 means anything more than the search for a general law according to which a phenomenon occurs1 then it is certainly the case that this question cannot be answered in physics and ought not to be asked4 ,in this sense1 the descriptive view is indubitably in the right4 ,but in using causal laws to support inferences from the observed to the unobserved1 physics ceases to be purely descriptive1 and it is these laws which give the scientifically useful part of the traditional notion of cause4 ,there is therefore something to preserve in this notion1 though it is a very tiny part of what is commonly assumed in orthodox metaphysics4 ,in order to understand the difference between the kind of cause which science uses and the kind which we naturally imagine1 it is necessary to shut out1 by an effort1 everything that differentiates between past and future4 ,this is an extraordinarily difficult thing to do1 because our mental life is so intimately bound up with difference4 ,not only do memory and hope make a difference in our feelings as regards past and future1 but almost our whole vocabulary is filled with the idea of activity1 of things done now for the sake of their future effects4 ,all transitive verbs involve the notion of cause as activity1 and would have to be replaced by some cumbrous periphrasis before this notion could be eliminated4 ,consider such a statement as1 ,brutus killed ,caesar4 ,on another occasion1 ,brutus and ,caesar might engage our attention1 but for the present it is the killing that we have to study4 ,we may say that to kill a person is to cause his death intentionally4 ,this means that desire for a person's death causes a certain act1 because it is believed that that act will cause the person's death2 or more accurately1 the desire and the belief jointly cause the act4 ,brutus desires that ,caesar should be dead1 and believes that he will be dead if he is stabbed2 ,brutus therefore stabs him1 and the stab causes ,caesar's death1 as ,brutus expected it would4 ,every act which realises a purpose involves two causal steps in this way3 ,c is desired1 and it is believed 7truly if the purpose is achieved7 that ,b will cause ,c2 the desire and the belief together cause ,b1 which in turn causes ,c4 ,thus we have first ,a1 which is a desire for ,c and a belief that ,b 7an act7 will cause ,c2 then we have ,b1 the act caused by ,a1 and believed to be a cause of ,c2 then1 if the belief was correct1 we have ,c1 caused by ,b1 and if the belief was incorrect we have disappointment4 ,regarded purely scientifically1 this series ,a1 ,b1 ,c may equally well be considered in the inverse order1 as they would be at a coroner's inquest4 ,but from the point of view of ,brutus1 the desire1 which comes at the beginning1 is what makes the whole series interesting4 ,we feel that if his desires had been different1 the effects which he in fact produced would not have occurred4 ,this is true1 and gives him a sense of power and freedom4 ,it is equally true that if the effects had not occurred1 his desires would have been different1 since being what they were the effects did occur4 ,thus the desires are determined by their consequences just as much as the consequences by the desires2 but as we cannot 7in general7 know in advance the consequences of our desires without knowing our desires1 this form of inference is uninteresting as applied to our own acts1 though quite vital as applied to those of others4 ,a cause1 considered scientifically1 has none of that analogy with volition which makes us imagine that the effect is compelled by it4 ,a cause is an event or group of events1 of some known general character1 and having a known relation to some other event1 called the effect2 the relation being of such a kind that only one event1 or at any rate only one well-defined sort of event1 can have the relation to a given cause4 ,it is customary only to give the name effect to an event which is later than the cause1 but there is no kind of reason for this restriction4 ,we shall do better to allow the effect to be before the cause or simultaneous with it1 because nothing of any scientific importance depends upon its being after the cause4 ,if the inference from cause to effect is to be indubitable1 it seems that the cause can hardly stop short of the whole universe4 ,so long as anything is left out1 something may be left out which alters the expected result4 ,but for practical and scientific purposes1 phenomena can be collected into groups which are causally self-contained1 or nearly so4 ,in the common notion of causation1 the cause is a single event--we say the lightning causes the thunder1 and so on4 ,but it is difficult to know what we mean by a single event2 and it generally appears that1 in order to have anything approaching certainty concerning the effect1 it is necessary to include many more circumstances in the cause than unscientific common sense would suppose4 ,but often a probable causal connection1 where the cause is fairly simple1 is of more practical importance than a more indubitable connection in which the cause is so complex as to be hard to ascertain4 ,to sum up3 the strict1 certain1 universal law of causation which philosophers advocate is an ideal1 possibly true1 but not known to be true in virtue of any available evidence4 ,what is actually known1 as a matter of empirical science1 is that certain constant relations are observed to hold between the members of a group of events at certain times1 and that when such relations fail1 as they sometimes do1 it is usually possible to discover a new1 more constant relation by enlarging the group4 ,any such constant relation between events of specified kinds with given intervals of time between them is a causal law4 ,but all causal laws are liable to exceptions1 if the cause is less than the whole state of the universe2 we believe1 on the basis of a good deal of experience1 that such exceptions can be dealt with by enlarging the group we call the cause1 but this belief1 wherever it is still unverified1 ought not to be regarded as certain1 but only as suggesting a direction for further inquiry4 ,a very common causal group consists of volitions and the consequent bodily acts1 though exceptions arise 7for example7 through sudden paralysis4 ,another very frequent connection 7though here the exceptions are much more numerous7 is between a bodily act and the realisation of the purpose which led to the act4 ,these connections are patent1 whereas the causes of desires are more obscure4 ,thus it is natural to begin causal series with desires1 to suppose that all causes are analogous to desires1 and that desires themselves arise spontaneously4 ,such a view1 however1 is not one which any serious psychologist would maintain4 ,but this brings us to the question of the application of our analysis of cause to the problem of free will4 ,v4 ,the problem of free will is so intimately bound up with the analysis of causation that1 old as it is1 we need not despair of obtaining new light on it by the help of new views on the notion of cause4 ,the free-will problem has1 at one time or another1 stirred men's passions profoundly1 and the fear that the will might not be free has been to some men a source of great unhappiness4 ,i believe that1 under the influence of a cool analysis1 the doubtful questions involved will be found to have no such emotional importance as is sometimes thought1 since the disagreeable consequences supposed to flow from a denial of free will do not flow from this denial in any form in which there is reason to make it4 ,it is not1 however1 on this account chiefly that ,i wish to discuss this problem1 but rather because it affords a good example of the clarifying effect of analysis and of the interminable controversies which may result from its neglect4 ,let us first try to discover what it is we really desire when we desire free will4 ,some of our reasons for desiring free will are profound1 some trivial4 ,to begin with the former3 we do not wish to feel ourselves in the hands of fate1 so that1 however much we may desire to will one thing1 we may nevertheless be compelled by an outside force to will another4 ,we do not wish to think that1 however much we may desire to act well1 heredity and surroundings may force us into acting ill4 ,we wish to feel that1 in cases of doubt1 our choice is momentous and lies within our power4 ,besides these desires1 which are worthy of all respect1 we have1 however1 others not so respectable1 which equally make us desire free will4 ,we do not like to think that other people1 if they knew enough1 could predict our actions1 though we know that we can often predict those of other people1 especially if they are elderly4 ,much as we esteem the old gentleman who is our neighbour in the country1 we know that when grouse are mentioned he will tell the story of the grouse in the gun-room4 ,but we ourselves are not so mechanical3 we never tell an anecdote to the same person twice1 or even once unless he is sure to enjoy it2 although we once met 7say7 ,bismarck1 we are quite capable of hearing him mentioned without relating the occasion when we met him4 ,in this sense1 everybody thinks that he himself has free will1 though he knows that no one else has4 ,the desire for this kind of free will seems to be no better than a form of vanity4 ,i do not believe that this desire can be gratified with any certainty2 but the other1 more respectable desires are1 ,i believe1 not inconsistent with any tenable form of determinism4 ,we have thus two questions to consider3 7#a7 ,are human actions theoretically predictable from a sufficient number of antecedents8 7#b7 ,are human actions subject to an external compulsion8 ,the two questions1 as ,i shall try to show1 are entirely distinct1 and we may answer the first in the affirmative without therefore being forced to give an affirmative answer to the second4 7#a7 ,are human actions theoretically predictable from a sufficient number of antecedents8 ,let us first endeavour to give precision to this question4 ,we may state the question thus3 ,is there some constant relation between an act and a certain number of earlier events1 such that1 when the earlier events are given1 only one act1 or at most only acts with some well-marked character1 can have this relation to the earlier events8 ,if this is the case1 then1 as soon as the earlier events are known1 it is theoretically possible to predict either the precise act1 or at least the character necessary to its fulfilling the constant relation4 ,to this question1 a negative answer has been given by ,bergson1 in a form which calls in question the general applicability of the law of causation4 ,he maintains that every event1 and more particularly every mental event1 embodies so much of the past that it could not possibly have occurred at any earlier time1 and is therefore necessarily quite different from all previous and subsequent events4 ,if1 for example1 ,i read a certain poem many times1 my experience on each occasion is modified by the previous readings1 and my emotions are never repeated exactly4 ,the principle of causation1 according to him1 asserts that the same cause1 if repeated1 will produce the same effect4 ,but owing to memory1 he contends1 this principle does not apply to mental events4 ,what is apparently the same cause1 if repeated1 is modified by the mere fact of repetition1 and cannot produce the same effect4 ,he infers that every mental event is a genuine novelty1 not predictable from the past1 because the past contains nothing exactly like it by which we could imagine it4 ,and on this ground he regards the freedom of the will as unassailable4 ,bergson's contention has undoubtedly a great deal of truth1 and ,i have no wish to deny its importance4 ,but ,i do not think its consequences are quite what he believes them to be4 ,it is not necessary for the determinist to maintain that he can foresee the whole particularity of the act which will be performed4 ,if he could foresee that ,a was going to murder ,b1 his foresight would not be invalidated by the fact that he could not know all the infinite complexity of ,a's state of mind in committing the murder1 nor whether the murder was to be performed with a knife or with a revolver4 ,if the kind of act which will be performed can be foreseen within narrow limits1 it is of little practical interest that there are fine shades which cannot be foreseen4 ,no doubt every time the story of the grouse in the gun-room is told1 there will be slight differences due to increasing habitualness1 but they do not invalidate the prediction that the story will be told4 ,and there is nothing in ,bergson's argument to show that we can never predict what kind of act will be performed4 ,again1 his statement of the law of causation is inadequate4 ,the law does not state merely that1 if the same cause is repeated1 the same effect will result4 ,it states rather that there is a constant relation between causes of certain kinds and effects of certain kinds4 ,for example1 if a body falls freely1 there is a constant relation between the height through which it falls and the time it takes in falling4 ,it is not necessary to have a body fall through the same height which has been previously observed1 in order to be able to foretell the length of time occupied in falling4 ,if this were necessary1 no prediction would be possible1 since it would be impossible to make the height exactly the same on two occasions4 ,similarly1 the attraction which the sun will exert on the earth is not only known at distances for which it has been observed1 but at all distances1 because it is known to vary as the inverse square of the distance4 ,in fact1 what is found to be repeated is always the relation of cause and effect1 not the cause itself2 all that is necessary as regards the cause is that it should be of the same kind 7in the relevant respect7 as earlier causes whose effects have been observed4 ,another respect in which ,bergson's statement of causation is inadequate is in its assumption that the cause must be one event1 whereas it may be two or more events1 or even some continuous process4 ,the substantive question at issue is whether mental events are determined by the past4 ,now in such a case as the repeated reading of a poem1 it is obvious that our feelings in reading the poem are most emphatically dependent upon the past1 but not upon one single event in the past4 ,all our previous readings of the poem must be included in the cause4 ,but we easily perceive a certain law according to which the effect varies as the previous readings increase in number1 and in fact ,bergson himself tacitly assumes such a law4 ,we decide at last not to read the poem again1 because we know that this time the effect would be boredom4 ,we may not know all the niceties and shades of the boredom we should feel1 but we know enough to guide our decision1 and the prophecy of boredom is none the less true for being more or less general4 ,thus the kinds of cases upon which ,bergson relies are insufficient to show the impossibility of prediction in the only sense in which prediction has practical or emotional interest4 ,we may therefore leave the consideration of his arguments and address ourselves to the problem directly4 ,the law of causation1 according to which later events can theoretically be predicted by means of earlier events1 has often been held to be a priori1 a necessity of thought1 a category without which science would be impossible4 ,these claims seem to me excessive4 ,in certain directions the law has been verified empirically1 and in other directions there is no positive evidence against it4 ,but science can use it where it has been found to be true1 without being forced into any assumption as to its truth in other fields4 ,we cannot1 therefore1 feel any a priori certainty that causation must apply to human volitions4 ,the question how far human volitions are subject to causal laws is a purely empirical one4 ,empirically it seems plain that the great majority of our volitions have causes1 but it cannot1 on this account1 be held necessarily certain that all have causes4 ,there are1 however1 precisely the same kinds of reasons for regarding it as probable that they all have causes as there are in the case of physical events4 ,we may suppose--though this is doubtful--that there are laws of correlation of the mental and the physical1 in virtue of which1 given the state of all the matter in the world1 and therefore of all the brains and living organisms1 the state of all the minds in the world could be inferred1 while conversely the state of all the matter in the world could be inferred if the state of all the minds were given4 ,it is obvious that there is some degree of correlation between brain and mind1 and it is impossible to say how complete it may be4 ,this1 however1 is not the point which ,i wish to elicit4 ,what ,i wish to urge is that1 even if we admit the most extreme claims of determinism and of correlation of mind and brain1 still the consequences inimical to what is worth preserving in free will do not follow4 ,the belief that they follow results1 ,i think1 entirely from the assimilation of causes to volitions1 and from the notion that causes compel their effects in some sense analogous to that in which a human authority can compel a man to do what he would rather not do4 ,this assimilation1 as soon as the true nature of scientific causal laws is realised1 is seen to be a sheer mistake4 ,but this brings us to the second of the two questions which we raised in regard to free will1 namely1 whether1 assuming determinism1 our actions can be in any proper sense regarded as compelled by outside forces4 7#b7 ,are human actions subject to an external compulsion8 ,we have1 in deliberation1 a subjective sense of freedom1 which is sometimes alleged against the view that volitions have causes4 ,this sense of freedom1 however1 is only a sense that we can choose which we please of a number of alternatives3 it does not show us that there is no causal connection between what we please to choose and our previous history4 ,the supposed inconsistency of these two springs from the habit of conceiving causes as analogous to volitions--a habit which often survives unconsciously in those who intend to conceive causes in a more scientific manner4 ,if a cause is analogous to a volition1 outside causes will be analogous to an alien will1 and acts predictable from outside causes will be subject to compulsion4 ,but this view of cause is one to which science lends no countenance4 ,causes1 we have seen1 do not compel their effects1 any more than effects compel their causes4 ,there is a mutual relation1 so that either can be inferred from the other4 ,when the geologist infers the past state of the earth from its present state1 we should not say that the present state compels the past state to have been what it was2 yet it renders it necessary as a consequence of the data1 in the only sense in which effects are rendered necessary by their causes4 ,the difference which we feel1 in this respect1 between causes and effects is a mere confusion due to the fact that we remember past events but do not happen to have memory of the future4 ,the apparent indeterminateness of the future1 upon which some advocates of free will rely1 is merely a result of our ignorance4 ,it is plain that no desirable kind of free will can be dependent simply upon our ignorance2 for if that were the case1 animals would be more free than men1 and savages than civilised people4 ,free will in any valuable sense must be compatible with the fullest knowledge4 ,now1 quite apart from any assumption as to causality1 it is obvious that complete knowledge would embrace the future as well as the past4 ,our knowledge of the past is not wholly based upon causal inferences1 but is partly derived from memory4 ,it is a mere accident that we have no memory of the future4 ,we might--as in the pretended visions of seers--see future events immediately1 in the way in which we see past events4 ,they certainly will be what they will be1 and are in this sense just as determined as the past4 ,if we saw future events in the same immediate way in which we see past events1 what kind of free will would still be possible8 ,such a kind would be wholly independent of determinism3 it could not be contrary to even the most entirely universal reign of causality4 ,and such a kind must contain whatever is worth having in free will1 since it is impossible to believe that mere ignorance can be the essential condition of any good thing4 ,let us therefore imagine a set of beings who know the whole future with absolute certainty1 and let us ask ourselves whether they could have anything that we should call free will4 ,such beings as we are imagining would not have to wait for the event in order to know what decision they were going to adopt on some future occasion4 ,they would know now what their volitions were going to be4 ,but would they have any reason to regret this knowledge8 ,surely not1 unless the foreseen volitions were in themselves regrettable4 ,and it is less likely that the foreseen volitions would be regrettable if the steps which would lead to them were also foreseen4 ,it is difficult not to suppose that what is foreseen is fated1 and must happen however much it may be dreaded4 ,but human actions are the outcome of desire1 and no foreseeing can be true unless it takes account of desire4 ,a foreseen volition will have to be one which does not become odious through being foreseen4 ,the beings we are imagining would easily come to know the causal connections of volitions1 and therefore their volitions would be better calculated to satisfy their desires than ours are4 ,since volitions are the outcome of desires1 a prevision of volitions contrary to desires could not be a true one4 ,it must be remembered that the supposed prevision would not create the future any more than memory creates the past4 ,we do not think we were necessarily not free in the past1 merely because we can now remember our past volitions4 ,similarly1 we might be free in the future1 even if we could now see what our future volitions were going to be4 ,freedom1 in short1 in any valuable sense1 demands only that our volitions shall be1 as they are1 the result of our own desires1 not of an outside force compelling us to will what we would rather not will4 ,everything else is confusion of thought1 due to the feeling that knowledge compels the happening of what it knows when this is future1 though it is at once obvious that knowledge has no such power in regard to the past4 ,free will1 therefore1 is true in the only form which is important2 and the desire for other forms is a mere effect of insufficient analysis4 99 99 99 99 99 ,what has been said on philosophical method in the foregoing lectures has been rather by means of illustrations in particular cases than by means of general precepts4 ,nothing of any value can be said on method except through examples2 but now1 at the end of our course1 we may collect certain general maxims which may possibly be a help in acquiring a philosophical habit of mind and a guide in looking for solutions of philosophic problems4 ,philosophy does not become scientific by making use of other sciences1 in the kind of way in which 7e4g47 ,herbert ,spencer does4 ,philosophy aims at what is general1 and the special sciences1 however they may suggest large generalisations1 cannot make them certain4 ,and a hasty generalisation1 such as ,spencer's generalisation of evolution1 is none the less hasty because what is generalised is the latest scientific theory4 ,philosophy is a study apart from the other sciences3 its results cannot be established by the other sciences1 and conversely must not be such as some other science might conceivably contradict4 ,prophecies as to the future of the universe1 for example1 are not the business of philosophy2 whether the universe is progressive1 retrograde1 or stationary1 it is not for the philosopher to say4 ,in order to become a scientific philosopher1 a certain peculiar mental discipline is required4 ,there must be present1 first of all1 the desire to know philosophical truth1 and this desire must be sufficiently strong to survive through years when there seems no hope of its finding any satisfaction4 ,the desire to know philosophical truth is very rare--in its purity1 it is not often found even among philosophers4 ,it is obscured sometimes--particularly after long periods of fruitless search--by the desire to think we know4 ,some plausible opinion presents itself1 and by turning our attention away from the objections to it1 or merely by not making great efforts to find objections to it1 we may obtain the comfort of believing it1 although1 if we had resisted the wish for comfort1 we should have come to see that the opinion was false4 ,again the desire for unadulterated truth is often obscured1 in professional philosophers1 by love of system3 the one little fact which will not come inside the philosopher's edifice has to be pushed and tortured until it seems to consent4 ,yet the one little fact is more likely to be important for the future than the system with which it is inconsistent4 ,pythagoras invented a system which fitted admirably with all the facts he knew1 except the incommensurability of the diagonal of a square and the side2 this one little fact stood out1 and remained a fact even after ,hippasos of ,metapontion was drowned for revealing it4 ,to us1 the discovery of this fact is the chief claim of ,pythagoras to immortality1 while his system has become a matter of merely historical curiosity4#eg ,love of system1 therefore1 and the system-maker's vanity which becomes associated with it1 are among the snares that the student of philosophy must guard against4 #eg ,the above remarks1 for purposes of illustration1 adopt one of several possible opinions on each of several disputed points4 ,the desire to establish this or that result1 or generally to discover evidence for agreeable results1 of whatever kind1 has of course been the chief obstacle to honest philosophising4 ,so strangely perverted do men become by unrecognised passions1 that a determination in advance to arrive at this or that conclusion is generally regarded as a mark of virtue1 and those whose studies lead to an opposite conclusion are thought to be wicked4 ,no doubt it is commoner to wish to arrive at an agreeable result than to wish to arrive at a true result4 ,but only those in whom the desire to arrive at a true result is paramount can hope to serve any good purpose by the study of philosophy4 ,but even when the desire to know exists in the requisite strength1 the mental vision by which abstract truth is recognised is hard to distinguish from vivid imaginability and consonance with mental habits4 ,it is necessary to practise methodological doubt1 like ,descartes1 in order to loosen the hold of mental habits2 and it is necessary to cultivate logical imagination1 in order to have a number of hypotheses at command1 and not to be the slave of the one which common sense has rendered easy to imagine4 ,these two processes1 of doubting the familiar and imagining the unfamiliar1 are correlative1 and form the chief part of the mental training required for a philosopher4 ,the naive beliefs which we find in ourselves when we first begin the process of philosophic reflection may turn out1 in the end1 to be almost all capable of a true interpretation2 but they ought all1 before being admitted into philosophy1 to undergo the ordeal of sceptical criticism4 ,until they have gone through this ordeal1 they are mere blind habits1 ways of behaving rather than intellectual convictions4 ,and although it may be that a majority will pass the test1 we may be pretty sure that some will not1 and that a serious readjustment of our outlook ought to result4 ,in order to break the dominion of habit1 we must do our best to doubt the senses1 reason1 morals1 everything in short4 ,in some directions1 doubt will be found possible2 in others1 it will be checked by that direct vision of abstract truth upon which the possibility of philosophical knowledge depends4 ,at the same time1 and as an essential aid to the direct perception of the truth1 it is necessary to acquire fertility in imagining abstract hypotheses4 ,this is1 ,i think1 what has most of all been lacking hitherto in philosophy4 ,so meagre was the logical apparatus that all the hypotheses philosophers could imagine were found to be inconsistent with the facts4 ,too often this state of things led to the adoption of heroic measures1 such as a wholesale denial of the facts1 when an imagination better stocked with logical tools would have found a key to unlock the mystery4 ,it is in this way that the study of logic becomes the central study in philosophy3 it gives the method of research in philosophy1 just as mathematics gives the method in physics4 ,and as physics1 which1 from ,plato to the ,renaissance1 was as unprogressive1 dim1 and superstitious as philosophy1 became a science through ,galileo's fresh observation of facts and subsequent mathematical manipulation1 so philosophy1 in our own day1 is becoming scientific through the simultaneous acquisition of new facts and logical methods4 ,in spite1 however1 of the new possibility of progress in philosophy1 the first effect1 as in the case of physics1 is to diminish very greatly the extent of what is thought to be known4 ,before ,galileo1 people believed themselves possessed of immense knowledge on all the most interesting questions in physics4 ,he established certain facts as to the way in which bodies fall1 not very interesting on their own account1 but of quite immeasurable interest as examples of real knowledge and of a new method whose future fruitfulness he himself divined4 ,but his few facts sufficed to destroy the whole vast system of supposed knowledge handed down from ,aristotle1 as even the palest morning sun suffices to extinguish the stars4 ,so in philosophy3 though some have believed one system1 and others another1 almost all have been of opinion that a great deal was known2 but all this supposed knowledge in the traditional systems must be swept away1 and a new beginning must be made1 which we shall esteem fortunate indeed if it can attain results comparable to ,galileo's law of falling bodies4 ,by the practice of methodological doubt1 if it is genuine and prolonged1 a certain humility as to our knowledge is induced3 we become glad to know anything in philosophy1 however seemingly trivial4 ,philosophy has suffered from the lack of this kind of modesty4 ,it has made the mistake of attacking the interesting problems at once1 instead of proceeding patiently and slowly1 accumulating whatever solid knowledge was obtainable1 and trusting the great problems to the future4 ,men of science are not ashamed of what is intrinsically trivial1 if its consequences are likely to be important2 the immediate outcome of an experiment is hardly ever interesting on its own account4 ,so in philosophy1 it is often desirable to expend time and care on matters which1 judged alone1 might seem frivolous1 for it is often only through the consideration of such matters that the greater problems can be approached4 ,when our problem has been selected1 and the necessary mental discipline has been acquired1 the method to be pursued is fairly uniform4 ,the big problems which provoke philosophical inquiry are found1 on examination1 to be complex1 and to depend upon a number of component problems1 usually more abstract than those of which they are the components4 ,it will generally be found that all our initial data1 all the facts that we seem to know to begin with1 suffer from vagueness1 confusion1 and complexity4 ,current philosophical ideas share these defects2 it is therefore necessary to create an apparatus of precise conceptions as general and as free from complexity as possible1 before the data can be analysed into the kind of premisses which philosophy aims at discovering4 ,in this process of analysis1 the source of difficulty is tracked further and further back1 growing at each stage more abstract1 more refined1 more difficult to apprehend4 ,usually it will be found that a number of these extraordinarily abstract questions underlie any one of the big obvious problems4 ,when everything has been done that can be done by method1 a stage is reached where only direct philosophic vision can carry matters further4 ,here only genius will avail4 ,what is wanted1 as a rule1 is some new effort of logical imagination1 some glimpse of a possibility never conceived before1 and then the direct perception that this possibility is realised in the case in question4 ,failure to think of the right possibility leaves insoluble difficulties1 balanced arguments pro and con1 utter bewilderment and despair4 ,but the right possibility1 as a rule1 when once conceived1 justifies itself swiftly by its astonishing power of absorbing apparently conflicting facts4 ,from this point onward1 the work of the philosopher is synthetic and comparatively easy2 it is in the very last stage of the analysis that the real difficulty consists4 ,of the prospect of progress in philosophy1 it would be rash to speak with confidence4 ,many of the traditional problems of philosophy1 perhaps most of those which have interested a wider circle than that of technical students1 do not appear to be soluble by scientific methods4 ,just as astronomy lost much of its human interest when it ceased to be astrology1 so philosophy must lose in attractiveness as it grows less prodigal of promises4 ,but to the large and still growing body of men engaged in the pursuit of science--men who hitherto1 not without justification1 have turned aside from philosophy with a certain contempt--the new method1 successful already in such time-honoured problems as number1 infinity1 continuity1 space and time1 should make an appeal which the older methods have wholly failed to make4 ,physics1 with its principle of relativity and its revolutionary investigations into the nature of matter1 is feeling the need for that kind of novelty in fundamental hypotheses which scientific philosophy aims at facilitating4 ,the one and only condition1 ,i believe1 which is necessary in order to secure for philosophy in the near future an achievement surpassing all that has hitherto been accomplished by philosophers1 is the creation of a school of men with scientific training and philosophical interests1 unhampered by the traditions of the past1 and not misled by the literary methods of those who copy the ancients in all except their merits4 ,,index ,absolute1 #f1 #ci4 ,abstraction1 principle of1 #db1 #abd ff4 ,achilles1 ,zeno's argument of1 #agc4 ,acquaintance1 #be1 #add4 ,activity1 #bbd ff4 ,allman1 #afa n4 ,analysis1 #ahe1 #bjd1 #baa1 #bda4 legitimacy of1 #aej4 ,anaximander1 #c4 ,antinomies1 ,kant's1 #aee ff4 ,aquinas1 #aj4 ,aristotle1 #dj1 #afj n41 #afa ff41 #bdj4 ,arrow1 ,zeno's argument of1 #agc4 ,assertion1 #eb4 ,atomism1 logical1 #d4 ,atomists1 #afj4 ,belief1 #eh4 primitive and derivative1 #fi ff4 ,bergson1 #d1 #aa1 #ac1 #bj ff41 #acg1 #ach1 #aej1 #aeh1 #afe1 #agd1 #agh1 #bbi ff4 ,berkeley1 #fc1 #fd1 #ajb4 ,bolzano1 #afe4 ,boole1 #dj4 ,bradley1 #f1 #ci1 #afe4 ,broad1 #agb n4 ,brochard1 #afi n4 ,burnet1 #ai n41 #afj n41 #afa n41 #agj n41 #aga ff4 ,calderon1 #ie4 ,cantor1 vi1 vii1 #aee1 #afe1 #aij1 #aid1 #aii4 ,categories1 #ch4 ,causal laws1 #aji1 #bab ff4 evidence for1 #baf ff4 in psychology1 #bai4 ,causation1 #cd ff41 #gi1 #bab ff4 law of1 #bba4 not a priori1 #bbc1 #bcb4 ,cause1 #bbj1 #bbc4 ,certainty1 degrees of1 #fg1 #fh1 #bab4 ,change1 demands analysis1 #aea4 ,cinematograph1 #adh1 #agd4 ,classes1 #bjb4 non-existence of1 #bje ff4 ,classical tradition1 #c ff41 #eh4 ,complexity1 #ade1 #aeg ff4 ,compulsion1 #bbi1 #bcc ff4 ,congruence1 #aie4 ,consecutiveness1 #acd4 ,conservation1 #aje4 ,constituents of facts1 #ea1 #ade4 ,construction v4 inference1 iv4 ,contemporaries1 initial1 #aai1 #abj n4 ,continuity1 #fd1 #abi ff41 #ada ff41 #aee ff4 of change1 #ajf1 #ajh1 #acj ff4 ,correlation of mental and physical1 #bcc4 ,counting1 #afd1 #aha1 #ahg ff41 #bjc4 ,couturat1 #dj n4 ,dante1 #aj4 ,darwin1 #d1 #aa1 #bc1 #cj4 ,data1 #fe ff41 #baa4 hard and soft1 #gj ff4 ,dates1 #aag4 ,definition1 #bjd4 ,descartes1 #e1 #gc1 #bch4 ,descriptions1 #bja1 #bad4 ,desire1 #bbg1 #bce4 ,determinism1 #bcc4 ,doubt1 #bcg4 ,dreams1 #he1 #ic4 ,duration1 #adf1 #adi4 ,earlier and later1 #aaf4 ,effect1 #bbj4 ,eleatics1 #ai4 ,empiricism1 #cg1 #bbb4 ,enclosure1 #aad ff41 #abj4 ,enumeration1 #bjb4 ,euclid1 #afj1 #afd4 ,evellin1 #afi4 ,evolutionism1 #d1 #aa ff4 ,extension1 #adf1 #adi4 ,external world1 knowledge of1 #fc ff4 ,fact1 #ea4 atomic1 #eb4 ,finalism1 #ac4 ,form1 logical1 #db ff41 #ahe1 #bjh4 ,fractions1 #acb1 #agi4 ,free will1 #bac1 #bbg ff4 ,frege1 #e1 #dj1 #aii ff4 ,galileo1 #d1 #ei1 #aib1 #aid1 #bci1 #bdj4 ,gaye1 #afi n41 #age1 #agg4 ,geometry1 #e4 ,giles1 #bjf n4 ,greater and less1 #aie4 ,harvard1 #d4 ,hegel1 #c1 #cg ff41 #df1 #aff4 ,here1 #gc1 #ib4 ,hereditary properties1 #aie4 ,hippasos1 #afc1 #bcg4 ,hui ,tzu1 #bjf4 ,hume1 #bag1 #bba4 ,hypotheses in philosophy1 #bci4 ,illusions1 #he4 ,incommensurables1 #afb ff41 #bcg4 ,independence1 #gc1 #gd4 causal and logical1 #gd1 #ge4 ,indiscernibility1 #ada1 #adh4 ,indivisibles1 #afj4 ,induction1 #cd1 #bbb4 mathematical1 #aie ff4 ,inductiveness1 #aij1 #aie ff4 ,inference1 #dd1 #ed4 ,infinite1 vi1 #fd1 #acc1 #adi4 historically considered1 #aee ff4 true1 #agi1 #ahj4 positive theory of1 #ahe ff4 ,infinitesimals1 #ace4 ,instants1 #aaf ff41 #abi1 #aea1 #baf4 defined1 #aah4 ,instinct v4 ,reason1 #bj ff4 ,intellect1 #bb ff4 ,intelligence1 how displayed by friends1 #ic4 inadequacy of display1 #if4 ,interpretation1 #add4 ,james1 #d1 #aj1 #ac4 ,jourdain1 #afe n4 ,jowett1 #afg4 ,judgment1 #eh4 ,kant1 #c1 #aab1 #aaf1 #aee ff41 #bjj4 ,knowledge about1 #add4 ,language1 bad1 #hb1 #ace4 ,laplace1 #ab4 ,laws of nature1 #bah ff4 ,leibniz1 #ac1 #dj1 #hg1 #ahf1 #aia4 ,logic1 #bja4 analytic not constructive1 #h4 ,aristotelian1 #e4 and fact1 #ec4 inductive1 #cd1 #bbb4 mathematical1 vi1 #dj ff4 mystical1 #df4 and philosophy1 #h1 #cc ff41 #bci4 ,logical constants1 #bjh1 #bac4 ,mach1 #abc1 #bbd4 ,macran1 #ci n4 ,mathematics1 #dj1 #eg4 ,matter1 #ge1 #aja ff4 permanence of1 #ajb ff4 ,measurement1 #afd4 ,memory1 #bcj1 #bcd1 #bcf4 ,method1 deductive1 #e4 logical-analytic1 v1 #fe1 #baa1 #bcf ff4 ,milhaud1 #afh n41 #afi n4 ,mill1 #cd1 #bjj4 ,montaigne1 #bh4 ,motion1 #acj1 #baf4 continuous1 #acc1 #acf4 mathematical theory of1 #acc4 perception of1 #acg ff4 ,zeno's arguments on1 #afh ff4 ,mysticism1 #ai1 #df1 #fc1 #ie4 ,newton1 #cj1 #adf4 ,nietzsche1 #aj1 #aa4 ,noel1 #afi4 ,number1 cardinal1 #aca1 #ahf ff4 defined1 #aii ff4 finite1 #afj1 #aij ff4 inductive1 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,synthesis1 #aeg1 #ahe4 ,tannery1 ,paul1 #afi n4 ,teleology1 #bbc4 ,testimony1 #fg1 #gb1 #hb1 #hg1 #if1 #bab4 ,thales1 #c4 ,thing-in-itself1 #ge1 #hd4 ,things1 #hi ff41 #ajd ff41 #bac4 ,time1 #ajc1 #aaf ff41 #acj1 #aee ff41 #aff1 #bae4 absolute or relative1 #adf4 local1 #ajc4 private1 #aba4 ,uniformities1 #bag4 ,unity1 organic1 #i4 ,universal and particular1 #ci n4 ,volition1 #bbc ff4 ,whitehead1 vi1 #bjg4 ,wittgenstein1 vii1 #bjh n4 ,worlds1 actual and ideal1 #aaa4 possible1 #ahf4 private1 #hh4 ,zeller1 #agc4 ,zeno1 #abi1 #acd1 #acf1 #afe ff4 ,,printed ,,by ,,neill ,,and ,,co41 ,,ltd41 ,,edinburgh4 ,transcriber's ,note3 ,the following is a list of corrections made to the original4 ,the first line is the original line1 the second the corrected one4 ,second ,impression ,second ,impression4 more often than the impossibility of alternatives which seemed prima more often than the impossibility of alternatives which seemed prima with it4 ,we will call these the initial contemporaries of the given with it4 ,we will call these the initial contemporaries of the given the next are liable to fail when we come to infinite number4 ,the first the next are liable to fail when we come to infinite numbers4 ,the first of psychology or an outcome of pscyhical processes as the ,north ,sea4444 of psychology or an outcome of psychical processes as the ,north ,sea4444 according to the length and direction of the interval4 ,a quarter of a according to the length and direction of the interval4 ,a quarter of an ,intelligence1 how displayed by friends1 #ic ,intelligence1 how displayed by friends1 #ic4 ,number1 cardinal1 #aca #ahf ff4 ,number1 cardinal1 #aca1 #ahf ff4