{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_37090/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_37090","id":"gutenberg_37090","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/37090","summary":null,"tags":["Category: Philosophy & Ethics","Knowledge, Theory of","Logical atomism","Nobel Prizes in Literature","Philosophy"],"text":"</pre>\n\n\n\nProduced by Jana Srna and the Online Distributed\nProofreading Team at https://www.pgdp.net\n\n\n\n\n\n\n  [ Transcriber's Notes:\n\n    Every effort has been made to replicate this text as faithfully\n    as possible, including inconsistencies in spelling and hyphenation.\n    Some corrections of spelling and punctuation have been made. They\n    are listed at the end of the text.\n\n    Italic text has been marked with _underscores_.\n    Greek text has been transliterated and marked with ~tildes~.\n  ]\n\n\n\n\n  OUR KNOWLEDGE OF THE\n  EXTERNAL WORLD\n\n  AS A FIELD FOR SCIENTIFIC METHOD\n  IN PHILOSOPHY\n\n\n\n\nBY THE SAME AUTHOR\n\n\n  INTRODUCTION TO MATHEMATICAL PHILOSOPHY\n  Second Edition.\n  Demy 8vo, 12s. 6d. net.\n\n  THE ANALYSIS OF MIND\n  Demy 8vo, 16s. net.\n\n  PRINCIPLES OF SOCIAL RECONSTRUCTION\n  Seventh Impression.\n  Cr. 8vo, 5s. net; Limp, 3s. 6d. net.\n\n  ROADS TO FREEDOM: SOCIALISM, ANARCHISM AND SYNDICALISM\n  Fourth Impression.\n  Cr. 8vo, 5s. net; Limp, 3s. 6d. net.\n\n  THE PRACTICE AND THEORY OF BOLSHEVISM\n  Second Impression.\n  Cr. 8vo, 6s. net.\n\n\n\n\n  OUR KNOWLEDGE OF\n  THE EXTERNAL WORLD\n  AS A FIELD FOR SCIENTIFIC METHOD\n  IN PHILOSOPHY\n\n  BY\n  BERTRAND RUSSELL, F.R.S\n\n  LONDON: GEORGE ALLEN & UNWIN LTD\n  RUSKIN HOUSE, 40 MUSEUM STREET, W.C. 1\n\n\n  First published in 1914 by\n  The Open Court Publishing Company\n\n  Reissued by George Allen & Unwin Ltd.\n  1922\n\n\n\n\nPREFACE\n\n\nThe following lectures[1] are an attempt to show, by means of examples,\nthe nature, capacity, and limitations of the logical-analytic method in\nphilosophy. This method, of which the first complete example is to be\nfound in the writings of Frege, has gradually, in the course of actual\nresearch, increasingly forced itself upon me as something perfectly\ndefinite, capable of embodiment in maxims, and adequate, in all branches\nof philosophy, to yield whatever objective scientific knowledge it is\npossible to obtain. Most of the methods hitherto practised have\nprofessed to lead to more ambitious results than any that logical\nanalysis can claim to reach, but unfortunately these results have always\nbeen such as many competent philosophers considered inadmissible.\nRegarded merely as hypotheses and as aids to imagination, the great\nsystems of the past serve a very useful purpose, and are abundantly\nworthy of study. But something different is required if philosophy is to\nbecome a science, and to aim at results independent of the tastes and\ntemperament of the philosopher who advocates them. In what follows, I\nhave endeavoured to show, however imperfectly, the way by which I\nbelieve that this _desideratum_ is to be found.\n\n  [1] Delivered as Lowell Lectures in Boston, in March and April 1914.\n\nThe central problem by which I have sought to illustrate method is the\nproblem of the relation between the crude data of sense and the space,\ntime, and matter of mathematical physics. I have been made aware of the\nimportance of this problem by my friend and collaborator Dr Whitehead,\nto whom are due almost all the differences between the views advocated\nhere and those suggested in _The Problems of Philosophy_.[2] I owe to\nhim the definition of points, the suggestion for the treatment of\ninstants and \"things,\" and the whole conception of the world of physics\nas a _construction_ rather than an _inference_. What is said on these\ntopics here is, in fact, a rough preliminary account of the more precise\nresults which he is giving in the fourth volume of our _Principia\nMathematica_.[3] It will be seen that if his way of dealing with these\ntopics is capable of being successfully carried through, a wholly new\nlight is thrown on the time-honoured controversies of realists and\nidealists, and a method is obtained of solving all that is soluble in\ntheir problem.\n\n  [2] London and New York, 1912 (\"Home University Library\").\n\n  [3] The first volume was published at Cambridge in 1910, the second in\n  1912, and the third in 1913.\n\nThe speculations of the past as to the reality or unreality of the world\nof physics were baffled, at the outset, by the absence of any\nsatisfactory theory of the mathematical infinite. This difficulty has\nbeen removed by the work of Georg Cantor. But the positive and detailed\nsolution of the problem by means of mathematical constructions based\nupon sensible objects as data has only been rendered possible by the\ngrowth of mathematical logic, without which it is practically impossible\nto manipulate ideas of the requisite abstractness and complexity. This\naspect, which is somewhat obscured in a merely popular outline such as\nis contained in the following lectures, will become plain as soon as Dr\nWhitehead's work is published. In pure logic, which, however, will be\nvery briefly discussed in these lectures, I have had the benefit of\nvitally important discoveries, not yet published, by my friend Mr Ludwig\nWittgenstein.\n\nSince my purpose was to illustrate method, I have included much that is\ntentative and incomplete, for it is not by the study of finished\nstructures alone that the manner of construction can be learnt. Except\nin regard to such matters as Cantor's theory of infinity, no finality is\nclaimed for the theories suggested; but I believe that where they are\nfound to require modification, this will be discovered by substantially\nthe same method as that which at present makes them appear probable, and\nit is on this ground that I ask the reader to be tolerant of their\nincompleteness.\n\n  Cambridge,\n    June 1914.\n\n\n\n\nCONTENTS\n\n\n  LECTURE                                                     PAGE\n\n       I. Current Tendencies                                     3\n\n      II. Logic as the Essence of Philosophy                    33\n\n     III. On our Knowledge of the External World                63\n\n      IV. The World of Physics and the World of Sense          101\n\n       V. The Theory of Continuity                             129\n\n      VI. The Problem of Infinity considered Historically      155\n\n     VII. The Positive Theory of Infinity                      185\n\n    VIII. On the Notion of Cause, with Applications to\n             the Free-will Problem                             211\n\n          Index                                                243\n\n\n\n\nLECTURE I\n\nCURRENT TENDENCIES\n\n\nPhilosophy, from the earliest times, has made greater claims, and\nachieved fewer results, than any other branch of learning. Ever since\nThales said that all is water, philosophers have been ready with glib\nassertions about the sum-total of things; and equally glib denials have\ncome from other philosophers ever since Thales was contradicted by\nAnaximander. I believe that the time has now arrived when this\nunsatisfactory state of things can be brought to an end. In the\nfollowing course of lectures I shall try, chiefly by taking certain\nspecial problems as examples, to indicate wherein the claims of\nphilosophers have been excessive, and why their achievements have not\nbeen greater. The problems and the method of philosophy have, I believe,\nbeen misconceived by all schools, many of its traditional problems being\ninsoluble with our means of knowledge, while other more neglected but\nnot less important problems can, by a more patient and more adequate\nmethod, be solved with all the precision and certainty to which the most\nadvanced sciences have attained.\n\nAmong present-day philosophies, we may distinguish three principal\ntypes, often combined in varying proportions by a single philosopher,\nbut in essence and tendency distinct. The first of these, which I shall\ncall the classical tradition, descends in the main from Kant and Hegel;\nit represents the attempt to adapt to present needs the methods and\nresults of the great constructive philosophers from Plato downwards. The\nsecond type, which may be called evolutionism, derived its predominance\nfrom Darwin, and must be reckoned as having had Herbert Spencer for its\nfirst philosophical representative; but in recent times it has become,\nchiefly through William James and M. Bergson, far bolder and far more\nsearching in its innovations than it was in the hands of Herbert\nSpencer. The third type, which may be called \"logical atomism\" for want\nof a better name, has gradually crept into philosophy through the\ncritical scrutiny of mathematics. This type of philosophy, which is the\none that I wish to advocate, has not as yet many whole-hearted\nadherents, but the \"new realism\" which owes its inception to Harvard is\nvery largely impregnated with its spirit. It represents, I believe, the\nsame kind of advance as was introduced into physics by Galileo: the\nsubstitution of piecemeal, detailed, and verifiable results for large\nuntested generalities recommended only by a certain appeal to\nimagination. But before we can understand the changes advocated by this\nnew philosophy, we must briefly examine and criticise the other two\ntypes with which it has to contend.\n\n\nA. The Classical Tradition\n\nTwenty years ago, the classical tradition, having vanquished the\nopposing tradition of the English empiricists, held almost unquestioned\nsway in all Anglo-Saxon universities. At the present day, though it is\nlosing ground, many of the most prominent teachers still adhere to it.\nIn academic France, in spite of M. Bergson, it is far stronger than all\nits opponents combined; and in Germany it has many vigorous advocates.\nNevertheless, it represents on the whole a decaying force, and it has\nfailed to adapt itself to the temper of the age. Its advocates are, in\nthe main, those whose extra-philosophical knowledge is literary, rather\nthan those who have felt the inspiration of science. There are, apart\nfrom reasoned arguments, certain general intellectual forces against\nit--the same general forces which are breaking down the other great\nsyntheses of the past, and making our age one of bewildered groping\nwhere our ancestors walked in the clear daylight of unquestioning\ncertainty.\n\nThe original impulse out of which the classical tradition developed was\nthe naive faith of the Greek philosophers in the omnipotence of\nreasoning. The discovery of geometry had intoxicated them, and its _a\npriori_ deductive method appeared capable of universal application. They\nwould prove, for instance, that all reality is one, that there is no\nsuch thing as change, that the world of sense is a world of mere\nillusion; and the strangeness of their results gave them no qualms\nbecause they believed in the correctness of their reasoning. Thus it\ncame to be thought that by mere thinking the most surprising and\nimportant truths concerning the whole of reality could be established\nwith a certainty which no contrary observations could shake. As the\nvital impulse of the early philosophers died away, its place was taken\nby authority and tradition, reinforced, in the Middle Ages and almost to\nour own day, by systematic theology. Modern philosophy, from Descartes\nonwards, though not bound by authority like that of the Middle Ages,\nstill accepted more or less uncritically the Aristotelian logic.\nMoreover, it still believed, except in Great Britain, that _a priori_\nreasoning could reveal otherwise undiscoverable secrets about the\nuniverse, and could prove reality to be quite different from what, to\ndirect observation, it appears to be. It is this belief, rather than any\nparticular tenets resulting from it, that I regard as the distinguishing\ncharacteristic of the classical tradition, and as hitherto the main\nobstacle to a scientific attitude in philosophy.\n\nThe nature of the philosophy embodied in the classical tradition may be\nmade clearer by taking a particular exponent as an illustration. For\nthis purpose, let us consider for a moment the doctrines of Mr Bradley,\nwho is probably the most distinguished living representative of this\nschool. Mr Bradley's _Appearance and Reality_ is a book consisting of\ntwo parts, the first called _Appearance_, the second _Reality_. The\nfirst part examines and condemns almost all that makes up our everyday\nworld: things and qualities, relations, space and time, change,\ncausation, activity, the self. All these, though in some sense facts\nwhich qualify reality, are not real as they appear. What is real is one\nsingle, indivisible, timeless whole, called the Absolute, which is in\nsome sense spiritual, but does not consist of souls, or of thought and\nwill as we know them. And all this is established by abstract logical\nreasoning professing to find self-contradictions in the categories\ncondemned as mere appearance, and to leave no tenable alternative to the\nkind of Absolute which is finally affirmed to be real.\n\nOne brief example may suffice to illustrate Mr Bradley's method. The\nworld appears to be full of many things with various relations to each\nother--right and left, before and after, father and son, and so on. But\nrelations, according to Mr Bradley, are found on examination to be\nself-contradictory and therefore impossible. He first argues that, if\nthere are relations, there must be qualities between which they hold.\nThis part of his argument need not detain us. He then proceeds:\n\n\"But how the relation can stand to the qualities is, on the other side,\nunintelligible. If it is nothing to the qualities, then they are not\nrelated at all; and, if so, as we saw, they have ceased to be qualities,\nand their relation is a nonentity. But if it is to be something to them,\nthen clearly we shall require a _new_ connecting relation. For the\nrelation hardly can be the mere adjective of one or both of its terms;\nor, at least, as such it seems indefensible. And, being something\nitself, if it does not itself bear a relation to the terms, in what\nintelligible way will it succeed in being anything to them? But here\nagain we are hurried off into the eddy of a hopeless process, since we\nare forced to go on finding new relations without end. The links are\nunited by a link, and this bond of union is a link which also has two\nends; and these require each a fresh link to connect them with the old.\nThe problem is to find how the relation can stand to its qualities, and\nthis problem is insoluble.\"[4]\n\n  [4] _Appearance and Reality_, pp. 32-33.\n\nI do not propose to examine this argument in detail, or to show the\nexact points where, in my opinion, it is fallacious. I have quoted it\nonly as an example of method. Most people will admit, I think, that it\nis calculated to produce bewilderment rather than conviction, because\nthere is more likelihood of error in a very subtle, abstract, and\ndifficult argument than in so patent a fact as the interrelatedness of\nthe things in the world. To the early Greeks, to whom geometry was\npractically the only known science, it was possible to follow reasoning\nwith assent even when it led to the strangest conclusions. But to us,\nwith our methods of experiment and observation, our knowledge of the\nlong history of _a priori_ errors refuted by empirical science, it has\nbecome natural to suspect a fallacy in any deduction of which the\nconclusion appears to contradict patent facts. It is easy to carry such\nsuspicion too far, and it is very desirable, if possible, actually to\ndiscover the exact nature of the error when it exists. But there is no\ndoubt that what we may call the empirical outlook has become part of\nmost educated people's habit of mind; and it is this, rather than any\ndefinite argument, that has diminished the hold of the classical\ntradition upon students of philosophy and the instructed public\ngenerally.\n\nThe function of logic in philosophy, as I shall try to show at a later\nstage, is all-important; but I do not think its function is that which\nit has in the classical tradition. In that tradition, logic becomes\nconstructive through negation. Where a number of alternatives seem, at\nfirst sight, to be equally possible, logic is made to condemn all of\nthem except one, and that one is then pronounced to be realised in the\nactual world. Thus the world is constructed by means of logic, with\nlittle or no appeal to concrete experience. The true function of logic\nis, in my opinion, exactly the opposite of this. As applied to matters\nof experience, it is analytic rather than constructive; taken _a\npriori_, it shows the possibility of hitherto unsuspected alternatives\nmore often than the impossibility of alternatives which seemed _prima\nfacie_ possible. Thus, while it liberates imagination as to what the\nworld _may_ be, it refuses to legislate as to what the world _is_. This\nchange, which has been brought about by an internal revolution in logic,\nhas swept away the ambitious constructions of traditional metaphysics,\neven for those whose faith in logic is greatest; while to the many who\nregard logic as a chimera the paradoxical systems to which it has given\nrise do not seem worthy even of refutation. Thus on all sides these\nsystems have ceased to attract, and even the philosophical world tends\nmore and more to pass them by.\n\nOne or two of the favourite doctrines of the school we are considering\nmay be mentioned to illustrate the nature of its claims. The universe,\nit tells us, is an \"organic unity,\" like an animal or a perfect work of\nart. By this it means, roughly speaking, that all the different parts\nfit together and co-operate, and are what they are because of their\nplace in the whole. This belief is sometimes advanced dogmatically,\nwhile at other times it is defended by certain logical arguments. If it\nis true, every part of the universe is a microcosm, a miniature\nreflection of the whole. If we knew ourselves thoroughly, according to\nthis doctrine, we should know everything. Common sense would naturally\nobject that there are people--say in China--with whom our relations are\nso indirect and trivial that we cannot infer anything important as to\nthem from any fact about ourselves. If there are living beings in Mars\nor in more distant parts of the universe, the same argument becomes even\nstronger. But further, perhaps the whole contents of the space and time\nin which we live form only one of many universes, each seeming to itself\ncomplete. And thus the conception of the necessary unity of all that is\nresolves itself into the poverty of imagination, and a freer logic\nemancipates us from the strait-waistcoated benevolent institution which\nidealism palms off as the totality of being.\n\nAnother very important doctrine held by most, though not all, of the\nschool we are examining is the doctrine that all reality is what is\ncalled \"mental\" or \"spiritual,\" or that, at any rate, all reality is\ndependent for its existence upon what is mental. This view is often\nparticularised into the form which states that the relation of knower\nand known is fundamental, and that nothing can exist unless it either\nknows or is known. Here again the same legislative function is ascribed\nto _a priori_ argumentation: it is thought that there are contradictions\nin an unknown reality. Again, if I am not mistaken, the argument is\nfallacious, and a better logic will show that no limits can be set to\nthe extent and nature of the unknown. And when I speak of the unknown, I\ndo not mean merely what we personally do not know, but what is not known\nto any mind. Here as elsewhere, while the older logic shut out\npossibilities and imprisoned imagination within the walls of the\nfamiliar, the newer logic shows rather what may happen, and refuses to\ndecide as to what _must_ happen.\n\nThe classical tradition in philosophy is the last surviving child of two\nvery diverse parents: the Greek belief in reason, and the mediaeval\nbelief in the tidiness of the universe. To the schoolmen, who lived amid\nwars, massacres, and pestilences, nothing appeared so delightful as\nsafety and order. In their idealising dreams, it was safety and order\nthat they sought: the universe of Thomas Aquinas or Dante is as small\nand neat as a Dutch interior. To us, to whom safety has become monotony,\nto whom the primeval savageries of nature are so remote as to become a\nmere pleasing condiment to our ordered routine, the world of dreams is\nvery different from what it was amid the wars of Guelf and Ghibelline.\nHence William James's protest against what he calls the \"block universe\"\nof the classical tradition; hence Nietzsche's worship of force; hence\nthe verbal bloodthirstiness of many quiet literary men. The barbaric\nsubstratum of human nature, unsatisfied in action, finds an outlet in\nimagination. In philosophy, as elsewhere, this tendency is visible; and\nit is this, rather than formal argument, that has thrust aside the\nclassical tradition for a philosophy which fancies itself more virile\nand more vital.\n\n\nB. Evolutionism\n\nEvolutionism, in one form or another, is the prevailing creed of our\ntime. It dominates our politics, our literature, and not least our\nphilosophy. Nietzsche, pragmatism, Bergson, are phases in its\nphilosophic development, and their popularity far beyond the circles of\nprofessional philosophers shows its consonance with the spirit of the\nage. It believes itself firmly based on science, a liberator of hopes,\nan inspirer of an invigorating faith in human power, a sure antidote to\nthe ratiocinative authority of the Greeks and the dogmatic authority of\nmediaeval systems. Against so fashionable and so agreeable a creed it may\nseem useless to raise a protest; and with much of its spirit every\nmodern man must be in sympathy. But I think that, in the intoxication of\na quick success, much that is important and vital to a true\nunderstanding of the universe has been forgotten. Something of Hellenism\nmust be combined with the new spirit before it can emerge from the\nardour of youth into the wisdom of manhood. And it is time to remember\nthat biology is neither the only science, nor yet the model to which all\nother sciences must adapt themselves. Evolutionism, as I shall try to\nshow, is not a truly scientific philosophy, either in its method or in\nthe problems which it considers. The true scientific philosophy is\nsomething more arduous and more aloof, appealing to less mundane hopes,\nand requiring a severer discipline for its successful practice.\n\nDarwin's _Origin of Species_ persuaded the world that the difference\nbetween different species of animals and plants is not the fixed,\nimmutable difference that it appears to be. The doctrine of natural\nkinds, which had rendered classification easy and definite, which was\nenshrined in the Aristotelian tradition, and protected by its supposed\nnecessity for orthodox dogma, was suddenly swept away for ever out of\nthe biological world. The difference between man and the lower animals,\nwhich to our human conceit appears enormous, was shown to be a gradual\nachievement, involving intermediate beings who could not with certainty\nbe placed either within or without the human family. The sun and planets\nhad already been shown by Laplace to be very probably derived from a\nprimitive more or less undifferentiated nebula. Thus the old fixed\nlandmarks became wavering and indistinct, and all sharp outlines were\nblurred. Things and species lost their boundaries, and none could say\nwhere they began or where they ended.\n\nBut if human conceit was staggered for a moment by its kinship with the\nape, it soon found a way to reassert itself, and that way is the\n\"philosophy\" of evolution. A process which led from the amoeba to man\nappeared to the philosophers to be obviously a progress--though whether\nthe amoeba would agree with this opinion is not known. Hence the cycle\nof changes which science had shown to be the probable history of the\npast was welcomed as revealing a law of development towards good in the\nuniverse--an evolution or unfolding of an ideal slowly embodying itself\nin the actual. But such a view, though it might satisfy Spencer and\nthose whom we may call Hegelian evolutionists, could not be accepted as\nadequate by the more whole-hearted votaries of change. An ideal to which\nthe world continuously approaches is, to these minds, too dead and\nstatic to be inspiring. Not only the aspirations, but the ideal too,\nmust change and develop with the course of evolution; there must be no\nfixed goal, but a continual fashioning of fresh needs by the impulse\nwhich is life and which alone gives unity to the process.\n\nEver since the seventeenth century, those whom William James described\nas the \"tender-minded\" have been engaged in a desperate struggle with\nthe mechanical view of the course of nature which physical science seems\nto impose. A great part of the attractiveness of the classical tradition\nwas due to the partial escape from mechanism which it provided. But now,\nwith the influence of biology, the \"tender-minded\" believe that a more\nradical escape is possible, sweeping aside not merely the laws of\nphysics, but the whole apparently immutable apparatus of logic, with its\nfixed concepts, its general principles, and its reasonings which seem\nable to compel even the most unwilling assent. The older kind of\nteleology, therefore, which regarded the End as a fixed goal, already\npartially visible, towards which we were gradually approaching, is\nrejected by M. Bergson as not allowing enough for the absolute dominion\nof change. After explaining why he does not accept mechanism, he\nproceeds:[5]\n\n\"But radical finalism is quite as unacceptable, and for the same reason.\nThe doctrine of teleology, in its extreme form, as we find it in Leibniz\nfor example, implies that things and beings merely realise a programme\npreviously arranged. But if there is nothing unforeseen, no invention or\ncreation in the universe, time is useless again. As in the mechanistic\nhypothesis, here again it is supposed that _all is given_. Finalism thus\nunderstood is only inverted mechanism. It springs from the same\npostulate, with this sole difference, that in the movement of our finite\nintellects along successive things, whose successiveness is reduced to a\nmere appearance, it holds in front of us the light with which it claims\nto guide us, instead of putting it behind. It substitutes the attraction\nof the future for the impulsion of the past. But succession remains none\nthe less a mere appearance, as indeed does movement itself. In the\ndoctrine of Leibniz, time is reduced to a confused perception, relative\nto the human standpoint, a perception which would vanish, like a rising\nmist, for a mind seated at the centre of things.\n\n\"Yet finalism is not, like mechanism, a doctrine with fixed rigid\noutlines. It admits of as many inflections as we like. The mechanistic\nphilosophy is to be taken or left: it must be left if the least grain of\ndust, by straying from the path foreseen by mechanics, should show the\nslightest trace of spontaneity. The doctrine of final causes, on the\ncontrary, will never be definitively refuted. If one form of it be put\naside, it will take another. Its principle, which is essentially\npsychological, is very flexible. It is so extensible, and thereby so\ncomprehensive, that one accepts something of it as soon as one rejects\npure mechanism. The theory we shall put forward in this book will\ntherefore necessarily partake of finalism to a certain extent.\"\n\n  [5] _Creative Evolution_, English translation, p. 41.\n\nM. Bergson's form of finalism depends upon his conception of life. Life,\nin his philosophy, is a continuous stream, in which all divisions are\nartificial and unreal. Separate things, beginnings and endings, are mere\nconvenient fictions: there is only smooth, unbroken transition. The\nbeliefs of to-day may count as true to-day, if they carry us along the\nstream; but to-morrow they will be false, and must be replaced by new\nbeliefs to meet the new situation. All our thinking consists of\nconvenient fictions, imaginary congealings of the stream: reality flows\non in spite of all our fictions, and though it can be lived, it cannot\nbe conceived in thought. Somehow, without explicit statement, the\nassurance is slipped in that the future, though we cannot foresee it,\nwill be better than the past or the present: the reader is like the\nchild who expects a sweet because it has been told to open its mouth and\nshut its eyes. Logic, mathematics, physics disappear in this philosophy,\nbecause they are too \"static\"; what is real is an impulse and movement\ntowards a goal which, like the rainbow, recedes as we advance, and makes\nevery place different when we reach it from what it appeared to be at a\ndistance.\n\nNow I do not propose at present to enter upon a technical examination of\nthis philosophy. At present I wish to make only two criticisms of\nit--first, that its truth does not follow from what science has rendered\nprobable concerning the facts of evolution, and secondly, that the\nmotives and interests which inspire it are so exclusively practical, and\nthe problems with which it deals are so special, that it can hardly be\nregarded as really touching any of the questions that to my mind\nconstitute genuine philosophy.\n\n(1) What biology has rendered probable is that the diverse species arose\nby adaptation from a less differentiated ancestry. This fact is in\nitself exceedingly interesting, but it is not the kind of fact from\nwhich philosophical consequences follow. Philosophy is general, and\ntakes an impartial interest in all that exists. The changes suffered by\nminute portions of matter on the earth's surface are very important to\nus as active sentient beings; but to us as philosophers they have no\ngreater interest than other changes in portions of matter elsewhere. And\nif the changes on the earth's surface during the last few millions of\nyears appear to our present ethical notions to be in the nature of a\nprogress, that gives no ground for believing that progress is a general\nlaw of the universe. Except under the influence of desire, no one would\nadmit for a moment so crude a generalisation from such a tiny selection\nof facts. What does result, not specially from biology, but from all the\nsciences which deal with what exists, is that we cannot understand the\nworld unless we can understand change and continuity. This is even more\nevident in physics than it is in biology. But the analysis of change and\ncontinuity is not a problem upon which either physics or biology throws\nany light: it is a problem of a new kind, belonging to a different kind\nof study. The question whether evolutionism offers a true or a false\nanswer to this problem is not, therefore, a question to be solved by\nappeals to particular facts, such as biology and physics reveal. In\nassuming dogmatically a certain answer to this question, evolutionism\nceases to be scientific, yet it is only in touching on this question\nthat evolutionism reaches the subject-matter of philosophy. Evolutionism\nthus consists of two parts: one not philosophical, but only a hasty\ngeneralisation of the kind which the special sciences might hereafter\nconfirm or confute; the other not scientific, but a mere unsupported\ndogma, belonging to philosophy by its subject-matter, but in no way\ndeducible from the facts upon which evolution relies.\n\n(2) The predominant interest of evolutionism is in the question of human\ndestiny, or at least of the destiny of Life. It is more interested in\nmorality and happiness than in knowledge for its own sake. It must be\nadmitted that the same may be said of many other philosophies, and that\na desire for the kind of knowledge which philosophy really can give is\nvery rare. But if philosophy is to become scientific--and it is our\nobject to discover how this can be achieved--it is necessary first and\nforemost that philosophers should acquire the disinterested intellectual\ncuriosity which characterises the genuine man of science. Knowledge\nconcerning the future--which is the kind of knowledge that must be\nsought if we are to know about human destiny--is possible within certain\nnarrow limits. It is impossible to say how much the limits may be\nenlarged with the progress of science. But what is evident is that any\nproposition about the future belongs by its subject-matter to some\nparticular science, and is to be ascertained, if at all, by the methods\nof that science. Philosophy is not a short cut to the same kind of\nresults as those of the other sciences: if it is to be a genuine study,\nit must have a province of its own, and aim at results which the other\nsciences can neither prove nor disprove.\n\nThe consideration that philosophy, if there is such a study, must\nconsist of propositions which could not occur in the other sciences, is\none which has very far-reaching consequences. All the questions which\nhave what is called a human interest--such, for example, as the question\nof a future life--belong, at least in theory, to special sciences, and\nare capable, at least in theory, of being decided by empirical evidence.\nPhilosophers have too often, in the past, permitted themselves to\npronounce on empirical questions, and found themselves, as a result, in\ndisastrous conflict with well-attested facts. We must, therefore,\nrenounce the hope that philosophy can promise satisfaction to our\nmundane desires. What it can do, when it is purified from all practical\ntaint, is to help us to understand the general aspects of the world and\nthe logical analysis of familiar but complex things. Through this\nachievement, by the suggestion of fruitful hypotheses, it may be\nindirectly useful in other sciences, notably mathematics, physics, and\npsychology. But a genuinely scientific philosophy cannot hope to appeal\nto any except those who have the wish to understand, to escape from\nintellectual bewilderment. It offers, in its own domain, the kind of\nsatisfaction which the other sciences offer. But it does not offer, or\nattempt to offer, a solution of the problem of human destiny, or of the\ndestiny of the universe.\n\nEvolutionism, if what has been said is true, is to be regarded as a\nhasty generalisation from certain rather special facts, accompanied by a\ndogmatic rejection of all attempts at analysis, and inspired by\ninterests which are practical rather than theoretical. In spite,\ntherefore, of its appeal to detailed results in various sciences, it\ncannot be regarded as any more genuinely scientific than the classical\ntradition which it has replaced. How philosophy is to be rendered\nscientific, and what is the true subject-matter of philosophy, I shall\ntry to show first by examples of certain achieved results, and then more\ngenerally. We will begin with the problem of the physical conceptions of\nspace and time and matter, which, as we have seen, are challenged by the\ncontentions of the evolutionists. That these conceptions stand in need\nof reconstruction will be admitted, and is indeed increasingly urged by\nphysicists themselves. It will also be admitted that the reconstruction\nmust take more account of change and the universal flux than is done in\nthe older mechanics with its fundamental conception of an indestructible\nmatter. But I do not think the reconstruction required is on Bergsonian\nlines, nor do I think that his rejection of logic can be anything but\nharmful. I shall not, however, adopt the method of explicit controversy,\nbut rather the method of independent inquiry, starting from what, in a\npre-philosophic stage, appear to be facts, and keeping always as close\nto these initial data as the requirements of consistency will permit.\n\nAlthough explicit controversy is almost always fruitless in philosophy,\nowing to the fact that no two philosophers ever understand one another,\nyet it seems necessary to say something at the outset in justification\nof the scientific as against the mystical attitude. Metaphysics, from\nthe first, has been developed by the union or the conflict of these two\nattitudes. Among the earliest Greek philosophers, the Ionians were more\nscientific and the Sicilians more mystical.[6] But among the latter,\nPythagoras, for example, was in himself a curious mixture of the two\ntendencies: the scientific attitude led him to his proposition on\nright-angled triangles, while his mystic insight showed him that it is\nwicked to eat beans. Naturally enough, his followers divided into two\nsects, the lovers of right-angled triangles and the abhorrers of beans;\nbut the former sect died out, leaving, however, a haunting flavour of\nmysticism over much Greek mathematical speculation, and in particular\nover Plato's views on mathematics. Plato, of course, embodies both the\nscientific and the mystical attitudes in a higher form than his\npredecessors, but the mystical attitude is distinctly the stronger of\nthe two, and secures ultimate victory whenever the conflict is sharp.\nPlato, moreover, adopted from the Eleatics the device of using logic to\ndefeat common sense, and thus to leave the field clear for mysticism--a\ndevice still employed in our own day by the adherents of the classical\ntradition.\n\n  [6] _Cf._ Burnet, _Early Greek Philosophy_, pp. 85 ff.\n\nThe logic used in defence of mysticism seems to me faulty as logic, and\nin a later lecture I shall criticise it on this ground. But the more\nthorough-going mystics do not employ logic, which they despise: they\nappeal instead directly to the immediate deliverance of their insight.\nNow, although fully developed mysticism is rare in the West, some\ntincture of it colours the thoughts of many people, particularly as\nregards matters on which they have strong convictions not based on\nevidence. In all who seek passionately for the fugitive and difficult\ngoods, the conviction is almost irresistible that there is in the world\nsomething deeper, more significant, than the multiplicity of little\nfacts chronicled and classified by science. Behind the veil of these\nmundane things, they feel, something quite different obscurely shimmers,\nshining forth clearly in the great moments of illumination, which alone\ngive anything worthy to be called real knowledge of truth. To seek such\nmoments, therefore, is to them the way of wisdom, rather than, like the\nman of science, to observe coolly, to analyse without emotion, and to\naccept without question the equal reality of the trivial and the\nimportant.\n\nOf the reality or unreality of the mystic's world I know nothing. I have\nno wish to deny it, nor even to declare that the insight which reveals\nit is not a genuine insight. What I do wish to maintain--and it is here\nthat the scientific attitude becomes imperative--is that insight,\nuntested and unsupported, is an insufficient guarantee of truth, in\nspite of the fact that much of the most important truth is first\nsuggested by its means. It is common to speak of an opposition between\ninstinct and reason; in the eighteenth century, the opposition was drawn\nin favour of reason, but under the influence of Rousseau and the\nromantic movement instinct was given the preference, first by those who\nrebelled against artificial forms of government and thought, and then,\nas the purely rationalistic defence of traditional theology became\nincreasingly difficult, by all who felt in science a menace to creeds\nwhich they associated with a spiritual outlook on life and the world.\nBergson, under the name of \"intuition,\" has raised instinct to the\nposition of sole arbiter of metaphysical truth. But in fact the\nopposition of instinct and reason is mainly illusory. Instinct,\nintuition, or insight is what first leads to the beliefs which\nsubsequent reason confirms or confutes; but the confirmation, where it\nis possible, consists, in the last analysis, of agreement with other\nbeliefs no less instinctive. Reason is a harmonising, controlling force\nrather than a creative one. Even in the most purely logical realms, it\nis insight that first arrives at what is new.\n\nWhere instinct and reason do sometimes conflict is in regard to single\nbeliefs, held instinctively, and held with such determination that no\ndegree of inconsistency with other beliefs leads to their abandonment.\nInstinct, like all human faculties, is liable to error. Those in whom\nreason is weak are often unwilling to admit this as regards themselves,\nthough all admit it in regard to others. Where instinct is least liable\nto error is in practical matters as to which right judgment is a help to\nsurvival; friendship and hostility in others, for instance, are often\nfelt with extraordinary discrimination through very careful disguises.\nBut even in such matters a wrong impression may be given by reserve or\nflattery; and in matters less directly practical, such as philosophy\ndeals with, very strong instinctive beliefs may be wholly mistaken, as\nwe may come to know through their perceived inconsistency with other\nequally strong beliefs. It is such considerations that necessitate the\nharmonising mediation of reason, which tests our beliefs by their mutual\ncompatibility, and examines, in doubtful cases, the possible sources of\nerror on the one side and on the other. In this there is no opposition\nto instinct as a whole, but only to blind reliance upon some one\ninteresting aspect of instinct to the exclusion of other more\ncommonplace but not less trustworthy aspects. It is such onesidedness,\nnot instinct itself, that reason aims at correcting.\n\nThese more or less trite maxims may be illustrated by application to\nBergson's advocacy of \"intuition\" as against \"intellect.\" There are, he\nsays, \"two profoundly different ways of knowing a thing. The first\nimplies that we move round the object; the second that we enter into it.\nThe first depends on the point of view at which we are placed and on the\nsymbols by which we express ourselves. The second neither depends on a\npoint of view nor relies on any symbol. The first kind of knowledge may\nbe said to stop at the _relative_; the second, in those cases where it\nis possible, to attain the _absolute_.\"[7] The second of these, which is\nintuition, is, he says, \"the kind of intellectual _sympathy_ by which\none places oneself within an object in order to coincide with what is\nunique in it and therefore inexpressible\" (p. 6). In illustration, he\nmentions self-knowledge: \"there is one reality, at least, which we all\nseize from within, by intuition and not by simple analysis. It is our\nown personality in its flowing through time--our self which endures\"\n(p. 8). The rest of Bergson's philosophy consists in reporting, through\nthe imperfect medium of words, the knowledge gained by intuition, and\nthe consequent complete condemnation of all the pretended knowledge\nderived from science and common sense.\n\n  [7] _Introduction to Metaphysics_, p. 1.\n\nThis procedure, since it takes sides in a conflict of instinctive\nbeliefs, stands in need of justification by proving the greater\ntrustworthiness of the beliefs on one side than of those on the other.\nBergson attempts this justification in two ways--first, by explaining\nthat intellect is a purely practical faculty designed to secure\nbiological success; secondly, by mentioning remarkable feats of instinct\nin animals, and by pointing out characteristics of the world which,\nthough intuition can apprehend them, are baffling to intellect as he\ninterprets it.\n\nOf Bergson's theory that intellect is a purely practical faculty\ndeveloped in the struggle for survival, and not a source of true\nbeliefs, we may say, first, that it is only through intellect that we\nknow of the struggle for survival and of the biological ancestry of man:\nif the intellect is misleading, the whole of this merely inferred\nhistory is presumably untrue. If, on the other hand, we agree with M.\nBergson in thinking that evolution took place as Darwin believed, then\nit is not only intellect, but all our faculties, that have been\ndeveloped under the stress of practical utility. Intuition is seen at\nits best where it is directly useful--for example, in regard to other\npeople's characters and dispositions. Bergson apparently holds that\ncapacity for this kind of knowledge is less explicable by the struggle\nfor existence than, for example, capacity for pure mathematics. Yet the\nsavage deceived by false friendship is likely to pay for his mistake\nwith his life; whereas even in the most civilised societies men are not\nput to death for mathematical incompetence. All the most striking of his\ninstances of intuition in animals have a very direct survival value. The\nfact is, of course, that both intuition and intellect have been\ndeveloped because they are useful, and that, speaking broadly, they are\nuseful when they give truth and become harmful when they give falsehood.\nIntellect, in civilised man, like artistic capacity, has occasionally\nbeen developed beyond the point where it is useful to the individual;\nintuition, on the other hand, seems on the whole to diminish as\ncivilisation increases. Speaking broadly, it is greater in children than\nin adults, in the uneducated than in the educated. Probably in dogs it\nexceeds anything to be found in human beings. But those who find in\nthese facts a recommendation of intuition ought to return to running\nwild in the woods, dyeing themselves with woad and living on hips and\nhaws.\n\nLet us next examine whether intuition possesses any such infallibility\nas Bergson claims for it. The best instance of it, according to him, is\nour acquaintance with ourselves; yet self-knowledge is proverbially rare\nand difficult. Most men, for example, have in their nature meannesses,\nvanities, and envies of which they are quite unconscious, though even\ntheir best friends can perceive them without any difficulty. It is true\nthat intuition has a convincingness which is lacking to intellect: while\nit is present, it is almost impossible to doubt its truth. But if it\nshould appear, on examination, to be at least as fallible as intellect,\nits greater subjective certainty becomes a demerit, making it only the\nmore irresistibly deceptive. Apart from self-knowledge, one of the most\nnotable examples of intuition is the knowledge people believe themselves\nto possess of those with whom they are in love: the wall between\ndifferent personalities seems to become transparent, and people think\nthey see into another soul as into their own. Yet deception in such\ncases is constantly practised with success; and even where there is no\nintentional deception, experience gradually proves, as a rule, that the\nsupposed insight was illusory, and that the slower, more groping methods\nof the intellect are in the long run more reliable.\n\nBergson maintains that intellect can only deal with things in so far as\nthey resemble what has been experienced in the past, while intuition has\nthe power of apprehending the uniqueness and novelty that always belong\nto each fresh moment. That there is something unique and new at every\nmoment, is certainly true; it is also true that this cannot be fully\nexpressed by means of intellectual concepts. Only direct acquaintance\ncan give knowledge of what is unique and new. But direct acquaintance of\nthis kind is given fully in sensation, and does not require, so far as I\ncan see, any special faculty of intuition for its apprehension. It is\nneither intellect nor intuition, but sensation, that supplies new data;\nbut when the data are new in any remarkable manner, intellect is much\nmore capable of dealing with them than intuition would be. The hen with\na brood of ducklings no doubt has intuitions which seem to place her\ninside them, and not merely to know them analytically; but when the\nducklings take to the water, the whole apparent intuition is seen to be\nillusory, and the hen is left helpless on the shore. Intuition, in fact,\nis an aspect and development of instinct, and, like all instinct, is\nadmirable in those customary surroundings which have moulded the habits\nof the animal in question, but totally incompetent as soon as the\nsurroundings are changed in a way which demands some non-habitual mode\nof action.\n\nThe theoretical understanding of the world, which is the aim of\nphilosophy, is not a matter of great practical importance to animals, or\nto savages, or even to most civilised men. It is hardly to be supposed,\ntherefore, that the rapid, rough and ready methods of instinct or\nintuition will find in this field a favourable ground for their\napplication. It is the older kinds of activity, which bring out our\nkinship with remote generations of animal and semi-human ancestors, that\nshow intuition at its best. In such matters as self-preservation and\nlove, intuition will act sometimes (though not always) with a swiftness\nand precision which are astonishing to the critical intellect. But\nphilosophy is not one of the pursuits which illustrate our affinity with\nthe past: it is a highly refined, highly civilised pursuit, demanding,\nfor its success, a certain liberation from the life of instinct, and\neven, at times, a certain aloofness from all mundane hopes and fears. It\nis not in philosophy, therefore, that we can hope to see intuition at\nits best. On the contrary, since the true objects of philosophy, and the\nhabits of thought demanded for their apprehension, are strange, unusual,\nand remote, it is here, more almost than anywhere else, that intellect\nproves superior to intuition, and that quick unanalysed convictions are\nleast deserving of uncritical acceptance.\n\nBefore embarking upon the somewhat difficult and abstract discussions\nwhich lie before us, it will be well to take a survey of the hopes we\nmay retain and the hopes we must abandon. The hope of satisfaction to\nour more human desires--the hope of demonstrating that the world has\nthis or that desirable ethical characteristic--is not one which, so far\nas I can see, philosophy can do anything whatever to satisfy. The\ndifference between a good world and a bad one is a difference in the\nparticular characteristics of the particular things that exist in these\nworlds: it is not a sufficiently abstract difference to come within the\nprovince of philosophy. Love and hate, for example, are ethical\nopposites, but to philosophy they are closely analogous attitudes\ntowards objects. The general form and structure of those attitudes\ntowards objects which constitute mental phenomena is a problem for\nphilosophy; but the difference between love and hate is not a difference\nof form or structure, and therefore belongs rather to the special\nscience of psychology than to philosophy. Thus the ethical interests\nwhich have often inspired philosophers must remain in the background:\nsome kind of ethical interest may inspire the whole study, but none must\nobtrude in the detail or be expected in the special results which are\nsought.\n\nIf this view seems at first sight disappointing, we may remind ourselves\nthat a similar change has been found necessary in all the other\nsciences. The physicist or chemist is not now required to prove the\nethical importance of his ions or atoms; the biologist is not expected\nto prove the utility of the plants or animals which he dissects. In\npre-scientific ages this was not the case. Astronomy, for example, was\nstudied because men believed in astrology: it was thought that the\nmovements of the planets had the most direct and important bearing upon\nthe lives of human beings. Presumably, when this belief decayed and the\ndisinterested study of astronomy began, many who had found astrology\nabsorbingly interesting decided that astronomy had too little human\ninterest to be worthy of study. Physics, as it appears in Plato's\n_Timaeus_ for example, is full of ethical notions: it is an essential\npart of its purpose to show that the earth is worthy of admiration. The\nmodern physicist, on the contrary, though he has no wish to deny that\nthe earth is admirable, is not concerned, as physicist, with its ethical\nattributes: he is merely concerned to find out facts, not to consider\nwhether they are good or bad. In psychology, the scientific attitude is\neven more recent and more difficult than in the physical sciences: it is\nnatural to consider that human nature is either good or bad, and to\nsuppose that the difference between good and bad, so all-important in\npractice, must be important in theory also. It is only during the last\ncentury that an ethically neutral science of psychology has grown up;\nand here too ethical neutrality has been essential to scientific\nsuccess.\n\nIn philosophy, hitherto, ethical neutrality has been seldom sought and\nhardly ever achieved. Men have remembered their wishes, and have judged\nphilosophies in relation to their wishes. Driven from the particular\nsciences, the belief that the notions of good and evil must afford a key\nto the understanding of the world has sought a refuge in philosophy. But\neven from this last refuge, if philosophy is not to remain a set of\npleasing dreams, this belief must be driven forth. It is a commonplace\nthat happiness is not best achieved by those who seek it directly; and\nit would seem that the same is true of the good. In thought, at any\nrate, those who forget good and evil and seek only to know the facts are\nmore likely to achieve good than those who view the world through the\ndistorting medium of their own desires.\n\nThe immense extension of our knowledge of facts in recent times has had,\nas it had in the Renaissance, two effects upon the general intellectual\noutlook. On the one hand, it has made men distrustful of the truth of\nwide, ambitious systems: theories come and go swiftly, each serving, for\na moment, to classify known facts and promote the search for new ones,\nbut each in turn proving inadequate to deal with the new facts when they\nhave been found. Even those who invent the theories do not, in science,\nregard them as anything but a temporary makeshift. The ideal of an\nall-embracing synthesis, such as the Middle Ages believed themselves to\nhave attained, recedes further and further beyond the limits of what\nseems feasible. In such a world, as in the world of Montaigne, nothing\nseems worth while except the discovery of more and more facts, each in\nturn the deathblow to some cherished theory; the ordering intellect\ngrows weary, and becomes slovenly through despair.\n\nOn the other hand, the new facts have brought new powers; man's physical\ncontrol over natural forces has been increasing with unexampled\nrapidity, and promises to increase in the future beyond all easily\nassignable limits. Thus alongside of despair as regards ultimate theory\nthere is an immense optimism as regards practice: what man can _do_\nseems almost boundless. The old fixed limits of human power, such as\ndeath, or the dependence of the race on an equilibrium of cosmic forces,\nare forgotten, and no hard facts are allowed to break in upon the dream\nof omnipotence. No philosophy is tolerated which sets bounds to man's\ncapacity of gratifying his wishes; and thus the very despair of theory\nis invoked to silence every whisper of doubt as regards the\npossibilities of practical achievement.\n\nIn the welcoming of new fact, and in the suspicion of dogmatism as\nregards the universe at large, the modern spirit should, I think, be\naccepted as wholly an advance. But both in its practical pretensions and\nin its theoretical despair it seems to me to go too far. Most of what is\ngreatest in man is called forth in response to the thwarting of his\nhopes by immutable natural obstacles; by the pretence of omnipotence, he\nbecomes trivial and a little absurd. And on the theoretical side,\nultimate metaphysical truth, though less all-embracing and harder of\nattainment than it appeared to some philosophers in the past, can, I\nbelieve, be discovered by those who are willing to combine the\nhopefulness, patience, and open-mindedness of science with something of\nthe Greek feeling for beauty in the abstract world of logic and for the\nultimate intrinsic value in the contemplation of truth.\n\nThe philosophy, therefore, which is to be genuinely inspired by the\nscientific spirit, must deal with somewhat dry and abstract matters, and\nmust not hope to find an answer to the practical problems of life. To\nthose who wish to understand much of what has in the past been most\ndifficult and obscure in the constitution of the universe, it has great\nrewards to offer--triumphs as noteworthy as those of Newton and Darwin,\nand as important in the long run, for the moulding of our mental habits.\nAnd it brings with it--as a new and powerful method of investigation\nalways does--a sense of power and a hope of progress more reliable and\nbetter grounded than any that rests on hasty and fallacious\ngeneralisation as to the nature of the universe at large. Many hopes\nwhich inspired philosophers in the past it cannot claim to fulfil; but\nother hopes, more purely intellectual, it can satisfy more fully than\nformer ages could have deemed possible for human minds.\n\n\n\n\nLECTURE II\n\nLOGIC AS THE ESSENCE OF PHILOSOPHY\n\n\nThe topics we discussed in our first lecture, and the topics we shall\ndiscuss later, all reduce themselves, in so far as they are genuinely\nphilosophical, to problems of logic. This is not due to any accident,\nbut to the fact that every philosophical problem, when it is subjected\nto the necessary analysis and purification, is found either to be not\nreally philosophical at all, or else to be, in the sense in which we are\nusing the word, logical. But as the word \"logic\" is never used in the\nsame sense by two different philosophers, some explanation of what I\nmean by the word is indispensable at the outset.\n\nLogic, in the Middle Ages, and down to the present day in teaching,\nmeant no more than a scholastic collection of technical terms and rules\nof syllogistic inference. Aristotle had spoken, and it was the part of\nhumbler men merely to repeat the lesson after him. The trivial nonsense\nembodied in this tradition is still set in examinations, and defended by\neminent authorities as an excellent \"propaedeutic,\" _i.e._ a training in\nthose habits of solemn humbug which are so great a help in later life.\nBut it is not this that I mean to praise in saying that all philosophy\nis logic. Ever since the beginning of the seventeenth century, all\nvigorous minds that have concerned themselves with inference have\nabandoned the mediaeval tradition, and in one way or other have widened\nthe scope of logic.\n\nThe first extension was the introduction of the inductive method by\nBacon and Galileo--by the former in a theoretical and largely mistaken\nform, by the latter in actual use in establishing the foundations of\nmodern physics and astronomy. This is probably the only extension of the\nold logic which has become familiar to the general educated public. But\ninduction, important as it is when regarded as a method of\ninvestigation, does not seem to remain when its work is done: in the\nfinal form of a perfected science, it would seem that everything ought\nto be deductive. If induction remains at all, which is a difficult\nquestion, it will remain merely as one of the principles according to\nwhich deductions are effected. Thus the ultimate result of the\nintroduction of the inductive method seems not the creation of a new\nkind of non-deductive reasoning, but rather the widening of the scope of\ndeduction by pointing out a way of deducing which is certainly not\nsyllogistic, and does not fit into the mediaeval scheme.\n\nThe question of the scope and validity of induction is of great\ndifficulty, and of great importance to our knowledge. Take such a\nquestion as, \"Will the sun rise to-morrow?\" Our first instinctive\nfeeling is that we have abundant reason for saying that it will, because\nit has risen on so many previous mornings. Now, I do not myself know\nwhether this does afford a ground or not, but I am willing to suppose\nthat it does. The question which then arises is: What is the principle\nof inference by which we pass from past sunrises to future ones? The\nanswer given by Mill is that the inference depends upon the law of\ncausation. Let us suppose this to be true; then what is the reason for\nbelieving in the law of causation? There are broadly three possible\nanswers: (1) that it is itself known _a priori_; (2) that it is a\npostulate; (3) that it is an empirical generalisation from past\ninstances in which it has been found to hold. The theory that causation\nis known _a priori_ cannot be definitely refuted, but it can be rendered\nvery unplausible by the mere process of formulating the law exactly, and\nthereby showing that it is immensely more complicated and less obvious\nthan is generally supposed. The theory that causation is a postulate,\n_i.e._ that it is something which we choose to assert although we know\nthat it is very likely false, is also incapable of refutation; but it is\nplainly also incapable of justifying any use of the law in inference. We\nare thus brought to the theory that the law is an empirical\ngeneralisation, which is the view held by Mill.\n\nBut if so, how are empirical generalisations to be justified? The\nevidence in their favour cannot be empirical, since we wish to argue\nfrom what has been observed to what has not been observed, which can\nonly be done by means of some known relation of the observed and the\nunobserved; but the unobserved, by definition, is not known empirically,\nand therefore its relation to the observed, if known at all, must be\nknown independently of empirical evidence. Let us see what Mill says on\nthis subject.\n\nAccording to Mill, the law of causation is proved by an admittedly\nfallible process called \"induction by simple enumeration.\" This process,\nhe says, \"consists in ascribing the nature of general truths to all\npropositions which are true in every instance that we happen to know\nof.\"[8] As regards its fallibility, he asserts that \"the precariousness\nof the method of simple enumeration is in an inverse ratio to the\nlargeness of the generalisation. The process is delusive and\ninsufficient, exactly in proportion as the subject-matter of the\nobservation is special and limited in extent. As the sphere widens, this\nunscientific method becomes less and less liable to mislead; and the\nmost universal class of truths, the law of causation for instance, and\nthe principles of number and of geometry, are duly and satisfactorily\nproved by that method alone, nor are they susceptible of any other\nproof.\"[9]\n\n  [8] _Logic_, book iii., chapter iii., Sec. 2.\n\n  [9] Book iii., chapter xxi., Sec. 3.\n\nIn the above statement, there are two obvious lacunae: (1) How is the\nmethod of simple enumeration itself justified? (2) What logical\nprinciple, if any, covers the same ground as this method, without being\nliable to its failures? Let us take the second question first.\n\nA method of proof which, when used as directed, gives sometimes truth\nand sometimes falsehood--as the method of simple enumeration does--is\nobviously not a valid method, for validity demands invariable truth.\nThus, if simple enumeration is to be rendered valid, it must not be\nstated as Mill states it. We shall have to say, at most, that the data\nrender the result _probable_. Causation holds, we shall say, in every\ninstance we have been able to test; therefore it _probably_ holds in\nuntested instances. There are terrible difficulties in the notion of\nprobability, but we may ignore them at present. We thus have what at\nleast _may_ be a logical principle, since it is without exception. If a\nproposition is true in every instance that we happen to know of, and if\nthe instances are very numerous, then, we shall say, it becomes very\nprobable, on the data, that it will be true in any further instance.\nThis is not refuted by the fact that what we declare to be probable does\nnot always happen, for an event may be probable on the data and yet not\noccur. It is, however, obviously capable of further analysis, and of\nmore exact statement. We shall have to say something like this: that\nevery instance of a proposition[10] being true increases the probability\nof its being true in a fresh instance, and that a sufficient number of\nfavourable instances will, in the absence of instances to the contrary,\nmake the probability of the truth of a fresh instance approach\nindefinitely near to certainty. Some such principle as this is required\nif the method of simple enumeration is to be valid.\n\n  [10] Or rather a propositional function.\n\nBut this brings us to our other question, namely, how is our principle\nknown to be true? Obviously, since it is required to justify induction,\nit cannot be proved by induction; since it goes beyond the empirical\ndata, it cannot be proved by them alone; since it is required to justify\nall inferences from empirical data to what goes beyond them, it cannot\nitself be even rendered in any degree probable by such data. Hence, _if_\nit is known, it is not known by experience, but independently of\nexperience. I do not say that any such principle is known: I only say\nthat it is required to justify the inferences from experience which\nempiricists allow, and that it cannot itself be justified\nempirically.[11]\n\n  [11] The subject of causality and induction will be discussed again in\n  Lecture VIII.\n\nA similar conclusion can be proved by similar arguments concerning any\nother logical principle. Thus logical knowledge is not derivable from\nexperience alone, and the empiricist's philosophy can therefore not be\naccepted in its entirety, in spite of its excellence in many matters\nwhich lie outside logic.\n\nHegel and his followers widened the scope of logic in quite a different\nway--a way which I believe to be fallacious, but which requires\ndiscussion if only to show how their conception of logic differs from\nthe conception which I wish to advocate. In their writings, logic is\npractically identical with metaphysics. In broad outline, the way this\ncame about is as follows. Hegel believed that, by means of _a priori_\nreasoning, it could be shown that the world _must_ have various\nimportant and interesting characteristics, since any world without these\ncharacteristics would be impossible and self-contradictory. Thus what he\ncalls \"logic\" is an investigation of the nature of the universe, in so\nfar as this can be inferred merely from the principle that the universe\nmust be logically self-consistent. I do not myself believe that from\nthis principle alone anything of importance can be inferred as regards\nthe existing universe. But, however that may be, I should not regard\nHegel's reasoning, even if it were valid, as properly belonging to\nlogic: it would rather be an application of logic to the actual world.\nLogic itself would be concerned rather with such questions as what\nself-consistency is, which Hegel, so far as I know, does not discuss.\nAnd though he criticises the traditional logic, and professes to replace\nit by an improved logic of his own, there is some sense in which the\ntraditional logic, with all its faults, is uncritically and\nunconsciously assumed throughout his reasoning. It is not in the\ndirection advocated by him, it seems to me, that the reform of logic is\nto be sought, but by a more fundamental, more patient, and less\nambitious investigation into the presuppositions which his system shares\nwith those of most other philosophers.\n\nThe way in which, as it seems to me, Hegel's system assumes the ordinary\nlogic which it subsequently criticises, is exemplified by the general\nconception of \"categories\" with which he operates throughout. This\nconception is, I think, essentially a product of logical confusion, but\nit seems in some way to stand for the conception of \"qualities of\nReality as a whole.\" Mr Bradley has worked out a theory according to\nwhich, in all judgment, we are ascribing a predicate to Reality as a\nwhole; and this theory is derived from Hegel. Now the traditional logic\nholds that every proposition ascribes a predicate to a subject, and from\nthis it easily follows that there can be only one subject, the Absolute,\nfor if there were two, the proposition that there were two would not\nascribe a predicate to either. Thus Hegel's doctrine, that philosophical\npropositions must be of the form, \"the Absolute is such-and-such,\"\ndepends upon the traditional belief in the universality of the\nsubject-predicate form. This belief, being traditional, scarcely\nself-conscious, and not supposed to be important, operates underground,\nand is assumed in arguments which, like the refutation of relations,\nappear at first sight such as to establish its truth. This is the most\nimportant respect in which Hegel uncritically assumes the traditional\nlogic. Other less important respects--though important enough to be the\nsource of such essentially Hegelian conceptions as the \"concrete\nuniversal\" and the \"union of identity in difference\"--will be found\nwhere he explicitly deals with formal logic.[12]\n\n  [12] See the translation by H. S. Macran, _Hegel's Doctrine of Formal\n  Logic_, Oxford, 1912. Hegel's argument in this portion of his \"Logic\"\n  depends throughout upon confusing the \"is\" of predication, as in\n  \"Socrates is mortal,\" with the \"is\" of identity, as in \"Socrates is\n  the philosopher who drank the hemlock.\" Owing to this confusion, he\n  thinks that \"Socrates\" and \"mortal\" must be identical. Seeing that\n  they are different, he does not infer, as others would, that there is\n  a mistake somewhere, but that they exhibit \"identity in difference.\"\n  Again, Socrates is particular, \"mortal\" is universal. Therefore, he\n  says, since Socrates is mortal, it follows that the particular is the\n  universal--taking the \"is\" to be throughout expressive of identity.\n  But to say \"the particular is the universal\" is self-contradictory.\n  Again Hegel does not suspect a mistake but proceeds to synthesise\n  particular and universal in the individual, or concrete universal.\n  This is an example of how, for want of care at the start, vast and\n  imposing systems of philosophy are built upon stupid and trivial\n  confusions, which, but for the almost incredible fact that they are\n  unintentional, one would be tempted to characterise as puns.\n\nThere is quite another direction in which a large technical development\nof logic has taken place: I mean the direction of what is called\nlogistic or mathematical logic. This kind of logic is mathematical in\ntwo different senses: it is itself a branch of mathematics, and it is\nthe logic which is specially applicable to other more traditional\nbranches of mathematics. Historically, it began as _merely_ a branch of\nmathematics: its special applicability to other branches is a more\nrecent development. In both respects, it is the fulfilment of a hope\nwhich Leibniz cherished throughout his life, and pursued with all the\nardour of his amazing intellectual energy. Much of his work on this\nsubject has been published recently, since his discoveries have been\nremade by others; but none was published by him, because his results\npersisted in contradicting certain points in the traditional doctrine of\nthe syllogism. We now know that on these points the traditional doctrine\nis wrong, but respect for Aristotle prevented Leibniz from realising\nthat this was possible.[13]\n\n  [13] _Cf._ Couturat, _La Logique de Leibniz_, pp. 361, 386.\n\nThe modern development of mathematical logic dates from Boole's _Laws of\nThought_ (1854). But in him and his successors, before Peano and Frege,\nthe only thing really achieved, apart from certain details, was the\ninvention of a mathematical symbolism for deducing consequences from the\npremisses which the newer methods shared with those of Aristotle. This\nsubject has considerable interest as an independent branch of\nmathematics, but it has very little to do with real logic. The first\nserious advance in real logic since the time of the Greeks was made\nindependently by Peano and Frege--both mathematicians. They both arrived\nat their logical results by an analysis of mathematics. Traditional\nlogic regarded the two propositions, \"Socrates is mortal\" and \"All men\nare mortal,\" as being of the same form;[14] Peano and Frege showed that\nthey are utterly different in form. The philosophical importance of\nlogic may be illustrated by the fact that this confusion--which is still\ncommitted by most writers--obscured not only the whole study of the\nforms of judgment and inference, but also the relations of things to\ntheir qualities, of concrete existence to abstract concepts, and of the\nworld of sense to the world of Platonic ideas. Peano and Frege, who\npointed out the error, did so for technical reasons, and applied their\nlogic mainly to technical developments; but the philosophical importance\nof the advance which they made is impossible to exaggerate.\n\n  [14] It was often recognised that there was _some_ difference between\n  them, but it was not recognised that the difference is fundamental,\n  and of very great importance.\n\nMathematical logic, even in its most modern form, is not _directly_ of\nphilosophical importance except in its beginnings. After the beginnings,\nit belongs rather to mathematics than to philosophy. Of its beginnings,\nwhich are the only part of it that can properly be called\n_philosophical_ logic, I shall speak shortly. But even the later\ndevelopments, though not directly philosophical, will be found of great\nindirect use in philosophising. They enable us to deal easily with more\nabstract conceptions than merely verbal reasoning can enumerate; they\nsuggest fruitful hypotheses which otherwise could hardly be thought of;\nand they enable us to see quickly what is the smallest store of\nmaterials with which a given logical or scientific edifice can be\nconstructed. Not only Frege's theory of number, which we shall deal with\nin Lecture VII., but the whole theory of physical concepts which will be\noutlined in our next two lectures, is inspired by mathematical logic,\nand could never have been imagined without it.\n\nIn both these cases, and in many others, we shall appeal to a certain\nprinciple called \"the principle of abstraction.\" This principle, which\nmight equally well be called \"the principle which dispenses with\nabstraction,\" and is one which clears away incredible accumulations of\nmetaphysical lumber, was directly suggested by mathematical logic, and\ncould hardly have been proved or practically used without its help. The\nprinciple will be explained in our fourth lecture, but its use may be\nbriefly indicated in advance. When a group of objects have that kind of\nsimilarity which we are inclined to attribute to possession of a common\nquality, the principle in question shows that membership of the group\nwill serve all the purposes of the supposed common quality, and that\ntherefore, unless some common quality is actually known, the group or\nclass of similar objects may be used to replace the common quality,\nwhich need not be assumed to exist. In this and other ways, the indirect\nuses of even the later parts of mathematical logic are very great; but\nit is now time to turn our attention to its philosophical foundations.\n\nIn every proposition and in every inference there is, besides the\nparticular subject-matter concerned, a certain _form_, a way in which\nthe constituents of the proposition or inference are put together. If I\nsay, \"Socrates is mortal,\" \"Jones is angry,\" \"The sun is hot,\" there is\nsomething in common in these three cases, something indicated by the\nword \"is.\" What is in common is the _form_ of the proposition, not an\nactual constituent. If I say a number of things about Socrates--that he\nwas an Athenian, that he married Xantippe, that he drank the\nhemlock--there is a common constituent, namely Socrates, in all the\npropositions I enunciate, but they have diverse forms. If, on the other\nhand, I take any one of these propositions and replace its constituents,\none at a time, by other constituents, the form remains constant, but no\nconstituent remains. Take (say) the series of propositions, \"Socrates\ndrank the hemlock,\" \"Coleridge drank the hemlock,\" \"Coleridge drank\nopium,\" \"Coleridge ate opium.\" The form remains unchanged throughout\nthis series, but all the constituents are altered. Thus form is not\nanother constituent, but is the way the constituents are put together.\nIt is forms, in this sense, that are the proper object of philosophical\nlogic.\n\nIt is obvious that the knowledge of logical forms is something quite\ndifferent from knowledge of existing things. The form of \"Socrates drank\nthe hemlock\" is not an existing thing like Socrates or the hemlock, nor\ndoes it even have that close relation to existing things that drinking\nhas. It is something altogether more abstract and remote. We might\nunderstand all the separate words of a sentence without understanding\nthe sentence: if a sentence is long and complicated, this is apt to\nhappen. In such a case we have knowledge of the constituents, but not of\nthe form. We may also have knowledge of the form without having\nknowledge of the constituents. If I say, \"Rorarius drank the hemlock,\"\nthose among you who have never heard of Rorarius (supposing there are\nany) will understand the form, without having knowledge of all the\nconstituents. In order to understand a sentence, it is necessary to have\nknowledge both of the constituents and of the particular instance of the\nform. It is in this way that a sentence conveys information, since it\ntells us that certain known objects are related according to a certain\nknown form. Thus some kind of knowledge of logical forms, though with\nmost people it is not explicit, is involved in all understanding of\ndiscourse. It is the business of philosophical logic to extract this\nknowledge from its concrete integuments, and to render it explicit and\npure.\n\nIn all inference, form alone is essential: the particular subject-matter\nis irrelevant except as securing the truth of the premisses. This is one\nreason for the great importance of logical form. When I say, \"Socrates\nwas a man, all men are mortal, therefore Socrates was mortal,\" the\nconnection of premisses and conclusion does not in any way depend upon\nits being Socrates and man and mortality that I am mentioning. The\ngeneral form of the inference may be expressed in some such words as,\n\"If a thing has a certain property, and whatever has this property has a\ncertain other property, then the thing in question also has that other\nproperty.\" Here no particular things or properties are mentioned: the\nproposition is absolutely general. All inferences, when stated fully,\nare instances of propositions having this kind of generality. If they\nseem to depend upon the subject-matter otherwise than as regards the\ntruth of the premisses, that is because the premisses have not been all\nexplicitly stated. In logic, it is a waste of time to deal with\ninferences concerning particular cases: we deal throughout with\ncompletely general and purely formal implications, leaving it to other\nsciences to discover when the hypotheses are verified and when they are\nnot.\n\nBut the forms of propositions giving rise to inferences are not the\nsimplest forms: they are always hypothetical, stating that if one\nproposition is true, then so is another. Before considering inference,\ntherefore, logic must consider those simpler forms which inference\npresupposes. Here the traditional logic failed completely: it believed\nthat there was only one form of simple proposition (_i.e._ of\nproposition not stating a relation between two or more other\npropositions), namely, the form which ascribes a predicate to a subject.\nThis is the appropriate form in assigning the qualities of a given\nthing--we may say \"this thing is round, and red, and so on.\" Grammar\nfavours this form, but philosophically it is so far from universal that\nit is not even very common. If we say \"this thing is bigger than that,\"\nwe are not assigning a mere quality of \"this,\" but a relation of \"this\"\nand \"that.\" We might express the same fact by saying \"that thing is\nsmaller than this,\" where grammatically the subject is changed. Thus\npropositions stating that two things have a certain relation have a\ndifferent form from subject-predicate propositions, and the failure to\nperceive this difference or to allow for it has been the source of many\nerrors in traditional metaphysics.\n\nThe belief or unconscious conviction that all propositions are of the\nsubject-predicate form--in other words, that every fact consists in some\nthing having some quality--has rendered most philosophers incapable of\ngiving any account of the world of science and daily life. If they had\nbeen honestly anxious to give such an account, they would probably have\ndiscovered their error very quickly; but most of them were less anxious\nto understand the world of science and daily life, than to convict it of\nunreality in the interests of a super-sensible \"real\" world. Belief in\nthe unreality of the world of sense arises with irresistible force in\ncertain moods--moods which, I imagine, have some simple physiological\nbasis, but are none the less powerfully persuasive. The conviction born\nof these moods is the source of most mysticism and of most metaphysics.\nWhen the emotional intensity of such a mood subsides, a man who is in\nthe habit of reasoning will search for logical reasons in favour of the\nbelief which he finds in himself. But since the belief already exists,\nhe will be very hospitable to any reason that suggests itself. The\nparadoxes apparently proved by his logic are really the paradoxes of\nmysticism, and are the goal which he feels his logic must reach if it is\nto be in accordance with insight. It is in this way that logic has been\npursued by those of the great philosophers who were mystics--notably\nPlato, Spinoza, and Hegel. But since they usually took for granted the\nsupposed insight of the mystic emotion, their logical doctrines were\npresented with a certain dryness, and were believed by their disciples\nto be quite independent of the sudden illumination from which they\nsprang. Nevertheless their origin clung to them, and they remained--to\nborrow a useful word from Mr Santayana--\"malicious\" in regard to the\nworld of science and common sense. It is only so that we can account for\nthe complacency with which philosophers have accepted the inconsistency\nof their doctrines with all the common and scientific facts which seem\nbest established and most worthy of belief.\n\nThe logic of mysticism shows, as is natural, the defects which are\ninherent in anything malicious. While the mystic mood is dominant, the\nneed of logic is not felt; as the mood fades, the impulse to logic\nreasserts itself, but with a desire to retain the vanishing insight, or\nat least to prove that it _was_ insight, and that what seems to\ncontradict it is illusion. The logic which thus arises is not quite\ndisinterested or candid, and is inspired by a certain hatred of the\ndaily world to which it is to be applied. Such an attitude naturally\ndoes not tend to the best results. Everyone knows that to read an author\nsimply in order to refute him is not the way to understand him; and to\nread the book of Nature with a conviction that it is all illusion is\njust as unlikely to lead to understanding. If our logic is to find the\ncommon world intelligible, it must not be hostile, but must be inspired\nby a genuine acceptance such as is not usually to be found among\nmetaphysicians.\n\nTraditional logic, since it holds that all propositions have the\nsubject-predicate form, is unable to admit the reality of relations: all\nrelations, it maintains, must be reduced to properties of the apparently\nrelated terms. There are many ways of refuting this opinion; one of the\neasiest is derived from the consideration of what are called\n\"asymmetrical\" relations. In order to explain this, I will first explain\ntwo independent ways of classifying relations.\n\nSome relations, when they hold between A and B, also hold between B and\nA. Such, for example, is the relation \"brother or sister.\" If A is a\nbrother or sister of B, then B is a brother or sister of A. Such again\nis any kind of similarity, say similarity of colour. Any kind of\ndissimilarity is also of this kind: if the colour of A is unlike the\ncolour of B, then the colour of B is unlike the colour of A. Relations\nof this sort are called _symmetrical_. Thus a relation is symmetrical\nif, whenever it holds between A and B, it also holds between B and A.\n\nAll relations that are not symmetrical are called _non-symmetrical_.\nThus \"brother\" is non-symmetrical, because, if A is a brother of B, it\nmay happen that B is a _sister_ of A.\n\nA relation is called _asymmetrical_ when, if it holds between A and B,\nit _never_ holds between B and A. Thus husband, father, grandfather,\netc., are asymmetrical relations. So are _before_, _after_, _greater_,\n_above_, _to the right of_, etc. All the relations that give rise to\nseries are of this kind.\n\nClassification into symmetrical, asymmetrical, and merely\nnon-symmetrical relations is the first of the two classifications we had\nto consider. The second is into transitive, intransitive, and merely\nnon-transitive relations, which are defined as follows.\n\nA relation is said to be _transitive_, if, whenever it holds between A\nand B and also between B and C, it holds between A and C. Thus _before_,\n_after_, _greater_, _above_ are transitive. All relations giving rise to\nseries are transitive, but so are many others. The transitive relations\njust mentioned were asymmetrical, but many transitive relations are\nsymmetrical--for instance, equality in any respect, exact identity of\ncolour, being equally numerous (as applied to collections), and so on.\n\nA relation is said to be _non-transitive_ whenever it is not transitive.\nThus \"brother\" is non-transitive, because a brother of one's brother may\nbe oneself. All kinds of dissimilarity are non-transitive.\n\nA relation is said to be _intransitive_ when, if A has the relation to\nB, and B to C, A never has it to C. Thus \"father\" is intransitive. So is\nsuch a relation as \"one inch taller\" or \"one year later.\"\n\nLet us now, in the light of this classification, return to the question\nwhether all relations can be reduced to predications.\n\nIn the case of symmetrical relations--_i.e._ relations which, if they\nhold between A and B, also hold between B and A--some kind of\nplausibility can be given to this doctrine. A symmetrical relation which\nis transitive, such as equality, can be regarded as expressing\npossession of some common property, while one which is not transitive,\nsuch as inequality, can be regarded as expressing possession of\ndifferent properties. But when we come to asymmetrical relations, such\nas before and after, greater and less, etc., the attempt to reduce them\nto properties becomes obviously impossible. When, for example, two\nthings are merely known to be unequal, without our knowing which is\ngreater, we may say that the inequality results from their having\ndifferent magnitudes, because inequality is a symmetrical relation; but\nto say that when one thing is _greater_ than another, and not merely\nunequal to it, that means that they have different magnitudes, is\nformally incapable of explaining the facts. For if the other thing had\nbeen greater than the one, the magnitudes would also have been\ndifferent, though the fact to be explained would not have been the same.\nThus mere _difference_ of magnitude is not _all_ that is involved,\nsince, if it were, there would be no difference between one thing being\ngreater than another, and the other being greater than the one. We shall\nhave to say that the one magnitude is _greater_ than the other, and thus\nwe shall have failed to get rid of the relation \"greater.\" In short,\nboth possession of the same property and possession of different\nproperties are _symmetrical_ relations, and therefore cannot account for\nthe existence of _asymmetrical_ relations.\n\nAsymmetrical relations are involved in all series--in space and time,\ngreater and less, whole and part, and many others of the most important\ncharacteristics of the actual world. All these aspects, therefore, the\nlogic which reduces everything to subjects and predicates is compelled\nto condemn as error and mere appearance. To those whose logic is not\nmalicious, such a wholesale condemnation appears impossible. And in fact\nthere is no reason except prejudice, so far as I can discover, for\ndenying the reality of relations. When once their reality is admitted,\nall _logical_ grounds for supposing the world of sense to be illusory\ndisappear. If this is to be supposed, it must be frankly and simply on\nthe ground of mystic insight unsupported by argument. It is impossible\nto argue against what professes to be insight, so long as it does not\nargue in its own favour. As logicians, therefore, we may admit the\npossibility of the mystic's world, while yet, so long as we do not have\nhis insight, we must continue to study the everyday world with which we\nare familiar. But when he contends that our world is impossible, then\nour logic is ready to repel his attack. And the first step in creating\nthe logic which is to perform this service is the recognition of the\nreality of relations.\n\nRelations which have two terms are only one kind of relations. A\nrelation may have three terms, or four, or any number. Relations of two\nterms, being the simplest, have received more attention than the others,\nand have generally been alone considered by philosophers, both those who\naccepted and those who denied the reality of relations. But other\nrelations have their importance, and are indispensable in the solution\nof certain problems. Jealousy, for example, is a relation between three\npeople. Professor Royce mentions the relation \"giving\": when A gives B\nto C, that is a relation of three terms.[15] When a man says to his\nwife: \"My dear, I wish you could induce Angelina to accept Edwin,\" his\nwish constitutes a relation between four people, himself, his wife,\nAngelina, and Edwin. Thus such relations are by no means recondite or\nrare. But in order to explain exactly how they differ from relations of\ntwo terms, we must embark upon a classification of the logical forms of\nfacts, which is the first business of logic, and the business in which\nthe traditional logic has been most deficient.\n\n  [15] _Encyclopaedia of the Philosophical Sciences_, vol. i. p. 97.\n\nThe existing world consists of many things with many qualities and\nrelations. A complete description of the existing world would require\nnot only a catalogue of the things, but also a mention of all their\nqualities and relations. We should have to know not only this, that, and\nthe other thing, but also which was red, which yellow, which was earlier\nthan which, which was between which two others, and so on. When I speak\nof a \"fact,\" I do not mean one of the simple things in the world; I mean\nthat a certain thing has a certain quality, or that certain things have\na certain relation. Thus, for example, I should not call Napoleon a\nfact, but I should call it a fact that he was ambitious, or that he\nmarried Josephine. Now a fact, in this sense, is never simple, but\nalways has two or more constituents. When it simply assigns a quality to\na thing, it has only two constituents, the thing and the quality. When\nit consists of a relation between two things, it has three constituents,\nthe things and the relation. When it consists of a relation between\nthree things, it has four constituents, and so on. The constituents of\nfacts, in the sense in which we are using the word \"fact,\" are not other\nfacts, but are things and qualities or relations. When we say that there\nare relations of more than two terms, we mean that there are single\nfacts consisting of a single relation and more than two things. I do not\nmean that one relation of two terms may hold between A and B, and also\nbetween A and C, as, for example, a man is the son of his father and\nalso the son of his mother. This constitutes two distinct facts: if we\nchoose to treat it as one fact, it is a fact which has facts for its\nconstituents. But the facts I am speaking of have no facts among their\nconstituents, but only things and relations. For example, when A is\njealous of B on account of C, there is only one fact, involving three\npeople; there are not two instances of jealousy, but only one. It is in\nsuch cases that I speak of a relation of three terms, where the simplest\npossible fact in which the relation occurs is one involving three things\nin addition to the relation. And the same applies to relations of four\nterms or five or any other number. All such relations must be admitted\nin our inventory of the logical forms of facts: two facts involving the\nsame number of things have the same form, and two which involve\ndifferent numbers of things have different forms.\n\nGiven any fact, there is an assertion which expresses the fact. The fact\nitself is objective, and independent of our thought or opinion about it;\nbut the assertion is something which involves thought, and may be either\ntrue or false. An assertion may be positive or negative: we may assert\nthat Charles I. was executed, or that he did _not_ die in his bed. A\nnegative assertion may be said to be a _denial_. Given a form of words\nwhich must be either true or false, such as \"Charles I. died in his\nbed,\" we may either assert or deny this form of words: in the one case\nwe have a positive assertion, in the other a negative one. A form of\nwords which must be either true or false I shall call a _proposition_.\nThus a proposition is the same as what may be significantly asserted or\ndenied. A proposition which expresses what we have called a fact, _i.e._\nwhich, when asserted, asserts that a certain thing has a certain\nquality, or that certain things have a certain relation, will be called\nan atomic proposition, because, as we shall see immediately, there are\nother propositions into which atomic propositions enter in a way\nanalogous to that in which atoms enter into molecules. Atomic\npropositions, although, like facts, they may have any one of an infinite\nnumber of forms, are only one kind of propositions. All other kinds are\nmore complicated. In order to preserve the parallelism in language as\nregards facts and propositions, we shall give the name \"atomic facts\" to\nthe facts we have hitherto been considering. Thus atomic facts are what\ndetermine whether atomic propositions are to be asserted or denied.\n\nWhether an atomic proposition, such as \"this is red,\" or \"this is before\nthat,\" is to be asserted or denied can only be known empirically.\nPerhaps one atomic fact may sometimes be capable of being inferred from\nanother, though this seems very doubtful; but in any case it cannot be\ninferred from premisses no one of which is an atomic fact. It follows\nthat, if atomic facts are to be known at all, some at least must be\nknown without inference. The atomic facts which we come to know in this\nway are the facts of sense-perception; at any rate, the facts of\nsense-perception are those which we most obviously and certainly come to\nknow in this way. If we knew all atomic facts, and also knew that there\nwere none except those we knew, we should, theoretically, be able to\ninfer all truths of whatever form.[16] Thus logic would then supply us\nwith the whole of the apparatus required. But in the first acquisition\nof knowledge concerning atomic facts, logic is useless. In pure logic,\nno atomic fact is ever mentioned: we confine ourselves wholly to forms,\nwithout asking ourselves what objects can fill the forms. Thus pure\nlogic is independent of atomic facts; but conversely, they are, in a\nsense, independent of logic. Pure logic and atomic facts are the two\npoles, the wholly _a priori_ and the wholly empirical. But between the\ntwo lies a vast intermediate region, which we must now briefly explore.\n\n  [16] This perhaps requires modification in order to include such facts\n  as beliefs and wishes, since such facts apparently contain\n  propositions as components. Such facts, though not strictly atomic,\n  must be supposed included if the statement in the text is to be true.\n\n\"Molecular\" propositions are such as contain conjunctions--_if_, _or_,\n_and_, _unless_, etc.--and such words are the marks of a molecular\nproposition. Consider such an assertion as, \"If it rains, I shall bring\nmy umbrella.\" This assertion is just as capable of truth or falsehood as\nthe assertion of an atomic proposition, but it is obvious that either\nthe corresponding fact, or the nature of the correspondence with fact,\nmust be quite different from what it is in the case of an atomic\nproposition. Whether it rains, and whether I bring my umbrella, are each\nseverally matters of atomic fact, ascertainable by observation. But the\nconnection of the two involved in saying that _if_ the one happens,\n_then_ the other will happen, is something radically different from\neither of the two separately. It does not require for its truth that it\nshould actually rain, or that I should actually bring my umbrella; even\nif the weather is cloudless, it may still be true that I should have\nbrought my umbrella if the weather had been different. Thus we have here\na connection of two propositions, which does not depend upon whether\nthey are to be asserted or denied, but only upon the second being\ninferable from the first. Such propositions, therefore, have a form\nwhich is different from that of any atomic proposition.\n\nSuch propositions are important to logic, because all inference depends\nupon them. If I have told you that if it rains I shall bring my\numbrella, and if you see that there is a steady downpour, you can infer\nthat I shall bring my umbrella. There can be no inference except where\npropositions are connected in some such way, so that from the truth or\nfalsehood of the one something follows as to the truth or falsehood of\nthe other. It seems to be the case that we can sometimes know molecular\npropositions, as in the above instance of the umbrella, when we do not\nknow whether the component atomic propositions are true or false. The\n_practical_ utility of inference rests upon this fact.\n\nThe next kind of propositions we have to consider are _general_\npropositions, such as \"all men are mortal,\" \"all equilateral triangles\nare equiangular.\" And with these belong propositions in which the word\n\"some\" occurs, such as \"some men are philosophers\" or \"some philosophers\nare not wise.\" These are the denials of general propositions, namely (in\nthe above instances), of \"all men are non-philosophers\" and \"all\nphilosophers are wise.\" We will call propositions containing the word\n\"some\" _negative_ general propositions, and those containing the word\n\"all\" _positive_ general propositions. These propositions, it will be\nseen, begin to have the appearance of the propositions in logical\ntext-books. But their peculiarity and complexity are not known to the\ntext-books, and the problems which they raise are only discussed in the\nmost superficial manner.\n\nWhen we were discussing atomic facts, we saw that we should be able,\ntheoretically, to infer all other truths by logic if we knew all atomic\nfacts and also knew that there were no other atomic facts besides those\nwe knew. The knowledge that there are no other atomic facts is positive\ngeneral knowledge; it is the knowledge that \"all atomic facts are known\nto me,\" or at least \"all atomic facts are in this collection\"--however\nthe collection may be given. It is easy to see that general\npropositions, such as \"all men are mortal,\" cannot be known by inference\nfrom atomic facts alone. If we could know each individual man, and know\nthat he was mortal, that would not enable us to know that all men are\nmortal, unless we _knew_ that those were all the men there are, which is\na general proposition. If we knew every other existing thing throughout\nthe universe, and knew that each separate thing was not an immortal man,\nthat would not give us our result unless we _knew_ that we had explored\nthe whole universe, _i.e._ unless we knew \"all things belong to this\ncollection of things I have examined.\" Thus general truths cannot be\ninferred from particular truths alone, but must, if they are to be\nknown, be either self-evident, or inferred from premisses of which at\nleast one is a general truth. But all _empirical_ evidence is of\n_particular_ truths. Hence, if there is any knowledge of general truths\nat all, there must be _some_ knowledge of general truths which is\nindependent of empirical evidence, _i.e._ does not depend upon the data\nof sense.\n\nThe above conclusion, of which we had an instance in the case of the\ninductive principle, is important, since it affords a refutation of the\nolder empiricists. They believed that all our knowledge is derived from\nthe senses and dependent upon them. We see that, if this view is to be\nmaintained, we must refuse to admit that we know any general\npropositions. It is perfectly possible logically that this should be the\ncase, but it does not appear to be so in fact, and indeed no one would\ndream of maintaining such a view except a theorist at the last\nextremity. We must therefore admit that there is general knowledge not\nderived from sense, and that some of this knowledge is not obtained by\ninference but is primitive.\n\nSuch general knowledge is to be found in logic. Whether there is any\nsuch knowledge not derived from logic, I do not know; but in logic, at\nany rate, we have such knowledge. It will be remembered that we excluded\nfrom pure logic such propositions as, \"Socrates is a man, all men are\nmortal, therefore Socrates is mortal,\" because Socrates and _man_ and\n_mortal_ are empirical terms, only to be understood through particular\nexperience. The corresponding proposition in pure logic is: \"If anything\nhas a certain property, and whatever has this property has a certain\nother property, then the thing in question has the other property.\" This\nproposition is absolutely general: it applies to all things and all\nproperties. And it is quite self-evident. Thus in such propositions of\npure logic we have the self-evident general propositions of which we\nwere in search.\n\nA proposition such as, \"If Socrates is a man, and all men are mortal,\nthen Socrates is mortal,\" is true in virtue of its _form_ alone. Its\ntruth, in this hypothetical form, does not depend upon whether Socrates\nactually is a man, nor upon whether in fact all men are mortal; thus it\nis equally true when we substitute other terms for Socrates and _man_\nand _mortal_. The general truth of which it is an instance is purely\nformal, and belongs to logic. Since it does not mention any particular\nthing, or even any particular quality or relation, it is wholly\nindependent of the accidental facts of the existent world, and can be\nknown, theoretically, without any experience of particular things or\ntheir qualities and relations.\n\nLogic, we may say, consists of two parts. The first part investigates\nwhat propositions are and what forms they may have; this part enumerates\nthe different kinds of atomic propositions, of molecular propositions,\nof general propositions, and so on. The second part consists of certain\nsupremely general propositions, which assert the truth of all\npropositions of certain forms. This second part merges into pure\nmathematics, whose propositions all turn out, on analysis, to be such\ngeneral formal truths. The first part, which merely enumerates forms, is\nthe more difficult, and philosophically the more important; and it is\nthe recent progress in this first part, more than anything else, that\nhas rendered a truly scientific discussion of many philosophical\nproblems possible.\n\nThe problem of the nature of judgment or belief may be taken as an\nexample of a problem whose solution depends upon an adequate inventory\nof logical forms. We have already seen how the supposed universality of\nthe subject-predicate form made it impossible to give a right analysis\nof serial order, and therefore made space and time unintelligible. But\nin this case it was only necessary to admit relations of two terms. The\ncase of judgment demands the admission of more complicated forms. If all\njudgments were true, we might suppose that a judgment consisted in\napprehension of a _fact_, and that the apprehension was a relation of a\nmind to the fact. From poverty in the logical inventory, this view has\noften been held. But it leads to absolutely insoluble difficulties in\nthe case of error. Suppose I believe that Charles I. died in his bed.\nThere is no objective fact \"Charles I.'s death in his bed\" to which I\ncan have a relation of apprehension. Charles I. and death and his bed\nare objective, but they are not, except in my thought, put together as\nmy false belief supposes. It is therefore necessary, in analysing a\nbelief, to look for some other logical form than a two-term relation.\nFailure to realise this necessity has, in my opinion, vitiated almost\neverything that has hitherto been written on the theory of knowledge,\nmaking the problem of error insoluble and the difference between belief\nand perception inexplicable.\n\nModern logic, as I hope is now evident, has the effect of enlarging our\nabstract imagination, and providing an infinite number of possible\nhypotheses to be applied in the analysis of any complex fact. In this\nrespect it is the exact opposite of the logic practised by the classical\ntradition. In that logic, hypotheses which seem _prima facie_ possible\nare professedly proved impossible, and it is decreed in advance that\nreality must have a certain special character. In modern logic, on the\ncontrary, while the _prima facie_ hypotheses as a rule remain\nadmissible, others, which only logic would have suggested, are added to\nour stock, and are very often found to be indispensable if a right\nanalysis of the facts is to be obtained. The old logic put thought in\nfetters, while the new logic gives it wings. It has, in my opinion,\nintroduced the same kind of advance into philosophy as Galileo\nintroduced into physics, making it possible at last to see what kinds of\nproblems may be capable of solution, and what kinds must be abandoned as\nbeyond human powers. And where a solution appears possible, the new\nlogic provides a method which enables us to obtain results that do not\nmerely embody personal idiosyncrasies, but must command the assent of\nall who are competent to form an opinion.\n\n\n\n\nLECTURE III\n\nON OUR KNOWLEDGE OF THE EXTERNAL WORLD\n\n\nPhilosophy may be approached by many roads, but one of the oldest and\nmost travelled is the road which leads through doubt as to the reality\nof the world of sense. In Indian mysticism, in Greek and modern monistic\nphilosophy from Parmenides onward, in Berkeley, in modern physics, we\nfind sensible appearance criticised and condemned for a bewildering\nvariety of motives. The mystic condemns it on the ground of immediate\nknowledge of a more real and significant world behind the veil;\nParmenides and Plato condemn it because its continual flux is thought\ninconsistent with the unchanging nature of the abstract entities\nrevealed by logical analysis; Berkeley brings several weapons, but his\nchief is the subjectivity of sense-data, their dependence upon the\norganisation and point of view of the spectator; while modern physics,\non the basis of sensible evidence itself, maintains a mad dance of\nelectrons which has, superficially at least, very little resemblance to\nthe immediate objects of sight or touch.\n\nEvery one of these lines of attack raises vital and interesting\nproblems.\n\nThe mystic, so long as he merely reports a positive revelation, cannot\nbe refuted; but when he _denies_ reality to objects of sense, he may be\nquestioned as to what he means by \"reality,\" and may be asked how their\nunreality follows from the supposed reality of his super-sensible world.\nIn answering these questions, he is led to a logic which merges into\nthat of Parmenides and Plato and the idealist tradition.\n\nThe logic of the idealist tradition has gradually grown very complex and\nvery abstruse, as may be seen from the Bradleian sample considered in\nour first lecture. If we attempted to deal fully with this logic, we\nshould not have time to reach any other aspect of our subject; we will\ntherefore, while acknowledging that it deserves a long discussion, pass\nby its central doctrines with only such occasional criticism as may\nserve to exemplify other topics, and concentrate our attention on such\nmatters as its objections to the continuity of motion and the infinity\nof space and time--objections which have been fully answered by modern\nmathematicians in a manner constituting an abiding triumph for the\nmethod of logical analysis in philosophy. These objections and the\nmodern answers to them will occupy our fifth, sixth, and seventh\nlectures.\n\nBerkeley's attack, as reinforced by the physiology of the sense-organs\nand nerves and brain, is very powerful. I think it must be admitted as\nprobable that the immediate objects of sense depend for their existence\nupon physiological conditions in ourselves, and that, for example, the\ncoloured surfaces which we see cease to exist when we shut our eyes. But\nit would be a mistake to infer that they are dependent upon mind, not\nreal while we see them, or not the sole basis for our knowledge of the\nexternal world. This line of argument will be developed in the present\nlecture.\n\nThe discrepancy between the world of physics and the world of sense,\nwhich we shall consider in our fourth lecture, will be found to be more\napparent than real, and it will be shown that whatever there is reason\nto believe in physics can probably be interpreted in terms of sense.\n\nThe instrument of discovery throughout is modern logic, a very different\nscience from the logic of the text-books and also from the logic of\nidealism. Our second lecture has given a short account of modern logic\nand of its points of divergence from the various traditional kinds of\nlogic.\n\nIn our last lecture, after a discussion of causality and free will, we\nshall try to reach a general account of the logical-analytic method of\nscientific philosophy, and a tentative estimate of the hopes of\nphilosophical progress which it allows us to entertain.\n\nIn this lecture, I wish to apply the logical-analytic method to one of\nthe oldest problems of philosophy, namely, the problem of our knowledge\nof the external world. What I have to say on this problem does not\namount to an answer of a definite and dogmatic kind; it amounts only to\nan analysis and statement of the questions involved, with an indication\nof the directions in which evidence may be sought. But although not yet\na definite solution, what can be said at present seems to me to throw a\ncompletely new light on the problem, and to be indispensable, not only\nin seeking the answer, but also in the preliminary question as to what\nparts of our problem may possibly have an ascertainable answer.\n\nIn every philosophical problem, our investigation starts from what may\nbe called \"data,\" by which I mean matters of common knowledge, vague,\ncomplex, inexact, as common knowledge always is, but yet somehow\ncommanding our assent as on the whole and in some interpretation pretty\ncertainly true. In the case of our present problem, the common knowledge\ninvolved is of various kinds. There is first our acquaintance with\nparticular objects of daily life--furniture, houses, towns, other\npeople, and so on. Then there is the extension of such particular\nknowledge to particular things outside our personal experience, through\nhistory and geography, newspapers, etc. And lastly, there is the\nsystematisation of all this knowledge of particulars by means of\nphysical science, which derives immense persuasive force from its\nastonishing power of foretelling the future. We are quite willing to\nadmit that there may be errors of detail in this knowledge, but we\nbelieve them to be discoverable and corrigible by the methods which have\ngiven rise to our beliefs, and we do not, as practical men, entertain\nfor a moment the hypothesis that the whole edifice may be built on\ninsecure foundations. In the main, therefore, and without absolute\ndogmatism as to this or that special portion, we may accept this mass of\ncommon knowledge as affording data for our philosophical analysis.\n\nIt may be said--and this is an objection which must be met at the\noutset--that it is the duty of the philosopher to call in question the\nadmittedly fallible beliefs of daily life, and to replace them by\nsomething more solid and irrefragable. In a sense this is true, and in a\nsense it is effected in the course of analysis. But in another sense,\nand a very important one, it is quite impossible. While admitting that\ndoubt is possible with regard to all our common knowledge, we must\nnevertheless accept that knowledge in the main if philosophy is to be\npossible at all. There is not any superfine brand of knowledge,\nobtainable by the philosopher, which can give us a standpoint from which\nto criticise the whole of the knowledge of daily life. The most that can\nbe done is to examine and purify our common knowledge by an internal\nscrutiny, assuming the canons by which it has been obtained, and\napplying them with more care and with more precision. Philosophy cannot\nboast of having achieved such a degree of certainty that it can have\nauthority to condemn the facts of experience and the laws of science.\nThe philosophic scrutiny, therefore, though sceptical in regard to every\ndetail, is not sceptical as regards the whole. That is to say, its\ncriticism of details will only be based upon their relation to other\ndetails, not upon some external criterion which can be applied to all\nthe details equally. The reason for this abstention from a universal\ncriticism is not any dogmatic confidence, but its exact opposite; it is\nnot that common knowledge _must_ be true, but that we possess no\nradically different kind of knowledge derived from some other source.\nUniversal scepticism, though logically irrefutable, is practically\nbarren; it can only, therefore, give a certain flavour of hesitancy to\nour beliefs, and cannot be used to substitute other beliefs for them.\n\nAlthough data can only be criticised by other data, not by an outside\nstandard, yet we may distinguish different grades of certainty in the\ndifferent kinds of common knowledge which we enumerated just now. What\ndoes not go beyond our own personal sensible acquaintance must be for us\nthe most certain: the \"evidence of the senses\" is proverbially the least\nopen to question. What depends on testimony, like the facts of history\nand geography which are learnt from books, has varying degrees of\ncertainty according to the nature and extent of the testimony. Doubts as\nto the existence of Napoleon can only be maintained for a joke, whereas\nthe historicity of Agamemnon is a legitimate subject of debate. In\nscience, again, we find all grades of certainty short of the highest.\nThe law of gravitation, at least as an approximate truth, has acquired\nby this time the same kind of certainty as the existence of Napoleon,\nwhereas the latest speculations concerning the constitution of matter\nwould be universally acknowledged to have as yet only a rather slight\nprobability in their favour. These varying degrees of certainty\nattaching to different data may be regarded as themselves forming part\nof our data; they, along with the other data, lie within the vague,\ncomplex, inexact body of knowledge which it is the business of the\nphilosopher to analyse.\n\nThe first thing that appears when we begin to analyse our common\nknowledge is that some of it is derivative, while some is primitive;\nthat is to say, there is some that we only believe because of something\nelse from which it has been inferred in some sense, though not\nnecessarily in a strict logical sense, while other parts are believed on\ntheir own account, without the support of any outside evidence. It is\nobvious that the senses give knowledge of the latter kind: the immediate\nfacts perceived by sight or touch or hearing do not need to be proved by\nargument, but are completely self-evident. Psychologists, however, have\nmade us aware that what is actually given in sense is much less than\nmost people would naturally suppose, and that much of what at first\nsight seems to be given is really inferred. This applies especially in\nregard to our space-perceptions. For instance, we instinctively infer\nthe \"real\" size and shape of a visible object from its apparent size and\nshape, according to its distance and our point of view. When we hear a\nperson speaking, our actual sensations usually miss a great deal of what\nhe says, and we supply its place by unconscious inference; in a foreign\nlanguage, where this process is more difficult, we find ourselves\napparently grown deaf, requiring, for example, to be much nearer the\nstage at a theatre than would be necessary in our own country. Thus the\nfirst step in the analysis of data, namely, the discovery of what is\nreally given in sense, is full of difficulty. We will, however, not\nlinger on this point; so long as its existence is realised, the exact\noutcome does not make any very great difference in our main problem.\n\nThe next step in our analysis must be the consideration of how the\nderivative parts of our common knowledge arise. Here we become involved\nin a somewhat puzzling entanglement of logic and psychology.\nPsychologically, a belief may be called derivative whenever it is caused\nby one or more other beliefs, or by some fact of sense which is not\nsimply what the belief asserts. Derivative beliefs in this sense\nconstantly arise without any process of logical inference, merely by\nassociation of ideas or some equally extra-logical process. From the\nexpression of a man's face we judge as to what he is feeling: we say we\n_see_ that he is angry, when in fact we only see a frown. We do not\njudge as to his state of mind by any logical process: the judgment grows\nup, often without our being able to say what physical mark of emotion we\nactually saw. In such a case, the knowledge is derivative\npsychologically; but logically it is in a sense primitive, since it is\nnot the result of any logical deduction. There may or may not be a\npossible deduction leading to the same result, but whether there is or\nnot, we certainly do not employ it. If we call a belief \"logically\nprimitive\" when it is not actually arrived at by a logical inference,\nthen innumerable beliefs are logically primitive which psychologically\nare derivative. The separation of these two kinds of primitiveness is\nvitally important to our present discussion.\n\nWhen we reflect upon the beliefs which are logically but not\npsychologically primitive, we find that, unless they can on reflection\nbe deduced by a logical process from beliefs which are also\npsychologically primitive, our confidence in their truth tends to\ndiminish the more we think about them. We naturally believe, for\nexample, that tables and chairs, trees and mountains, are still there\nwhen we turn our backs upon them. I do not wish for a moment to maintain\nthat this is certainly not the case, but I do maintain that the question\nwhether it is the case is not to be settled off-hand on any supposed\nground of obviousness. The belief that they persist is, in all men\nexcept a few philosophers, logically primitive, but it is not\npsychologically primitive; psychologically, it arises only through our\nhaving seen those tables and chairs, trees and mountains. As soon as the\nquestion is seriously raised whether, because we have seen them, we have\na right to suppose that they are there still, we feel that some kind of\nargument must be produced, and that if none is forthcoming, our belief\ncan be no more than a pious opinion. We do not feel this as regards the\nimmediate objects of sense: there they are, and as far as their\nmomentary existence is concerned, no further argument is required. There\nis accordingly more need of justifying our psychologically derivative\nbeliefs than of justifying those that are primitive.\n\nWe are thus led to a somewhat vague distinction between what we may call\n\"hard\" data and \"soft\" data. This distinction is a matter of degree, and\nmust not be pressed; but if not taken too seriously it may help to make\nthe situation clear. I mean by \"hard\" data those which resist the\nsolvent influence of critical reflection, and by \"soft\" data those\nwhich, under the operation of this process, become to our minds more or\nless doubtful. The hardest of hard data are of two sorts: the particular\nfacts of sense, and the general truths of logic. The more we reflect\nupon these, the more we realise exactly what they are, and exactly what\na doubt concerning them really means, the more luminously certain do\nthey become. _Verbal_ doubt concerning even these is possible, but\nverbal doubt may occur when what is nominally being doubted is not\nreally in our thoughts, and only words are actually present to our\nminds. Real doubt, in these two cases, would, I think, be pathological.\nAt any rate, to me they seem quite certain, and I shall assume that you\nagree with me in this. Without this assumption, we are in danger of\nfalling into that universal scepticism which, as we saw, is as barren as\nit is irrefutable. If we are to continue philosophising, we must make\nour bow to the sceptical hypothesis, and, while admitting the elegant\nterseness of its philosophy, proceed to the consideration of other\nhypotheses which, though perhaps not certain, have at least as good a\nright to our respect as the hypothesis of the sceptic.\n\nApplying our distinction of \"hard\" and \"soft\" data to psychologically\nderivative but logically primitive beliefs, we shall find that most, if\nnot all, are to be classed as soft data. They may be found, on\nreflection, to be capable of logical proof, and they then again become\nbelieved, but no longer as data. As data, though entitled to a certain\nlimited respect, they cannot be placed on a level with the facts of\nsense or the laws of logic. The kind of respect which they deserve seems\nto me such as to warrant us in hoping, though not too confidently, that\nthe hard data may prove them to be at least probable. Also, if the hard\ndata are found to throw no light whatever upon their truth or falsehood,\nwe are justified, I think, in giving rather more weight to the\nhypothesis of their truth than to the hypothesis of their falsehood. For\nthe present, however, let us confine ourselves to the hard data, with a\nview to discovering what sort of world can be constructed by their means\nalone.\n\nOur data now are primarily the facts of sense (_i.e._ of _our own_\nsense-data) and the laws of logic. But even the severest scrutiny will\nallow some additions to this slender stock. Some facts of\nmemory--especially of recent memory--seem to have the highest degree of\ncertainty. Some introspective facts are as certain as any facts of\nsense. And facts of sense themselves must, for our present purposes, be\ninterpreted with a certain latitude. Spatial and temporal relations must\nsometimes be included, for example in the case of a swift motion falling\nwholly within the specious present. And some facts of comparison, such\nas the likeness or unlikeness of two shades of colour, are certainly to\nbe included among hard data. Also we must remember that the distinction\nof hard and soft data is psychological and subjective, so that, if there\nare other minds than our own--which at our present stage must be held\ndoubtful--the catalogue of hard data may be different for them from what\nit is for us.\n\nCertain common beliefs are undoubtedly excluded from hard data. Such is\nthe belief which led us to introduce the distinction, namely, that\nsensible objects in general persist when we are not perceiving them.\nSuch also is the belief in other people's minds: this belief is\nobviously derivative from our perception of their bodies, and is felt to\ndemand logical justification as soon as we become aware of its\nderivativeness. Belief in what is reported by the testimony of others,\nincluding all that we learn from books, is of course involved in the\ndoubt as to whether other people have minds at all. Thus the world from\nwhich our reconstruction is to begin is very fragmentary. The best we\ncan say for it is that it is slightly more extensive than the world at\nwhich Descartes arrived by a similar process, since that world contained\nnothing except himself and his thoughts.\n\nWe are now in a position to understand and state the problem of our\nknowledge of the external world, and to remove various misunderstandings\nwhich have obscured the meaning of the problem. The problem really is:\nCan the existence of anything other than our own hard data be inferred\nfrom the existence of those data? But before considering this problem,\nlet us briefly consider what the problem is _not_.\n\nWhen we speak of the \"external\" world in this discussion, we must not\nmean \"spatially external,\" unless \"space\" is interpreted in a peculiar\nand recondite manner. The immediate objects of sight, the coloured\nsurfaces which make up the visible world, are spatially external in the\nnatural meaning of this phrase. We feel them to be \"there\" as opposed to\n\"here\"; without making any assumption of an existence other than hard\ndata, we can more or less estimate the distance of a coloured surface.\nIt seems probable that distances, provided they are not too great, are\nactually given more or less roughly in sight; but whether this is the\ncase or not, ordinary distances can certainly be estimated approximately\nby means of the data of sense alone. The immediately given world is\nspatial, and is further not wholly contained within our own bodies. Thus\nour knowledge of what is external in this sense is not open to doubt.\n\nAnother form in which the question is often put is: \"Can we know of the\nexistence of any reality which is independent of ourselves?\" This form\nof the question suffers from the ambiguity of the two words\n\"independent\" and \"self.\" To take the Self first: the question as to\nwhat is to be reckoned part of the Self and what is not, is a very\ndifficult one. Among many other things which we may mean by the Self,\ntwo may be selected as specially important, namely, (1) the bare subject\nwhich thinks and is aware of objects, (2) the whole assemblage of things\nthat would necessarily cease to exist if our lives came to an end. The\nbare subject, if it exists at all, is an inference, and is not part of\nthe data; therefore this meaning of Self may be ignored in our present\ninquiry. The second meaning is difficult to make precise, since we\nhardly know what things depend upon our lives for their existence. And\nin this form, the definition of Self introduces the word \"depend,\" which\nraises the same questions as are raised by the word \"independent.\" Let\nus therefore take up the word \"independent,\" and return to the Self\nlater.\n\nWhen we say that one thing is \"independent\" of another, we may mean\neither that it is logically possible for the one to exist without the\nother, or that there is no causal relation between the two such that the\none only occurs as the effect of the other. The only way, so far as I\nknow, in which one thing can be _logically_ dependent upon another is\nwhen the other is _part_ of the one. The existence of a book, for\nexample, is logically dependent upon that of its pages: without the\npages there would be no book. Thus in this sense the question, \"Can we\nknow of the existence of any reality which is independent of ourselves?\"\nreduces to the question, \"Can we know of the existence of any reality of\nwhich our Self is not part?\" In this form, the question brings us back\nto the problem of defining the Self; but I think, however the Self may\nbe defined, even when it is taken as the bare subject, it cannot be\nsupposed to be part of the immediate object of sense; thus in this form\nof the question we must admit that we can know of the existence of\nrealities independent of ourselves.\n\nThe question of causal dependence is much more difficult. To know that\none kind of thing is causally independent of another, we must know that\nit actually occurs without the other. Now it is fairly obvious that,\nwhatever legitimate meaning we give to the Self, our thoughts and\nfeelings are causally dependent upon ourselves, _i.e._ do not occur when\nthere is no Self for them to belong to. But in the case of objects of\nsense this is not obvious; indeed, as we saw, the common-sense view is\nthat such objects persist in the absence of any percipient. If this is\nthe case, then they are causally independent of ourselves; if not, not.\nThus in this form the question reduces to the question whether we can\nknow that objects of sense, or any other objects not our own thoughts\nand feelings, exist at times when we are not perceiving them. This form,\nin which the difficult word \"independent\" no longer occurs, is the form\nin which we stated the problem a minute ago.\n\nOur question in the above form raises two distinct problems, which it is\nimportant to keep separate. First, can we know that objects of sense, or\nvery similar objects, exist at times when we are not perceiving them?\nSecondly, if this cannot be known, can we know that other objects,\ninferable from objects of sense but not necessarily resembling them,\nexist either when we are perceiving the objects of sense or at any other\ntime? This latter problem arises in philosophy as the problem of the\n\"thing in itself,\" and in science as the problem of matter as assumed in\nphysics. We will consider this latter problem first.\n\nOwing to the fact that we feel passive in sensation, we naturally\nsuppose that our sensations have outside causes. Now it is necessary\nhere first of all to distinguish between (1) our sensation, which is a\nmental event consisting in our being aware of a sensible object, and (2)\nthe sensible object of which we are aware in sensation. When I speak of\nthe sensible object, it must be understood that I do not mean such a\nthing as a table, which is both visible and tangible, can be seen by\nmany people at once, and is more or less permanent. What I mean is just\nthat patch of colour which is momentarily seen when we look at the\ntable, or just that particular hardness which is felt when we press it,\nor just that particular sound which is heard when we rap it. Each of\nthese I call a sensible object, and our awareness of it I call a\nsensation. Now our sense of passivity, if it really afforded any\nargument, would only tend to show that the _sensation_ has an outside\ncause; this cause we should naturally seek in the sensible object. Thus\nthere is no good reason, so far, for supposing that sensible objects\nmust have outside causes. But both the thing-in-itself of philosophy and\nthe matter of physics present themselves as outside causes of the\nsensible object as much as of the sensation. What are the grounds for\nthis common opinion?\n\nIn each case, I think, the opinion has resulted from the combination of\na belief that _something_ which can persist independently of our\nconsciousness makes itself known in sensation, with the fact that our\nsensations often change in ways which seem to depend upon us rather than\nupon anything which would be supposed to persist independently of us. At\nfirst, we believe unreflectingly that everything is as it seems to be,\nand that, if we shut our eyes, the objects we had been seeing remain as\nthey were though we no longer see them. But there are arguments against\nthis view, which have generally been thought conclusive. It is\nextraordinarily difficult to see just what the arguments prove; but if\nwe are to make any progress with the problem of the external world, we\nmust try to make up our minds as to these arguments.\n\nA table viewed from one place presents a different appearance from that\nwhich it presents from another place. This is the language of common\nsense, but this language already assumes that there is a real table of\nwhich we see the appearances. Let us try to state what is known in terms\nof sensible objects alone, without any element of hypothesis. We find\nthat as we walk round the table, we perceive a series of gradually\nchanging visible objects. But in speaking of \"walking round the table,\"\nwe have still retained the hypothesis that there is a single table\nconnected with all the appearances. What we ought to say is that, while\nwe have those muscular and other sensations which make us say we are\nwalking, our visual sensations change in a continuous way, so that, for\nexample, a striking patch of colour is not suddenly replaced by\nsomething wholly different, but is replaced by an insensible gradation\nof slightly different colours with slightly different shapes. This is\nwhat we really know by experience, when we have freed our minds from the\nassumption of permanent \"things\" with changing appearances. What is\nreally known is a correlation of muscular and other bodily sensations\nwith changes in visual sensations.\n\nBut walking round the table is not the only way of altering its\nappearance. We can shut one eye, or put on blue spectacles, or look\nthrough a microscope. All these operations, in various ways, alter the\nvisual appearance which we call that of the table. More distant objects\nwill also alter their appearance if (as we say) the state of the\natmosphere changes--if there is fog or rain or sunshine. Physiological\nchanges also alter the appearances of things. If we assume the world of\ncommon sense, all these changes, including those attributed to\nphysiological causes, are changes in the intervening medium. It is not\nquite so easy as in the former case to reduce this set of facts to a\nform in which nothing is assumed beyond sensible objects. Anything\nintervening between ourselves and what we see must be invisible: our\nview in every direction is bounded by the nearest visible object. It\nmight be objected that a dirty pane of glass, for example, is visible\nalthough we can see things through it. But in this case we really see a\nspotted patchwork: the dirtier specks in the glass are visible, while\nthe cleaner parts are invisible and allow us to see what is beyond. Thus\nthe discovery that the intervening medium affects the appearances of\nthings cannot be made by means of the sense of sight alone.\n\nLet us take the case of the blue spectacles, which is the simplest, but\nmay serve as a type for the others. The frame of the spectacles is of\ncourse visible, but the blue glass, if it is clean, is not visible. The\nblueness, which we say is in the glass, appears as being in the objects\nseen through the glass. The glass itself is known by means of the sense\nof touch. In order to know that it is between us and the objects seen\nthrough it, we must know how to correlate the space of touch with the\nspace of sight. This correlation itself, when stated in terms of the\ndata of sense alone, is by no means a simple matter. But it presents no\ndifficulties of principle, and may therefore be supposed accomplished.\nWhen it has been accomplished, it becomes possible to attach a meaning\nto the statement that the blue glass, which we can touch, is between us\nand the object seen, as we say, \"through\" it.\n\nBut we have still not reduced our statement completely to what is\nactually given in sense. We have fallen into the assumption that the\nobject of which we are conscious when we touch the blue spectacles still\nexists after we have ceased to touch them. So long as we are touching\nthem, nothing except our finger can be seen through the part touched,\nwhich is the only part where we immediately know that there is\nsomething. If we are to account for the blue appearance of objects other\nthan the spectacles, when seen through them, it might seem as if we must\nassume that the spectacles still exist when we are not touching them;\nand if this assumption really is necessary, our main problem is\nanswered: we have means of knowing of the present existence of objects\nnot given in sense, though of the same kind as objects formerly given in\nsense.\n\nIt may be questioned, however, whether this assumption is actually\nunavoidable, though it is unquestionably the most natural one to make.\nWe may say that the object of which we become aware when we touch the\nspectacles continues to have effects afterwards, though perhaps it no\nlonger exists. In this view, the supposed continued existence of\nsensible objects after they have ceased to be sensible will be a\nfallacious inference from the fact that they still have effects. It is\noften supposed that nothing which has ceased to exist can continue to\nhave effects, but this is a mere prejudice, due to a wrong conception of\ncausality. We cannot, therefore, dismiss our present hypothesis on the\nground of _a priori_ impossibility, but must examine further whether it\ncan really account for the facts.\n\nIt may be said that our hypothesis is useless in the case when the blue\nglass is never touched at all. How, in that case, are we to account for\nthe blue appearance of objects? And more generally, what are we to make\nof the hypothetical sensations of touch which we associate with\nuntouched visible objects, which we know would be verified if we chose,\nthough in fact we do not verify them? Must not these be attributed to\npermanent possession, by the objects, of the properties which touch\nwould reveal?\n\nLet us consider the more general question first. Experience has taught\nus that where we see certain kinds of coloured surfaces we can, by\ntouch, obtain certain expected sensations of hardness or softness,\ntactile shape, and so on. This leads us to believe that what is seen is\nusually tangible, and that it has, whether we touch it or not, the\nhardness or softness which we should expect to feel if we touched it.\nBut the mere fact that we are able to infer what our tactile sensations\nwould be shows that it is not logically necessary to assume tactile\nqualities before they are felt. All that is really known is that the\nvisual appearance in question, together with touch, will lead to certain\nsensations, which can necessarily be determined in terms of the visual\nappearance, since otherwise they could not be inferred from it.\n\nWe can now give a statement of the experienced facts concerning the blue\nspectacles, which will supply an interpretation of common-sense beliefs\nwithout assuming anything beyond the existence of sensible objects at\nthe times when they are sensible. By experience of the correlation of\ntouch and sight sensations, we become able to associate a certain place\nin touch-space with a certain corresponding place in sight-space.\nSometimes, namely in the case of transparent things, we find that there\nis a tangible object in a touch-place without there being any visible\nobject in the corresponding sight-place. But in such a case as that of\nthe blue spectacles, we find that whatever object is visible beyond the\nempty sight-place in the same line of sight has a different colour from\nwhat it has when there is no tangible object in the intervening\ntouch-place; and as we move the tangible object in touch-space, the blue\npatch moves in sight-space. If now we find a blue patch moving in this\nway in sight-space, when we have no sensible experience of an\nintervening tangible object, we nevertheless infer that, if we put our\nhand at a certain place in touch-space, we should experience a certain\ntouch-sensation. If we are to avoid non-sensible objects, this must be\ntaken as the whole of our meaning when we say that the blue spectacles\nare in a certain place, though we have not touched them, and have only\nseen other things rendered blue by their interposition.\n\nI think it may be laid down quite generally that, _in so far_ as physics\nor common sense is verifiable, it must be capable of interpretation in\nterms of actual sense-data alone. The reason for this is simple.\nVerification consists always in the occurrence of an expected\nsense-datum. Astronomers tell us there will be an eclipse of the moon:\nwe look at the moon, and find the earth's shadow biting into it, that is\nto say, we see an appearance quite different from that of the usual full\nmoon. Now if an expected sense-datum constitutes a verification, what\nwas asserted must have been about sense-data; or, at any rate, if part\nof what was asserted was not about sense-data, then only the other part\nhas been verified. There is in fact a certain regularity or conformity\nto law about the occurrence of sense-data, but the sense-data that occur\nat one time are often causally connected with those that occur at quite\nother times, and not, or at least not very closely, with those that\noccur at neighbouring times. If I look at the moon and immediately\nafterwards hear a train coming, there is no very close causal connection\nbetween my two sense-data; but if I look at the moon on two nights a\nweek apart, there is a very close causal connection between the two\nsense-data. The simplest, or at least the easiest, statement of the\nconnection is obtained by imagining a \"real\" moon which goes on whether\nI look at it or not, providing a series of _possible_ sense-data of\nwhich only those are actual which belong to moments when I choose to\nlook at the moon.\n\nBut the degree of verification obtainable in this way is very small. It\nmust be remembered that, at our present level of doubt, we are not at\nliberty to accept testimony. When we hear certain noises, which are\nthose we should utter if we wished to express a certain thought, we\nassume that that thought, or one very like it, has been in another mind,\nand has given rise to the expression which we hear. If at the same time\nwe see a body resembling our own, moving its lips as we move ours when\nwe speak, we cannot resist the belief that it is alive, and that the\nfeelings inside it continue when we are not looking at it. When we see\nour friend drop a weight upon his toe, and hear him say--what we should\nsay in similar circumstances, the phenomena _can_ no doubt be explained\nwithout assuming that he is anything but a series of shapes and noises\nseen and heard by us, but practically no man is so infected with\nphilosophy as not to be quite certain that his friend has felt the same\nkind of pain as he himself would feel. We will consider the legitimacy\nof this belief presently; for the moment, I only wish to point out that\nit needs the same kind of justification as our belief that the moon\nexists when we do not see it, and that, without it, testimony heard or\nread is reduced to noises and shapes, and cannot be regarded as evidence\nof the facts which it reports. The verification of physics which is\npossible at our present level is, therefore, only that degree of\nverification which is possible by one man's unaided observations, which\nwill not carry us very far towards the establishment of a whole science.\n\nBefore proceeding further, let us summarise the argument so far as it\nhas gone. The problem is: \"Can the existence of anything other than our\nown hard data be inferred from these data?\" It is a mistake to state the\nproblem in the form: \"Can we know of the existence of anything other\nthan ourselves and our states?\" or: \"Can we know of the existence of\nanything independent of ourselves?\" because of the extreme difficulty of\ndefining \"self\" and \"independent\" precisely. The felt passivity of\nsensation is irrelevant, since, even if it proved anything, it could\nonly prove that sensations are caused by sensible objects. The natural\n_naive_ belief is that things seen persist, when unseen, exactly or\napproximately as they appeared when seen; but this belief tends to be\ndispelled by the fact that what common sense regards as the appearance\nof one object changes with what common sense regards as changes in the\npoint of view and in the intervening medium, including in the latter our\nown sense-organs and nerves and brain. This fact, as just stated,\nassumes, however, the common-sense world of stable objects which it\nprofesses to call in question; hence, before we can discover its precise\nbearing on our problem, we must find a way of stating it which does not\ninvolve any of the assumptions which it is designed to render doubtful.\nWhat we then find, as the bare outcome of experience, is that gradual\nchanges in certain sense-data are correlated with gradual changes in\ncertain others, or (in the case of bodily motions) with the other\nsense-data themselves.\n\nThe assumption that sensible objects persist after they have ceased to\nbe sensible--for example, that the hardness of a visible body, which has\nbeen discovered by touch, continues when the body is no longer\ntouched--may be replaced by the statement that the _effects_ of sensible\nobjects persist, _i.e._ that what happens now can only be accounted for,\nin many cases, by taking account of what happened at an earlier time.\nEverything that one man, by his own personal experience, can verify in\nthe account of the world given by common sense and physics, will be\nexplicable by some such means, since verification consists merely in the\noccurrence of an expected sense-datum. But what depends upon testimony,\nwhether heard or read, cannot be explained in this way, since testimony\ndepends upon the existence of minds other than our own, and thus\nrequires a knowledge of something not given in sense. But before\nexamining the question of our knowledge of other minds, let us return to\nthe question of the thing-in-itself, namely, to the theory that what\nexists at times when we are not perceiving a given sensible object is\nsomething quite unlike that object, something which, together with us\nand our sense-organs, causes our sensations, but is never itself given\nin sensation.\n\nThe thing-in-itself, when we start from common-sense assumptions, is a\nfairly natural outcome of the difficulties due to the changing\nappearances of what is supposed to be one object. It is supposed that\nthe table (for example) causes our sense-data of sight and touch, but\nmust, since these are altered by the point of view and the intervening\nmedium, be quite different from the sense-data to which it gives rise.\nThere is, in this theory, a tendency to a confusion from which it\nderives some of its plausibility, namely, the confusion between a\nsensation as a psychical occurrence and its object. A patch of colour,\neven if it only exists when it is seen, is still something quite\ndifferent from the seeing of it: the seeing of it is mental, but the\npatch of colour is not. This confusion, however, can be avoided without\nour necessarily abandoning the theory we are examining. The objection to\nit, I think, lies in its failure to realise the radical nature of the\nreconstruction demanded by the difficulties to which it points. We\ncannot speak legitimately of changes in the point of view and the\nintervening medium until we have already constructed some world more\nstable than that of momentary sensation. Our discussion of the blue\nspectacles and the walk round the table has, I hope, made this clear.\nBut what remains far from clear is the nature of the reconstruction\nrequired.\n\nAlthough we cannot rest content with the above theory, in the terms in\nwhich it is stated, we must nevertheless treat it with a certain\nrespect, for it is in outline the theory upon which physical science and\nphysiology are built, and it must, therefore, be susceptible of a true\ninterpretation. Let us see how this is to be done.\n\nThe first thing to realise is that there are no such things as\n\"illusions of sense.\" Objects of sense, even when they occur in dreams,\nare the most indubitably real objects known to us. What, then, makes us\ncall them unreal in dreams? Merely the unusual nature of their\nconnection with other objects of sense. I dream that I am in America,\nbut I wake up and find myself in England without those intervening days\non the Atlantic which, alas! are inseparably connected with a \"real\"\nvisit to America. Objects of sense are called \"real\" when they have the\nkind of connection with other objects of sense which experience has led\nus to regard as normal; when they fail in this, they are called\n\"illusions.\" But what is illusory is only the inferences to which they\ngive rise; in themselves, they are every bit as real as the objects of\nwaking life. And conversely, the sensible objects of waking life must\nnot be expected to have any more intrinsic reality than those of dreams.\nDreams and waking life, in our first efforts at construction, must be\ntreated with equal respect; it is only by some reality not _merely_\nsensible that dreams can be condemned.\n\nAccepting the indubitable momentary reality of objects of sense, the\nnext thing to notice is the confusion underlying objections derived from\ntheir changeableness. As we walk round the table, its aspect changes;\nbut it is thought impossible to maintain either that the table changes,\nor that its various aspects can all \"really\" exist in the same place. If\nwe press one eyeball, we shall see two tables; but it is thought\npreposterous to maintain that there are \"really\" two tables. Such\narguments, however, seem to involve the assumption that there can be\nsomething more real than objects of sense. If we see two tables, then\nthere _are_ two visual tables. It is perfectly true that, at the same\nmoment, we may discover by touch that there is only one tactile table.\nThis makes us declare the two visual tables an illusion, because usually\none visual object corresponds to one tactile object. But all that we are\nwarranted in saying is that, in this case, the manner of correlation of\ntouch and sight is unusual. Again, when the aspect of the table changes\nas we walk round it, and we are told there cannot be so many different\naspects in the same place, the answer is simple: what does the critic of\nthe table mean by \"the same place\"? The use of such a phrase presupposes\nthat all our difficulties have been solved; as yet, we have no right to\nspeak of a \"place\" except with reference to one given set of momentary\nsense-data. When all are changed by a bodily movement, no place remains\nthe same as it was. Thus the difficulty, if it exists, has at least not\nbeen rightly stated.\n\nWe will now make a new start, adopting a different method. Instead of\ninquiring what is the minimum of assumption by which we can explain the\nworld of sense, we will, in order to have a model hypothesis as a help\nfor the imagination, construct one possible (not necessary) explanation\nof the facts. It may perhaps then be possible to pare away what is\nsuperfluous in our hypothesis, leaving a residue which may be regarded\nas the abstract answer to our problem.\n\nLet us imagine that each mind looks out upon the world, as in Leibniz's\nmonadology, from a point of view peculiar to itself; and for the sake of\nsimplicity let us confine ourselves to the sense of sight, ignoring\nminds which are devoid of this sense. Each mind sees at each moment an\nimmensely complex three-dimensional world; but there is absolutely\nnothing which is seen by two minds simultaneously. When we say that two\npeople see the same thing, we always find that, owing to difference of\npoint of view, there are differences, however slight, between their\nimmediate sensible objects. (I am here assuming the validity of\ntestimony, but as we are only constructing a _possible_ theory, that is\na legitimate assumption.) The three-dimensional world seen by one mind\ntherefore contains no place in common with that seen by another, for\nplaces can only be constituted by the things in or around them. Hence we\nmay suppose, in spite of the differences between the different worlds,\nthat each exists entire exactly as it is perceived, and might be exactly\nas it is even if it were not perceived. We may further suppose that\nthere are an infinite number of such worlds which are in fact\nunperceived. If two men are sitting in a room, two somewhat similar\nworlds are perceived by them; if a third man enters and sits between\nthem, a third world, intermediate between the two previous worlds,\nbegins to be perceived. It is true that we cannot reasonably suppose\njust this world to have existed before, because it is conditioned by the\nsense-organs, nerves, and brain of the newly arrived man; but we can\nreasonably suppose that _some_ aspect of the universe existed from that\npoint of view, though no one was perceiving it. The system consisting of\nall views of the universe perceived and unperceived, I shall call the\nsystem of \"perspectives\"; I shall confine the expression \"private\nworlds\" to such views of the universe as are actually perceived. Thus a\n\"private world\" is a perceived \"perspective\"; but there may be any\nnumber of unperceived perspectives.\n\nTwo men are sometimes found to perceive very similar perspectives, so\nsimilar that they can use the same words to describe them. They say they\nsee the same table, because the differences between the two tables they\nsee are slight and not practically important. Thus it is possible,\nsometimes, to establish a correlation by similarity between a great many\nof the things of one perspective, and a great many of the things of\nanother. In case the similarity is very great, we say the points of view\nof the two perspectives are near together in space; but this space in\nwhich they are near together is totally different from the spaces inside\nthe two perspectives. It is a relation between the perspectives, and is\nnot in either of them; no one can perceive it, and if it is to be known\nit can be only by inference. Between two perceived perspectives which\nare similar, we can imagine a whole series of other perspectives, some\nat least unperceived, and such that between any two, however similar,\nthere are others still more similar. In this way the space which\nconsists of relations between perspectives can be rendered continuous,\nand (if we choose) three-dimensional.\n\nWe can now define the momentary common-sense \"thing,\" as opposed to its\nmomentary appearances. By the similarity of neighbouring perspectives,\nmany objects in the one can be correlated with objects in the other,\nnamely, with the similar objects. Given an object in one perspective,\nform the system of all the objects correlated with it in all the\nperspectives; that system may be identified with the momentary\ncommon-sense \"thing.\" Thus an aspect of a \"thing\" is a member of the\nsystem of aspects which _is_ the \"thing\" at that moment. (The\ncorrelation of the times of different perspectives raises certain\ncomplications, of the kind considered in the theory of relativity; but\nwe may ignore these at present.) All the aspects of a thing are real,\nwhereas the thing is a mere logical construction. It has, however, the\nmerit of being neutral as between different points of view, and of being\nvisible to more than one person, in the only sense in which it can ever\nbe visible, namely, in the sense that each sees one of its aspects.\n\nIt will be observed that, while each perspective contains its own space,\nthere is only one space in which the perspectives themselves are the\nelements. There are as many private spaces as there are perspectives;\nthere are therefore at least as many as there are percipients, and there\nmay be any number of others which have a merely material existence and\nare not seen by anyone. But there is only one perspective-space, whose\nelements are single perspectives, each with its own private space. We\nhave now to explain how the private space of a single perspective is\ncorrelated with part of the one all-embracing perspective space.\n\nPerspective space is the system of \"points of view\" of private spaces\n(perspectives), or, since \"points of view\" have not been defined, we may\nsay it is the system of the private spaces themselves. These private\nspaces will each count as one point, or at any rate as one element, in\nperspective space. They are ordered by means of their similarities.\nSuppose, for example, that we start from one which contains the\nappearance of a circular disc, such as would be called a penny, and\nsuppose this appearance, in the perspective in question, is circular,\nnot elliptic. We can then form a whole series of perspectives containing\na graduated series of circular aspects of varying sizes: for this\npurpose we only have to move (as we say) towards the penny or away from\nit. The perspectives in which the penny looks circular will be said to\nlie on a straight line in perspective space, and their order on this\nline will be that of the sizes of the circular aspects. Moreover--though\nthis statement must be noticed and subsequently examined--the\nperspectives in which the penny looks big will be said to be nearer to\nthe penny than those in which it looks small. It is to be remarked also\nthat any other \"thing\" than our penny might have been chosen to define\nthe relations of our perspectives in perspective space, and that\nexperience shows that the same spatial order of perspectives would have\nresulted.\n\nIn order to explain the correlation of private spaces with perspective\nspace, we have first to explain what is meant by \"the place (in\nperspective space) where a thing is.\" For this purpose, let us again\nconsider the penny which appears in many perspectives. We formed a\nstraight line of perspectives in which the penny looked circular, and we\nagreed that those in which it looked larger were to be considered as\nnearer to the penny. We can form another straight line of perspectives\nin which the penny is seen end-on and looks like a straight line of a\ncertain thickness. These two lines will meet in a certain place in\nperspective space, _i.e._ in a certain perspective, which may be defined\nas \"the place (in perspective space) where the penny is.\" It is true\nthat, in order to prolong our lines until they reach this place, we\nshall have to make use of other things besides the penny, because, so\nfar as experience goes, the penny ceases to present any appearance after\nwe have come so near to it that it touches the eye. But this raises no\nreal difficulty, because the spatial order of perspectives is found\nempirically to be independent of the particular \"things\" chosen for\ndefining the order. We can, for example, remove our penny and prolong\neach of our two straight lines up to their intersection by placing other\npennies further off in such a way that the aspects of the one are\ncircular where those of our original penny were circular, and the\naspects of the other are straight where those of our original penny were\nstraight. There will then be just one perspective in which one of the\nnew pennies looks circular and the other straight. This will be, by\ndefinition, the place where the original penny was in perspective space.\n\nThe above is, of course, only a first rough sketch of the way in which\nour definition is to be reached. It neglects the size of the penny, and\nit assumes that we can remove the penny without being disturbed by any\nsimultaneous changes in the positions of other things. But it is plain\nthat such niceties cannot affect the principle, and can only introduce\ncomplications in its application.\n\nHaving now defined the perspective which is the place where a given\nthing is, we can understand what is meant by saying that the\nperspectives in which a thing looks large are nearer to the thing than\nthose in which it looks small: they are, in fact, nearer to the\nperspective which is the place where the thing is.\n\nWe can now also explain the correlation between a private space and\nparts of perspective space. If there is an aspect of a given thing in a\ncertain private space, then we correlate the place where this aspect is\nin the private space with the place where the thing is in perspective\nspace.\n\nWe may define \"here\" as the place, in perspective space, which is\noccupied by our private world. Thus we can now understand what is meant\nby speaking of a thing as near to or far from \"here.\" A thing is near to\n\"here\" if the place where it is is near to my private world. We can also\nunderstand what is meant by saying that our private world is inside our\nhead; for our private world is a place in perspective space, and may be\npart of the place where our head is.\n\nIt will be observed that _two_ places in perspective space are\nassociated with every aspect of a thing: namely, the place where the\nthing is, and the place which is the perspective of which the aspect in\nquestion forms part. Every aspect of a thing is a member of two\ndifferent classes of aspects, namely: (1) the various aspects of the\nthing, of which at most one appears in any given perspective; (2) the\nperspective of which the given aspect is a member, _i.e._ that in which\nthe thing has the given aspect. The physicist naturally classifies\naspects in the first way, the psychologist in the second. The two places\nassociated with a single aspect correspond to the two ways of\nclassifying it. We may distinguish the two places as that _at_ which,\nand that _from_ which, the aspect appears. The \"place at which\" is the\nplace of the thing to which the aspect belongs; the \"place from which\"\nis the place of the perspective to which the aspect belongs.\n\nLet us now endeavour to state the fact that the aspect which a thing\npresents at a given place is affected by the intervening medium. The\naspects of a thing in different perspectives are to be conceived as\nspreading outwards from the place where the thing is, and undergoing\nvarious changes as they get further away from this place. The laws\naccording to which they change cannot be stated if we only take account\nof the aspects that are near the thing, but require that we should also\ntake account of the things that are at the places from which these\naspects appear. This empirical fact can, therefore, be interpreted in\nterms of our construction.\n\nWe have now constructed a largely hypothetical picture of the world,\nwhich contains and places the experienced facts, including those derived\nfrom testimony. The world we have constructed can, with a certain amount\nof trouble, be used to interpret the crude facts of sense, the facts of\nphysics, and the facts of physiology. It is therefore a world which\n_may_ be actual. It fits the facts, and there is no empirical evidence\nagainst it; it also is free from logical impossibilities. But have we\nany good reason to suppose that it is real? This brings us back to our\noriginal problem, as to the grounds for believing in the existence of\nanything outside my private world. What we have derived from our\nhypothetical construction is that there are no grounds _against_ the\ntruth of this belief, but we have not derived any positive grounds in\nits favour. We will resume this inquiry by taking up again the question\nof testimony and the evidence for the existence of other minds.\n\nIt must be conceded to begin with that the argument in favour of the\nexistence of other people's minds cannot be conclusive. A phantasm of\nour dreams will appear to have a mind--a mind to be annoying, as a rule.\nIt will give unexpected answers, refuse to conform to our desires, and\nshow all those other signs of intelligence to which we are accustomed in\nthe acquaintances of our waking hours. And yet, when we are awake, we do\nnot believe that the phantasm was, like the appearances of people in\nwaking life, representative of a private world to which we have no\ndirect access. If we are to believe this of the people we meet when we\nare awake, it must be on some ground short of demonstration, since it is\nobviously possible that what we call waking life may be only an\nunusually persistent and recurrent nightmare. It may be that our\nimagination brings forth all that other people seem to say to us, all\nthat we read in books, all the daily, weekly, monthly, and quarterly\njournals that distract our thoughts, all the advertisements of soap and\nall the speeches of politicians. This _may_ be true, since it cannot be\nshown to be false, yet no one can really believe it. Is there any\n_logical_ ground for regarding this possibility as improbable? Or is\nthere nothing beyond habit and prejudice?\n\nThe minds of other people are among our data, in the very wide sense in\nwhich we used the word at first. That is to say, when we first begin to\nreflect, we find ourselves already believing in them, not because of any\nargument, but because the belief is natural to us. It is, however, a\npsychologically derivative belief, since it results from observation of\npeople's bodies; and along with other such beliefs, it does not belong\nto the hardest of hard data, but becomes, under the influence of\nphilosophic reflection, just sufficiently questionable to make us desire\nsome argument connecting it with the facts of sense.\n\nThe obvious argument is, of course, derived from analogy. Other people's\nbodies behave as ours do when we have certain thoughts and feelings;\nhence, by analogy, it is natural to suppose that such behaviour is\nconnected with thoughts and feelings like our own. Someone says, \"Look\nout!\" and we find we are on the point of being killed by a motor-car; we\ntherefore attribute the words we heard to the person in question having\nseen the motor-car first, in which case there are existing things of\nwhich we are not directly conscious. But this whole scene, with our\ninference, may occur in a dream, in which case the inference is\ngenerally considered to be mistaken. Is there anything to make the\nargument from analogy more cogent when we are (as we think) awake?\n\nThe analogy in waking life is only to be preferred to that in dreams on\nthe ground of its greater extent and consistency. If a man were to dream\nevery night about a set of people whom he never met by day, who had\nconsistent characters and grew older with the lapse of years, he might,\nlike the man in Calderon's play, find it difficult to decide which was\nthe dream-world and which was the so-called \"real\" world. It is only the\nfailure of our dreams to form a consistent whole either with each other\nor with waking life that makes us condemn them. Certain uniformities are\nobserved in waking life, while dreams seem quite erratic. The natural\nhypothesis would be that demons and the spirits of the dead visit us\nwhile we sleep; but the modern mind, as a rule, refuses to entertain\nthis view, though it is hard to see what could be said against it. On\nthe other hand, the mystic, in moments of illumination, seems to awaken\nfrom a sleep which has filled all his mundane life: the whole world of\nsense becomes phantasmal, and he sees, with the clarity and\nconvincingness that belongs to our morning realisation after dreams, a\nworld utterly different from that of our daily cares and troubles. Who\nshall condemn him? Who shall justify him? Or who shall justify the\nseeming solidity of the common objects among which we suppose ourselves\nto live?\n\nThe hypothesis that other people have minds must, I think, be allowed to\nbe not susceptible of any very strong support from the analogical\nargument. At the same time, it is a hypothesis which systematises a vast\nbody of facts and never leads to any consequences which there is reason\nto think false. There is therefore nothing to be said against its truth,\nand good reason to use it as a working hypothesis. When once it is\nadmitted, it enables us to extend our knowledge of the sensible world by\ntestimony, and thus leads to the system of private worlds which we\nassumed in our hypothetical construction. In actual fact, whatever we\nmay try to think as philosophers, we cannot help believing in the minds\nof other people, so that the question whether our belief is justified\nhas a merely speculative interest. And if it is justified, then there is\nno further difficulty of principle in that vast extension of our\nknowledge, beyond our own private data, which we find in science and\ncommon sense.\n\nThis somewhat meagre conclusion must not be regarded as the whole\noutcome of our long discussion. The problem of the connection of sense\nwith objective reality has commonly been dealt with from a standpoint\nwhich did not carry initial doubt so far as we have carried it; most\nwriters, consciously or unconsciously, have assumed that the testimony\nof others is to be admitted, and therefore (at least by implication)\nthat others have minds. Their difficulties have arisen after this\nadmission, from the differences in the appearance which one physical\nobject presents to two people at the same time, or to one person at two\ntimes between which it cannot be supposed to have changed. Such\ndifficulties have made people doubtful how far objective reality could\nbe known by sense at all, and have made them suppose that there were\npositive arguments against the view that it can be so known. Our\nhypothetical construction meets these arguments, and shows that the\naccount of the world given by common sense and physical science can be\ninterpreted in a way which is logically unobjectionable, and finds a\nplace for all the data, both hard and soft. It is this hypothetical\nconstruction, with its reconciliation of psychology and physics, which\nis the chief outcome of our discussion. Probably the construction is\nonly in part necessary as an initial assumption, and can be obtained\nfrom more slender materials by the logical methods of which we shall\nhave an example in the definitions of points, instants, and particles;\nbut I do not yet know to what lengths this diminution in our initial\nassumptions can be carried.\n\n\n\n\nLECTURE IV\n\nTHE WORLD OF PHYSICS AND THE WORLD OF SENSE\n\n\nAmong the objections to the reality of objects of sense, there is one\nwhich is derived from the apparent difference between matter as it\nappears in physics and things as they appear in sensation. Men of\nscience, for the most part, are willing to condemn immediate data as\n\"merely subjective,\" while yet maintaining the truth of the physics\ninferred from those data. But such an attitude, though it may be\n_capable_ of justification, obviously stands in need of it; and the only\njustification possible must be one which exhibits matter as a logical\nconstruction from sense-data--unless, indeed, there were some wholly _a\npriori_ principle by which unknown entities could be inferred from such\nas are known. It is therefore necessary to find some way of bridging the\ngulf between the world of physics and the world of sense, and it is this\nproblem which will occupy us in the present lecture. Physicists appear\nto be unconscious of the gulf, while psychologists, who are conscious of\nit, have not the mathematical knowledge required for spanning it. The\nproblem is difficult, and I do not know its solution in detail. All that\nI can hope to do is to make the problem felt, and to indicate the kind\nof methods by which a solution is to be sought.\n\nLet us begin by a brief description of the two contrasted worlds. We\nwill take first the world of physics, for, though the other world is\ngiven while the physical world is inferred, to us now the world of\nphysics is the more familiar, the world of pure sense having become\nstrange and difficult to rediscover. Physics started from the\ncommon-sense belief in fairly permanent and fairly rigid bodies--tables\nand chairs, stones, mountains, the earth and moon and sun. This\ncommon-sense belief, it should be noticed, is a piece of audacious\nmetaphysical theorising; objects are not continually present to\nsensation, and it may be doubted whether they are there when they are\nnot seen or felt. This problem, which has been acute since the time of\nBerkeley, is ignored by common sense, and has therefore hitherto been\nignored by physicists. We have thus here a first departure from the\nimmediate data of sensation, though it is a departure merely by way of\nextension, and was probably made by our savage ancestors in some very\nremote prehistoric epoch.\n\nBut tables and chairs, stones and mountains, are not _quite_ permanent\nor _quite_ rigid. Tables and chairs lose their legs, stones are split by\nfrost, and mountains are cleft by earthquakes and eruptions. Then there\nare other things, which seem material, and yet present almost no\npermanence or rigidity. Breath, smoke, clouds, are examples of such\nthings--so, in a lesser degree, are ice and snow; and rivers and seas,\nthough fairly permanent, are not in any degree rigid. Breath, smoke,\nclouds, and generally things that can be seen but not touched, were\nthought to be hardly real; to this day the usual mark of a ghost is that\nit can be seen but not touched. Such objects were peculiar in the fact\nthat they seemed to disappear completely, not merely to be transformed\ninto something else. Ice and snow, when they disappear, are replaced by\nwater; and it required no great theoretical effort to invent the\nhypothesis that the water was the same thing as the ice and snow, but in\na new form. Solid bodies, when they break, break into parts which are\npractically the same in shape and size as they were before. A stone can\nbe hammered into a powder, but the powder consists of grains which\nretain the character they had before the pounding. Thus the ideal of\nabsolutely rigid and absolutely permanent bodies, which early physicists\npursued throughout the changing appearances, seemed attainable by\nsupposing ordinary bodies to be composed of a vast number of tiny atoms.\nThis billiard-ball view of matter dominated the imagination of\nphysicists until quite modern times, until, in fact, it was replaced by\nthe electromagnetic theory, which in its turn is developing into a new\natomism. Apart from the special form of the atomic theory which was\ninvented for the needs of chemistry, some kind of atomism dominated the\nwhole of traditional dynamics, and was implied in every statement of its\nlaws and axioms.\n\nThe pictorial accounts which physicists give of the material world as\nthey conceive it undergo violent changes under the influence of\nmodifications in theory which are much slighter than the layman might\nsuppose from the alterations of the description. Certain features,\nhowever, have remained fairly stable. It is always assumed that there is\n_something_ indestructible which is capable of motion in space; what is\nindestructible is always very small, but does not always occupy a mere\npoint in space. There is supposed to be one all-embracing space in which\nthe motion takes place, and until lately we might have assumed one\nall-embracing time also. But the principle of relativity has given\nprominence to the conception of \"local time,\" and has somewhat\ndiminished men's confidence in the one even-flowing stream of time.\nWithout dogmatising as to the ultimate outcome of the principle of\nrelativity, however, we may safely say, I think, that it does not\ndestroy the possibility of correlating different local times, and does\nnot therefore have such far-reaching philosophical consequences as is\nsometimes supposed. In fact, in spite of difficulties as to measurement,\nthe one all-embracing time still, I think, underlies all that physics\nhas to say about motion. We thus have still in physics, as we had in\nNewton's time, a set of indestructible entities which may be called\nparticles, moving relatively to each other in a single space and a\nsingle time.\n\nThe world of immediate data is quite different from this. Nothing is\npermanent; even the things that we think are fairly permanent, such as\nmountains, only become data when we see them, and are not immediately\ngiven as existing at other moments. So far from one all-embracing space\nbeing given, there are several spaces for each person, according to the\ndifferent senses which give relations that may be called spatial.\nExperience teaches us to obtain one space from these by correlation, and\nexperience, together with instinctive theorising, teaches us to\ncorrelate our spaces with those which we believe to exist in the\nsensible worlds of other people. The construction of a single time\noffers less difficulty so long as we confine ourselves to one person's\nprivate world, but the correlation of one private time with another is a\nmatter of great difficulty. Thus, apart from any of the fluctuating\nhypotheses of physics, three main problems arise in connecting the world\nof physics with the world of sense, namely (1) the construction of\npermanent \"things,\" (2) the construction of a single space, and (3) the\nconstruction of a single time. We will consider these three problems in\nsuccession.\n\n(1) The belief in indestructible \"things\" very early took the form of\natomism. The underlying motive in atomism was not, I think, any\nempirical success in interpreting phenomena, but rather an instinctive\nbelief that beneath all the changes of the sensible world there must be\nsomething permanent and unchanging. This belief was, no doubt, fostered\nand nourished by its practical successes, culminating in the\nconservation of mass; but it was not produced by these successes. On the\ncontrary, they were produced by it. Philosophical writers on physics\nsometimes speak as though the conservation of something or other were\nessential to the possibility of science, but this, I believe, is an\nentirely erroneous opinion. If the _a priori_ belief in permanence had\nnot existed, the same laws which are now formulated in terms of this\nbelief might just as well have been formulated without it. Why should we\nsuppose that, when ice melts, the water which replaces it is the same\nthing in a new form? Merely because this supposition enables us to state\nthe phenomena in a way which is consonant with our prejudices. What we\nreally know is that, under certain conditions of temperature, the\nappearance we call ice is replaced by the appearance we call water. We\ncan give laws according to which the one appearance will be succeeded by\nthe other, but there is no reason except prejudice for regarding both as\nappearances of the same substance.\n\nOne task, if what has just been said is correct, which confronts us in\ntrying to connect the world of sense with the world of physics, is the\ntask of reconstructing the conception of matter without the _a priori_\nbeliefs which historically gave rise to it. In spite of the\nrevolutionary results of modern physics, the empirical successes of the\nconception of matter show that there must be some legitimate conception\nwhich fulfils roughly the same functions. The time has hardly come when\nwe can state precisely what this legitimate conception is, but we can\nsee in a general way what it must be like. For this purpose, it is only\nnecessary to take our ordinary common-sense statements and reword them\nwithout the assumption of permanent substance. We say, for example, that\nthings change gradually--sometimes very quickly, but not without passing\nthrough a continuous series of intermediate states. What this means is\nthat, given any sensible appearance, there will usually be, _if we\nwatch_, a continuous series of appearances connected with the given one,\nleading on by imperceptible gradations to the new appearances which\ncommon-sense regards as those of the same thing. Thus a thing may be\ndefined as a certain series of appearances, connected with each other by\ncontinuity and by certain causal laws. In the case of slowly changing\nthings, this is easily seen. Consider, say, a wall-paper which fades in\nthe course of years. It is an effort not to conceive of it as one\n\"thing\" whose colour is slightly different at one time from what it is\nat another. But what do we really _know_ about it? We know that under\nsuitable circumstances--_i.e._ when we are, as is said, \"in the\nroom\"--we perceive certain colours in a certain pattern: not always\nprecisely the same colours, but sufficiently similar to feel familiar.\nIf we can state the laws according to which the colour varies, we can\nstate all that is empirically verifiable; the assumption that there is a\nconstant entity, the wall-paper, which \"has\" these various colours at\nvarious times, is a piece of gratuitous metaphysics. We may, if we like,\n_define_ the wall-paper as the series of its aspects. These are\ncollected together by the same motives which led us to regard the\nwall-paper as one thing, namely a combination of sensible continuity and\ncausal connection. More generally, a \"thing\" will be defined as a\ncertain series of aspects, namely those which would commonly be said to\nbe _of_ the thing. To say that a certain aspect is an aspect _of_ a\ncertain thing will merely mean that it is one of those which, taken\nserially, _are_ the thing. Everything will then proceed as before:\nwhatever was verifiable is unchanged, but our language is so interpreted\nas to avoid an unnecessary metaphysical assumption of permanence.\n\nThe above extrusion of permanent things affords an example of the maxim\nwhich inspires all scientific philosophising, namely \"Occam's razor\":\n_Entities are not to be multiplied without necessity._ In other words,\nin dealing with any subject-matter, find out what entities are\nundeniably involved, and state everything in terms of these entities.\nVery often the resulting statement is more complicated and difficult\nthan one which, like common sense and most philosophy, assumes\nhypothetical entities whose existence there is no good reason to believe\nin. We find it easier to imagine a wall-paper with changing colours than\nto think merely of the series of colours; but it is a mistake to suppose\nthat what is easy and natural in thought is what is most free from\nunwarrantable assumptions, as the case of \"things\" very aptly\nillustrates.\n\nThe above summary account of the genesis of \"things,\" though it may be\ncorrect in outline, has omitted some serious difficulties which it is\nnecessary briefly to consider. Starting from a world of helter-skelter\nsense-data, we wish to collect them into series, each of which can be\nregarded as consisting of the successive appearances of one \"thing.\"\nThere is, to begin with, some conflict between what common sense regards\nas one thing, and what physics regards an unchanging collection of\nparticles. To common sense, a human body is one thing, but to science\nthe matter composing it is continually changing. This conflict, however,\nis not very serious, and may, for our rough preliminary purpose, be\nlargely ignored. The problem is: by what principles shall we select\ncertain data from the chaos, and call them all appearances of the same\nthing?\n\nA rough and approximate answer to this question is not very difficult.\nThere are certain fairly stable collections of appearances, such as\nlandscapes, the furniture of rooms, the faces of acquaintances. In these\ncases, we have little hesitation in regarding them on successive\noccasions as appearances of one thing or collection of things. But, as\nthe _Comedy of Errors_ illustrates, we may be led astray if we judge by\nmere resemblance. This shows that something more is involved, for two\ndifferent things may have any degree of likeness up to exact similarity.\n\nAnother insufficient criterion of one thing is _continuity_. As we have\nalready seen, if we watch what we regard as one changing thing, we\nusually find its changes to be continuous so far as our senses can\nperceive. We are thus led to assume that, if we see two finitely\ndifferent appearances at two different times, and if we have reason to\nregard them as belonging to the same thing, then there was a continuous\nseries of intermediate states of that thing during the time when we were\nnot observing it. And so it comes to be thought that continuity of\nchange is necessary and sufficient to constitute one thing. But in fact\nit is neither. It is not _necessary_, because the unobserved states, in\nthe case where our attention has not been concentrated on the thing\nthroughout, are purely hypothetical, and cannot possibly be our ground\nfor supposing the earlier and later appearances to belong to the same\nthing; on the contrary, it is because we suppose this that we assume\nintermediate unobserved states. Continuity is also not sufficient, since\nwe can, for example, pass by sensibly continuous gradations from any one\ndrop of the sea to any other drop. The utmost we can say is that\ndiscontinuity during uninterrupted observation is as a rule a mark of\ndifference between things, though even this cannot be said in such cases\nas sudden explosions.\n\nThe assumption of continuity is, however, successfully made in physics.\nThis proves something, though not anything of very obvious utility to\nour present problem: it proves that nothing in the known world is\ninconsistent with the hypothesis that all changes are really continuous,\nthough from too great rapidity or from our lack of observation they may\nnot always appear continuous. In this hypothetical sense, continuity may\nbe allowed to be a _necessary_ condition if two appearances are to be\nclassed as appearances of the same thing. But it is not a _sufficient_\ncondition, as appears from the instance of the drops in the sea. Thus\nsomething more must be sought before we can give even the roughest\ndefinition of a \"thing.\"\n\nWhat is wanted further seems to be something in the nature of fulfilment\nof causal laws. This statement, as it stands, is very vague, but we will\nendeavour to give it precision. When I speak of \"causal laws,\" I mean\nany laws which connect events at different times, or even, as a limiting\ncase, events at the same time provided the connection is not logically\ndemonstrable. In this very general sense, the laws of dynamics are\ncausal laws, and so are the laws correlating the simultaneous\nappearances of one \"thing\" to different senses. The question is: How do\nsuch laws help in the definition of a \"thing\"?\n\nTo answer this question, we must consider what it is that is proved by\nthe empirical success of physics. What is proved is that its hypotheses,\nthough unverifiable where they go beyond sense-data, are at no point in\ncontradiction with sense-data, but, on the contrary, are ideally such as\nto render all sense-data calculable from a sufficient collection of data\nall belonging to a given period of time. Now physics has found it\nempirically possible to collect sense-data into series, each series\nbeing regarded as belonging to one \"thing,\" and behaving, with regard to\nthe laws of physics, in a way in which series not belonging to one thing\nwould in general not behave. If it is to be unambiguous whether two\nappearances belong to the same thing or not, there must be only one way\nof grouping appearances so that the resulting things obey the laws of\nphysics. It would be very difficult to prove that this is the case, but\nfor our present purposes we may let this point pass, and assume that\nthere is only one way. We must include in our definition of a \"thing\"\nthose of its aspects, if any, which are not observed. Thus we may lay\ndown the following definition: _Things are those series of aspects which\nobey the laws of physics._ That such series exist is an empirical fact,\nwhich constitutes the verifiability of physics.\n\nIt may still be objected that the \"matter\" of physics is something other\nthan series of sense-data. Sense-data, it may be said, belong to\npsychology and are, at any rate in some sense, subjective, whereas\nphysics is quite independent of psychological considerations, and does\nnot assume that its matter only exists when it is perceived.\n\nTo this objection there are two answers, both of some importance.\n\n(a) We have been considering, in the above account, the question of the\n_verifiability_ of physics. Now verifiability is by no means the same\nthing as truth; it is, in fact, something far more subjective and\npsychological. For a proposition to be verifiable, it is not enough that\nit should be true, but it must also be such as we can _discover_ to be\ntrue. Thus verifiability depends upon our capacity for acquiring\nknowledge, and not only upon the objective truth. In physics, as\nordinarily set forth, there is much that is unverifiable: there are\nhypotheses as to (A) how things would appear to a spectator in a place\nwhere, as it happens, there is no spectator; (B) how things would appear\nat times when, in fact, they are not appearing to anyone; (C) things\nwhich never appear at all. All these are introduced to simplify the\nstatement of the causal laws, but none of them form an integral part of\nwhat is _known_ to be true in physics. This brings us to our second\nanswer.\n\n(b) If physics is to consist wholly of propositions known to be true, or\nat least capable of being proved or disproved, the three kinds of\nhypothetical entities we have just enumerated must all be capable of\nbeing exhibited as logical functions of sense-data. In order to show how\nthis might possibly be done, let us recall the hypothetical Leibnizian\nuniverse of Lecture III. In that universe, we had a number of\nperspectives, two of which never had any entity in common, but often\ncontained entities which could be sufficiently correlated to be regarded\nas belonging to the same thing. We will call one of these an \"actual\"\nprivate world when there is an actual spectator to which it appears, and\n\"ideal\" when it is merely constructed on principles of continuity. A\nphysical thing consists, at each instant, of the whole set of its\naspects at that instant, in all the different worlds; thus a momentary\nstate of a thing is a whole set of aspects. An \"ideal\" appearance will\nbe an aspect merely calculated, but not actually perceived by any\nspectator. An \"ideal\" state of a thing will be a state at a moment when\nall its appearances are ideal. An ideal thing will be one whose states\nat all times are ideal. Ideal appearances, states, and things, since\nthey are calculated, must be functions of actual appearances, states,\nand things; in fact, ultimately, they must be functions of actual\nappearances. Thus it is unnecessary, for the enunciation of the laws of\nphysics, to assign any reality to ideal elements: it is enough to accept\nthem as logical constructions, provided we have means of knowing how to\ndetermine when they become actual. This, in fact, we have with some\ndegree of approximation; the starry heaven, for instance, becomes actual\nwhenever we choose to look at it. It is open to us to believe that the\nideal elements exist, and there can be no reason for _dis_believing\nthis; but unless in virtue of some _a priori_ law we cannot _know_ it,\nfor empirical knowledge is confined to what we actually observe.\n\n(2) The three main conceptions of physics are space, time, and matter.\nSome of the problems raised by the conception of matter have been\nindicated in the above discussion of \"things.\" But space and time also\nraise difficult problems of much the same kind, namely, difficulties in\nreducing the haphazard untidy world of immediate sensation to the smooth\norderly world of geometry and kinematics. Let us begin with the\nconsideration of space.\n\nPeople who have never read any psychology seldom realise how much mental\nlabour has gone into the construction of the one all-embracing space\ninto which all sensible objects are supposed to fit. Kant, who was\nunusually ignorant of psychology, described space as \"an infinite given\nwhole,\" whereas a moment's psychological reflection shows that a space\nwhich is infinite is not given, while a space which can be called given\nis not infinite. What the nature of \"given\" space really is, is a\ndifficult question, upon which psychologists are by no means agreed. But\nsome general remarks may be made, which will suffice to show the\nproblems, without taking sides on any psychological issue still in\ndebate.\n\nThe first thing to notice is that different senses have different\nspaces. The space of sight is quite different from the space of touch:\nit is only by experience in infancy that we learn to correlate them. In\nlater life, when we see an object within reach, we know how to touch it,\nand more or less what it will feel like; if we touch an object with our\neyes shut, we know where we should have to look for it, and more or less\nwhat it would look like. But this knowledge is derived from early\nexperience of the correlation of certain kinds of touch-sensations with\ncertain kinds of sight-sensations. The one space into which both kinds\nof sensations fit is an intellectual construction, not a datum. And\nbesides touch and sight, there are other kinds of sensation which give\nother, though less important spaces: these also have to be fitted into\nthe one space by means of experienced correlations. And as in the case\nof things, so here: the one all-embracing space, though convenient as a\nway of speaking, need not be supposed really to exist. All that\nexperience makes certain is the several spaces of the several senses,\ncorrelated by empirically discovered laws. The one space may turn out to\nbe valid as a logical construction, compounded of the several spaces,\nbut there is no good reason to assume its independent metaphysical\nreality.\n\nAnother respect in which the spaces of immediate experience differ from\nthe space of geometry and physics is in regard to _points_. The space of\ngeometry and physics consists of an infinite number of points, but no\none has ever seen or touched a point. If there are points in a sensible\nspace, they must be an inference. It is not easy to see any way in\nwhich, as independent entities, they could be validly inferred from the\ndata; thus here again, we shall have, if possible, to find some logical\nconstruction, some complex assemblage of immediately given objects,\nwhich will have the geometrical properties required of points. It is\ncustomary to think of points as simple and infinitely small, but\ngeometry in no way demands that we should think of them in this way. All\nthat is necessary for geometry is that they should have mutual relations\npossessing certain enumerated abstract properties, and it may be that an\nassemblage of data of sensation will serve this purpose. Exactly how\nthis is to be done, I do not yet know, but it seems fairly certain that\nit can be done.\n\nThe following illustrative method, simplified so as to be easily\nmanipulated, has been invented by Dr Whitehead for the purpose of\nshowing how points might be manufactured from sense-data. We have first\nof all to observe that there are no infinitesimal sense-data: any\nsurface we can see, for example, must be of some finite extent. But what\nat first appears as one undivided whole is often found, under the\ninfluence of attention, to split up into parts contained within the\nwhole. Thus one spatial object may be contained within another, and\nentirely enclosed by the other. This relation of enclosure, by the help\nof some very natural hypotheses, will enable us to define a \"point\" as a\ncertain class of spatial objects, namely all those (as it will turn out\nin the end) which would naturally be said to contain the point. In order\nto obtain a definition of a \"point\" in this way, we proceed as follows:\n\nGiven any set of volumes or surfaces, they will not in general converge\ninto one point. But if they get smaller and smaller, while of any two of\nthe set there is always one that encloses the other, then we begin to\nhave the kind of conditions which would enable us to treat them as\nhaving a point for their limit. The hypotheses required for the relation\nof enclosure are that (1) it must be transitive; (2) of two _different_\nspatial objects, it is impossible for each to enclose the other, but a\nsingle spatial object always encloses itself; (3) any set of spatial\nobjects such that there is at least one spatial object enclosed by them\nall has a lower limit or minimum, _i.e._ an object enclosed by all of\nthem and enclosing all objects which are enclosed by all of them; (4) to\nprevent trivial exceptions, we must add that there are to be instances\nof enclosure, _i.e._ there are really to be objects of which one\nencloses the other. When an enclosure-relation has these properties, we\nwill call it a \"point-producer.\" Given any relation of enclosure, we\nwill call a set of objects an \"enclosure-series\" if, of any two of them,\none is contained in the other. We require a condition which shall secure\nthat an enclosure-series converges to a point, and this is obtained as\nfollows: Let our enclosure-series be such that, given any other\nenclosure-series of which there are members enclosed in any arbitrarily\nchosen member of our first series, then there are members of our first\nseries enclosed in any arbitrarily chosen member of our second series.\nIn this case, our first enclosure-series may be called a \"punctual\nenclosure-series.\" Then a \"point\" is all the objects which enclose\nmembers of a given punctual enclosure-series. In order to ensure\ninfinite divisibility, we require one further property to be added to\nthose defining point-producers, namely that any object which encloses\nitself also encloses an object other than itself. The \"points\" generated\nby point-producers with this property will be found to be such as\ngeometry requires.\n\n(3) The question of time, so long as we confine ourselves to one private\nworld, is rather less complicated than that of space, and we can see\npretty clearly how it might be dealt with by such methods as we have\nbeen considering. Events of which we are conscious do not last merely\nfor a mathematical instant, but always for some finite time, however\nshort. Even if there be a physical world such as the mathematical theory\nof motion supposes, impressions on our sense-organs produce sensations\nwhich are not merely and strictly instantaneous, and therefore the\nobjects of sense of which we are immediately conscious are not strictly\ninstantaneous. Instants, therefore, are not among the data of\nexperience, and, if legitimate, must be either inferred or constructed.\nIt is difficult to see how they can be validly inferred; thus we are\nleft with the alternative that they must be constructed. How is this to\nbe done?\n\nImmediate experience provides us with two time-relations among events:\nthey may be simultaneous, or one may be earlier and the other later.\nThese two are both part of the crude data; it is not the case that only\nthe events are given, and their time-order is added by our subjective\nactivity. The time-order, within certain limits, is as much given as the\nevents. In any story of adventure you will find such passages as the\nfollowing: \"With a cynical smile he pointed the revolver at the breast\nof the dauntless youth. 'At the word _three_ I shall fire,' he said. The\nwords one and two had already been spoken with a cool and deliberate\ndistinctness. The word _three_ was forming on his lips. At this moment a\nblinding flash of lightning rent the air.\" Here we have\nsimultaneity--not due, as Kant would have us believe, to the subjective\nmental apparatus of the dauntless youth, but given as objectively as the\nrevolver and the lightning. And it is equally given in immediate\nexperience that the words _one_ and _two_ come earlier than the flash.\nThese time-relations hold between events which are not strictly\ninstantaneous. Thus one event may begin sooner than another, and\ntherefore be before it, but may continue after the other has begun, and\ntherefore be also simultaneous with it. If it persists after the other\nis over, it will also be later than the other. Earlier, simultaneous,\nand later, are not inconsistent with each other when we are concerned\nwith events which last for a finite time, however short; they only\nbecome inconsistent when we are dealing with something instantaneous.\n\nIt is to be observed that we cannot give what may be called _absolute_\ndates, but only dates determined by events. We cannot point to a time\nitself, but only to some event occurring at that time. There is\ntherefore no reason in experience to suppose that there are times as\nopposed to events: the events, ordered by the relations of simultaneity\nand succession, are all that experience provides. Hence, unless we are\nto introduce superfluous metaphysical entities, we must, in defining\nwhat mathematical physics can regard as an instant, proceed by means of\nsome construction which assumes nothing beyond events and their temporal\nrelations.\n\nIf we wish to assign a date exactly by means of events, how shall we\nproceed? If we take any one event, we cannot assign our date exactly,\nbecause the event is not instantaneous, that is to say, it may be\nsimultaneous with two events which are not simultaneous with each other.\nIn order to assign a date exactly, we must be able, theoretically, to\ndetermine whether any given event is before, at, or after this date, and\nwe must know that any other date is either before or after this date,\nbut not simultaneous with it. Suppose, now, instead of taking one event\nA, we take two events A and B, and suppose A and B partly overlap, but B\nends before A ends. Then an event which is simultaneous with both A and\nB must exist during the time when A and B overlap; thus we have come\nrather nearer to a precise date than when we considered A and B alone.\nLet C be an event which is simultaneous with both A and B, but which\nends before either A or B has ended. Then an event which is simultaneous\nwith A and B and C must exist during the time when all three overlap,\nwhich is a still shorter time. Proceeding in this way, by taking more\nand more events, a new event which is dated as simultaneous with all of\nthem becomes gradually more and more accurately dated. This suggests a\nway by which a completely accurate date can be defined.\n\n        A____________________\n\n    B____________________\n\n            C________\n\nLet us take a group of events of which any two overlap, so that there is\nsome time, however short, when they all exist. If there is any other\nevent which is simultaneous with all of these, let us add it to the\ngroup; let us go on until we have constructed a group such that no event\noutside the group is simultaneous with all of them, but all the events\ninside the group are simultaneous with each other. Let us define this\nwhole group as an instant of time. It remains to show that it has the\nproperties we expect of an instant.\n\nWhat are the properties we expect of instants? First, they must form a\nseries: of any two, one must be before the other, and the other must be\nnot before the one; if one is before another, and the other before a\nthird, the first must be before the third. Secondly, every event must be\nat a certain number of instants; two events are simultaneous if they are\nat the same instant, and one is before the other if there is an instant,\nat which the one is, which is earlier than some instant at which the\nother is. Thirdly, if we assume that there is always some change going\non somewhere during the time when any given event persists, the series\nof instants ought to be compact, _i.e._ given any two instants, there\nought to be other instants between them. Do instants, as we have defined\nthem, have these properties?\n\nWe shall say that an event is \"at\" an instant when it is a member of the\ngroup by which the instant is constituted; and we shall say that one\ninstant is before another if the group which is the one instant contains\nan event which is earlier than, but not simultaneous with, some event in\nthe group which is the other instant. When one event is earlier than,\nbut not simultaneous with another, we shall say that it \"wholly\nprecedes\" the other. Now we know that of two events which are not\nsimultaneous, there must be one which wholly precedes the other, and in\nthat case the other cannot also wholly precede the one; we also know\nthat, if one event wholly precedes another, and the other wholly\nprecedes a third, then the first wholly precedes the third. From these\nfacts it is easy to deduce that the instants as we have defined them\nform a series.\n\nWe have next to show that every event is \"at\" at least one instant,\n_i.e._ that, given any event, there is at least one class, such as we\nused in defining instants, of which it is a member. For this purpose,\nconsider all the events which are simultaneous with a given event, and\ndo not begin later, _i.e._ are not wholly after anything simultaneous\nwith it. We will call these the \"initial contemporaries\" of the given\nevent. It will be found that this class of events is the first instant\nat which the given event exists, provided every event wholly after some\ncontemporary of the given event is wholly after some _initial_\ncontemporary of it.\n\nFinally, the series of instants will be compact if, given any two events\nof which one wholly precedes the other, there are events wholly after\nthe one and simultaneous with something wholly before the other. Whether\nthis is the case or not, is an empirical question; but if it is not,\nthere is no reason to expect the time-series to be compact.[17]\n\n  [17] The assumptions made concerning time-relations in the above are\n  as follows:--\n\n    I. In order to secure that instants form a series, we assume:\n\n      (a) No event wholly precedes itself. (An \"event\" is defined as\n      whatever is simultaneous with something or other.)\n\n      (b) If one event wholly precedes another, and the other wholly\n      precedes a third, then the first wholly precedes the third.\n\n      (c) If one event wholly precedes another, it is not simultaneous\n      with it.\n\n      (d) Of two events which are not simultaneous, one must wholly\n      precede the other.\n\n    II. In order to secure that the initial contemporaries of a given\n    event should form an instant, we assume:\n\n      (e) An event wholly after some contemporary of a given event is\n      wholly after some _initial_ contemporary of the given event.\n\n    III. In order to secure that the series of instants shall be\n    compact, we assume:\n\n      (f) If one event wholly precedes another, there is an event wholly\n      after the one and simultaneous with something wholly before the\n      other.\n\n  This assumption entails the consequence that if one event covers the\n  whole of a stretch of time immediately preceding another event, then\n  it must have at least one instant in common with the other event;\n  _i.e._ it is impossible for one event to cease just before another\n  begins. I do not know whether this should be regarded as inadmissible.\n  For a mathematico-logical treatment of the above topics, _cf._ N.\n  Wilner, \"A Contribution to the Theory of Relative Position,\" _Proc.\n  Camb. Phil. Soc._, xvii. 5, pp. 441-449.\n\nThus our definition of instants secures all that mathematics requires,\nwithout having to assume the existence of any disputable metaphysical\nentities.\n\nInstants may also be defined by means of the enclosure-relation, exactly\nas was done in the case of points. One object will be temporally\nenclosed by another when it is simultaneous with the other, but not\nbefore or after it. Whatever encloses temporally or is enclosed\ntemporally we shall call an \"event.\" In order that the relation of\ntemporal enclosure may be a \"point-producer,\" we require (1) that it\nshould be transitive, _i.e._ that if one event encloses another, and the\nother a third, then the first encloses the third; (2) that every event\nencloses itself, but if one event encloses another different event, then\nthe other does not enclose the one; (3) that given any set of events\nsuch that there is at least one event enclosed by all of them, then\nthere is an event enclosing all that they all enclose, and itself\nenclosed by all of them; (4) that there is at least one event. To ensure\ninfinite divisibility, we require also that every event should enclose\nevents other than itself. Assuming these characteristics, temporal\nenclosure is an infinitely divisible point-producer. We can now form an\n\"enclosure-series\" of events, by choosing a group of events such that of\nany two there is one which encloses the other; this will be a \"punctual\nenclosure-series\" if, given any other enclosure-series such that every\nmember of our first series encloses some member of our second, then\nevery member of our second series encloses some member of our first.\nThen an \"instant\" is the class of all events which enclose members of a\ngiven punctual enclosure-series.\n\nThe correlation of the times of different private worlds so as to\nproduce the one all-embracing time of physics is a more difficult\nmatter. We saw, in Lecture III., that different private worlds often\ncontain correlated appearances, such as common sense would regard as\nappearances of the same \"thing.\" When two appearances in different\nworlds are so correlated as to belong to one momentary \"state\" of a\nthing, it would be natural to regard them as simultaneous, and as thus\naffording a simple means of correlating different private times. But\nthis can only be regarded as a first approximation. What we call one\nsound will be heard sooner by people near the source of the sound than\nby people further from it, and the same applies, though in a less\ndegree, to light. Thus two correlated appearances in different worlds\nare not necessarily to be regarded as occurring at the same date in\nphysical time, though they will be parts of one momentary state of a\nthing. The correlation of different private times is regulated by the\ndesire to secure the simplest possible statement of the laws of physics,\nand thus raises rather complicated technical problems; but from the\npoint of view of philosophical theory, there is no very serious\ndifficulty of principle involved.\n\nThe above brief outline must not be regarded as more than tentative and\nsuggestive. It is intended merely to show the kind of way in which,\ngiven a world with the kind of properties that psychologists find in the\nworld of sense, it may be possible, by means of purely logical\nconstructions, to make it amenable to mathematical treatment by defining\nseries or classes of sense-data which can be called respectively\nparticles, points, and instants. If such constructions are possible,\nthen mathematical physics is applicable to the real world, in spite of\nthe fact that its particles, points, and instants are not to be found\namong actually existing entities.\n\nThe problem which the above considerations are intended to elucidate is\none whose importance and even existence has been concealed by the\nunfortunate separation of different studies which prevails throughout\nthe civilised world. Physicists, ignorant and contemptuous of\nphilosophy, have been content to assume their particles, points, and\ninstants in practice, while conceding, with ironical politeness, that\ntheir concepts laid no claim to metaphysical validity. Metaphysicians,\nobsessed by the idealistic opinion that only mind is real, and the\nParmenidean belief that the real is unchanging, repeated one after\nanother the supposed contradictions in the notions of matter, space, and\ntime, and therefore naturally made no endeavour to invent a tenable\ntheory of particles, points, and instants. Psychologists, who have done\ninvaluable work in bringing to light the chaotic nature of the crude\nmaterials supplied by unmanipulated sensation, have been ignorant of\nmathematics and modern logic, and have therefore been content to say\nthat matter, space, and time are \"intellectual constructions,\" without\nmaking any attempt to show in detail either how the intellect can\nconstruct them, or what secures the practical validity which physics\nshows them to possess. Philosophers, it is to be hoped, will come to\nrecognise that they cannot achieve any solid success in such problems\nwithout some slight knowledge of logic, mathematics, and physics;\nmeanwhile, for want of students with the necessary equipment, this vital\nproblem remains unattempted and unknown.\n\nThere are, it is true, two authors, both physicists, who have done\nsomething, though not much, to bring about a recognition of the problem\nas one demanding study. These two authors are Poincare and Mach,\nPoincare especially in his _Science and Hypothesis_, Mach especially in\nhis _Analysis of Sensations_. Both of them, however, admirable as their\nwork is, seem to me to suffer from a general philosophical bias.\nPoincare is Kantian, while Mach is ultra-empiricist; with Poincare\nalmost all the mathematical part of physics is merely conventional,\nwhile with Mach the sensation as a mental event is identified with its\nobject as a part of the physical world. Nevertheless, both these\nauthors, and especially Mach, deserve mention as having made serious\ncontributions to the consideration of our problem.\n\nWhen a point or an instant is defined as a class of sensible qualities,\nthe first impression produced is likely to be one of wild and wilful\nparadox. Certain considerations apply here, however, which will again be\nrelevant when we come to the definition of numbers. There is a whole\ntype of problems which can be solved by such definitions, and almost\nalways there will be at first an effect of paradox. Given a set of\nobjects any two of which have a relation of the sort called \"symmetrical\nand transitive,\" it is almost certain that we shall come to regard them\nas all having some common quality, or as all having the same relation to\nsome one object outside the set. This kind of case is important, and I\nshall therefore try to make it clear even at the cost of some repetition\nof previous definitions.\n\nA relation is said to be \"symmetrical\" when, if one term has this\nrelation to another, then the other also has it to the one. Thus\n\"brother or sister\" is a \"symmetrical\" relation: if one person is a\nbrother or a sister of another, then the other is a brother or sister of\nthe one. Simultaneity, again, is a symmetrical relation; so is equality\nin size. A relation is said to be \"transitive\" when, if one term has\nthis relation to another, and the other to a third, then the one has it\nto the third. The symmetrical relations mentioned just now are also\ntransitive--provided, in the case of \"brother or sister,\" we allow a\nperson to be counted as his or her own brother or sister, and provided,\nin the case of simultaneity, we mean complete simultaneity, _i.e._\nbeginning and ending together.\n\nBut many relations are transitive without being symmetrical--for\ninstance, such relations as \"greater,\" \"earlier,\" \"to the right of,\"\n\"ancestor of,\" in fact all such relations as give rise to series. Other\nrelations are symmetrical without being transitive--for example,\ndifference in any respect. If A is of a different age from B, and B of a\ndifferent age from C, it does not follow that A is of a different age\nfrom C. Simultaneity, again, in the case of events which last for a\nfinite time, will not necessarily be transitive if it only means that\nthe times of the two events overlap. If A ends just after B has begun,\nand B ends just after C has begun, A and B will be simultaneous in this\nsense, and so will B and C, but A and C may well not be simultaneous.\n\nAll the relations which can naturally be represented as equality in any\nrespect, or as possession of a common property, are transitive and\nsymmetrical--this applies, for example, to such relations as being of\nthe same height or weight or colour. Owing to the fact that possession\nof a common property gives rise to a transitive symmetrical relation, we\ncome to imagine that wherever such a relation occurs it must be due to a\ncommon property. \"Being equally numerous\" is a transitive symmetrical\nrelation of two collections; hence we imagine that both have a common\nproperty, called their number. \"Existing at a given instant\" (in the\nsense in which we defined an instant) is a transitive symmetrical\nrelation; hence we come to think that there really is an instant which\nconfers a common property on all the things existing at that instant.\n\"Being states of a given thing\" is a transitive symmetrical relation;\nhence we come to imagine that there really is a thing, other than the\nseries of states, which accounts for the transitive symmetrical\nrelation. In all such cases, the class of terms that have the given\ntransitive symmetrical relation to a given term will fulfil all the\nformal requisites of a common property of all the members of the class.\nSince there certainly is the class, while any other common property may\nbe illusory, it is prudent, in order to avoid needless assumptions, to\nsubstitute the class for the common property which would be ordinarily\nassumed. This is the reason for the definitions we have adopted, and\nthis is the source of the apparent paradoxes. No harm is done if there\nare such common properties as language assumes, since we do not deny\nthem, but merely abstain from asserting them. But if there are not such\ncommon properties in any given case, then our method has secured us\nagainst error. In the absence of special knowledge, therefore, the\nmethod we have adopted is the only one which is safe, and which avoids\nthe risk of introducing fictitious metaphysical entities.\n\n\n\n\nLECTURE V\n\nTHE THEORY OF CONTINUITY\n\n\nThe theory of continuity, with which we shall be occupied in the present\nlecture, is, in most of its refinements and developments, a purely\nmathematical subject--very beautiful, very important, and very\ndelightful, but not, strictly speaking, a part of philosophy. The\nlogical basis of the theory alone belongs to philosophy, and alone will\noccupy us to-night. The way the problem of continuity enters into\nphilosophy is, broadly speaking, the following: Space and time are\ntreated by mathematicians as consisting of points and instants, but they\nalso have a property, easier to feel than to define, which is called\ncontinuity, and is thought by many philosophers to be destroyed when\nthey are resolved into points and instants. Zeno, as we shall see,\nproved that analysis into points and instants was impossible if we\nadhered to the view that the number of points or instants in a finite\nspace or time must be finite. Later philosophers, believing infinite\nnumber to be self-contradictory, have found here an antinomy: Spaces and\ntimes could not consist of a _finite_ number of points and instants, for\nsuch reasons as Zeno's; they could not consist of an _infinite_ number\nof points and instants, because infinite numbers were supposed to be\nself-contradictory. Therefore spaces and times, if real at all, must not\nbe regarded as composed of points and instants.\n\nBut even when points and instants, as independent entities, are\ndiscarded, as they were by the theory advocated in our last lecture, the\nproblems of continuity, as I shall try to show presently, remain, in a\npractically unchanged form. Let us therefore, to begin with, admit\npoints and instants, and consider the problems in connection with this\nsimpler or at least more familiar hypothesis.\n\nThe argument against continuity, in so far as it rests upon the supposed\ndifficulties of infinite numbers, has been disposed of by the positive\ntheory of the infinite, which will be considered in Lecture VII. But\nthere remains a feeling--of the kind that led Zeno to the contention\nthat the arrow in its flight is at rest--which suggests that points and\ninstants, even if they are infinitely numerous, can only give a jerky\nmotion, a succession of different immobilities, not the smooth\ntransitions with which the senses have made us familiar. This feeling is\ndue, I believe, to a failure to realise imaginatively, as well as\nabstractly, the nature of continuous series as they appear in\nmathematics. When a theory has been apprehended logically, there is\noften a long and serious labour still required in order to _feel_ it: it\nis necessary to dwell upon it, to thrust out from the mind, one by one,\nthe misleading suggestions of false but more familiar theories, to\nacquire the kind of intimacy which, in the case of a foreign language,\nwould enable us to think and dream in it, not merely to construct\nlaborious sentences by the help of grammar and dictionary. It is, I\nbelieve, the absence of this kind of intimacy which makes many\nphilosophers regard the mathematical doctrine of continuity as an\ninadequate explanation of the continuity which we experience in the\nworld of sense.\n\nIn the present lecture, I shall first try to explain in outline what the\nmathematical theory of continuity is in its philosophically important\nessentials. The application to actual space and time will not be in\nquestion to begin with. I do not see any reason to suppose that the\npoints and instants which mathematicians introduce in dealing with space\nand time are actual physically existing entities, but I do see reason to\nsuppose that the continuity of actual space and time may be more or less\nanalogous to mathematical continuity. The theory of mathematical\ncontinuity is an abstract logical theory, not dependent for its validity\nupon any properties of actual space and time. What is claimed for it is\nthat, when it is understood, certain characteristics of space and time,\npreviously very hard to analyse, are found not to present any logical\ndifficulty. What we know empirically about space and time is\ninsufficient to enable us to decide between various mathematically\npossible alternatives, but these alternatives are all fully intelligible\nand fully adequate to the observed facts. For the present, however, it\nwill be well to forget space and time and the continuity of sensible\nchange, in order to return to these topics equipped with the weapons\nprovided by the abstract theory of continuity.\n\nContinuity, in mathematics, is a property only possible to a _series_ of\nterms, _i.e._ to terms arranged in an order, so that we can say of any\ntwo that one comes _before_ the other. Numbers in order of magnitude,\nthe points on a line from left to right, the moments of time from\nearlier to later, are instances of series. The notion of order, which is\nhere introduced, is one which is not required in the theory of cardinal\nnumber. It is possible to know that two classes have the same number of\nterms without knowing any order in which they are to be taken. We have\nan instance of this in such a case as English husbands and English\nwives: we can see that there must be the same number of husbands as of\nwives, without having to arrange them in a series. But continuity, which\nwe are now to consider, is essentially a property of an order: it does\nnot belong to a set of terms in themselves, but only to a set in a\ncertain order. A set of terms which can be arranged in one order can\nalways also be arranged in other orders, and a set of terms which can be\narranged in a continuous order can always also be arranged in orders\nwhich are not continuous. Thus the essence of continuity must not be\nsought in the nature of the set of terms, but in the nature of their\narrangement in a series.\n\nMathematicians have distinguished different degrees of continuity, and\nhave confined the word \"continuous,\" for technical purposes, to series\nhaving a certain high degree of continuity. But for philosophical\npurposes, all that is important in continuity is introduced by the\nlowest degree of continuity, which is called \"compactness.\" A series is\ncalled \"compact\" when no two terms are consecutive, but between any two\nthere are others. One of the simplest examples of a compact series is\nthe series of fractions in order of magnitude. Given any two fractions,\nhowever near together, there are other fractions greater than the one\nand smaller than the other, and therefore no two fractions are\nconsecutive. There is no fraction, for example, which is next after 1/2:\nif we choose some fraction which is very little greater than 1/2, say\n51/100 we can find others, such as 101/200, which are nearer to 1/2.\nThus between any two fractions, however little they differ, there are an\ninfinite number of other fractions. Mathematical space and time also\nhave this property of compactness, though whether actual space and time\nhave it is a further question, dependent upon empirical evidence, and\nprobably incapable of being answered with certainty.\n\nIn the case of abstract objects such as fractions, it is perhaps not\nvery difficult to realise the logical possibility of their forming a\ncompact series. The difficulties that might be felt are those of\ninfinity, for in a compact series the number of terms between any two\ngiven terms must be infinite. But when these difficulties have been\nsolved, the mere compactness in itself offers no great obstacle to the\nimagination. In more concrete cases, however, such as motion,\ncompactness becomes much more repugnant to our habits of thought. It\nwill therefore be desirable to consider explicitly the mathematical\naccount of motion, with a view to making its logical possibility felt.\nThe mathematical account of motion is perhaps artificially simplified\nwhen regarded as describing what actually occurs in the physical world;\nbut what actually occurs must be capable, by a certain amount of logical\nmanipulation, of being brought within the scope of the mathematical\naccount, and must, in its analysis, raise just such problems as are\nraised in their simplest form by this account. Neglecting, therefore,\nfor the present, the question of its physical adequacy, let us devote\nourselves merely to considering its possibility as a formal statement of\nthe nature of motion.\n\nIn order to simplify our problem as much as possible, let us imagine a\ntiny speck of light moving along a scale. What do we mean by saying that\nthe motion is continuous? It is not necessary for our purposes to\nconsider the whole of what the mathematician means by this statement:\nonly part of what he means is philosophically important. One part of\nwhat he means is that, if we consider any two positions of the speck\noccupied at any two instants, there will be other intermediate positions\noccupied at intermediate instants. However near together we take the two\npositions, the speck will not jump suddenly from the one to the other,\nbut will pass through an infinite number of other positions on the way.\nEvery distance, however small, is traversed by passing through all the\ninfinite series of positions between the two ends of the distance.\n\nBut at this point imagination suggests that we may describe the\ncontinuity of motion by saying that the speck always passes from one\nposition at one instant to _the next_ position at _the next_ instant. As\nsoon as we say this or imagine it, we fall into error, because there is\nno _next_ point or _next_ instant. If there were, we should find Zeno's\nparadoxes, in some form, unavoidable, as will appear in our next\nlecture. One simple paradox may serve as an illustration. If our speck\nis in motion along the scale throughout the whole of a certain time, it\ncannot be at the same point at two consecutive instants. But it cannot,\nfrom one instant to the next, travel further than from one point to the\nnext, for if it did, there would be no instant at which it was in the\npositions intermediate between that at the first instant and that at the\nnext, and we agreed that the continuity of motion excludes the\npossibility of such sudden jumps. It follows that our speck must, so\nlong as it moves, pass from one point at one instant to the next point\nat the next instant. Thus there will be just one perfectly definite\nvelocity with which all motions must take place: no motion can be faster\nthan this, and no motion can be slower. Since this conclusion is false,\nwe must reject the hypothesis upon which it is based, namely that there\nare consecutive points and instants.[18] Hence the continuity of motion\nmust not be supposed to consist in a body's occupying consecutive\npositions at consecutive times.\n\n  [18] The above paradox is essentially the same as Zeno's argument of\n  the stadium which will be considered in our next lecture.\n\nThe difficulty to imagination lies chiefly, I think, in keeping out the\nsuggestion of _infinitesimal_ distances and times. Suppose we halve a\ngiven distance, and then halve the half, and so on, we can continue the\nprocess as long as we please, and the longer we continue it, the smaller\nthe resulting distance becomes. This infinite divisibility seems, at\nfirst sight, to imply that there are infinitesimal distances, _i.e._\ndistances so small that any finite fraction of an inch would be greater.\nThis, however, is an error. The continued bisection of our distance,\nthough it gives us continually smaller distances, gives us always\n_finite_ distances. If our original distance was an inch, we reach\nsuccessively half an inch, a quarter of an inch, an eighth, a sixteenth,\nand so on; but every one of this infinite series of diminishing\ndistances is finite. \"But,\" it may be said, \"_in the end_ the distance\nwill grow infinitesimal.\" No, because there is no end. The process of\nbisection is one which can, theoretically, be carried on for ever,\nwithout any last term being attained. Thus infinite divisibility of\ndistances, which must be admitted, does not imply that there are\ndistances so small that any finite distance would be larger.\n\nIt is easy, in this kind of question, to fall into an elementary logical\nblunder. Given any finite distance, we can find a smaller distance; this\nmay be expressed in the ambiguous form \"there is a distance smaller than\nany finite distance.\" But if this is then interpreted as meaning \"there\nis a distance such that, whatever finite distance may be chosen, the\ndistance in question is smaller,\" then the statement is false. Common\nlanguage is ill adapted to expressing matters of this kind, and\nphilosophers who have been dependent on it have frequently been misled\nby it.\n\nIn a continuous motion, then, we shall say that at any given instant the\nmoving body occupies a certain position, and at other instants it\noccupies other positions; the interval between any two instants and\nbetween any two positions is always finite, but the continuity of the\nmotion is shown in the fact that, however near together we take the two\npositions and the two instants, there are an infinite number of\npositions still nearer together, which are occupied at instants that are\nalso still nearer together. The moving body never jumps from one\nposition to another, but always passes by a gradual transition through\nan infinite number of intermediaries. At a given instant, it is where it\nis, like Zeno's arrow;[19] but we cannot say that it is at rest at the\ninstant, since the instant does not last for a finite time, and there is\nnot a beginning and end of the instant with an interval between them.\nRest consists in being in the same position at all the instants\nthroughout a certain finite period, however short; it does not consist\nsimply in a body's being where it is at a given instant. This whole\ntheory, as is obvious, depends upon the nature of compact series, and\ndemands, for its full comprehension, that compact series should have\nbecome familiar and easy to the imagination as well as to deliberate\nthought.\n\n  [19] See next lecture.\n\nWhat is required may be expressed in mathematical language by saying\nthat the position of a moving body must be a continuous function of the\ntime. To define accurately what this means, we proceed as follows.\nConsider a particle which, at the moment _t_, is at the point P. Choose\nnow any small portion P1P2 of the path of the particle, this portion\nbeing one which contains P. We say then that, if the motion of the\nparticle is continuous at the time _t_, it must be possible to find two\ninstants _t_1, _t_2, one earlier than _t_ and one later, such that\nthroughout the whole time from _t_1 to _t_2 (both included), the\nparticle lies between P1 and P2. And we say that this must still hold\nhowever small we make the portion P1P2. When this is the case, we say\nthat the motion is continuous at the time _t_; and when the motion is\ncontinuous at all times, we say that the motion as a whole is\ncontinuous. It is obvious that if the particle were to jump suddenly\nfrom P to some other point Q, our definition would fail for all\nintervals P1P2 which were too small to include Q. Thus our definition\naffords an analysis of the continuity of motion, while admitting points\nand instants and denying infinitesimal distances in space or periods in\ntime.\n\n        P1   P    P2   Q\n  ------|----|----|----|------>\n\nPhilosophers, mostly in ignorance of the mathematician's analysis, have\nadopted other and more heroic methods of dealing with the _prima facie_\ndifficulties of continuous motion. A typical and recent example of\nphilosophic theories of motion is afforded by Bergson, whose views on\nthis subject I have examined elsewhere.[20]\n\n  [20] _Monist_, July 1912, pp. 337-341.\n\nApart from definite arguments, there are certain feelings, rather than\nreasons, which stand in the way of an acceptance of the mathematical\naccount of motion. To begin with, if a body is moving at all fast, we\n_see_ its motion just as we see its colour. A _slow_ motion, like that\nof the hour-hand of a watch, is only known in the way which mathematics\nwould lead us to expect, namely by observing a change of position after\na lapse of time; but, when we observe the motion of the second-hand, we\ndo not merely see first one position and then another--we see something\nas directly sensible as colour. What is this something that we see, and\nthat we call visible motion? Whatever it is, it is _not_ the successive\noccupation of successive positions: something beyond the mathematical\ntheory of motion is required to account for it. Opponents of the\nmathematical theory emphasise this fact. \"Your theory,\" they say, \"may\nbe very logical, and might apply admirably to some other world; but in\nthis actual world, actual motions are quite different from what your\ntheory would declare them to be, and require, therefore, some different\nphilosophy from yours for their adequate explanation.\"\n\nThe objection thus raised is one which I have no wish to underrate, but\nI believe it can be fully answered without departing from the methods\nand the outlook which have led to the mathematical theory of motion. Let\nus, however, first try to state the objection more fully.\n\nIf the mathematical theory is adequate, nothing happens when a body\nmoves except that it is in different places at different times. But in\nthis sense the hour-hand and the second-hand are equally in motion, yet\nin the second-hand there is something perceptible to our senses which is\nabsent in the hour-hand. We can see, at each moment, that the\nsecond-hand _is moving_, which is different from seeing it first in one\nplace and then in another. This seems to involve our seeing it\nsimultaneously in a number of places, although it must also involve our\nseeing that it is in some of these places earlier than in others. If,\nfor example, I move my hand quickly from left to right, you seem to see\nthe whole movement at once, in spite of the fact that you know it begins\nat the left and ends at the right. It is this kind of consideration, I\nthink, which leads Bergson and many others to regard a movement as\nreally one indivisible whole, not the series of separate states imagined\nby the mathematician.\n\nTo this objection there are three supplementary answers, physiological,\npsychological, and logical. We will consider them successively.\n\n(1) The physiological answer merely shows that, if the physical world is\nwhat the mathematician supposes, its sensible appearance may\nnevertheless be expected to be what it is. The aim of this answer is\nthus the modest one of showing that the mathematical account is not\nimpossible as applied to the physical world; it does not even attempt to\nshow that this account is necessary, or that an analogous account\napplies in psychology.\n\nWhen any nerve is stimulated, so as to cause a sensation, the sensation\ndoes not cease instantaneously with the cessation of the stimulus, but\ndies away in a short finite time. A flash of lightning, brief as it is\nto our sight, is briefer still as a physical phenomenon: we continue to\nsee it for a few moments after the light-waves have ceased to strike the\neye. Thus in the case of a physical motion, if it is sufficiently swift,\nwe shall actually at one instant see the moving body throughout a finite\nportion of its course, and not only at the exact spot where it is at\nthat instant. Sensations, however, as they die away, grow gradually\nfainter; thus the sensation due to a stimulus which is recently past is\nnot exactly like the sensation due to a present stimulus. It follows\nfrom this that, when we see a rapid motion, we shall not only see a\nnumber of positions of the moving body simultaneously, but we shall see\nthem with different degrees of intensity--the present position most\nvividly, and the others with diminishing vividness, until sensation\nfades away into immediate memory. This state of things accounts fully\nfor the perception of motion. A motion is _perceived_, not merely\n_inferred_, when it is sufficiently swift for many positions to be\nsensible at one time; and the earlier and later parts of one perceived\nmotion are distinguished by the less and greater vividness of the\nsensations.\n\nThis answer shows that physiology can account for our perception of\nmotion. But physiology, in speaking of stimulus and sense-organs and a\nphysical motion distinct from the immediate object of sense, is assuming\nthe truth of physics, and is thus only capable of showing the physical\naccount to be possible, not of showing it to be _necessary_. This\nconsideration brings us to the psychological answer.\n\n(2) The psychological answer to our difficulty about motion is part of a\nvast theory, not yet worked out, and only capable, at present, of being\nvaguely outlined. We considered this theory in the third and fourth\nlectures; for the present, a mere sketch of its application to our\npresent problem must suffice. The world of physics, which was assumed in\nthe physiological answer, is obviously inferred from what is given in\nsensation; yet as soon as we seriously consider what is actually given\nin sensation, we find it apparently very different from the world of\nphysics. The question is thus forced upon us: Is the inference from\nsense to physics a valid one? I believe the answer to be affirmative,\nfor reasons which I suggested in the third and fourth lectures; but the\nanswer cannot be either short or easy. It consists, broadly speaking, in\nshowing that, although the particles, points, and instants with which\nphysics operates are not themselves given in experience, and are very\nlikely not actually existing things, yet, out of the materials provided\nin sensation, it is possible to make logical constructions having the\nmathematical properties which physics assigns to particles, points, and\ninstants. If this can be done, then all the propositions of physics can\nbe translated, by a sort of dictionary, into propositions about the\nkinds of objects which are given in sensation.\n\nApplying these general considerations to the case of motion, we find\nthat, even within the sphere of immediate sense-data, it is necessary,\nor at any rate more consonant with the facts than any other equally\nsimple view, to distinguish instantaneous states of objects, and to\nregard such states as forming a compact series. Let us consider a body\nwhich is moving swiftly enough for its motion to be perceptible, and\nlong enough for its motion to be not wholly comprised in one sensation.\nThen, in spite of the fact that we see a finite extent of the motion at\none instant, the extent which we see at one instant is different from\nthat which we see at another. Thus we are brought back, after all, to a\nseries of momentary views of the moving body, and this series will be\ncompact, like the former physical series of points. In fact, though the\n_terms_ of the series seem different, the mathematical character of the\nseries is unchanged, and the whole mathematical theory of motion will\napply to it _verbatim_.\n\nWhen we are considering the actual data of sensation in this connection,\nit is important to realise that two sense-data may be, and _must_\nsometimes be, really different when we cannot perceive any difference\nbetween them. An old but conclusive reason for believing this was\nemphasised by Poincare.[21] In all cases of sense-data capable of\ngradual change, we may find one sense-datum indistinguishable from\nanother, and that other indistinguishable from a third, while yet the\nfirst and third are quite easily distinguishable. Suppose, for example,\na person with his eyes shut is holding a weight in his hand, and someone\nnoiselessly adds a small extra weight. If the extra weight is small\nenough, no difference will be perceived in the sensation. After a time,\nanother small extra weight may be added, and still no change will be\nperceived; but if both extra weights had been added at once, it may be\nthat the change would be quite easily perceptible. Or, again, take\nshades of colour. It would be easy to find three stuffs of such closely\nsimilar shades that no difference could be perceived between the first\nand second, nor yet between the second and third, while yet the first\nand third would be distinguishable. In such a case, the second shade\ncannot be the same as the first, or it would be distinguishable from the\nthird; nor the same as the third, or it would be distinguishable from\nthe first. It must, therefore, though indistinguishable from both, be\nreally intermediate between them.\n\n  [21] \"Le continu mathematique,\" _Revue de Metaphysique et de Morale_,\n  vol. i. p. 29.\n\nSuch considerations as the above show that, although we cannot\ndistinguish sense-data unless they differ by more than a certain amount,\nit is perfectly reasonable to suppose that sense-data of a given kind,\nsuch as weights or colours, really form a compact series. The objections\nwhich may be brought from a psychological point of view against the\nmathematical theory of motion are not, therefore, objections to this\ntheory properly understood, but only to a quite unnecessary assumption\nof simplicity in the momentary object of sense. Of the immediate object\nof sense, in the case of a visible motion, we may say that at each\ninstant it is in all the positions which remain sensible at that\ninstant; but this set of positions changes continuously from moment to\nmoment, and is amenable to exactly the same mathematical treatment as if\nit were a mere point. When we assert that some mathematical account of\nphenomena is correct, all that we primarily assert is that _something_\ndefinable in terms of the crude phenomena satisfies our formulae; and in\nthis sense the mathematical theory of motion is applicable to the data\nof sensation as well as to the supposed particles of abstract physics.\n\nThere are a number of distinct questions which are apt to be confused\nwhen the mathematical continuum is said to be inadequate to the facts of\nsense. We may state these, in order of diminishing generality, as\nfollows:--\n\n    (a) Are series possessing mathematical continuity logically\n    possible?\n\n    (b) Assuming that they are possible logically, are they not\n    impossible as applied to actual sense-data, because, among actual\n    sense-data, there are no such fixed mutually external terms as are\n    to be found, _e.g._, in the series of fractions?\n\n    (c) Does not the assumption of points and instants make the whole\n    mathematical account fictitious?\n\n    (d) Finally, assuming that all these objections have been answered,\n    is there, in actual empirical fact, any sufficient reason to believe\n    the world of sense continuous?\n\nLet us consider these questions in succession.\n\n(a) The question of the logical possibility of the mathematical\ncontinuum turns partly on the elementary misunderstandings we considered\nat the beginning of the present lecture, partly on the possibility of\nthe mathematical infinite, which will occupy our next two lectures, and\npartly on the logical form of the answer to the Bergsonian objection\nwhich we stated a few minutes ago. I shall say no more on this topic at\npresent, since it is desirable first to complete the psychological\nanswer.\n\n(b) The question whether sense-data are composed of mutually external\nunits is not one which can be decided by empirical evidence. It is often\nurged that, as a matter of immediate experience, the sensible flux is\ndevoid of divisions, and is falsified by the dissections of the\nintellect. Now I have no wish to argue that this view is _contrary_ to\nimmediate experience: I wish only to maintain that it is essentially\nincapable of being _proved_ by immediate experience. As we saw, there\nmust be among sense-data differences so slight as to be imperceptible:\nthe fact that sense-data are immediately given does not mean that their\ndifferences also _must_ be immediately given (though they _may_ be).\nSuppose, for example, a coloured surface on which the colour changes\ngradually--so gradually that the difference of colour in two very\nneighbouring portions is imperceptible, while the difference between\nmore widely separated portions is quite noticeable. The effect produced,\nin such a case, will be precisely that of \"interpenetration,\" of\ntransition which is not a matter of discrete units. And since it tends\nto be supposed that the colours, being immediate data, must _appear_\ndifferent if they _are_ different, it seems easily to follow that\n\"interpenetration\" must be the ultimately right account. But this does\nnot follow. It is unconsciously assumed, as a premiss for a _reductio ad\nabsurdum_ of the analytic view, that, if A and B are immediate data, and\nA differs from B, then the fact that they differ must also be an\nimmediate datum. It is difficult to say how this assumption arose, but I\nthink it is to be connected with the confusion between \"acquaintance\"\nand \"knowledge about.\" Acquaintance, which is what we derive from sense,\ndoes not, theoretically at least, imply even the smallest \"knowledge\nabout,\" _i.e._ it does not imply knowledge of any proposition concerning\nthe object with which we are acquainted. It is a mistake to speak as if\nacquaintance had degrees: there is merely acquaintance and\nnon-acquaintance. When we speak of becoming \"better acquainted,\" as for\ninstance with a person, what we must mean is, becoming acquainted with\nmore parts of a certain whole; but the acquaintance with each part is\neither complete or nonexistent. Thus it is a mistake to say that if we\nwere perfectly acquainted with an object we should know all about it.\n\"Knowledge about\" is knowledge of propositions, which is not involved\nnecessarily in acquaintance with the constituents of the propositions.\nTo know that two shades of colour are different is knowledge about them;\nhence acquaintance with the two shades does not in any way necessitate\nthe knowledge that they are different.\n\nFrom what has just been said it follows that the nature of sense-data\ncannot be validly used to prove that they are not composed of mutually\nexternal units. It may be admitted, on the other hand, that nothing in\ntheir empirical character specially necessitates the view that they are\ncomposed of mutually external units. This view, if it is held, must be\nheld on logical, not on empirical, grounds. I believe that the logical\ngrounds are adequate to the conclusion. They rest, at bottom, upon the\nimpossibility of explaining complexity without assuming constituents. It\nis undeniable that the visual field, for example, is complex; and so far\nas I can see, there is always self-contradiction in the theories which,\nwhile admitting this complexity, attempt to deny that it results from a\ncombination of mutually external units. But to pursue this topic would\nlead us too far from our theme, and I shall therefore say no more about\nit at present.\n\n(c) It is sometimes urged that the mathematical account of motion is\nrendered fictitious by its assumption of points and instants. Now there\nare here two different questions to be distinguished. There is the\nquestion of absolute or relative space and time, and there is the\nquestion whether what occupies space and time must be composed of\nelements which have no extension or duration. And each of these\nquestions in turn may take two forms, namely: (A) is the hypothesis\n_consistent_ with the facts and with logic? (B) is it _necessitated_ by\nthe facts or by logic? I wish to answer, in each case, yes to the first\nform of the question, and no to the second. But in any case the\nmathematical account of motion will not be fictitious, provided a right\ninterpretation is given to the words \"point\" and \"instant.\" A few words\non each alternative will serve to make this clear.\n\nFormally, mathematics adopts an absolute theory of space and time,\n_i.e._ it assumes that, besides the things which are in space and time,\nthere are also entities, called \"points\" and \"instants,\" which are\noccupied by things. This view, however, though advocated by Newton, has\nlong been regarded by mathematicians as merely a convenient fiction.\nThere is, so far as I can see, no conceivable evidence either for or\nagainst it. It is logically possible, and it is consistent with the\nfacts. But the facts are also consistent with the denial of spatial and\ntemporal entities over and above things with spatial and temporal\nrelations. Hence, in accordance with Occam's razor, we shall do well to\nabstain from either assuming or denying points and instants. This means,\nso far as practical working out is concerned, that we adopt the\nrelational theory; for in practice the refusal to assume points and\ninstants has the same effect as the denial of them. But in strict theory\nthe two are quite different, since the denial introduces an element of\nunverifiable dogma which is wholly absent when we merely refrain from\nthe assertion. Thus, although we shall derive points and instants from\nthings, we shall leave the bare possibility open that they may also have\nan independent existence as simple entities.\n\nWe come now to the question whether the things in space and time are to\nbe conceived as composed of elements without extension or duration,\n_i.e._ of elements which only occupy a point and an instant. Physics,\nformally, assumes in its differential equations that things consist of\nelements which occupy only a point at each instant, but persist\nthroughout time. For reasons explained in Lecture IV., the persistence\nof things through time is to be regarded as the formal result of a\nlogical construction, not as necessarily implying any actual\npersistence. The same motives, in fact, which lead to the division of\nthings into point-particles, ought presumably to lead to their division\ninto instant-particles, so that the ultimate _formal_ constituent of the\nmatter in physics will be a point-instant-particle. But such objects, as\nwell as the particles of physics, are not data. The same economy of\nhypothesis, which dictates the practical adoption of a relative rather\nthan an absolute space and time, also dictates the practical adoption of\nmaterial elements which have a finite extension and duration. Since, as\nwe saw in Lecture IV., points and instants can be constructed as logical\nfunctions of such elements, the mathematical account of motion, in which\na particle passes continuously through a continuous series of points,\ncan be interpreted in a form which assumes only elements which agree\nwith our actual data in having a finite extension and duration. Thus, so\nfar as the use of points and instants is concerned, the mathematical\naccount of motion can be freed from the charge of employing fictions.\n\n(d) But we must now face the question: Is there, in actual empirical\nfact, any sufficient reason to believe the world of sense continuous?\nThe answer here must, I think, be in the negative. We may say that the\nhypothesis of continuity is perfectly consistent with the facts and with\nlogic, and that it is technically simpler than any other tenable\nhypothesis. But since our powers of discrimination among very similar\nsensible objects are not infinitely precise, it is quite impossible to\ndecide between different theories which only differ in regard to what is\nbelow the margin of discrimination. If, for example, a coloured surface\nwhich we see consists of a finite number of very small surfaces, and if\na motion which we see consists, like a cinematograph, of a large finite\nnumber of successive positions, there will be nothing empirically\ndiscoverable to show that objects of sense are not continuous. In what\nis called _experienced_ continuity, such as is said to be given in\nsense, there is a large negative element: absence of perception of\ndifference occurs in cases which are _thought_ to give perception of\nabsence of difference. When, for example, we cannot distinguish a colour\nA from a colour B, nor a colour B from a colour C, but can distinguish A\nfrom C, the indistinguishability is a purely negative fact, namely, that\nwe do not _perceive_ a difference. Even in regard to immediate data,\nthis is no reason for denying that there is a difference. Thus, if we\nsee a coloured surface whose colour changes gradually, its sensible\nappearance if the change is continuous will be indistinguishable from\nwhat it would be if the change were by small finite jumps. If this is\ntrue, as it seems to be, it follows that there can never be any\nempirical evidence to demonstrate that the sensible world is continuous,\nand not a collection of a very large finite number of elements of which\neach differs from its neighbour in a finite though very small degree.\nThe continuity of space and time, the infinite number of different\nshades in the spectrum, and so on, are all in the nature of unverifiable\nhypotheses--perfectly possible logically, perfectly consistent with the\nknown facts, and simpler technically than any other tenable hypotheses,\nbut not the sole hypotheses which are logically and empirically\nadequate.\n\nIf a relational theory of instants is constructed, in which an \"instant\"\nis defined as a group of events simultaneous with each other and not all\nsimultaneous with any event outside the group, then if our resulting\nseries of instants is to be compact, it must be possible, if _x_ wholly\nprecedes _y_, to find an event _z_, simultaneous with part of _x_, which\nwholly precedes some event which wholly precedes _y_. Now this requires\nthat the number of events concerned should be infinite in any finite\nperiod of time. If this is to be the case in the world of one man's\nsense-data, and if each sense-datum is to have not less than a certain\nfinite temporal extension, it will be necessary to assume that we always\nhave an infinite number of sense-data simultaneous with any given\nsense-datum. Applying similar considerations to space, and assuming that\nsense-data are to have not less than a certain spatial extension, it\nwill be necessary to suppose that an infinite number of sense-data\noverlap spatially with any given sense-datum. This hypothesis is\npossible, if we suppose a single sense-datum, _e.g._ in sight, to be a\nfinite surface, enclosing other surfaces which are also single\nsense-data. But there are difficulties in such a hypothesis, and I do\nnot know whether these difficulties could be successfully met. If they\ncannot, we must do one of two things: either declare that the world of\none man's sense-data is not continuous, or else refuse to admit that\nthere is any lower limit to the duration and extension of a single\nsense-datum. I do not know what is the right course to adopt as regards\nthese alternatives. The logical analysis we have been considering\nprovides the apparatus for dealing with the various hypotheses, and the\nempirical decision between them is a problem for the psychologist.\n\n(3) We have now to consider the _logical_ answer to the alleged\ndifficulties of the mathematical theory of motion, or rather to the\npositive theory which is urged on the other side. The view urged\nexplicitly by Bergson, and implied in the doctrines of many\nphilosophers, is, that a motion is something indivisible, not validly\nanalysable into a series of states. This is part of a much more general\ndoctrine, which holds that analysis always falsifies, because the parts\nof a complex whole are different, as combined in that whole, from what\nthey would otherwise be. It is very difficult to state this doctrine in\nany form which has a precise meaning. Often arguments are used which\nhave no bearing whatever upon the question. It is urged, for example,\nthat when a man becomes a father, his nature is altered by the new\nrelation in which he finds himself, so that he is not strictly identical\nwith the man who was previously not a father. This may be true, but it\nis a causal psychological fact, not a logical fact. The doctrine would\nrequire that a man who is a father cannot be strictly identical with a\nman who is a son, because he is modified in one way by the relation of\nfatherhood and in another by that of sonship. In fact, we may give a\nprecise statement of the doctrine we are combating in the form: _There\ncan never be two facts concerning the same thing._ A fact concerning a\nthing always is or involves a relation to one or more entities; thus two\nfacts concerning the same thing would involve two relations of the same\nthing. But the doctrine in question holds that a thing is so modified by\nits relations that it cannot be the same in one relation as in another.\nHence, if this doctrine is true, there can never be more than one fact\nconcerning any one thing. I do not think the philosophers in question\nhave realised that this is the precise statement of the view they\nadvocate, because in this form the view is so contrary to plain truth\nthat its falsehood is evident as soon as it is stated. The discussion of\nthis question, however, involves so many logical subtleties, and is so\nbeset with difficulties, that I shall not pursue it further at present.\n\nWhen once the above general doctrine is rejected, it is obvious that,\nwhere there is change, there must be a succession of states. There\ncannot be change--and motion is only a particular case of change--unless\nthere is something different at one time from what there is at some\nother time. Change, therefore, must involve relations and complexity,\nand must demand analysis. So long as our analysis has only gone as far\nas other smaller changes, it is not complete; if it is to be complete,\nit must end with terms that are not changes, but are related by a\nrelation of earlier and later. In the case of changes which appear\ncontinuous, such as motions, it seems to be impossible to find anything\nother than change so long as we deal with finite periods of time,\nhowever short. We are thus driven back, by the logical necessities of\nthe case, to the conception of instants without duration, or at any rate\nwithout any duration which even the most delicate instruments can\nreveal. This conception, though it can be made to seem difficult, is\nreally easier than any other that the facts allow. It is a kind of\nlogical framework into which any tenable theory must fit--not\nnecessarily itself the statement of the crude facts, but a form in which\nstatements which are true of the crude facts can be made by a suitable\ninterpretation. The direct consideration of the crude facts of the\nphysical world has been undertaken in earlier lectures; in the present\nlecture, we have only been concerned to show that nothing in the crude\nfacts is inconsistent with the mathematical doctrine of continuity, or\ndemands a continuity of a radically different kind from that of\nmathematical motion.\n\n\n\n\nLECTURE VI\n\nTHE PROBLEM OF INFINITY CONSIDERED HISTORICALLY\n\n\nIt will be remembered that, when we enumerated the grounds upon which\nthe reality of the sensible world has been questioned, one of those\nmentioned was the supposed impossibility of infinity and continuity. In\nview of our earlier discussion of physics, it would seem that no\n_conclusive_ empirical evidence exists in favour of infinity or\ncontinuity in objects of sense or in matter. Nevertheless, the\nexplanation which assumes infinity and continuity remains incomparably\neasier and more natural, from a scientific point of view, than any\nother, and since Georg Cantor has shown that the supposed contradictions\nare illusory, there is no longer any reason to struggle after a finitist\nexplanation of the world.\n\nThe supposed difficulties of continuity all have their source in the\nfact that a continuous series must have an infinite number of terms, and\nare in fact difficulties concerning infinity. Hence, in freeing the\ninfinite from contradiction, we are at the same time showing the logical\npossibility of continuity as assumed in science.\n\nThe kind of way in which infinity has been used to discredit the world\nof sense may be illustrated by Kant's first two antinomies. In the\nfirst, the thesis states: \"The world has a beginning in time, and as\nregards space is enclosed within limits\"; the antithesis states: \"The\nworld has no beginning and no limits in space, but is infinite in\nrespect of both time and space.\" Kant professes to prove both these\npropositions, whereas, if what we have said on modern logic has any\ntruth, it must be impossible to prove either. In order, however, to\nrescue the world of sense, it is enough to destroy the proof of _one_ of\nthe two. For our present purpose, it is the proof that the world is\n_finite_ that interests us. Kant's argument as regards space here rests\nupon his argument as regards time. We need therefore only examine the\nargument as regards time. What he says is as follows:\n\n\"For let us assume that the world has no beginning as regards time, so\nthat up to every given instant an eternity has elapsed, and therefore an\ninfinite series of successive states of the things in the world has\npassed by. But the infinity of a series consists just in this, that it\ncan never be completed by successive synthesis. Therefore an infinite\npast world-series is impossible, and accordingly a beginning of the\nworld is a necessary condition of its existence; which was the first\nthing to be proved.\"\n\nMany different criticisms might be passed on this argument, but we will\ncontent ourselves with a bare minimum. To begin with, it is a mistake to\ndefine the infinity of a series as \"impossibility of completion by\nsuccessive synthesis.\" The notion of infinity, as we shall see in the\nnext lecture, is primarily a property of _classes_, and only\nderivatively applicable to series; classes which are infinite are given\nall at once by the defining property of their members, so that there is\nno question of \"completion\" or of \"successive synthesis.\" And the word\n\"synthesis,\" by suggesting the mental activity of synthesising,\nintroduces, more or less surreptitiously, that reference to mind by\nwhich all Kant's philosophy was infected. In the second place, when Kant\nsays that an infinite series can \"never\" be completed by successive\nsynthesis, all that he has even conceivably a right to say is that it\ncannot be completed _in a finite time_. Thus what he really proves is,\nat most, that if the world had no beginning, it must have already\nexisted for an infinite time. This, however, is a very poor conclusion,\nby no means suitable for his purposes. And with this result we might, if\nwe chose, take leave of the first antinomy.\n\nIt is worth while, however, to consider how Kant came to make such an\nelementary blunder. What happened in his imagination was obviously\nsomething like this: Starting from the present and going backwards in\ntime, we have, if the world had no beginning, an infinite series of\nevents. As we see from the word \"synthesis,\" he imagined a mind trying\nto grasp these successively, _in the reverse order_ to that in which\nthey had occurred, _i.e._ going from the present backwards. _This_\nseries is obviously one which has no end. But the series of events up to\nthe present has an end, since it ends with the present. Owing to the\ninveterate subjectivism of his mental habits, he failed to notice that\nhe had reversed the sense of the series by substituting backward\nsynthesis for forward happening, and thus he supposed that it was\nnecessary to identify the mental series, which had no end, with the\nphysical series, which had an end but no beginning. It was this mistake,\nI think, which, operating unconsciously, led him to attribute validity\nto a singularly flimsy piece of fallacious reasoning.\n\nThe second antinomy illustrates the dependence of the problem of\ncontinuity upon that of infinity. The thesis states: \"Every complex\nsubstance in the world consists of simple parts, and there exists\neverywhere nothing but the simple or what is composed of it.\" The\nantithesis states: \"No complex thing in the world consists of simple\nparts, and everywhere in it there exists nothing simple.\" Here, as\nbefore, the proofs of both thesis and antithesis are open to criticism,\nbut for the purpose of vindicating physics and the world of sense it is\nenough to find a fallacy in _one_ of the proofs. We will choose for this\npurpose the proof of the antithesis, which begins as follows:\n\n\"Assume that a complex thing (as substance) consists of simple parts.\nSince all external relation, and therefore all composition out of\nsubstances, is only possible in space, the space occupied by a complex\nthing must consist of as many parts as the thing consists of. Now space\ndoes not consist of simple parts, but of spaces.\"\n\nThe rest of his argument need not concern us, for the nerve of the proof\nlies in the one statement: \"Space does not consist of simple parts, but\nof spaces.\" This is like Bergson's objection to \"the absurd proposition\nthat motion is made up of immobilities.\" Kant does not tell us why he\nholds that a space must consist of spaces rather than of simple parts.\nGeometry regards space as made up of points, which are simple; and\nalthough, as we have seen, this view is not scientifically or logically\n_necessary_, it remains _prima facie_ possible, and its mere possibility\nis enough to vitiate Kant's argument. For, if his proof of the thesis of\nthe antinomy were valid, and if the antithesis could only be avoided by\nassuming points, then the antinomy itself would afford a conclusive\nreason in favour of points. Why, then, did Kant think it impossible that\nspace should be composed of points?\n\nI think two considerations probably influenced him. In the first place,\nthe essential thing about space is spatial order, and mere points, by\nthemselves, will not account for spatial order. It is obvious that his\nargument assumes absolute space; but it is spatial _relations_ that are\nalone important, and they cannot be reduced to points. This ground for\nhis view depends, therefore, upon his ignorance of the logical theory of\norder and his oscillations between absolute and relative space. But\nthere is also another ground for his opinion, which is more relevant to\nour present topic. This is the ground derived from infinite\ndivisibility. A space may be halved, and then halved again, and so on\n_ad infinitum_, and at every stage of the process the parts are still\nspaces, not points. In order to reach points by such a method, it would\nbe necessary to come to the end of an unending process, which is\nimpossible. But just as an infinite class can be given all at once by\nits defining concept, though it cannot be reached by successive\nenumeration, so an infinite set of points can be given all at once as\nmaking up a line or area or volume, though they can never be reached by\nthe process of successive division. Thus the infinite divisibility of\nspace gives no ground for denying that space is composed of points. Kant\ndoes not give his grounds for this denial, and we can therefore only\nconjecture what they were. But the above two grounds, which we have seen\nto be fallacious, seem sufficient to account for his opinion, and we may\ntherefore conclude that the antithesis of the second antinomy is\nunproved.\n\nThe above illustration of Kant's antinomies has only been introduced in\norder to show the relevance of the problem of infinity to the problem of\nthe reality of objects of sense. In the remainder of the present\nlecture, I wish to state and explain the problem of infinity, to show\nhow it arose, and to show the irrelevance of all the solutions proposed\nby philosophers. In the following lecture, I shall try to explain the\ntrue solution, which has been discovered by the mathematicians, but\nnevertheless belongs essentially to philosophy. The solution is\ndefinitive, in the sense that it entirely satisfies and convinces all\nwho study it carefully. For over two thousand years the human intellect\nwas baffled by the problem; its many failures and its ultimate success\nmake this problem peculiarly apt for the illustration of method.\n\nThe problem appears to have first arisen in some such way as the\nfollowing.[22] Pythagoras and his followers, who were interested, like\nDescartes, in the application of number to geometry, adopted in that\nscience more arithmetical methods than those with which Euclid has made\nus familiar. They, or their contemporaries the atomists, believed,\napparently, that space is composed of indivisible points, while time is\ncomposed of indivisible instants.[23] This belief would not, by itself,\nhave raised the difficulties which they encountered, but it was\npresumably accompanied by another belief, that the number of points in\nany finite area or of instants in any finite period must be finite. I do\nnot suppose that this latter belief was a conscious one, because\nprobably no other possibility had occurred to them. But the belief\nnevertheless operated, and very soon brought them into conflict with\nfacts which they themselves discovered. Before explaining how this\noccurred, however, it is necessary to say one word in explanation of the\nphrase \"finite number.\" The _exact_ explanation is a matter for our next\nlecture; for the present, it must suffice to say that I mean 0 and 1 and\n2 and 3 and so on, for ever--in other words, any number that can be\nobtained by successively adding ones. This includes all the numbers that\ncan be expressed by means of our ordinary numerals, and since such\nnumbers can be made greater and greater, without ever reaching an\nunsurpassable maximum, it is easy to suppose that there are no other\nnumbers. But this supposition, natural as it is, is mistaken.\n\n  [22] In what concerns the early Greek philosophers, my knowledge is\n  largely derived from Burnet's valuable work, _Early Greek Philosophy_\n  (2nd ed., London, 1908). I have also been greatly assisted by Mr D. S.\n  Robertson of Trinity College, who has supplied the deficiencies of my\n  knowledge of Greek, and brought important references to my notice.\n\n  [23] _Cf._ Aristotle, _Metaphysics_, M. 6, 1080b, 18 _sqq._, and\n  1083b, 8 _sqq._\n\nWhether the Pythagoreans themselves believed space and time to be\ncomposed of indivisible points and instants is a debatable question.[24]\nIt would seem that the distinction between space and matter had not yet\nbeen clearly made, and that therefore, when an atomistic view is\nexpressed, it is difficult to decide whether particles of matter or\npoints of space are intended. There is an interesting passage[25] in\nAristotle's _Physics_,[26] where he says:\n\n\"The Pythagoreans all maintained the existence of the void, and said\nthat it enters into the heaven itself from the boundless breath,\ninasmuch as the heaven breathes in the void also; and the void\ndifferentiates natures, as if it were a sort of separation of\nconsecutives, and as if it were their differentiation; and that this\nalso is what is first in numbers, for it is the void which\ndifferentiates them.\"\n\n  [24] There is some reason to think that the Pythagoreans distinguished\n  between discrete and continuous quantity. G. J. Allman, in his _Greek\n  Geometry from Thales to Euclid_, says (p. 23): \"The Pythagoreans made\n  a fourfold division of mathematical science, attributing one of its\n  parts to the how many, ~to poson~, and the other to the how much, ~to\n  pelikon~; and they assigned to each of these parts a twofold division.\n  For they said that discrete quantity, or the _how many_, either\n  subsists by itself or must be considered with relation to some other;\n  but that continued quantity, or the _how much_, is either stable or in\n  motion. Hence they affirmed that arithmetic contemplates that discrete\n  quantity which subsists by itself, but music that which is related to\n  another; and that geometry considers continued quantity so far as it\n  is immovable; but astronomy (~ten sphairiken~) contemplates continued\n  quantity so far as it is of a self-motive nature. (Proclus, ed.\n  Friedlein, p. 35. As to the distinction between ~to pelikon~,\n  continuous, and ~to poson~, discrete quantity, see Iambl., _in\n  Nicomachi Geraseni Arithmeticam introductionem_, ed. Tennulius,\n  p. 148.)\" _Cf._ p. 48.\n\n  [25] Referred to by Burnet, _op. cit._, p. 120.\n\n  [26] iv., 6. 213b, 22; H. Ritter and L. Preller, _Historia Philosophiae\n  Graecae_, 8th ed., Gotha, 1898, p. 75 (this work will be referred to in\n  future as \"R. P.\").\n\nThis seems to imply that they regarded matter as consisting of atoms\nwith empty space in between. But if so, they must have thought space\ncould be studied by only paying attention to the atoms, for otherwise it\nwould be hard to account for their arithmetical methods in geometry, or\nfor their statement that \"things are numbers.\"\n\nThe difficulty which beset the Pythagoreans in their attempts to apply\nnumbers arose through their discovery of incommensurables, and this, in\nturn, arose as follows. Pythagoras, as we all learnt in youth,\ndiscovered the proposition that the sum of the squares on the sides of a\nright-angled triangle is equal to the square on the hypotenuse. It is\nsaid that he sacrificed an ox when he discovered this theorem; if so,\nthe ox was the first martyr to science. But the theorem, though it has\nremained his chief claim to immortality, was soon found to have a\nconsequence fatal to his whole philosophy. Consider the case of a\nright-angled triangle whose two sides are equal, such a triangle as is\nformed by two sides of a square and a diagonal. Here, in virtue of the\ntheorem, the square on the diagonal is double of the square on either of\nthe sides. But Pythagoras or his early followers easily proved that the\nsquare of one whole number cannot be double of the square of\nanother.[27] Thus the length of the side and the length of the diagonal\nare incommensurable; that is to say, however small a unit of length you\ntake, if it is contained an exact number of times in the side, it is not\ncontained any exact number of times in the diagonal, and _vice versa_.\n\n  [27] The Pythagorean proof is roughly as follows. If possible, let the\n  ratio of the diagonal to the side of a square be _m_/_n_, where _m_\n  and _n_ are whole numbers having no common factor. Then we must have\n  _m_2 = 2_n_2. Now the square of an odd number is odd, but _m_2, being\n  equal to 2_n_2, is even. Hence _m_ must be even. But the square of an\n  even number divides by 4, therefore _n_2, which is half of _m_2, must\n  be even. Therefore _n_ must be even. But, since _m_ is even, and _m_\n  and _n_ have no common factor, _n_ must be odd. Thus _n_ must be both\n  odd and even, which is impossible; and therefore the diagonal and the\n  side cannot have a rational ratio.\n\nNow this fact might have been assimilated by some philosophies without\nany great difficulty, but to the philosophy of Pythagoras it was\nabsolutely fatal. Pythagoras held that number is the constitutive\nessence of all things, yet no two numbers could express the ratio of the\nside of a square to the diagonal. It would seem probable that we may\nexpand his difficulty, without departing from his thought, by assuming\nthat he regarded the length of a line as determined by the number of\natoms contained in it--a line two inches long would contain twice as\nmany atoms as a line one inch long, and so on. But if this were the\ntruth, then there must be a definite numerical ratio between any two\nfinite lengths, because it was supposed that the number of atoms in\neach, however large, must be finite. Here there was an insoluble\ncontradiction. The Pythagoreans, it is said, resolved to keep the\nexistence of incommensurables a profound secret, revealed only to a few\nof the supreme heads of the sect; and one of their number, Hippasos of\nMetapontion, is even said to have been shipwrecked at sea for impiously\ndisclosing the terrible discovery to their enemies. It must be\nremembered that Pythagoras was the founder of a new religion as well as\nthe teacher of a new science: if the science came to be doubted, the\ndisciples might fall into sin, and perhaps even eat beans, which\naccording to Pythagoras is as bad as eating parents' bones.\n\nThe problem first raised by the discovery of incommensurables proved, as\ntime went on, to be one of the most severe and at the same time most\nfar-reaching problems that have confronted the human intellect in its\nendeavour to understand the world. It showed at once that numerical\nmeasurement of lengths, if it was to be made accurate, must require an\narithmetic more advanced and more difficult than any that the ancients\npossessed. They therefore set to work to reconstruct geometry on a basis\nwhich did not assume the universal possibility of numerical\nmeasurement--a reconstruction which, as may be seen in Euclid, they\neffected with extraordinary skill and with great logical acumen. The\nmoderns, under the influence of Cartesian geometry, have reasserted the\nuniversal possibility of numerical measurement, extending arithmetic,\npartly for that purpose, so as to include what are called \"irrational\"\nnumbers, which give the ratios of incommensurable lengths. But although\nirrational numbers have long been used without a qualm, it is only in\nquite recent years that logically satisfactory definitions of them have\nbeen given. With these definitions, the first and most obvious form of\nthe difficulty which confronted the Pythagoreans has been solved; but\nother forms of the difficulty remain to be considered, and it is these\nthat introduce us to the problem of infinity in its pure form.\n\nWe saw that, accepting the view that a length is composed of points, the\nexistence of incommensurables proves that every finite length must\ncontain an infinite number of points. In other words, if we were to take\naway points one by one, we should never have taken away all the points,\nhowever long we continued the process. The number of points, therefore,\ncannot be _counted_, for counting is a process which enumerates things\none by one. The property of being unable to be counted is characteristic\nof infinite collections, and is a source of many of their paradoxical\nqualities. So paradoxical are these qualities that until our own day\nthey were thought to constitute logical contradictions. A long line of\nphilosophers, from Zeno[28] to M. Bergson, have based much of their\nmetaphysics upon the supposed impossibility of infinite collections.\nBroadly speaking, the difficulties were stated by Zeno, and nothing\nmaterial was added until we reach Bolzano's _Paradoxien des\nUnendlichen_, a little work written in 1847-8, and published\nposthumously in 1851. Intervening attempts to deal with the problem are\nfutile and negligible. The definitive solution of the difficulties is\ndue, not to Bolzano, but to Georg Cantor, whose work on this subject\nfirst appeared in 1882.\n\n  [28] In regard to Zeno and the Pythagoreans, I have derived much\n  valuable information and criticism from Mr P. E. B. Jourdain.\n\nIn order to understand Zeno, and to realise how little modern orthodox\nmetaphysics has added to the achievements of the Greeks, we must\nconsider for a moment his master Parmenides, in whose interest the\nparadoxes were invented.[29] Parmenides expounded his views in a poem\ndivided into two parts, called \"the way of truth\" and \"the way of\nopinion\"--like Mr Bradley's \"Appearance\" and \"Reality,\" except that\nParmenides tells us first about reality and then about appearance. \"The\nway of opinion,\" in his philosophy, is, broadly speaking,\nPythagoreanism; it begins with a warning: \"Here I shall close my\ntrustworthy speech and thought about the truth. Henceforward learn the\nopinions of mortals, giving ear to the deceptive ordering of my words.\"\nWhat has gone before has been revealed by a goddess, who tells him what\nreally _is_. Reality, she says, is uncreated, indestructible,\nunchanging, indivisible; it is \"immovable in the bonds of mighty chains,\nwithout beginning and without end; since coming into being and passing\naway have been driven afar, and true belief has cast them away.\" The\nfundamental principle of his inquiry is stated in a sentence which would\nnot be out of place in Hegel:[30] \"Thou canst not know what is not--that\nis impossible--nor utter it; for it is the same thing that can be\nthought and that can be.\" And again: \"It needs must be that what can be\nthought and spoken of is; for it is possible for it to be, and it is not\npossible for what is nothing to be.\" The impossibility of change follows\nfrom this principle; for what is past can be spoken of, and therefore,\nby the principle, still is.\n\n  [29] So Plato makes Zeno say in the _Parmenides_, apropos of his\n  philosophy as a whole; and all internal and external evidence supports\n  this view.\n\n  [30] \"With Parmenides,\" Hegel says, \"philosophising proper began.\"\n  _Werke_ (edition of 1840), vol. xiii. p. 274.\n\nThe great conception of a reality behind the passing illusions of sense,\na reality one, indivisible, and unchanging, was thus introduced into\nWestern philosophy by Parmenides, not, it would seem, for mystical or\nreligious reasons, but on the basis of a logical argument as to the\nimpossibility of not-being. All the great metaphysical systems--notably\nthose of Plato, Spinoza, and Hegel--are the outcome of this fundamental\nidea. It is difficult to disentangle the truth and the error in this\nview. The contention that time is unreal and that the world of sense is\nillusory must, I think, be regarded as based upon fallacious reasoning.\nNevertheless, there is some sense--easier to feel than to state--in\nwhich time is an unimportant and superficial characteristic of reality.\nPast and future must be acknowledged to be as real as the present, and a\ncertain emancipation from slavery to time is essential to philosophic\nthought. The importance of time is rather practical than theoretical,\nrather in relation to our desires than in relation to truth. A truer\nimage of the world, I think, is obtained by picturing things as entering\ninto the stream of time from an eternal world outside, than from a view\nwhich regards time as the devouring tyrant of all that is. Both in\nthought and in feeling, to realise the unimportance of time is the gate\nof wisdom. But unimportance is not unreality; and therefore what we\nshall have to say about Zeno's arguments in support of Parmenides must\nbe mainly critical.\n\nThe relation of Zeno to Parmenides is explained by Plato[31] in the\ndialogue in which Socrates, as a young man, learns logical acumen and\nphilosophic disinterestedness from their dialectic. I quote from\nJowett's translation:\n\n\"I see, Parmenides, said Socrates, that Zeno is your second self in his\nwritings too; he puts what you say in another way, and would fain\ndeceive us into believing that he is telling us what is new. For you, in\nyour poems, say All is one, and of this you adduce excellent proofs; and\nhe on the other hand says There is no Many; and on behalf of this he\noffers overwhelming evidence. To deceive the world, as you have done, by\nsaying the same thing in different ways, one of you affirming the one,\nand the other denying the many, is a strain of art beyond the reach of\nmost of us.\n\n\"Yes, Socrates, said Zeno. But although you are as keen as a Spartan\nhound in pursuing the track, you do not quite apprehend the true motive\nof the composition, which is not really such an ambitious work as you\nimagine; for what you speak of was an accident; I had no serious\nintention of deceiving the world. The truth is, that these writings of\nmine were meant to protect the arguments of Parmenides against those who\nscoff at him and show the many ridiculous and contradictory results\nwhich they suppose to follow from the affirmation of the one. My answer\nis an address to the partisans of the many, whose attack I return with\ninterest by retorting upon them that their hypothesis of the being of\nthe many if carried out appears in a still more ridiculous light than\nthe hypothesis of the being of the one.\"\n\n  [31] _Parmenides_, 128 A-D.\n\nZeno's four arguments against motion were intended to exhibit the\ncontradictions that result from supposing that there is such a thing as\nchange, and thus to support the Parmenidean doctrine that reality is\nunchanging.[32] Unfortunately, we only know his arguments through\nAristotle,[33] who stated them in order to refute them. Those\nphilosophers in the present day who have had their doctrines stated by\nopponents will realise that a just or adequate presentation of Zeno's\nposition is hardly to be expected from Aristotle; but by some care in\ninterpretation it seems possible to reconstruct the so-called \"sophisms\"\nwhich have been \"refuted\" by every tyro from that day to this.\n\n  [32] This interpretation is combated by Milhaud, _Les\n  philosophes-geometres de la Grece_, p. 140 n., but his reasons do not\n  seem to me convincing. All the interpretations in what follows are\n  open to question, but all have the support of reputable authorities.\n\n  [33] _Physics_, vi. 9. 2396 (R.P. 136-139).\n\nZeno's arguments would seem to be \"ad hominem\"; that is to say, they\nseem to assume premisses granted by his opponents, and to show that,\ngranting these premisses, it is possible to deduce consequences which\nhis opponents must deny. In order to decide whether they are valid\narguments or \"sophisms,\" it is necessary to guess at the tacit\npremisses, and to decide who was the \"homo\" at whom they were aimed.\nSome maintain that they were aimed at the Pythagoreans,[34] while others\nhave held that they were intended to refute the atomists.[35] M.\nEvellin, on the contrary, holds that they constitute a refutation of\ninfinite divisibility,[36] while M. G. Noel, in the interests of Hegel,\nmaintains that the first two arguments refute infinite divisibility,\nwhile the next two refute indivisibles.[37] Amid such a bewildering\nvariety of interpretations, we can at least not complain of any\nrestrictions on our liberty of choice.\n\n  [34] _Cf._ Gaston Milhaud, _Les philosophes-geometres de la Grece_,\n  p. 140 n.; Paul Tannery, _Pour l'histoire de la science hellene_,\n  p. 249; Burnet, _op. cit._, p. 362.\n\n  [35] _Cf._ R. K. Gaye, \"On Aristotle, _Physics_, Z ix.\" _Journal of\n  Philology_, vol. xxxi., esp. p. 111. Also Moritz Cantor, _Vorlesungen\n  ueber Geschichte der Mathematik_, 1st ed., vol. i., 1880, p. 168, who,\n  however, subsequently adopted Paul Tannery's opinion, _Vorlesungen_,\n  3rd ed. (vol. i. p. 200).\n\n  [36] \"Le mouvement et les partisans des indivisibles,\" _Revue de\n  Metaphysique et de Morale_, vol. i. pp. 382-395.\n\n  [37] \"Le mouvement et les arguments de Zenon d'Elee,\" _Revue de\n  Metaphysique et de Morale_, vol. i. pp. 107-125.\n\nThe historical questions raised by the above-mentioned discussions are\nno doubt largely insoluble, owing to the very scanty material from which\nour evidence is derived. The points which seem fairly clear are the\nfollowing: (1) That, in spite of MM. Milhaud and Paul Tannery, Zeno is\nanxious to prove that motion is really impossible, and that he desires\nto prove this because he follows Parmenides in denying plurality;[38]\n(2) that the third and fourth arguments proceed on the hypothesis of\nindivisibles, a hypothesis which, whether adopted by the Pythagoreans or\nnot, was certainly much advocated, as may be seen from the treatise _On\nIndivisible Lines_ attributed to Aristotle. As regards the first two\narguments, they would seem to be valid on the hypothesis of\nindivisibles, and also, without this hypothesis, to be such as would be\nvalid if the traditional contradictions in infinite numbers were\ninsoluble, which they are not.\n\n  [38] _Cf._ M. Brochard, \"Les pretendus sophismes de Zenon d'Elee,\"\n  _Revue de Metaphysique et de Morale_, vol. i. pp. 209-215.\n\nWe may conclude, therefore, that Zeno's polemic is directed against the\nview that space and time consist of points and instants; and that as\nagainst the view that a finite stretch of space or time consists of a\nfinite number of points and instants, his arguments are not sophisms,\nbut perfectly valid.\n\nThe conclusion which Zeno wishes us to draw is that plurality is a\ndelusion, and spaces and times are really indivisible. The other\nconclusion which is possible, namely, that the number of points and\ninstants is infinite, was not tenable so long as the infinite was\ninfected with contradictions. In a fragment which is not one of the four\nfamous arguments against motion, Zeno says:\n\n\"If things are a many, they must be just as many as they are, and\nneither more nor less. Now, if they are as many as they are, they will\nbe finite in number.\n\n\"If things are a many, they will be infinite in number; for there will\nalways be other things between them, and others again between these. And\nso things are infinite in number.\"[39]\n\n  [39] Simplicius, _Phys._, 140, 28 D (R.P. 133); Burnet, _op. cit._,\n  pp. 364-365.\n\nThis argument attempts to prove that, if there are many things, the\nnumber of them must be both finite and infinite, which is impossible;\nhence we are to conclude that there is only one thing. But the weak\npoint in the argument is the phrase: \"If they are just as many as they\nare, they will be finite in number.\" This phrase is not very clear, but\nit is plain that it assumes the impossibility of definite infinite\nnumbers. Without this assumption, which is now known to be false, the\narguments of Zeno, though they suffice (on certain very reasonable\nassumptions) to dispel the hypothesis of finite indivisibles, do not\nsuffice to prove that motion and change and plurality are impossible.\nThey are not, however, on any view, mere foolish quibbles: they are\nserious arguments, raising difficulties which it has taken two thousand\nyears to answer, and which even now are fatal to the teachings of most\nphilosophers.\n\nThe first of Zeno's arguments is the argument of the race-course, which\nis paraphrased by Burnet as follows:[40]\n\n\"You cannot get to the end of a race-course. You cannot traverse an\ninfinite number of points in a finite time. You must traverse the half\nof any given distance before you traverse the whole, and the half of\nthat again before you can traverse it. This goes on _ad infinitum_, so\nthat there are an infinite number of points in any given space, and you\ncannot touch an infinite number one by one in a finite time.\"[41]\n\n  [40] _Op. cit._, p. 367.\n\n  [41] Aristotle's words are: \"The first is the one on the non-existence\n  of motion on the ground that what is moved must always attain the\n  middle point sooner than the end-point, on which we gave our opinion\n  in the earlier part of our discourse.\" _Phys._, vi. 9. 939B (R.P.\n  136). Aristotle seems to refer to _Phys._, vi. 2. 223AB [R.P. 136A]:\n  \"All space is continuous, for time and space are divided into the same\n  and equal divisions.... Wherefore also Zeno's argument is fallacious,\n  that it is impossible to go through an infinite collection or to touch\n  an infinite collection one by one in a finite time. For there are two\n  senses in which the term 'infinite' is applied both to length and to\n  time, and in fact to all continuous things, either in regard to\n  divisibility, or in regard to the ends. Now it is not possible to\n  touch things infinite in regard to number in a finite time, but it is\n  possible to touch things infinite in regard to divisibility: for time\n  itself also is infinite in this sense. So that in fact we go through\n  an infinite, [space] in an infinite [time] and not in a finite [time],\n  and we touch infinite things with infinite things, not with finite\n  things.\" Philoponus, a sixth-century commentator (R.P. 136A, _Exc.\n  Paris Philop. in Arist. Phys._, 803, 2. Vit.), gives the following\n  illustration: \"For if a thing were moved the space of a cubit in one\n  hour, since in every space there are an infinite number of points, the\n  thing moved must needs touch all the points of the space: it will then\n  go through an infinite collection in a finite time, which is\n  impossible.\"\n\nZeno appeals here, in the first place, to the fact that any distance,\nhowever small, can be halved. From this it follows, of course, that\nthere must be an infinite number of points in a line. But, Aristotle\nrepresents him as arguing, you cannot touch an infinite number of points\n_one by one_ in a finite time. The words \"one by one\" are important. (1)\nIf _all_ the points touched are concerned, then, though you pass through\nthem continuously, you do not touch them \"one by one.\" That is to say,\nafter touching one, there is not another which you touch next: no two\npoints are next each other, but between any two there are always an\ninfinite number of others, which cannot be enumerated one by one. (2)\nIf, on the other hand, only the successive middle points are concerned,\nobtained by always halving what remains of the course, then the points\nare reached one by one, and, though they are infinite in number, they\nare in fact all reached in a finite time. His argument to the contrary\nmay be supposed to appeal to the view that a finite time must consist of\na finite number of instants, in which case what he says would be\nperfectly true on the assumption that the possibility of continued\ndichotomy is undeniable. If, on the other hand, we suppose the argument\ndirected against the partisans of infinite divisibility, we must suppose\nit to proceed as follows:[42] \"The points given by successive halving of\nthe distances still to be traversed are infinite in number, and are\nreached in succession, each being reached a finite time later than its\npredecessor; but the sum of an infinite number of finite times must be\ninfinite, and therefore the process will never be completed.\" It is very\npossible that this is historically the right interpretation, but in this\nform the argument is invalid. If half the course takes half a minute,\nand the next quarter takes a quarter of a minute, and so on, the whole\ncourse will take a minute. The apparent force of the argument, on this\ninterpretation, lies solely in the mistaken supposition that there\ncannot be anything beyond the whole of an infinite series, which can be\nseen to be false by observing that 1 is beyond the whole of the infinite\nseries 1/2, 3/4, 7/8, 15/16, ...\n\n  [42] _Cf._ Mr C. D. Broad, \"Note on Achilles and the Tortoise,\"\n  _Mind_, N.S., vol. xxii. pp. 318-9.\n\nThe second of Zeno's arguments is the one concerning Achilles and the\ntortoise, which has achieved more notoriety than the others. It is\nparaphrased by Burnet as follows:[43]\n\n\"Achilles will never overtake the tortoise. He must first reach the\nplace from which the tortoise started. By that time the tortoise will\nhave got some way ahead. Achilles must then make up that, and again the\ntortoise will be ahead. He is always coming nearer, but he never makes\nup to it.\"[44]\n\n  [43] _Op. cit._\n\n  [44] Aristotle's words are: \"The second is the so-called Achilles. It\n  consists in this, that the slower will never be overtaken in its\n  course by the quickest, for the pursuer must always come first to the\n  point from which the pursued has just departed, so that the slower\n  must necessarily be always still more or less in advance.\" _Phys._,\n  vi. 9. 239B (R.P. 137).\n\nThis argument is essentially the same as the previous one. It shows\nthat, if Achilles ever overtakes the tortoise, it must be after an\ninfinite number of instants have elapsed since he started. This is in\nfact true; but the view that an infinite number of instants make up an\ninfinitely long time is not true, and therefore the conclusion that\nAchilles will never overtake the tortoise does not follow.\n\nThe third argument,[45] that of the arrow, is very interesting. The text\nhas been questioned. Burnet accepts the alterations of Zeller, and\nparaphrases thus:\n\n\"The arrow in flight is at rest. For, if everything is at rest when it\noccupies a space equal to itself, and what is in flight at any given\nmoment always occupies a space equal to itself, it cannot move.\"\n\n  [45] _Phys._, vi. 9. 239B (R.P. 138).\n\nBut according to Prantl, the literal translation of the unemended text\nof Aristotle's statement of the argument is as follows: \"If everything,\nwhen it is behaving in a uniform manner, is continually either moving or\nat rest, but what is moving is always in the _now_, then the moving\narrow is motionless.\" This form of the argument brings out its force\nmore clearly than Burnet's paraphrase.\n\nHere, if not in the first two arguments, the view that a finite part of\ntime consists of a finite series of successive instants seems to be\nassumed; at any rate the plausibility of the argument seems to depend\nupon supposing that there are consecutive instants. Throughout an\ninstant, it is said, a moving body is where it is: it cannot move during\nthe instant, for that would require that the instant should have parts.\nThus, suppose we consider a period consisting of a thousand instants,\nand suppose the arrow is in flight throughout this period. At each of\nthe thousand instants, the arrow is where it is, though at the next\ninstant it is somewhere else. It is never moving, but in some miraculous\nway the change of position has to occur _between_ the instants, that is\nto say, not at any time whatever. This is what M. Bergson calls the\ncinematographic representation of reality. The more the difficulty is\nmeditated, the more real it becomes. The solution lies in the theory of\ncontinuous series: we find it hard to avoid supposing that, when the\narrow is in flight, there is a _next_ position occupied at the _next_\nmoment; but in fact there is no next position and no next moment, and\nwhen once this is imaginatively realised, the difficulty is seen to\ndisappear.\n\nThe fourth and last of Zeno's arguments is[46] the argument of the\nstadium.\n\n  [46] _Phys._, vi. 9. 239B (R.P. 139).\n\nThe argument as stated by Burnet is as follows:\n\n  First Position.  Second Position.\n      A ....           A  ....\n      B ....           B ....\n      C ....           C   ....\n\n\"Half the time may be equal to double the time. Let us suppose three\nrows of bodies, one of which (A) is at rest while the other two (B, C)\nare moving with equal velocity in opposite directions. By the time they\nare all in the same part of the course, B will have passed twice as many\nof the bodies in C as in A. Therefore the time which it takes to pass C\nis twice as long as the time it takes to pass A. But the time which B\nand C take to reach the position of A is the same. Therefore double the\ntime is equal to the half.\"\n\nGaye[47] devoted an interesting article to the interpretation of this\nargument. His translation of Aristotle's statement is as follows:\n\n\"The fourth argument is that concerning the two rows of bodies, each row\nbeing composed of an equal number of bodies of equal size, passing each\nother on a race-course as they proceed with equal velocity in opposite\ndirections, the one row originally occupying the space between the goal\nand the middle point of the course, and the other that between the\nmiddle point and the starting-post. This, he thinks, involves the\nconclusion that half a given time is equal to double the time. The\nfallacy of the reasoning lies in the assumption that a body occupies an\nequal time in passing with equal velocity a body that is in motion and a\nbody of equal size that is at rest, an assumption which is false. For\ninstance (so runs the argument), let A A ... be the stationary bodies of\nequal size, B B ... the bodies, equal in number and in size to A A ...,\noriginally occupying the half of the course from the starting-post to\nthe middle of the A's, and C C ... those originally occupying the other\nhalf from the goal to the middle of the A's, equal in number, size, and\nvelocity, to B B ... Then three consequences follow. First, as the B's\nand C's pass one another, the first B reaches the last C at the same\nmoment at which the first C reaches the last B. Secondly, at this moment\nthe first C has passed all the A's, whereas the first B has passed only\nhalf the A's and has consequently occupied only half the time occupied\nby the first C, since each of the two occupies an equal time in passing\neach A. Thirdly, at the same moment all the B's have passed all the C's:\nfor the first C and the first B will simultaneously reach the opposite\nends of the course, since (so says Zeno) the time occupied by the first\nC in passing each of the B's is equal to that occupied by it in passing\neach of the A's, because an equal time is occupied by both the first B\nand the first C in passing all the A's. This is the argument: but it\npresupposes the aforesaid fallacious assumption.\"\n\n  [47] _Loc. cit._\n\n  First Position.     Second Position.\n  B  B' B\"              B  B' B\"\n  .  .  .               .  .  .\n\n  A  A' A\"           A  A' A\"\n  .  .  .            .  .  .\n\n  C  C' C\"        C  C' C\"\n  .  .  .         .  .  .\n\nThis argument is not quite easy to follow, and it is only valid as\nagainst the assumption that a finite time consists of a finite number of\ninstants. We may re-state it in different language. Let us suppose three\ndrill-sergeants, A, A', and A\", standing in a row, while the two files\nof soldiers march past them in opposite directions. At the first moment\nwhich we consider, the three men B, B', B\" in one row, and the three men\nC, C', C\" in the other row, are respectively opposite to A, A', and A\".\nAt the very next moment, each row has moved on, and now B and C\" are\nopposite A'. Thus B and C\" are opposite each other. When, then, did B\npass C'? It must have been somewhere between the two moments which we\nsupposed consecutive, and therefore the two moments cannot really have\nbeen consecutive. It follows that there must be other moments between\nany two given moments, and therefore that there must be an infinite\nnumber of moments in any given interval of time.\n\nThe above difficulty, that B must have passed C' at some time between\ntwo consecutive moments, is a genuine one, but is not precisely the\ndifficulty raised by Zeno. What Zeno professes to prove is that \"half of\na given time is equal to double that time.\" The most intelligible\nexplanation of the argument known to me is that of Gaye.[48] Since,\nhowever, his explanation is not easy to set forth shortly, I will\nre-state what seems to me to be the logical essence of Zeno's\ncontention. If we suppose that time consists of a series of consecutive\ninstants, and that motion consists in passing through a series of\nconsecutive points, then the fastest possible motion is one which, at\neach instant, is at a point consecutive to that at which it was at the\nprevious instant. Any slower motion must be one which has intervals of\nrest interspersed, and any faster motion must wholly omit some points.\nAll this is evident from the fact that we cannot have more than one\nevent for each instant. But now, in the case of our A's and B's and C's,\nB is opposite a fresh A every instant, and therefore the number of A's\npassed gives the number of instants since the beginning of the motion.\nBut during the motion B has passed twice as many C's, and yet cannot\nhave passed more than one each instant. Hence the number of instants\nsince the motion began is twice the number of A's passed, though we\npreviously found it was equal to this number. From this result, Zeno's\nconclusion follows.\n\n  [48] _Loc. cit._, p. 105.\n\nZeno's arguments, in some form, have afforded grounds for almost all the\ntheories of space and time and infinity which have been constructed from\nhis day to our own. We have seen that all his arguments are valid (with\ncertain reasonable hypotheses) on the assumption that finite spaces and\ntimes consist of a finite number of points and instants, and that the\nthird and fourth almost certainly in fact proceeded on this assumption,\nwhile the first and second, which were perhaps intended to refute the\nopposite assumption, were in that case fallacious. We may therefore\nescape from his paradoxes either by maintaining that, though space and\ntime do consist of points and instants, the number of them in any finite\ninterval is infinite; or by denying that space and time consist of\npoints and instants at all; or lastly, by denying the reality of space\nand time altogether. It would seem that Zeno himself, as a supporter of\nParmenides, drew the last of these three possible deductions, at any\nrate in regard to time. In this a very large number of philosophers have\nfollowed him. Many others, like M. Bergson, have preferred to deny that\nspace and time consist of points and instants. Either of these solutions\nwill meet the difficulties in the form in which Zeno raised them. But,\nas we saw, the difficulties can also be met if infinite numbers are\nadmissible. And on grounds which are independent of space and time,\ninfinite numbers, and series in which no two terms are consecutive, must\nin any case be admitted. Consider, for example, all the fractions less\nthan 1, arranged in order of magnitude. Between any two of them, there\nare others, for example, the arithmetical mean of the two. Thus no two\nfractions are consecutive, and the total number of them is infinite. It\nwill be found that much of what Zeno says as regards the series of\npoints on a line can be equally well applied to the series of fractions.\nAnd we cannot deny that there are fractions, so that two of the above\nways of escape are closed to us. It follows that, if we are to solve the\nwhole class of difficulties derivable from Zeno's by analogy, we must\ndiscover some tenable theory of infinite numbers. What, then, are the\ndifficulties which, until the last thirty years, led philosophers to the\nbelief that infinite numbers are impossible?\n\nThe difficulties of infinity are of two kinds, of which the first may be\ncalled sham, while the others involve, for their solution, a certain\namount of new and not altogether easy thinking. The sham difficulties\nare those suggested by the etymology, and those suggested by confusion\nof the mathematical infinite with what philosophers impertinently call\nthe \"true\" infinite. Etymologically, \"infinite\" should mean \"having no\nend.\" But in fact some infinite series have ends, some have not; while\nsome collections are infinite without being serial, and can therefore\nnot properly be regarded as either endless or having ends. The series of\ninstants from any earlier one to any later one (both included) is\ninfinite, but has two ends; the series of instants from the beginning of\ntime to the present moment has one end, but is infinite. Kant, in his\nfirst antinomy, seems to hold that it is harder for the past to be\ninfinite than for the future to be so, on the ground that the past is\nnow completed, and that nothing infinite can be completed. It is very\ndifficult to see how he can have imagined that there was any sense in\nthis remark; but it seems most probable that he was thinking of the\ninfinite as the \"unended.\" It is odd that he did not see that the future\ntoo has one end at the present, and is precisely on a level with the\npast. His regarding the two as different in this respect illustrates\njust that kind of slavery to time which, as we agreed in speaking of\nParmenides, the true philosopher must learn to leave behind him.\n\nThe confusions introduced into the notions of philosophers by the\nso-called \"true\" infinite are curious. They see that this notion is not\nthe same as the mathematical infinite, but they choose to believe that\nit is the notion which the mathematicians are vainly trying to reach.\nThey therefore inform the mathematicians, kindly but firmly, that they\nare mistaken in adhering to the \"false\" infinite, since plainly the\n\"true\" infinite is something quite different. The reply to this is that\nwhat they call the \"true\" infinite is a notion totally irrelevant to the\nproblem of the mathematical infinite, to which it has only a fanciful\nand verbal analogy. So remote is it that I do not propose to confuse the\nissue by even mentioning what the \"true\" infinite is. It is the \"false\"\ninfinite that concerns us, and we have to show that the epithet \"false\"\nis undeserved.\n\nThere are, however, certain genuine difficulties in understanding the\ninfinite, certain habits of mind derived from the consideration of\nfinite numbers, and easily extended to infinite numbers under the\nmistaken notion that they represent logical necessities. For example,\nevery number that we are accustomed to, except 0, has another number\nimmediately before it, from which it results by adding 1; but the first\ninfinite number does not have this property. The numbers before it form\nan infinite series, containing all the ordinary finite numbers, having\nno maximum, no last finite number, after which one little step would\nplunge us into the infinite. If it is assumed that the first infinite\nnumber is reached by a succession of small steps, it is easy to show\nthat it is self-contradictory. The first infinite number is, in fact,\nbeyond the whole unending series of finite numbers. \"But,\" it will be\nsaid, \"there cannot be anything beyond the whole of an unending series.\"\nThis, we may point out, is the very principle upon which Zeno relies in\nthe arguments of the race-course and the Achilles. Take the race-course:\nthere is the moment when the runner still has half his distance to run,\nthen the moment when he still has a quarter, then when he still has an\neighth, and so on in a strictly unending series. Beyond the whole of\nthis series is the moment when he reaches the goal. Thus there certainly\ncan be something beyond the whole of an unending series. But it remains\nto show that this fact is only what might have been expected.\n\nThe difficulty, like most of the vaguer difficulties besetting the\nmathematical infinite, is derived, I think, from the more or less\nunconscious operation of the idea of _counting_. If you set to work to\ncount the terms in an infinite collection, you will never have completed\nyour task. Thus, in the case of the runner, if half, three-quarters,\nseven-eighths, and so on of the course were marked, and the runner was\nnot allowed to pass any of the marks until the umpire said \"Now,\" then\nZeno's conclusion would be true in practice, and he would never reach\nthe goal.\n\nBut it is not essential to the existence of a collection, or even to\nknowledge and reasoning concerning it, that we should be able to pass\nits terms in review one by one. This may be seen in the case of finite\ncollections; we can speak of \"mankind\" or \"the human race,\" though many\nof the individuals in this collection are not personally known to us. We\ncan do this because we know of various characteristics which every\nindividual has if he belongs to the collection, and not if he does not.\nAnd exactly the same happens in the case of infinite collections: they\nmay be known by their characteristics although their terms cannot be\nenumerated. In this sense, an unending series may nevertheless form a\nwhole, and there may be new terms beyond the whole of it.\n\nSome purely arithmetical peculiarities of infinite numbers have also\ncaused perplexity. For instance, an infinite number is not increased by\nadding one to it, or by doubling it. Such peculiarities have seemed to\nmany to contradict logic, but in fact they only contradict confirmed\nmental habits. The whole difficulty of the subject lies in the necessity\nof thinking in an unfamiliar way, and in realising that many properties\nwhich we have thought inherent in number are in fact peculiar to finite\nnumbers. If this is remembered, the positive theory of infinity, which\nwill occupy the next lecture, will not be found so difficult as it is to\nthose who cling obstinately to the prejudices instilled by the\narithmetic which is learnt in childhood.\n\n\n\n\nLECTURE VII\n\nTHE POSITIVE THEORY OF INFINITY\n\n\nThe positive theory of infinity, and the general theory of number to\nwhich it has given rise, are among the triumphs of scientific method in\nphilosophy, and are therefore specially suitable for illustrating the\nlogical-analytic character of that method. The work in this subject has\nbeen done by mathematicians, and its results can be expressed in\nmathematical symbolism. Why, then, it may be said, should the subject be\nregarded as philosophy rather than as mathematics? This raises a\ndifficult question, partly concerned with the use of words, but partly\nalso of real importance in understanding the function of philosophy.\nEvery subject-matter, it would seem, can give rise to philosophical\ninvestigations as well as to the appropriate science, the difference\nbetween the two treatments being in the direction of movement and in the\nkind of truths which it is sought to establish. In the special sciences,\nwhen they have become fully developed, the movement is forward and\nsynthetic, from the simpler to the more complex. But in philosophy we\nfollow the inverse direction: from the complex and relatively concrete\nwe proceed towards the simple and abstract by means of analysis,\nseeking, in the process, to eliminate the particularity of the original\nsubject-matter, and to confine our attention entirely to the logical\n_form_ of the facts concerned.\n\nBetween philosophy and pure mathematics there is a certain affinity, in\nthe fact that both are general and _a priori_. Neither of them asserts\npropositions which, like those of history and geography, depend upon the\nactual concrete facts being just what they are. We may illustrate this\ncharacteristic by means of Leibniz's conception of many _possible_\nworlds, of which one only is _actual_. In all the many possible worlds,\nphilosophy and mathematics will be the same; the differences will only\nbe in respect of those particular facts which are chronicled by the\ndescriptive sciences. Any quality, therefore, by which our actual world\nis distinguished from other abstractly possible worlds, must be ignored\nby mathematics and philosophy alike. Mathematics and philosophy differ,\nhowever, in their manner of treating the general properties in which all\npossible worlds agree; for while mathematics, starting from\ncomparatively simple propositions, seeks to build up more and more\ncomplex results by deductive synthesis, philosophy, starting from data\nwhich are common knowledge, seeks to purify and generalise them into the\nsimplest statements of abstract form that can be obtained from them by\nlogical analysis.\n\nThe difference between philosophy and mathematics may be illustrated by\nour present problem, namely, the nature of number. Both start from\ncertain facts about numbers which are evident to inspection. But\nmathematics uses these facts to deduce more and more complicated\ntheorems, while philosophy seeks, by analysis, to go behind these facts\nto others, simpler, more fundamental, and inherently more fitted to form\nthe premisses of the science of arithmetic. The question, \"What is a\nnumber?\" is the pre-eminent philosophic question in this subject, but it\nis one which the mathematician as such need not ask, provided he knows\nenough of the properties of numbers to enable him to deduce his\ntheorems. We, since our object is philosophical, must grapple with the\nphilosopher's question. The answer to the question, \"What is a number?\"\nwhich we shall reach in this lecture, will be found to give also, by\nimplication, the answer to the difficulties of infinity which we\nconsidered in the previous lecture.\n\nThe question \"What is a number?\" is one which, until quite recent times,\nwas never considered in the kind of way that is capable of yielding a\nprecise answer. Philosophers were content with some vague dictum such\nas, \"Number is unity in plurality.\" A typical definition of the kind\nthat contented philosophers is the following from Sigwart's _Logic_\n(Sec. 66, section 3): \"Every number is not merely a _plurality_, but a\nplurality thought _as held together and closed, and to that extent as a\nunity_.\" Now there is in such definitions a very elementary blunder, of\nthe same kind that would be committed if we said \"yellow is a flower\"\nbecause some flowers are yellow. Take, for example, the number 3. A\nsingle collection of three things might conceivably be described as \"a\nplurality thought as held together and closed, and to that extent as a\nunity\"; but a collection of three things is not the number 3. The number\n3 is something which all collections of three things have in common, but\nis not itself a collection of three things. The definition, therefore,\napart from any other defects, has failed to reach the necessary degree\nof abstraction: the number 3 is something more abstract than any\ncollection of three things.\n\nSuch vague philosophic definitions, however, remained inoperative\nbecause of their very vagueness. What most men who thought about numbers\nreally had in mind was that numbers are the result of _counting_. \"On\nthe consciousness of the law of counting,\" says Sigwart at the beginning\nof his discussion of number, \"rests the possibility of spontaneously\nprolonging the series of numbers _ad infinitum_.\" It is this view of\nnumber as generated by counting which has been the chief psychological\nobstacle to the understanding of infinite numbers. Counting, because it\nis familiar, is erroneously supposed to be simple, whereas it is in fact\na highly complex process, which has no meaning unless the numbers\nreached in counting have some significance independent of the process by\nwhich they are reached. And infinite numbers cannot be reached at all in\nthis way. The mistake is of the same kind as if cows were defined as\nwhat can be bought from a cattle-merchant. To a person who knew several\ncattle-merchants, but had never seen a cow, this might seem an admirable\ndefinition. But if in his travels he came across a herd of wild cows, he\nwould have to declare that they were not cows at all, because no\ncattle-merchant could sell them. So infinite numbers were declared not\nto be numbers at all, because they could not be reached by counting.\n\nIt will be worth while to consider for a moment what counting actually\nis. We count a set of objects when we let our attention pass from one to\nanother, until we have attended once to each, saying the names of the\nnumbers in order with each successive act of attention. The last number\nnamed in this process is the number of the objects, and therefore\ncounting is a method of finding out what the number of the objects is.\nBut this operation is really a very complicated one, and those who\nimagine that it is the logical source of number show themselves\nremarkably incapable of analysis. In the first place, when we say \"one,\ntwo, three ...\" as we count, we cannot be said to be discovering the\nnumber of the objects counted unless we attach some meaning to the words\none, two, three, ... A child may learn to know these words in order, and\nto repeat them correctly like the letters of the alphabet, without\nattaching any meaning to them. Such a child may count correctly from the\npoint of view of a grown-up listener, without having any idea of numbers\nat all. The operation of counting, in fact, can only be intelligently\nperformed by a person who already has some idea what the numbers are;\nand from this it follows that counting does not give the logical basis\nof number.\n\nAgain, how do we know that the last number reached in the process of\ncounting is the number of the objects counted? This is just one of those\nfacts that are too familiar for their significance to be realised; but\nthose who wish to be logicians must acquire the habit of dwelling upon\nsuch facts. There are two propositions involved in this fact: first,\nthat the number of numbers from 1 up to any given number is that given\nnumber--for instance, the number of numbers from 1 to 100 is a hundred;\nsecondly, that if a set of numbers can be used as names of a set of\nobjects, each number occurring only once, then the number of numbers\nused as names is the same as the number of objects. The first of these\npropositions is capable of an easy arithmetical proof so long as finite\nnumbers are concerned; but with infinite numbers, after the first, it\nceases to be true. The second proposition remains true, and is in fact,\nas we shall see, an immediate consequence of the definition of number.\nBut owing to the falsehood of the first proposition where infinite\nnumbers are concerned, counting, even if it were practically possible,\nwould not be a valid method of discovering the number of terms in an\ninfinite collection, and would in fact give different results according\nto the manner in which it was carried out.\n\nThere are two respects in which the infinite numbers that are known\ndiffer from finite numbers: first, infinite numbers have, while finite\nnumbers have not, a property which I shall call _reflexiveness_;\nsecondly, finite numbers have, while infinite numbers have not, a\nproperty which I shall call _inductiveness_. Let us consider these two\nproperties successively.\n\n(1) _Reflexiveness._--A number is said to be _reflexive_ when it is not\nincreased by adding 1 to it. It follows at once that any finite number\ncan be added to a reflexive number without increasing it. This property\nof infinite numbers was always thought, until recently, to be\nself-contradictory; but through the work of Georg Cantor it has come to\nbe recognised that, though at first astonishing, it is no more\nself-contradictory than the fact that people at the antipodes do not\ntumble off. In virtue of this property, given any infinite collection of\nobjects, any finite number of objects can be added or taken away without\nincreasing or diminishing the number of the collection. Even an infinite\nnumber of objects may, under certain conditions, be added or taken away\nwithout altering the number. This may be made clearer by the help of\nsome examples.\n\nImagine all the natural numbers 0, 1, 2, 3, ... to be written down in a\nrow, and immediately beneath them write down the numbers 1, 2, 3,\n4, ..., so that 1 is under 0, 2 is under 1, and so on. Then every number\nin the top row has a number directly under it in the bottom row, and no\nnumber occurs twice in either row. It follows that the number of numbers\nin the two rows must be the same. But all the numbers that occur in the\nbottom row also occur in the top row, and one more, namely 0; thus the\nnumber of terms in the top row is obtained by adding one to the number\nof the bottom row. So long, therefore, as it was supposed that a number\nmust be increased by adding 1 to it, this state of things constituted a\ncontradiction, and led to the denial that there are infinite numbers.\n\n  0, 1, 2, 3, ... _n_ ...\n  1, 2, 3, 4, ... _n_ + 1 ...\n\nThe following example is even more surprising. Write the natural numbers\n1, 2, 3, 4, ... in the top row, and the even numbers 2, 4, 6, 8, ... in\nthe bottom row, so that under each number in the top row stands its\ndouble in the bottom row. Then, as before, the number of numbers in the\ntwo rows is the same, yet the second row results from taking away all\nthe odd numbers--an infinite collection--from the top row. This example\nis given by Leibniz to prove that there can be no infinite numbers. He\nbelieved in infinite collections, but, since he thought that a number\nmust always be increased when it is added to and diminished when it is\nsubtracted from, he maintained that infinite collections do not have\nnumbers. \"The number of all numbers,\" he says, \"implies a contradiction,\nwhich I show thus: To any number there is a corresponding number equal\nto its double. Therefore the number of all numbers is not greater than\nthe number of even numbers, _i.e._ the whole is not greater than its\npart.\"[49] In dealing with this argument, we ought to substitute \"the\nnumber of all finite numbers\" for \"the number of all numbers\"; we then\nobtain exactly the illustration given by our two rows, one containing\nall the finite numbers, the other only the even finite numbers. It will\nbe seen that Leibniz regards it as self-contradictory to maintain that\nthe whole is not greater than its part. But the word \"greater\" is one\nwhich is capable of many meanings; for our purpose, we must substitute\nthe less ambiguous phrase \"containing a greater number of terms.\" In\nthis sense, it is not self-contradictory for whole and part to be equal;\nit is the realisation of this fact which has made the modern theory of\ninfinity possible.\n\n  [49] _Phil. Werke_, Gerhardt's edition, vol. i. p. 338.\n\nThere is an interesting discussion of the reflexiveness of infinite\nwholes in the first of Galileo's Dialogues on Motion. I quote from a\ntranslation published in 1730.[50] The personages in the dialogue are\nSalviati, Sagredo, and Simplicius, and they reason as follows:\n\n\"_Simp._ Here already arises a Doubt which I think is not to be\nresolv'd; and that is this: Since 'tis plain that one Line is given\ngreater than another, and since both contain infinite Points, we must\nsurely necessarily infer, that we have found in the same Species\nsomething greater than Infinite, since the Infinity of Points of the\ngreater Line exceeds the Infinity of Points of the lesser. But now, to\nassign an Infinite greater than an Infinite, is what I can't possibly\nconceive.\n\n\"_Salv._ These are some of those Difficulties which arise from\nDiscourses which our finite Understanding makes about Infinites, by\nascribing to them Attributes which we give to Things finite and\nterminate, which I think most improper, because those Attributes of\nMajority, Minority, and Equality, agree not with Infinities, of which we\ncan't say that one is greater than, less than, or equal to another. For\nProof whereof I have something come into my Head, which (that I may be\nthe better understood) I will propose by way of Interrogatories to\n_Simplicius_, who started this Difficulty. To begin then: I suppose you\nknow which are square Numbers, and which not?\n\n\"_Simp._ I know very well that a square Number is that which arises from\nthe Multiplication of any Number into itself; thus 4 and 9 are square\nNumbers, that arising from 2, and this from 3, multiplied by themselves.\n\n\"_Salv._ Very well; And you also know, that as the Products are call'd\nSquares, the Factors are call'd Roots: And that the other Numbers, which\nproceed not from Numbers multiplied into themselves, are not Squares.\nWhence taking in all Numbers, both Squares and Not Squares, if I should\nsay, that the Not Squares are more than the Squares, should I not be in\nthe right?\n\n\"_Simp._ Most certainly.\n\n\"_Salv._ If I go on with you then, and ask you, How many squar'd Numbers\nthere are? you may truly answer, That there are as many as are their\nproper Roots, since every Square has its own Root, and every Root its\nown Square, and since no Square has more than one Root, nor any Root\nmore than one Square.\n\n\"_Simp._ Very true.\n\n\"_Salv._ But now, if I should ask how many Roots there are, you can't\ndeny but there are as many as there are Numbers, since there's no Number\nbut what's the Root to some Square. And this being granted, we may\nlikewise affirm, that there are as many square Numbers, as there are\nNumbers; for there are as many Squares as there are Roots, and as many\nRoots as Numbers. And yet in the Beginning of this, we said, there were\nmany more Numbers than Squares, the greater Part of Numbers being not\nSquares: And tho' the Number of Squares decreases in a greater\nproportion, as we go on to bigger Numbers, for count to an Hundred\nyou'll find 10 Squares, viz. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, which\nis the same as to say the 10th Part are Squares; in Ten thousand only\nthe 100th Part are Squares; in a Million only the 1000th: And yet in an\ninfinite Number, if we can but comprehend it, we may say the Squares are\nas many as all the Numbers taken together.\n\n\"_Sagr._ What must be determin'd then in this Case?\n\n\"_Salv._ I see no other way, but by saying that all Numbers are\ninfinite; Squares are Infinite, their Roots Infinite, and that the\nNumber of Squares is not less than the Number of Numbers, nor this less\nthan that: and then by concluding that the Attributes or Terms of\nEquality, Majority, and Minority, have no Place in Infinites, but are\nconfin'd to terminate Quantities.\"\n\n  [50] _Mathematical Discourses concerning two new sciences relating to\n  mechanics and local motion, in four dialogues._ By Galileo Galilei,\n  Chief Philosopher and Mathematician to the Grand Duke of Tuscany. Done\n  into English from the Italian, by Tho. Weston, late Master, and now\n  published by John Weston, present Master, of the Academy at Greenwich.\n  See pp. 46 ff.\n\nThe way in which the problem is expounded in the above discussion is\nworthy of Galileo, but the solution suggested is not the right one. It\nis actually the case that the number of square (finite) numbers is the\nsame as the number of (finite) numbers. The fact that, so long as we\nconfine ourselves to numbers less than some given finite number, the\nproportion of squares tends towards zero as the given finite number\nincreases, does not contradict the fact that the number of all finite\nsquares is the same as the number of all finite numbers. This is only an\ninstance of the fact, now familiar to mathematicians, that the _limit_\nof a function as the variable _approaches_ a given point may not be the\nsame as its _value_ when the variable actually _reaches_ the given\npoint. But although the infinite numbers which Galileo discusses are\nequal, Cantor has shown that what Simplicius could not conceive is true,\nnamely, that there are an infinite number of different infinite numbers,\nand that the conception of _greater_ and _less_ can be perfectly well\napplied to them. The whole of Simplicius's difficulty comes, as is\nevident, from his belief that, if _greater_ and _less_ can be applied, a\npart of an infinite collection must have fewer terms than the whole; and\nwhen this is denied, all contradictions disappear. As regards greater\nand less lengths of lines, which is the problem from which the above\ndiscussion starts, that involves a meaning of _greater_ and _less_ which\nis not arithmetical. The number of points is the same in a long line and\nin a short one, being in fact the same as the number of points in all\nspace. The _greater_ and _less_ of metrical geometry involves the new\nmetrical conception of _congruence_, which cannot be developed out of\narithmetical considerations alone. But this question has not the\nfundamental importance which belongs to the arithmetical theory of\ninfinity.\n\n(2) _Non-inductiveness._--The second property by which infinite numbers\nare distinguished from finite numbers is the property of\nnon-inductiveness. This will be best explained by defining the positive\nproperty of inductiveness which characterises the finite numbers, and\nwhich is named after the method of proof known as \"mathematical\ninduction.\"\n\nLet us first consider what is meant by calling a property \"hereditary\"\nin a given series. Take such a property as being named Jones. If a man\nis named Jones, so is his son; we will therefore call the property of\nbeing called Jones hereditary with respect to the relation of father and\nson. If a man is called Jones, all his descendants in the direct male\nline are called Jones; this follows from the fact that the property is\nhereditary. Now, instead of the relation of father and son, consider the\nrelation of a finite number to its immediate successor, that is, the\nrelation which holds between 0 and 1, between 1 and 2, between 2 and 3,\nand so on. If a property of numbers is hereditary with respect to this\nrelation, then if it belongs to (say) 100, it must belong also to all\nfinite numbers greater than 100; for, being hereditary, it belongs to\n101 because it belongs to 100, and it belongs to 102 because it belongs\nto 101, and so on--where the \"and so on\" will take us, sooner or later,\nto any finite number greater than 100. Thus, for example, the property\nof being greater than 99 is hereditary in the series of finite numbers;\nand generally, a property is hereditary in this series when, given any\nnumber that possesses the property, the next number must always also\npossess it.\n\nIt will be seen that a hereditary property, though it must belong to all\nthe finite numbers greater than a given number possessing the property,\nneed not belong to all the numbers less than this number. For example,\nthe hereditary property of being greater than 99 belongs to 100 and all\ngreater numbers, but not to any smaller number. Similarly, the\nhereditary property of being called Jones belongs to all the descendants\n(in the direct male line) of those who have this property, but not to\nall their ancestors, because we reach at last a first Jones, before whom\nthe ancestors have no surname. It is obvious, however, that any\nhereditary property possessed by Adam must belong to all men; and\nsimilarly any hereditary property possessed by 0 must belong to all\nfinite numbers. This is the principle of what is called \"mathematical\ninduction.\" It frequently happens, when we wish to prove that all finite\nnumbers have some property, that we have first to prove that 0 has the\nproperty, and then that the property is hereditary, _i.e._ that, if it\nbelongs to a given number, then it belongs to the next number. Owing to\nthe fact that such proofs are called \"inductive,\" I shall call the\nproperties to which they are applicable \"inductive\" properties. Thus an\ninductive property of numbers is one which is hereditary and belongs to\n0.\n\nTaking any one of the natural numbers, say 29, it is easy to see that it\nmust have all inductive properties. For since such properties belong to\n0 and are hereditary, they belong to 1; therefore, since they are\nhereditary, they belong to 2, and so on; by twenty-nine repetitions of\nsuch arguments we show that they belong to 29. We may _define_ the\n\"inductive\" numbers as _all those that possess all inductive\nproperties_; they will be the same as what are called the \"natural\"\nnumbers, _i.e._ the ordinary finite whole numbers. To all such numbers,\nproofs by mathematical induction can be validly applied. They are those\nnumbers, we may loosely say, which can be reached from 0 by successive\nadditions of 1; in other words, they are all the numbers that can be\nreached by counting.\n\nBut beyond all these numbers, there are the infinite numbers, and\ninfinite numbers do not have all inductive properties. Such numbers,\ntherefore, may be called non-inductive. All those properties of numbers\nwhich are proved by an imaginary step-by-step process from one number to\nthe next are liable to fail when we come to infinite numbers. The first\nof the infinite numbers has no immediate predecessor, because there is\nno greatest finite number; thus no succession of steps from one number\nto the next will ever reach from a finite number to an infinite one, and\nthe step-by-step method of proof fails. This is another reason for the\nsupposed self-contradictions of infinite numbers. Many of the most\nfamiliar properties of numbers, which custom had led people to regard as\nlogically necessary, are in fact only demonstrable by the step-by-step\nmethod, and fail to be true of infinite numbers. But so soon as we\nrealise the necessity of proving such properties by mathematical\ninduction, and the strictly limited scope of this method of proof, the\nsupposed contradictions are seen to contradict, not logic, but only our\nprejudices and mental habits.\n\nThe property of being increased by the addition of 1--_i.e._ the\nproperty of non-reflexiveness--may serve to illustrate the limitations\nof mathematical induction. It is easy to prove that 0 is increased by\nthe addition of 1, and that, if a given number is increased by the\naddition of 1, so is the next number, _i.e._ the number obtained by the\naddition of 1. It follows that each of the natural numbers is increased\nby the addition of 1. This follows generally from the general argument,\nand follows for each particular case by a sufficient number of\napplications of the argument. We first prove that 0 is not equal to 1;\nthen, since the property of being increased by 1 is hereditary, it\nfollows that 1 is not equal to 2; hence it follows that 2 is not equal\nto 3; if we wish to prove that 30,000 is not equal to 30,001, we can do\nso by repeating this reasoning 30,000 times. But we cannot prove in this\nway that _all_ numbers are increased by the addition of 1; we can only\nprove that this holds of the numbers attainable by successive additions\nof 1 starting from 0. The reflexive numbers, which lie beyond all those\nattainable in this way, are as a matter of fact not increased by the\naddition of 1.\n\nThe two properties of reflexiveness and non-inductiveness, which we have\nconsidered as characteristics of infinite numbers, have not so far been\nproved to be always found together. It is known that all reflexive\nnumbers are non-inductive, but it is not known that all non-inductive\nnumbers are reflexive. Fallacious proofs of this proposition have been\npublished by many writers, including myself, but up to the present no\nvalid proof has been discovered. The infinite numbers actually known,\nhowever, are all reflexive as well as non-inductive; thus, in\nmathematical practice, if not in theory, the two properties are always\nassociated. For our purposes, therefore, it will be convenient to ignore\nthe bare possibility that there may be non-inductive non-reflexive\nnumbers, since all known numbers are either inductive or reflexive.\n\nWhen infinite numbers are first introduced to people, they are apt to\nrefuse the name of numbers to them, because their behaviour is so\ndifferent from that of finite numbers that it seems a wilful misuse of\nterms to call them numbers at all. In order to meet this feeling, we\nmust now turn to the logical basis of arithmetic, and consider the\nlogical definition of numbers.\n\nThe logical definition of numbers, though it seems an essential support\nto the theory of infinite numbers, was in fact discovered independently\nand by a different man. The theory of infinite numbers--that is to say,\nthe arithmetical as opposed to the logical part of the theory--was\ndiscovered by Georg Cantor, and published by him in 1882-3.[51] The\ndefinition of number was discovered about the same time by a man whose\ngreat genius has not received the recognition it deserves--I mean\nGottlob Frege of Jena. His first work, _Begriffsschrift_, published in\n1879, contained the very important theory of hereditary properties in a\nseries to which I alluded in connection with inductiveness. His\ndefinition of number is contained in his second work, published in 1884,\nand entitled _Die Grundlagen der Arithmetik, eine logisch-mathematische\nUntersuchung ueber den Begriff der Zahl_.[52] It is with this book that\nthe logical theory of arithmetic begins, and it will repay us to\nconsider Frege's analysis in some detail.\n\n  [51] In his _Grundlagen einer allgemeinen Mannichfaltigkeitslehre_ and\n  in articles in _Acta Mathematica_, vol. ii.\n\n  [52] The definition of number contained in this book, and elaborated\n  in the _Grundgesetze der Arithmetik_ (vol. i., 1893; vol. ii., 1903),\n  was rediscovered by me in ignorance of Frege's work. I wish to state\n  as emphatically as possible--what seems still often ignored--that his\n  discovery antedated mine by eighteen years.\n\nFrege begins by noting the increased desire for logical strictness in\nmathematical demonstrations which distinguishes modern mathematicians\nfrom their predecessors, and points out that this must lead to a\ncritical investigation of the definition of number. He proceeds to show\nthe inadequacy of previous philosophical theories, especially of the\n\"synthetic _a priori_\" theory of Kant and the empirical theory of Mill.\nThis brings him to the question: What kind of object is it that number\ncan properly be ascribed to? He points out that physical things may be\nregarded as one or many: for example, if a tree has a thousand leaves,\nthey may be taken altogether as constituting its foliage, which would\ncount as one, not as a thousand; and _one_ pair of boots is the same\nobject as _two_ boots. It follows that physical things are not the\nsubjects of which number is properly predicated; for when we have\ndiscovered the proper subjects, the number to be ascribed must be\nunambiguous. This leads to a discussion of the very prevalent view that\nnumber is really something psychological and subjective, a view which\nFrege emphatically rejects. \"Number,\" he says, \"is as little an object\nof psychology or an outcome of psychical processes as the North Sea....\nThe botanist wishes to state something which is just as much a fact when\nhe gives the number of petals in a flower as when he gives its colour.\nThe one depends as little as the other upon our caprice. There is\ntherefore a certain similarity between number and colour; but this does\nnot consist in the fact that both are sensibly perceptible in external\nthings, but in the fact that both are objective\" (p. 34).\n\n\"I distinguish the objective,\" he continues, \"from the palpable, the\nspatial, the actual. The earth's axis, the centre of mass of the solar\nsystem, are objective, but I should not call them actual, like the earth\nitself\" (p. 35). He concludes that number is neither spatial and\nphysical, nor subjective, but non-sensible and objective. This\nconclusion is important, since it applies to all the subject-matter of\nmathematics and logic. Most philosophers have thought that the physical\nand the mental between them exhausted the world of being. Some have\nargued that the objects of mathematics were obviously not subjective,\nand therefore must be physical and empirical; others have argued that\nthey were obviously not physical, and therefore must be subjective and\nmental. Both sides were right in what they denied, and wrong in what\nthey asserted; Frege has the merit of accepting both denials, and\nfinding a third assertion by recognising the world of logic, which is\nneither mental nor physical.\n\nThe fact is, as Frege points out, that no number, not even 1, is\napplicable to physical things, but only to general terms or\ndescriptions, such as \"man,\" \"satellite of the earth,\" \"satellite of\nVenus.\" The general term \"man\" is applicable to a certain number of\nobjects: there are in the world so and so many men. The unity which\nphilosophers rightly feel to be necessary for the assertion of a number\nis the unity of the general term, and it is the general term which is\nthe proper subject of number. And this applies equally when there is one\nobject or none which falls under the general term. \"Satellite of the\nearth\" is a term only applicable to one object, namely, the moon. But\n\"one\" is not a property of the moon itself, which may equally well be\nregarded as many molecules: it is a property of the general term\n\"earth's satellite.\" Similarly, 0 is a property of the general term\n\"satellite of Venus,\" because Venus has no satellite. Here at last we\nhave an intelligible theory of the number 0. This was impossible if\nnumbers applied to physical objects, because obviously no physical\nobject could have the number 0. Thus, in seeking our definition of\nnumber we have arrived so far at the result that numbers are properties\nof general terms or general descriptions, not of physical things or of\nmental occurrences.\n\nInstead of speaking of a general term, such as \"man,\" as the subject of\nwhich a number can be asserted, we may, without making any serious\nchange, take the subject as the class or collection of objects--_i.e._\n\"mankind\" in the above instance--to which the general term in question\nis applicable. Two general terms, such as \"man\" and \"featherless biped,\"\nwhich are applicable to the same collection of objects, will obviously\nhave the same number of instances; thus the number depends upon the\nclass, not upon the selection of this or that general term to describe\nit, provided several general terms can be found to describe the same\nclass. But some general term is always necessary in order to describe a\nclass. Even when the terms are enumerated, as \"this and that and the\nother,\" the collection is constituted by the general property of being\neither this, or that, or the other, and only so acquires the unity which\nenables us to speak of it as _one_ collection. And in the case of an\ninfinite class, enumeration is impossible, so that description by a\ngeneral characteristic common and peculiar to the members of the class\nis the only possible description. Here, as we see, the theory of number\nto which Frege was led by purely logical considerations becomes of use\nin showing how infinite classes can be amenable to number in spite of\nbeing incapable of enumeration.\n\nFrege next asks the question: When do two collections have the same\nnumber of terms? In ordinary life, we decide this question by counting;\nbut counting, as we saw, is impossible in the case of infinite\ncollections, and is not logically fundamental with finite collections.\nWe want, therefore, a different method of answering our question. An\nillustration may help to make the method clear. I do not know how many\nmarried men there are in England, but I do know that the number is the\nsame as the number of married women. The reason I know this is that the\nrelation of husband and wife relates one man to one woman and one woman\nto one man. A relation of this sort is called a one-one relation. The\nrelation of father to son is called a one-many relation, because a man\ncan have only one father but may have many sons; conversely, the\nrelation of son to father is called a many-one relation. But the\nrelation of husband to wife (in Christian countries) is called one-one,\nbecause a man cannot have more than one wife, or a woman more than one\nhusband. Now, whenever there is a one-one relation between all the terms\nof one collection and all the terms of another severally, as in the case\nof English husbands and English wives, the number of terms in the one\ncollection is the same as the number in the other; but when there is not\nsuch a relation, the number is different. This is the answer to the\nquestion: When do two collections have the same number of terms?\n\nWe can now at last answer the question: What is meant by the number of\nterms in a given collection? When there is a one-one relation between\nall the terms of one collection and all the terms of another severally,\nwe shall say that the two collections are \"similar.\" We have just seen\nthat two similar collections have the same number of terms. This leads\nus to define the number of a given collection as the class of all\ncollections that are similar to it; that is to say, we set up the\nfollowing formal definition:\n\n\"The number of terms in a given class\" is defined as meaning \"the class\nof all classes that are similar to the given class.\"\n\nThis definition, as Frege (expressing it in slightly different terms)\nshowed, yields the usual arithmetical properties of numbers. It is\napplicable equally to finite and infinite numbers, and it does not\nrequire the admission of some new and mysterious set of metaphysical\nentities. It shows that it is not physical objects, but classes or the\ngeneral terms by which they are defined, of which numbers can be\nasserted; and it applies to 0 and 1 without any of the difficulties\nwhich other theories find in dealing with these two special cases.\n\nThe above definition is sure to produce, at first sight, a feeling of\noddity, which is liable to cause a certain dissatisfaction. It defines\nthe number 2, for instance, as the class of all couples, and the number\n3 as the class of all triads. This does not _seem_ to be what we have\nhitherto been meaning when we spoke of 2 and 3, though it would be\ndifficult to say _what_ we had been meaning. The answer to a feeling\ncannot be a logical argument, but nevertheless the answer in this case\nis not without importance. In the first place, it will be found that\nwhen an idea which has grown familiar as an unanalysed whole is first\nresolved accurately into its component parts--which is what we do when\nwe define it--there is almost always a feeling of unfamiliarity produced\nby the analysis, which tends to cause a protest against the definition.\nIn the second place, it may be admitted that the definition, like all\ndefinitions, is to a certain extent arbitrary. In the case of the small\nfinite numbers, such as 2 and 3, it would be possible to frame\ndefinitions more nearly in accordance with our unanalysed feeling of\nwhat we mean; but the method of such definitions would lack uniformity,\nand would be found to fail sooner or later--at latest when we reached\ninfinite numbers.\n\nIn the third place, the real desideratum about such a definition as that\nof number is not that it should represent as nearly as possible the\nideas of those who have not gone through the analysis required in order\nto reach a definition, but that it should give us objects having the\nrequisite properties. Numbers, in fact, must satisfy the formulae of\narithmetic; any indubitable set of objects fulfilling this requirement\nmay be called numbers. So far, the simplest set known to fulfil this\nrequirement is the set introduced by the above definition. In comparison\nwith this merit, the question whether the objects to which the\ndefinition applies are like or unlike the vague ideas of numbers\nentertained by those who cannot give a definition, is one of very little\nimportance. All the important requirements are fulfilled by the above\ndefinition, and the sense of oddity which is at first unavoidable will\nbe found to wear off very quickly with the growth of familiarity.\n\nThere is, however, a certain logical doctrine which may be thought to\nform an objection to the above definition of numbers as classes of\nclasses--I mean the doctrine that there are no such objects as classes\nat all. It might be thought that this doctrine would make havoc of a\ntheory which reduces numbers to classes, and of the many other theories\nin which we have made use of classes. This, however, would be a mistake:\nnone of these theories are any the worse for the doctrine that classes\nare fictions. What the doctrine is, and why it is not destructive, I\nwill try briefly to explain.\n\nOn account of certain rather complicated difficulties, culminating in\ndefinite contradictions, I was led to the view that nothing that can be\nsaid significantly about things, _i.e._ particulars, can be said\nsignificantly (_i.e._ either truly or falsely) about classes of things.\nThat is to say, if, in any sentence in which a thing is mentioned, you\nsubstitute a class for the thing, you no longer have a sentence that has\nany meaning: the sentence is no longer either true or false, but a\nmeaningless collection of words. Appearances to the contrary can be\ndispelled by a moment's reflection. For example, in the sentence, \"Adam\nis fond of apples,\" you may substitute _mankind_, and say, \"Mankind is\nfond of apples.\" But obviously you do not mean that there is one\nindividual, called \"mankind,\" which munches apples: you mean that the\nseparate individuals who compose mankind are each severally fond of\napples.\n\nNow, if nothing that can be said significantly about a thing can be said\nsignificantly about a class of things, it follows that classes of things\ncannot have the same kind of reality as things have; for if they had, a\nclass could be substituted for a thing in a proposition predicating the\nkind of reality which would be common to both. This view is really\nconsonant to common sense. In the third or fourth century B.C. there\nlived a Chinese philosopher named Hui Tzu, who maintained that \"a bay\nhorse and a dun cow are three; because taken separately they are two,\nand taken together they are one: two and one make three.\"[53] The author\nfrom whom I quote says that Hui Tzu \"was particularly fond of the\nquibbles which so delighted the sophists or unsound reasoners of ancient\nGreece,\" and this no doubt represents the judgment of common sense upon\nsuch arguments. Yet if collections of things were things, his contention\nwould be irrefragable. It is only because the bay horse and the dun cow\ntaken together are not a new thing that we can escape the conclusion\nthat there are three things wherever there are two.\n\n  [53] Giles, _The Civilisation of China_ (Home University Library),\n  p. 147.\n\nWhen it is admitted that classes are not things, the question arises:\nWhat do we mean by statements which are nominally about classes? Take\nsuch a statement as, \"The class of people interested in mathematical\nlogic is not very numerous.\" Obviously this reduces itself to, \"Not very\nmany people are interested in mathematical logic.\" For the sake of\ndefiniteness, let us substitute some particular number, say 3, for \"very\nmany.\" Then our statement is, \"Not three people are interested in\nmathematical logic.\" This may be expressed in the form: \"If _x_ is\ninterested in mathematical logic, and also _y_ is interested, and also\n_z_ is interested, then _x_ is identical with _y_, or _x_ is identical\nwith _z_, or _y_ is identical with _z_.\" Here there is no longer any\nreference at all to a \"class.\" In some such way, all statements\nnominally about a class can be reduced to statements about what follows\nfrom the hypothesis of anything's having the defining property of the\nclass. All that is wanted, therefore, in order to render the _verbal_\nuse of classes legitimate, is a uniform method of interpreting\npropositions in which such a use occurs, so as to obtain propositions in\nwhich there is no longer any such use. The definition of such a method\nis a technical matter, which Dr Whitehead and I have dealt with\nelsewhere, and which we need not enter into on this occasion.[54]\n\n  [54] Cf. _Principia Mathematica_, Sec. 20, and Introduction, chapter iii.\n\nIf the theory that classes are merely symbolic is accepted, it follows\nthat numbers are not actual entities, but that propositions in which\nnumbers verbally occur have not really any constituents corresponding to\nnumbers, but only a certain logical form which is not a part of\npropositions having this form. This is in fact the case with all the\napparent objects of logic and mathematics. Such words as _or_, _not_,\n_if_, _there is_, _identity_, _greater_, _plus_, _nothing_,\n_everything_, _function_, and so on, are not names of definite objects,\nlike \"John\" or \"Jones,\" but are words which require a context in order\nto have meaning. All of them are _formal_, that is to say, their\noccurrence indicates a certain form of proposition, not a certain\nconstituent. \"Logical constants,\" in short, are not entities; the words\nexpressing them are not names, and cannot significantly be made into\nlogical subjects except when it is the words themselves, as opposed to\ntheir meanings, that are being discussed.[55] This fact has a very\nimportant bearing on all logic and philosophy, since it shows how they\ndiffer from the special sciences. But the questions raised are so large\nand so difficult that it is impossible to pursue them further on this\noccasion.\n\n  [55] In the above remarks I am making use of unpublished work by my\n  friend Ludwig Wittgenstein.\n\n\n\n\nLECTURE VIII\n\nON THE NOTION OF CAUSE, WITH APPLICATIONS TO THE FREE-WILL PROBLEM\n\n\nThe nature of philosophic analysis, as illustrated in our previous\nlectures, can now be stated in general terms. We start from a body of\ncommon knowledge, which constitutes our data. On examination, the data\nare found to be complex, rather vague, and largely interdependent\nlogically. By analysis we reduce them to propositions which are as\nnearly as possible simple and precise, and we arrange them in deductive\nchains, in which a certain number of initial propositions form a logical\nguarantee for all the rest. These initial propositions are _premisses_\nfor the body of knowledge in question. Premisses are thus quite\ndifferent from data--they are simpler, more precise, and less infected\nwith logical redundancy. If the work of analysis has been performed\ncompletely, they will be wholly free from logical redundancy, wholly\nprecise, and as simple as is logically compatible with their leading to\nthe given body of knowledge. The discovery of these premisses belongs to\nphilosophy; but the work of deducing the body of common knowledge from\nthem belongs to mathematics, if \"mathematics\" is interpreted in a\nsomewhat liberal sense.\n\nBut besides the logical analysis of the common knowledge which forms our\ndata, there is the consideration of its degree of certainty. When we\nhave arrived at its premisses, we may find that some of them seem open\nto doubt, and we may find further that this doubt extends to those of\nour original data which depend upon these doubtful premisses. In our\nthird lecture, for example, we saw that the part of physics which\ndepends upon testimony, and thus upon the existence of other minds than\nour own, does not seem so certain as the part which depends exclusively\nupon our own sense-data and the laws of logic. Similarly, it used to be\nfelt that the parts of geometry which depend upon the axiom of parallels\nhave less certainty than the parts which are independent of this\npremiss. We may say, generally, that what commonly passes as knowledge\nis not all equally certain, and that, when analysis into premisses has\nbeen effected, the degree of certainty of any consequence of the\npremisses will depend upon that of the most doubtful premiss employed in\nproving this consequence. Thus analysis into premisses serves not only a\nlogical purpose, but also the purpose of facilitating an estimate as to\nthe degree of certainty to be attached to this or that derivative\nbelief. In view of the fallibility of all human beliefs, this service\nseems at least as important as the purely logical services rendered by\nphilosophical analysis.\n\nIn the present lecture, I wish to apply the analytic method to the\nnotion of \"cause,\" and to illustrate the discussion by applying it to\nthe problem of free will. For this purpose I shall inquire: I., what is\nmeant by a causal law; II., what is the evidence that causal laws have\nheld hitherto; III., what is the evidence that they will continue to\nhold in the future; IV., how the causality which is used in science\ndiffers from that of common sense and traditional philosophy; V., what\nnew light is thrown on the question of free will by our analysis of the\nnotion of \"cause.\"\n\nI. By a \"causal law\" I mean any general proposition in virtue of which\nit is possible to infer the existence of one thing or event from the\nexistence of another or of a number of others. If you hear thunder\nwithout having seen lightning, you infer that there nevertheless was a\nflash, because of the general proposition, \"All thunder is preceded by\nlightning.\" When Robinson Crusoe sees a footprint, he infers a human\nbeing, and he might justify his inference by the general proposition,\n\"All marks in the ground shaped like a human foot are subsequent to a\nhuman being's standing where the marks are.\" When we see the sun set, we\nexpect that it will rise again the next day. When we hear a man\nspeaking, we infer that he has certain thoughts. All these inferences\nare due to causal laws.\n\nA causal law, we said, allows us to infer the existence of one _thing_\n(or _event_) from the existence of one or more others. The word \"thing\"\nhere is to be understood as only applying to particulars, _i.e._ as\nexcluding such logical objects as numbers or classes or abstract\nproperties and relations, and including sense-data, with whatever is\nlogically of the same type as sense-data.[56] In so far as a causal law\nis directly verifiable, the thing inferred and the thing from which it\nis inferred must both be data, though they need not both be data at the\nsame time. In fact, a causal law which is being used to extend our\nknowledge of existence must be applied to what, at the moment, is not a\ndatum; it is in the possibility of such application that the practical\nutility of a causal law consists. The important point, for our present\npurpose, however, is that what is inferred is a \"thing,\" a \"particular,\"\nan object having the kind of reality that belongs to objects of sense,\nnot an abstract object such as virtue or the square root of two.\n\n  [56] Thus we are not using \"thing\" here in the sense of a class of\n  correlated \"aspects,\" as we did in Lecture III. Each \"aspect\" will\n  count separately in stating causal laws.\n\nBut we cannot become acquainted with a particular except by its being\nactually given. Hence the particular inferred by a causal law must be\nonly _described_ with more or less exactness; it cannot be _named_ until\nthe inference is verified. Moreover, since the causal law is _general_,\nand capable of applying to many cases, the given particular from which\nwe infer must allow the inference in virtue of some general\ncharacteristic, not in virtue of its being just the particular that it\nis. This is obvious in all our previous instances: we infer the\nunperceived lightning from the thunder, not in virtue of any peculiarity\nof the thunder, but in virtue of its resemblance to other claps of\nthunder. Thus a causal law must state that the existence of a thing of a\ncertain sort (or of a number of things of a number of assigned sorts)\nimplies the existence of another thing having a relation to the first\nwhich remains invariable so long as the first is of the kind in\nquestion.\n\nIt is to be observed that what is constant in a causal law is not the\nobject or objects given, nor yet the object inferred, both of which may\nvary within wide limits, but the _relation_ between what is given and\nwhat is inferred. The principle, \"same cause, same effect,\" which is\nsometimes said to be the principle of causality, is much narrower in its\nscope than the principle which really occurs in science; indeed, if\nstrictly interpreted, it has no scope at all, since the \"same\" cause\nnever recurs exactly. We shall return to this point at a later stage of\nthe discussion.\n\nThe particular which is inferred may be uniquely determined by the\ncausal law, or may be only described in such general terms that many\ndifferent particulars might satisfy the description. This depends upon\nwhether the constant relation affirmed by the causal law is one which\nonly one term can have to the data, or one which many terms may have. If\nmany terms may have the relation in question, science will not be\nsatisfied until it has found some more stringent law, which will enable\nus to determine the inferred things uniquely.\n\nSince all known things are in time, a causal law must take account of\ntemporal relations. It will be part of the causal law to state a\nrelation of succession or coexistence between the thing given and the\nthing inferred. When we hear thunder and infer that there was lightning,\nthe law states that the thing inferred is earlier than the thing given.\nConversely, when we see lightning and wait expectantly for the thunder,\nthe law states that the thing given is earlier than the thing inferred.\nWhen we infer a man's thoughts from his words, the law states that the\ntwo are (at least approximately) simultaneous.\n\nIf a causal law is to achieve the precision at which science aims, it\nmust not be content with a vague _earlier_ or _later_, but must state\nhow much earlier or how much later. That is to say, the time-relation\nbetween the thing given and the thing inferred ought to be capable of\nexact statement; and usually the inference to be drawn is different\naccording to the length and direction of the interval. \"A quarter of an\nhour ago this man was alive; an hour hence he will be cold.\" Such a\nstatement involves two causal laws, one inferring from a datum something\nwhich existed a quarter of an hour ago, the other inferring from the\nsame datum something which will exist an hour hence.\n\nOften a causal law involves not one datum, but many, which need not be\nall simultaneous with each other, though their time-relations must be\ngiven. The general scheme of a causal law will be as follows:\n\n\"Whenever things occur in certain relations to each other (among which\ntheir time-relations must be included), then a thing having a fixed\nrelation to these things will occur at a date fixed relatively to their\ndates.\"\n\nThe things given will not, in practice, be things that only exist for an\ninstant, for such things, if there are any, can never be data. The\nthings given will each occupy some finite time. They may be not static\nthings, but processes, especially motions. We have considered in an\nearlier lecture the sense in which a motion may be a datum, and need not\nnow recur to this topic.\n\nIt is not essential to a causal law that the object inferred should be\nlater than some or all of the data. It may equally well be earlier or at\nthe same time. The only thing essential is that the law should be such\nas to enable us to infer the existence of an object which we can more or\nless accurately describe in terms of the data.\n\nII. I come now to our second question, namely: What is the nature of the\nevidence that causal laws have held hitherto, at least in the observed\nportions of the past? This question must not be confused with the\nfurther question: Does this evidence warrant us in assuming the truth of\ncausal laws in the future and in unobserved portions of the past? For\nthe present, I am only asking what are the grounds which lead to a\nbelief in causal laws, not whether these grounds are adequate to support\nthe belief in universal causation.\n\nThe first step is the discovery of approximate unanalysed uniformities\nof sequence or coexistence. After lightning comes thunder, after a blow\nreceived comes pain, after approaching a fire comes warmth; again, there\nare uniformities of coexistence, for example between touch and sight,\nbetween certain sensations in the throat and the sound of one's own\nvoice, and so on. Every such uniformity of sequence or coexistence,\nafter it has been experienced a certain number of times, is followed by\nan expectation that it will be repeated on future occasions, _i.e._ that\nwhere one of the correlated events is found, the other will be found\nalso. The connection of experienced past uniformity with expectation as\nto the future is just one of those uniformities of sequence which we\nhave observed to be true hitherto. This affords a psychological account\nof what may be called the animal belief in causation, because it is\nsomething which can be observed in horses and dogs, and is rather a\nhabit of acting than a real belief. So far, we have merely repeated\nHume, who carried the discussion of cause up to this point, but did not,\napparently, perceive how much remained to be said.\n\nIs there, in fact, any characteristic, such as might be called causality\nor uniformity, which is found to hold throughout the observed past? And\nif so, how is it to be stated?\n\nThe particular uniformities which we mentioned before, such as lightning\nbeing followed by thunder, are not found to be free from exceptions. We\nsometimes see lightning without hearing thunder; and although, in such a\ncase, we suppose that thunder might have been heard if we had been\nnearer to the lightning, that is a supposition based on theory, and\ntherefore incapable of being invoked to support the theory. What does\nseem, however, to be shown by scientific experience is this: that where\nan observed uniformity fails, some wider uniformity can be found,\nembracing more circumstances, and subsuming both the successes and the\nfailures of the previous uniformity. Unsupported bodies in air fall,\nunless they are balloons or aeroplanes; but the principles of mechanics\ngive uniformities which apply to balloons and aeroplanes just as\naccurately as to bodies that fall. There is much that is hypothetical\nand more or less artificial in the uniformities affirmed by mechanics,\nbecause, when they cannot otherwise be made applicable, unobserved\nbodies are inferred in order to account for observed peculiarities.\nStill, it is an empirical fact that it is possible to preserve the laws\nby assuming such bodies, and that they never have to be assumed in\ncircumstances in which they ought to be observable. Thus the empirical\nverification of mechanical laws may be admitted, although we must also\nadmit that it is less complete and triumphant than is sometimes\nsupposed.\n\nAssuming now, what must be admitted to be doubtful, that the whole of\nthe past has proceeded according to invariable laws, what can we say as\nto the nature of these laws? They will not be of the simple type which\nasserts that the same cause always produces the same effect. We may take\nthe law of gravitation as a sample of the kind of law that appears to be\nverified without exception. In order to state this law in a form which\nobservation can confirm, we will confine it to the solar system. It then\nstates that the motions of planets and their satellites have at every\ninstant an acceleration compounded of accelerations towards all the\nother bodies in the solar system, proportional to the masses of those\nbodies and inversely proportional to the squares of their distances. In\nvirtue of this law, given the state of the solar system throughout any\nfinite time, however short, its state at all earlier and later times is\ndeterminate except in so far as other forces than gravitation or other\nbodies than those in the solar system have to be taken into\nconsideration. But other forces, so far as science can discover, appear\nto be equally regular, and equally capable of being summed up in single\ncausal laws. If the mechanical account of matter were complete, the\nwhole physical history of the universe, past and future, could be\ninferred from a sufficient number of data concerning an assigned finite\ntime, however short.\n\nIn the mental world, the evidence for the universality of causal laws is\nless complete than in the physical world. Psychology cannot boast of any\ntriumph comparable to gravitational astronomy. Nevertheless, the\nevidence is not very greatly less than in the physical world. The crude\nand approximate causal laws from which science starts are just as easy\nto discover in the mental sphere as in the physical. In the world of\nsense, there are to begin with the correlations of sight and touch and\nso on, and the facts which lead us to connect various kinds of\nsensations with eyes, ears, nose, tongue, etc. Then there are such facts\nas that our body moves in answer to our volitions. Exceptions exist, but\nare capable of being explained as easily as the exceptions to the rule\nthat unsupported bodies in air fall. There is, in fact, just such a\ndegree of evidence for causal laws in psychology as will warrant the\npsychologist in assuming them as a matter of course, though not such a\ndegree as will suffice to remove all doubt from the mind of a sceptical\ninquirer. It should be observed that causal laws in which the given term\nis mental and the inferred term physical, or _vice versa_, are at least\nas easy to discover as causal laws in which both terms are mental.\n\nIt will be noticed that, although we have spoken of causal laws, we have\nnot hitherto introduced the word \"cause.\" At this stage, it will be well\nto say a few words on legitimate and illegitimate uses of this word. The\nword \"cause,\" in the scientific account of the world, belongs only to\nthe early stages, in which small preliminary, approximate\ngeneralisations are being ascertained with a view to subsequent larger\nand more invariable laws. We may say, \"Arsenic causes death,\" so long as\nwe are ignorant of the precise process by which the result is brought\nabout. But in a sufficiently advanced science, the word \"cause\" will not\noccur in any statement of invariable laws. There is, however, a somewhat\nrough and loose use of the word \"cause\" which may be preserved. The\napproximate uniformities which lead to its pre-scientific employment may\nturn out to be true in all but very rare and exceptional circumstances,\nperhaps in all circumstances that actually occur. In such cases, it is\nconvenient to be able to speak of the antecedent event as the \"cause\"\nand the subsequent event as the \"effect.\" In this sense, provided it is\nrealised that the sequence is not necessary and may have exceptions, it\nis still possible to employ the words \"cause\" and \"effect.\" It is in\nthis sense, and in this sense only, that we shall intend the words when\nwe speak of one particular event \"causing\" another particular event, as\nwe must sometimes do if we are to avoid intolerable circumlocution.\n\nIII. We come now to our third question, namely: What reason can be given\nfor believing that causal laws will hold in future, or that they have\nheld in unobserved portions of the past?\n\nWhat we have said so far is that there have been hitherto certain\nobserved causal laws, and that all the empirical evidence we possess is\ncompatible with the view that everything, both mental and physical, so\nfar as our observation has extended, has happened in accordance with\ncausal laws. The law of universal causation, suggested by these facts,\nmay be enunciated as follows:\n\n\"There are such invariable relations between different events at the\nsame or different times that, given the state of the whole universe\nthroughout any finite time, however short, every previous and subsequent\nevent can theoretically be determined as a function of the given events\nduring that time.\"\n\nHave we any reason to believe this universal law? Or, to ask a more\nmodest question, have we any reason to believe that a particular causal\nlaw, such as the law of gravitation, will continue to hold in the\nfuture?\n\nAmong observed causal laws is this, that observation of uniformities is\nfollowed by expectation of their recurrence. A horse who has been driven\nalways along a certain road expects to be driven along that road again;\na dog who is always fed at a certain hour expects food at that hour and\nnot at any other. Such expectations, as Hume pointed out, explain only\ntoo well the common-sense belief in uniformities of sequence, but they\nafford absolutely no logical ground for beliefs as to the future, not\neven for the belief that we shall continue to expect the continuation of\nexperienced uniformities, for that is precisely one of those causal laws\nfor which a ground has to be sought. If Hume's account of causation is\nthe last word, we have not only no reason to suppose that the sun will\nrise to-morrow, but no reason to suppose that five minutes hence we\nshall still expect it to rise to-morrow.\n\nIt may, of course, be said that all inferences as to the future are in\nfact invalid, and I do not see how such a view could be disproved. But,\nwhile admitting the legitimacy of such a view, we may nevertheless\ninquire: If inferences as to the future _are_ valid, what principle must\nbe involved in making them?\n\nThe principle involved is the principle of induction, which, if it is\ntrue, must be an _a priori_ logical law, not capable of being proved or\ndisproved by experience. It is a difficult question how this principle\nought to be formulated; but if it is to warrant the inferences which we\nwish to make by its means, it must lead to the following proposition:\n\"If, in a great number of instances, a thing of a certain kind is\nassociated in a certain way with a thing of a certain other kind, it is\nprobable that a thing of the one kind is always similarly associated\nwith a thing of the other kind; and as the number of instances\nincreases, the probability approaches indefinitely near to certainty.\"\nIt may well be questioned whether this proposition is true; but if we\nadmit it, we can infer that any characteristic of the whole of the\nobserved past is likely to apply to the future and to the unobserved\npast. This proposition, therefore, if it is true, will warrant the\ninference that causal laws probably hold at all times, future as well as\npast; but without this principle, the observed cases of the truth of\ncausal laws afford no presumption as to the unobserved cases, and\ntherefore the existence of a thing not directly observed can never be\nvalidly inferred.\n\nIt is thus the principle of induction, rather than the law of causality,\nwhich is at the bottom of all inferences as to the existence of things\nnot immediately given. With the principle of induction, all that is\nwanted for such inferences can be proved; without it, all such\ninferences are invalid. This principle has not received the attention\nwhich its great importance deserves. Those who were interested in\ndeductive logic naturally enough ignored it, while those who emphasised\nthe scope of induction wished to maintain that all logic is empirical,\nand therefore could not be expected to realise that induction itself,\ntheir own darling, required a logical principle which obviously could\nnot be proved inductively, and must therefore be _a priori_ if it could\nbe known at all.\n\nThe view that the law of causality itself is _a priori_ cannot, I think,\nbe maintained by anyone who realises what a complicated principle it is.\nIn the form which states that \"every event has a cause\" it looks simple;\nbut on examination, \"cause\" is merged in \"causal law,\" and the\ndefinition of a \"causal law\" is found to be far from simple. There must\nnecessarily be _some a priori_ principle involved in inference from\nthe existence of one thing to that of another, if such inference is ever\nvalid; but it would appear from the above analysis that the principle in\nquestion is induction, not causality. Whether inferences from past to\nfuture are valid depends wholly, if our discussion has been sound, upon\nthe inductive principle: if it is true, such inferences are valid, and\nif it is false, they are invalid.\n\nIV. I come now to the question how the conception of causal laws which\nwe have arrived at is related to the traditional conception of cause as\nit occurs in philosophy and common sense.\n\nHistorically, the notion of cause has been bound up with that of human\nvolition. The typical cause would be the fiat of a king. The cause is\nsupposed to be \"active,\" the effect \"passive.\" From this it is easy to\npass on to the suggestion that a \"true\" cause must contain some\nprevision of the effect; hence the effect becomes the \"end\" at which the\ncause aims, and teleology replaces causation in the explanation of\nnature. But all such ideas, as applied to physics, are mere\nanthropomorphic superstitions. It is as a reaction against these errors\nthat Mach and others have urged a purely \"descriptive\" view of physics:\nphysics, they say, does not aim at telling us \"why\" things happen, but\nonly \"how\" they happen. And if the question \"why?\" means anything more\nthan the search for a general law according to which a phenomenon\noccurs, then it is certainly the case that this question cannot be\nanswered in physics and ought not to be asked. In this sense, the\ndescriptive view is indubitably in the right. But in using causal laws\nto support inferences from the observed to the unobserved, physics\nceases to be _purely_ descriptive, and it is these laws which give the\nscientifically useful part of the traditional notion of \"cause.\" There\nis therefore _something_ to preserve in this notion, though it is a very\ntiny part of what is commonly assumed in orthodox metaphysics.\n\nIn order to understand the difference between the kind of cause which\nscience uses and the kind which we naturally imagine, it is necessary to\nshut out, by an effort, everything that differentiates between past and\nfuture. This is an extraordinarily difficult thing to do, because our\nmental life is so intimately bound up with difference. Not only do\nmemory and hope make a difference in our feelings as regards past and\nfuture, but almost our whole vocabulary is filled with the idea of\nactivity, of things done now for the sake of their future effects. All\ntransitive verbs involve the notion of cause as activity, and would have\nto be replaced by some cumbrous periphrasis before this notion could be\neliminated.\n\nConsider such a statement as, \"Brutus killed Caesar.\" On another\noccasion, Brutus and Caesar might engage our attention, but for the\npresent it is the killing that we have to study. We may say that to kill\na person is to cause his death intentionally. This means that desire for\na person's death causes a certain act, because it is believed that that\nact will cause the person's death; or more accurately, the desire and\nthe belief jointly cause the act. Brutus desires that Caesar should be\ndead, and believes that he will be dead if he is stabbed; Brutus\ntherefore stabs him, and the stab causes Caesar's death, as Brutus\nexpected it would. Every act which realises a purpose involves two\ncausal steps in this way: C is desired, and it is believed (truly if the\npurpose is achieved) that B will cause C; the desire and the belief\ntogether cause B, which in turn causes C. Thus we have first A, which is\na desire for C and a belief that B (an act) will cause C; then we have\nB, the act caused by A, and believed to be a cause of C; then, if the\nbelief was correct, we have C, caused by B, and if the belief was\nincorrect we have disappointment. Regarded purely scientifically, this\nseries A, B, C may equally well be considered in the inverse order, as\nthey would be at a coroner's inquest. But from the point of view of\nBrutus, the desire, which comes at the beginning, is what makes the\nwhole series interesting. We feel that if his desires had been\ndifferent, the effects which he in fact produced would not have\noccurred. This is true, and gives him a sense of power and freedom. It\nis equally true that if the effects had not occurred, his desires would\nhave been different, since being what they were the effects did occur.\nThus the desires are determined by their consequences just as much as\nthe consequences by the desires; but as we cannot (in general) know in\nadvance the consequences of our desires without knowing our desires,\nthis form of inference is uninteresting as applied to our own acts,\nthough quite vital as applied to those of others.\n\nA cause, considered scientifically, has none of that analogy with\nvolition which makes us imagine that the effect is _compelled_ by it. A\ncause is an event or group of events, of some known general character,\nand having a known relation to some other event, called the effect; the\nrelation being of such a kind that only one event, or at any rate only\none well-defined sort of event, can have the relation to a given cause.\nIt is customary only to give the name \"effect\" to an event which is\nlater than the cause, but there is no kind of reason for this\nrestriction. We shall do better to allow the effect to be before the\ncause or simultaneous with it, because nothing of any scientific\nimportance depends upon its being after the cause.\n\nIf the inference from cause to effect is to be indubitable, it seems\nthat the cause can hardly stop short of the whole universe. So long as\nanything is left out, something may be left out which alters the\nexpected result. But for practical and scientific purposes, phenomena\ncan be collected into groups which are causally self-contained, or\nnearly so. In the common notion of causation, the cause is a single\nevent--we say the lightning causes the thunder, and so on. But it is\ndifficult to know what we mean by a single event; and it generally\nappears that, in order to have anything approaching certainty concerning\nthe effect, it is necessary to include many more circumstances in the\ncause than unscientific common sense would suppose. But often a probable\ncausal connection, where the cause is fairly simple, is of more\npractical importance than a more indubitable connection in which the\ncause is so complex as to be hard to ascertain.\n\nTo sum up: the strict, certain, universal law of causation which\nphilosophers advocate is an ideal, possibly true, but not _known_ to be\ntrue in virtue of any available evidence. What is actually known, as a\nmatter of empirical science, is that certain constant relations are\nobserved to hold between the members of a group of events at certain\ntimes, and that when such relations fail, as they sometimes do, it is\nusually possible to discover a new, more constant relation by enlarging\nthe group. Any such constant relation between events of specified kinds\nwith given intervals of time between them is a \"causal law.\" But all\ncausal laws are liable to exceptions, if the cause is less than the\nwhole state of the universe; we believe, on the basis of a good deal of\nexperience, that such exceptions can be dealt with by enlarging the\ngroup we call the cause, but this belief, wherever it is still\nunverified, ought not to be regarded as certain, but only as suggesting\na direction for further inquiry.\n\nA very common causal group consists of volitions and the consequent\nbodily acts, though exceptions arise (for example) through sudden\nparalysis. Another very frequent connection (though here the exceptions\nare much more numerous) is between a bodily act and the realisation of\nthe purpose which led to the act. These connections are patent, whereas\nthe causes of desires are more obscure. Thus it is natural to begin\ncausal series with desires, to suppose that all causes are analogous to\ndesires, and that desires themselves arise spontaneously. Such a view,\nhowever, is not one which any serious psychologist would maintain. But\nthis brings us to the question of the application of our analysis of\ncause to the problem of free will.\n\nV. The problem of free will is so intimately bound up with the analysis\nof causation that, old as it is, we need not despair of obtaining new\nlight on it by the help of new views on the notion of cause. The\nfree-will problem has, at one time or another, stirred men's passions\nprofoundly, and the fear that the will might not be free has been to\nsome men a source of great unhappiness. I believe that, under the\ninfluence of a cool analysis, the doubtful questions involved will be\nfound to have no such emotional importance as is sometimes thought,\nsince the disagreeable consequences supposed to flow from a denial of\nfree will do not flow from this denial in any form in which there is\nreason to make it. It is not, however, on this account chiefly that I\nwish to discuss this problem, but rather because it affords a good\nexample of the clarifying effect of analysis and of the interminable\ncontroversies which may result from its neglect.\n\nLet us first try to discover what it is we really desire when we desire\nfree will. Some of our reasons for desiring free will are profound, some\ntrivial. To begin with the former: we do not wish to feel ourselves in\nthe hands of fate, so that, however much we may desire to will one\nthing, we may nevertheless be compelled by an outside force to will\nanother. We do not wish to think that, however much we may desire to act\nwell, heredity and surroundings may force us into acting ill. We wish to\nfeel that, in cases of doubt, our choice is momentous and lies within\nour power. Besides these desires, which are worthy of all respect, we\nhave, however, others not so respectable, which equally make us desire\nfree will. We do not like to think that other people, if they knew\nenough, could predict our actions, though we know that we can often\npredict those of other people, especially if they are elderly. Much as\nwe esteem the old gentleman who is our neighbour in the country, we know\nthat when grouse are mentioned he will tell the story of the grouse in\nthe gun-room. But we ourselves are not so mechanical: we never tell an\nanecdote to the same person twice, or even once unless he is sure to\nenjoy it; although we once met (say) Bismarck, we are quite capable of\nhearing him mentioned without relating the occasion when we met him. In\nthis sense, everybody thinks that he himself has free will, though he\nknows that no one else has. The desire for this kind of free will seems\nto be no better than a form of vanity. I do not believe that this desire\ncan be gratified with any certainty; but the other, more respectable\ndesires are, I believe, not inconsistent with any tenable form of\ndeterminism.\n\nWe have thus two questions to consider: (1) Are human actions\ntheoretically predictable from a sufficient number of antecedents? (2)\nAre human actions subject to an external compulsion? The two questions,\nas I shall try to show, are entirely distinct, and we may answer the\nfirst in the affirmative without therefore being forced to give an\naffirmative answer to the second.\n\n(1) _Are human actions theoretically predictable from a sufficient\nnumber of antecedents?_ Let us first endeavour to give precision to this\nquestion. We may state the question thus: Is there some constant\nrelation between an act and a certain number of earlier events, such\nthat, when the earlier events are given, only one act, or at most only\nacts with some well-marked character, can have this relation to the\nearlier events? If this is the case, then, as soon as the earlier events\nare known, it is theoretically possible to predict either the precise\nact, or at least the character necessary to its fulfilling the constant\nrelation.\n\nTo this question, a negative answer has been given by Bergson, in a form\nwhich calls in question the general applicability of the law of\ncausation. He maintains that every event, and more particularly every\nmental event, embodies so much of the past that it could not possibly\nhave occurred at any earlier time, and is therefore necessarily quite\ndifferent from all previous and subsequent events. If, for example, I\nread a certain poem many times, my experience on each occasion is\nmodified by the previous readings, and my emotions are never repeated\nexactly. The principle of causation, according to him, asserts that the\nsame cause, if repeated, will produce the same effect. But owing to\nmemory, he contends, this principle does not apply to mental events.\nWhat is apparently the same cause, if repeated, is modified by the mere\nfact of repetition, and cannot produce the same effect. He infers that\nevery mental event is a genuine novelty, not predictable from the past,\nbecause the past contains nothing exactly like it by which we could\nimagine it. And on this ground he regards the freedom of the will as\nunassailable.\n\nBergson's contention has undoubtedly a great deal of truth, and I have\nno wish to deny its importance. But I do not think its consequences are\nquite what he believes them to be. It is not necessary for the\ndeterminist to maintain that he can foresee the whole particularity of\nthe act which will be performed. If he could foresee that A was going to\nmurder B, his foresight would not be invalidated by the fact that he\ncould not know all the infinite complexity of A's state of mind in\ncommitting the murder, nor whether the murder was to be performed with a\nknife or with a revolver. If the _kind_ of act which will be performed\ncan be foreseen within narrow limits, it is of little practical interest\nthat there are fine shades which cannot be foreseen. No doubt every time\nthe story of the grouse in the gun-room is told, there will be slight\ndifferences due to increasing habitualness, but they do not invalidate\nthe prediction that the story will be told. And there is nothing in\nBergson's argument to show that we can never predict what _kind_ of act\nwill be performed.\n\nAgain, his statement of the law of causation is inadequate. The law does\nnot state merely that, if the _same_ cause is repeated, the _same_\neffect will result. It states rather that there is a constant relation\nbetween causes of certain kinds and effects of certain kinds. For\nexample, if a body falls freely, there is a constant relation between\nthe height through which it falls and the time it takes in falling. It\nis not necessary to have a body fall through the _same_ height which has\nbeen previously observed, in order to be able to foretell the length of\ntime occupied in falling. If this were necessary, no prediction would be\npossible, since it would be impossible to make the height exactly the\nsame on two occasions. Similarly, the attraction which the sun will\nexert on the earth is not only known at distances for which it has been\nobserved, but at all distances, because it is known to vary as the\ninverse square of the distance. In fact, what is found to be repeated is\nalways the _relation_ of cause and effect, not the cause itself; all\nthat is necessary as regards the cause is that it should be of the same\n_kind_ (in the relevant respect) as earlier causes whose effects have\nbeen observed.\n\nAnother respect in which Bergson's statement of causation is inadequate\nis in its assumption that the cause must be _one_ event, whereas it may\nbe two or more events, or even some continuous process. The substantive\nquestion at issue is whether mental events are determined by the past.\nNow in such a case as the repeated reading of a poem, it is obvious that\nour feelings in reading the poem are most emphatically dependent upon\nthe past, but not upon one single event in the past. All our previous\nreadings of the poem must be included in the cause. But we easily\nperceive a certain law according to which the effect varies as the\nprevious readings increase in number, and in fact Bergson himself\ntacitly assumes such a law. We decide at last not to read the poem\nagain, because we know that this time the effect would be boredom. We\nmay not know all the niceties and shades of the boredom we should feel,\nbut we know enough to guide our decision, and the prophecy of boredom is\nnone the less true for being more or less general. Thus the kinds of\ncases upon which Bergson relies are insufficient to show the\nimpossibility of prediction in the only sense in which prediction has\npractical or emotional interest. We may therefore leave the\nconsideration of his arguments and address ourselves to the problem\ndirectly.\n\nThe law of causation, according to which later events can theoretically\nbe predicted by means of earlier events, has often been held to be _a\npriori_, a necessity of thought, a category without which science would\nbe impossible. These claims seem to me excessive. In certain directions\nthe law has been verified empirically, and in other directions there is\nno positive evidence against it. But science can use it where it has\nbeen found to be true, without being forced into any assumption as to\nits truth in other fields. We cannot, therefore, feel any _a priori_\ncertainty that causation must apply to human volitions.\n\nThe question how far human volitions are subject to causal laws is a\npurely empirical one. Empirically it seems plain that the great majority\nof our volitions have causes, but it cannot, on this account, be held\nnecessarily certain that all have causes. There are, however, precisely\nthe same kinds of reasons for regarding it as probable that they all\nhave causes as there are in the case of physical events.\n\nWe may suppose--though this is doubtful--that there are laws of\ncorrelation of the mental and the physical, in virtue of which, given\nthe state of all the matter in the world, and therefore of all the\nbrains and living organisms, the state of all the minds in the world\ncould be inferred, while conversely the state of all the matter in the\nworld could be inferred if the state of all the minds were given. It is\nobvious that there is _some_ degree of correlation between brain and\nmind, and it is impossible to say how complete it may be. This, however,\nis not the point which I wish to elicit. What I wish to urge is that,\neven if we admit the most extreme claims of determinism and of\ncorrelation of mind and brain, still the consequences inimical to what\nis worth preserving in free will do not follow. The belief that they\nfollow results, I think, entirely from the assimilation of causes to\nvolitions, and from the notion that causes _compel_ their effects in\nsome sense analogous to that in which a human authority can compel a man\nto do what he would rather not do. This assimilation, as soon as the\ntrue nature of scientific causal laws is realised, is seen to be a sheer\nmistake. But this brings us to the second of the two questions which we\nraised in regard to free will, namely, whether, assuming determinism,\nour actions can be in any proper sense regarded as compelled by outside\nforces.\n\n(2) _Are human actions subject to an external compulsion?_ We have, in\ndeliberation, a subjective sense of freedom, which is sometimes alleged\nagainst the view that volitions have causes. This sense of freedom,\nhowever, is only a sense that we can choose which we please of a number\nof alternatives: it does not show us that there is no causal connection\nbetween what we please to choose and our previous history. The supposed\ninconsistency of these two springs from the habit of conceiving causes\nas analogous to volitions--a habit which often survives unconsciously in\nthose who intend to conceive causes in a more scientific manner. If a\ncause is analogous to a volition, outside causes will be analogous to an\nalien will, and acts predictable from outside causes will be subject to\ncompulsion. But this view of cause is one to which science lends no\ncountenance. Causes, we have seen, do not _compel_ their effects, any\nmore than effects _compel_ their causes. There is a mutual relation, so\nthat either can be inferred from the other. When the geologist infers\nthe past state of the earth from its present state, we should not say\nthat the present state _compels_ the past state to have been what it\nwas; yet it renders it necessary as a consequence of the data, in the\nonly sense in which effects are rendered necessary by their causes. The\ndifference which we _feel_, in this respect, between causes and effects\nis a mere confusion due to the fact that we remember past events but do\nnot happen to have memory of the future.\n\nThe apparent indeterminateness of the future, upon which some advocates\nof free will rely, is merely a result of our ignorance. It is plain that\nno desirable kind of free will can be dependent simply upon our\nignorance; for if that were the case, animals would be more free than\nmen, and savages than civilised people. Free will in any valuable sense\nmust be compatible with the fullest knowledge. Now, quite apart from any\nassumption as to causality, it is obvious that complete knowledge would\nembrace the future as well as the past. Our knowledge of the past is not\nwholly based upon causal inferences, but is partly derived from memory.\nIt is a mere accident that we have no memory of the future. We might--as\nin the pretended visions of seers--see future events immediately, in the\nway in which we see past events. They certainly will be what they will\nbe, and are in this sense just as determined as the past. If we saw\nfuture events in the same immediate way in which we see past events,\nwhat kind of free will would still be possible? Such a kind would be\nwholly independent of determinism: it could not be contrary to even the\nmost entirely universal reign of causality. And such a kind must contain\nwhatever is worth having in free will, since it is impossible to believe\nthat mere ignorance can be the essential condition of any good thing.\nLet us therefore imagine a set of beings who know the whole future with\nabsolute certainty, and let us ask ourselves whether they could have\nanything that we should call free will.\n\nSuch beings as we are imagining would not have to wait for the event in\norder to know what decision they were going to adopt on some future\noccasion. They would know now what their volitions were going to be. But\nwould they have any reason to regret this knowledge? Surely not, unless\nthe foreseen volitions were in themselves regrettable. And it is less\nlikely that the foreseen volitions would be regrettable if the steps\nwhich would lead to them were also foreseen. It is difficult not to\nsuppose that what is foreseen is fated, and must happen however much it\nmay be dreaded. But human actions are the outcome of desire, and no\nforeseeing can be true unless it takes account of desire. A foreseen\nvolition will have to be one which does not become odious through being\nforeseen. The beings we are imagining would easily come to know the\ncausal connections of volitions, and therefore their volitions would be\nbetter calculated to satisfy their desires than ours are. Since\nvolitions are the outcome of desires, a prevision of volitions contrary\nto desires could not be a true one. It must be remembered that the\nsupposed prevision would not create the future any more than memory\ncreates the past. We do not think we were necessarily not free in the\npast, merely because we can now remember our past volitions. Similarly,\nwe might be free in the future, even if we could now see what our future\nvolitions were going to be. Freedom, in short, in any valuable sense,\ndemands only that our volitions shall be, as they are, the result of our\nown desires, not of an outside force compelling us to will what we would\nrather not will. Everything else is confusion of thought, due to the\nfeeling that knowledge _compels_ the happening of what it knows when\nthis is future, though it is at once obvious that knowledge has no such\npower in regard to the past. Free will, therefore, is true in the only\nform which is important; and the desire for other forms is a mere effect\nof insufficient analysis.\n\n                   *       *       *       *       *\n\nWhat has been said on philosophical method in the foregoing lectures has\nbeen rather by means of illustrations in particular cases than by means\nof general precepts. Nothing of any value can be said on method except\nthrough examples; but now, at the end of our course, we may collect\ncertain general maxims which may possibly be a help in acquiring a\nphilosophical habit of mind and a guide in looking for solutions of\nphilosophic problems.\n\nPhilosophy does not become scientific by making use of other sciences,\nin the kind of way in which (_e.g._) Herbert Spencer does. Philosophy\naims at what is _general_, and the special sciences, however they may\n_suggest_ large generalisations, cannot make them certain. And a hasty\ngeneralisation, such as Spencer's generalisation of evolution, is none\nthe less hasty because what is generalised is the latest scientific\ntheory. Philosophy is a study apart from the other sciences: its results\ncannot be established by the other sciences, and conversely must not be\nsuch as some other science might conceivably contradict. Prophecies as\nto the future of the universe, for example, are not the business of\nphilosophy; whether the universe is progressive, retrograde, or\nstationary, it is not for the philosopher to say.\n\nIn order to become a scientific philosopher, a certain peculiar mental\ndiscipline is required. There must be present, first of all, the desire\nto know philosophical truth, and this desire must be sufficiently strong\nto survive through years when there seems no hope of its finding any\nsatisfaction. The desire to know philosophical truth is very rare--in\nits purity, it is not often found even among philosophers. It is\nobscured sometimes--particularly after long periods of fruitless\nsearch--by the desire to _think_ we know. Some plausible opinion\npresents itself, and by turning our attention away from the objections\nto it, or merely by not making great efforts to find objections to it,\nwe may obtain the comfort of believing it, although, if we had resisted\nthe wish for comfort, we should have come to see that the opinion was\nfalse. Again the desire for unadulterated truth is often obscured, in\nprofessional philosophers, by love of system: the one little fact which\nwill not come inside the philosopher's edifice has to be pushed and\ntortured until it seems to consent. Yet the one little fact is more\nlikely to be important for the future than the system with which it is\ninconsistent. Pythagoras invented a system which fitted admirably with\nall the facts he knew, except the incommensurability of the diagonal of\na square and the side; this one little fact stood out, and remained a\nfact even after Hippasos of Metapontion was drowned for revealing it. To\nus, the discovery of this fact is the chief claim of Pythagoras to\nimmortality, while his system has become a matter of merely historical\ncuriosity.[57] Love of system, therefore, and the system-maker's vanity\nwhich becomes associated with it, are among the snares that the student\nof philosophy must guard against.\n\n  [57] The above remarks, for purposes of illustration, adopt one of\n  several possible opinions on each of several disputed points.\n\nThe desire to establish this or that result, or generally to discover\nevidence for agreeable results, of whatever kind, has of course been the\nchief obstacle to honest philosophising. So strangely perverted do men\nbecome by unrecognised passions, that a determination in advance to\narrive at this or that conclusion is generally regarded as a mark of\nvirtue, and those whose studies lead to an opposite conclusion are\nthought to be wicked. No doubt it is commoner to wish to arrive at an\nagreeable result than to wish to arrive at a true result. But only those\nin whom the desire to arrive at a _true_ result is paramount can hope to\nserve any good purpose by the study of philosophy.\n\nBut even when the desire to know exists in the requisite strength, the\nmental vision by which abstract truth is recognised is hard to\ndistinguish from vivid imaginability and consonance with mental habits.\nIt is necessary to practise methodological doubt, like Descartes, in\norder to loosen the hold of mental habits; and it is necessary to\ncultivate logical imagination, in order to have a number of hypotheses\nat command, and not to be the slave of the one which common sense has\nrendered easy to imagine. These two processes, of doubting the familiar\nand imagining the unfamiliar, are correlative, and form the chief part\nof the mental training required for a philosopher.\n\nThe naive beliefs which we find in ourselves when we first begin the\nprocess of philosophic reflection may turn out, in the end, to be almost\nall capable of a true interpretation; but they ought all, before being\nadmitted into philosophy, to undergo the ordeal of sceptical criticism.\nUntil they have gone through this ordeal, they are mere blind habits,\nways of behaving rather than intellectual convictions. And although it\nmay be that a majority will pass the test, we may be pretty sure that\nsome will not, and that a serious readjustment of our outlook ought to\nresult. In order to break the dominion of habit, we must do our best to\ndoubt the senses, reason, morals, everything in short. In some\ndirections, doubt will be found possible; in others, it will be checked\nby that direct vision of abstract truth upon which the possibility of\nphilosophical knowledge depends.\n\nAt the same time, and as an essential aid to the direct perception of\nthe truth, it is necessary to acquire fertility in imagining abstract\nhypotheses. This is, I think, what has most of all been lacking hitherto\nin philosophy. So meagre was the logical apparatus that all the\nhypotheses philosophers could imagine were found to be inconsistent with\nthe facts. Too often this state of things led to the adoption of heroic\nmeasures, such as a wholesale denial of the facts, when an imagination\nbetter stocked with logical tools would have found a key to unlock the\nmystery. It is in this way that the study of logic becomes the central\nstudy in philosophy: it gives the method of research in philosophy, just\nas mathematics gives the method in physics. And as physics, which, from\nPlato to the Renaissance, was as unprogressive, dim, and superstitious\nas philosophy, became a science through Galileo's fresh observation of\nfacts and subsequent mathematical manipulation, so philosophy, in our\nown day, is becoming scientific through the simultaneous acquisition of\nnew facts and logical methods.\n\nIn spite, however, of the new possibility of progress in philosophy, the\nfirst effect, as in the case of physics, is to diminish very greatly the\nextent of what is thought to be known. Before Galileo, people believed\nthemselves possessed of immense knowledge on all the most interesting\nquestions in physics. He established certain facts as to the way in\nwhich bodies fall, not very interesting on their own account, but of\nquite immeasurable interest as examples of real knowledge and of a new\nmethod whose future fruitfulness he himself divined. But his few facts\nsufficed to destroy the whole vast system of supposed knowledge handed\ndown from Aristotle, as even the palest morning sun suffices to\nextinguish the stars. So in philosophy: though some have believed one\nsystem, and others another, almost all have been of opinion that a great\ndeal was known; but all this supposed knowledge in the traditional\nsystems must be swept away, and a new beginning must be made, which we\nshall esteem fortunate indeed if it can attain results comparable to\nGalileo's law of falling bodies.\n\nBy the practice of methodological doubt, if it is genuine and prolonged,\na certain humility as to our knowledge is induced: we become glad to\nknow _anything_ in philosophy, however seemingly trivial. Philosophy has\nsuffered from the lack of this kind of modesty. It has made the mistake\nof attacking the interesting problems at once, instead of proceeding\npatiently and slowly, accumulating whatever solid knowledge was\nobtainable, and trusting the great problems to the future. Men of\nscience are not ashamed of what is intrinsically trivial, if its\nconsequences are likely to be important; the _immediate_ outcome of an\nexperiment is hardly ever interesting on its own account. So in\nphilosophy, it is often desirable to expend time and care on matters\nwhich, judged alone, might seem frivolous, for it is often only through\nthe consideration of such matters that the greater problems can be\napproached.\n\nWhen our problem has been selected, and the necessary mental discipline\nhas been acquired, the method to be pursued is fairly uniform. The big\nproblems which provoke philosophical inquiry are found, on examination,\nto be complex, and to depend upon a number of component problems,\nusually more abstract than those of which they are the components. It\nwill generally be found that all our initial data, all the facts that we\nseem to know to begin with, suffer from vagueness, confusion, and\ncomplexity. Current philosophical ideas share these defects; it is\ntherefore necessary to create an apparatus of precise conceptions as\ngeneral and as free from complexity as possible, before the data can be\nanalysed into the kind of premisses which philosophy aims at\ndiscovering. In this process of analysis, the source of difficulty is\ntracked further and further back, growing at each stage more abstract,\nmore refined, more difficult to apprehend. Usually it will be found that\na number of these extraordinarily abstract questions underlie any one of\nthe big obvious problems. When everything has been done that can be done\nby method, a stage is reached where only direct philosophic vision can\ncarry matters further. Here only genius will avail. What is wanted, as a\nrule, is some new effort of logical imagination, some glimpse of a\npossibility never conceived before, and then the direct perception that\nthis possibility is realised in the case in question. Failure to think\nof the right possibility leaves insoluble difficulties, balanced\narguments pro and con, utter bewilderment and despair. But the right\npossibility, as a rule, when once conceived, justifies itself swiftly by\nits astonishing power of absorbing apparently conflicting facts. From\nthis point onward, the work of the philosopher is synthetic and\ncomparatively easy; it is in the very last stage of the analysis that\nthe real difficulty consists.\n\nOf the prospect of progress in philosophy, it would be rash to speak\nwith confidence. Many of the traditional problems of philosophy, perhaps\nmost of those which have interested a wider circle than that of\ntechnical students, do not appear to be soluble by scientific methods.\nJust as astronomy lost much of its human interest when it ceased to be\nastrology, so philosophy must lose in attractiveness as it grows less\nprodigal of promises. But to the large and still growing body of men\nengaged in the pursuit of science--men who hitherto, not without\njustification, have turned aside from philosophy with a certain\ncontempt--the new method, successful already in such time-honoured\nproblems as number, infinity, continuity, space and time, should make an\nappeal which the older methods have wholly failed to make. Physics, with\nits principle of relativity and its revolutionary investigations into\nthe nature of matter, is feeling the need for that kind of novelty in\nfundamental hypotheses which scientific philosophy aims at facilitating.\nThe one and only condition, I believe, which is necessary in order to\nsecure for philosophy in the near future an achievement surpassing all\nthat has hitherto been accomplished by philosophers, is the creation of\na school of men with scientific training and philosophical interests,\nunhampered by the traditions of the past, and not misled by the literary\nmethods of those who copy the ancients in all except their merits.\n\n\n\n\nINDEX\n\n\n  Absolute, 6, 39.\n\n  Abstraction, principle of, 42, 124 ff.\n\n  Achilles, Zeno's argument of, 173.\n\n  Acquaintance, 25, 144.\n\n  Activity, 224 ff.\n\n  Allman, 161 n.\n\n  Analysis, 185, 204, 211, 241.\n    legitimacy of, 150.\n\n  Anaximander, 3.\n\n  Antinomies, Kant's, 155 ff.\n\n  Aquinas, 10.\n\n  Aristotle, 40, 160 n., 161 ff., 240.\n\n  Arrow, Zeno's argument of, 173.\n\n  Assertion, 52.\n\n  Atomism, logical, 4.\n\n  Atomists, 160.\n\n\n  Belief, 58.\n    primitive and derivative, 69 ff.\n\n  Bergson, 4, 11, 13, 20 ff., 137, 138, 150, 158, 165, 174, 178, 229 ff.\n\n  Berkeley, 63, 64, 102.\n\n  Bolzano, 165.\n\n  Boole, 40.\n\n  Bradley, 6, 39, 165.\n\n  Broad, 172 n.\n\n  Brochard, 169 n.\n\n  Burnet, 19 n., 160 n., 161 n., 170 n., 171 ff.\n\n\n  Calderon, 95.\n\n  Cantor, vi, vii, 155, 165, 190, 194, 199.\n\n  Categories, 38.\n\n  Causal laws, 109, 212 ff.\n    evidence for, 216 ff.\n    in psychology, 219.\n\n  Causation, 34 ff., 79, 212 ff.\n    law of, 221.\n    not _a priori_, 223, 232.\n\n  Cause, 220, 223.\n\n  Certainty, degrees of, 67, 68, 212.\n\n  Change,\n    demands analysis, 151.\n\n  Cinematograph, 148, 174.\n\n  Classes, 202.\n    non-existence of, 205 ff.\n\n  Classical tradition, 3 ff., 58.\n\n  Complexity, 145, 157 ff.\n\n  Compulsion, 229, 233 ff.\n\n  Congruence, 195.\n\n  Consecutiveness, 134.\n\n  Conservation, 105.\n\n  Constituents of facts, 51, 145.\n\n  Construction _v._ inference, iv.\n\n  Contemporaries, initial, 119, 120 n.\n\n  Continuity, 64, 129 ff., 141 ff., 155 ff.\n    of change, 106, 108, 130 ff.\n\n  Correlation of mental and physical, 233.\n\n  Counting, 164, 181, 187 ff., 203.\n\n  Couturat, 40 n.\n\n\n  Dante, 10.\n\n  Darwin, 4, 11, 23, 30.\n\n  Data, 65 ff., 211.\n    \"hard\" and \"soft,\" 70 ff.\n\n  Dates, 117.\n\n  Definition, 204.\n\n  Descartes, 5, 73, 238.\n\n  Descriptions, 201, 214.\n\n  Desire, 227, 235.\n\n  Determinism, 233.\n\n  Doubt, 237.\n\n  Dreams, 85, 93.\n\n  Duration, 146, 149.\n\n\n  Earlier and later, 116.\n\n  Effect, 220.\n\n  Eleatics, 19.\n\n  Empiricism, 37, 222.\n\n  Enclosure, 114 ff., 120.\n\n  Enumeration, 202.\n\n  Euclid, 160, 164.\n\n  Evellin, 169.\n\n  Evolutionism, 4, 11 ff.\n\n  Extension, 146, 149.\n\n  External world, knowledge of, 63 ff.\n\n\n  Fact, 51.\n    atomic, 52.\n\n  Finalism, 13.\n\n  Form, logical, 42 ff., 185, 208.\n\n  Fractions, 132, 179.\n\n  Free will, 213, 227 ff.\n\n  Frege, 5, 40, 199 ff.\n\n\n  Galileo, 4, 59, 192, 194, 239, 240.\n\n  Gaye, 169 n., 175, 177.\n\n  Geometry, 5.\n\n  Giles, 206 n.\n\n  Greater and less, 195.\n\n\n  Harvard, 4.\n\n  Hegel, 3, 37 ff., 46, 166.\n\n  \"Here,\" 73, 92.\n\n  Hereditary properties, 195.\n\n  Hippasos, 163, 237.\n\n  Hui Tzu, 206.\n\n  Hume, 217, 221.\n\n  Hypotheses in philosophy, 239.\n\n\n  Illusions, 85.\n\n  Incommensurables, 162 ff., 237.\n\n  Independence, 73, 74.\n    causal and logical, 74, 75.\n\n  Indiscernibility, 141, 148.\n\n  Indivisibles, 160.\n\n  Induction, 34, 222.\n    mathematical, 195 ff.\n\n  Inductiveness, 190, 195 ff.\n\n  Inference, 44, 54.\n\n  Infinite, vi, 64, 133, 149.\n    historically considered, 155 ff.\n    \"true,\" 179, 180.\n    positive theory of, 185 ff.\n\n  Infinitesimals, 135.\n\n  Instants, 116 ff., 129, 151, 216.\n    defined, 118.\n\n  Instinct _v._ Reason, 20 ff.\n\n  Intellect, 22 ff.\n\n  Intelligence, how displayed by friends, 93.\n    inadequacy of display, 96.\n\n  Interpretation, 144.\n\n\n  James, 4, 10, 13.\n\n  Jourdain, 165 n.\n\n  Jowett, 167.\n\n  Judgment, 58.\n\n\n  Kant, 3, 112, 116, 155 ff., 200.\n\n  Knowledge about, 144.\n\n\n  Language, bad, 82, 135.\n\n  Laplace, 12.\n\n  Laws of nature, 218 ff.\n\n  Leibniz, 13, 40, 87, 186, 191.\n\n  Logic, 201.\n    analytic not constructive, 8.\n    Aristotelian, 5.\n    and fact, 53.\n    inductive, 34, 222.\n    mathematical, vi, 40 ff.\n    mystical, 46.\n    and philosophy, 8, 33 ff., 239.\n\n  Logical constants, 208, 213.\n\n\n  Mach, 123, 224.\n\n  Macran, 39 n.\n\n  Mathematics, 40, 57.\n\n  Matter, 75, 101 ff.\n    permanence of, 102 ff.\n\n  Measurement, 164.\n\n  Memory, 230, 234, 236.\n\n  Method, deductive, 5.\n    logical-analytic, v, 65, 211, 236 ff.\n\n  Milhaud, 168 n., 169 n.\n\n  Mill, 34, 200.\n\n  Montaigne, 28.\n\n  Motion, 130, 216.\n    continuous, 133, 136.\n    mathematical theory of, 133.\n    perception of, 137 ff.\n    Zeno's arguments on, 168 ff.\n\n  Mysticism, 19, 46, 63, 95.\n\n\n  Newton, 30, 146.\n\n  Nietzsche, 10, 11.\n\n  Noel, 169.\n\n  Number, cardinal, 131, 186 ff.\n    defined, 199 ff.\n    finite, 160, 190 ff.\n    inductive, 197.\n    infinite, 178, 180, 188 ff., 197.\n    reflexive, 190 ff.\n\n\n  Occam, 107, 146.\n\n  One and many, 167, 170.\n\n  Order, 131.\n\n\n  Parmenides, 63, 165 ff., 178.\n\n  Past and future, 224, 234 ff.\n\n  Peano, 40.\n\n  Perspectives, 88 ff., 111.\n\n  Philoponus, 171 n.\n\n  Philosophy and ethics, 26 ff.\n    and mathematics, 185 ff.\n    province of, 17, 26, 185, 236.\n    scientific, 11, 16, 18, 29, 236 ff.\n\n  Physics, 101 ff., 147, 239, 242.\n    descriptive, 224.\n    verifiability of, 81, 110.\n\n  Place, 86, 90.\n    _at_ and _from_, 92.\n\n  Plato, 4, 19, 27, 46, 63, 165 n., 166, 167.\n\n  Poincare, 123, 141.\n\n  Points, 113 ff., 129, 158.\n    definition of, vi, 115.\n\n  Pragmatism, 11.\n\n  Prantl, 174.\n\n  Predictability, 229 ff.\n\n  Premisses, 211.\n\n  Probability, 36.\n\n  Propositions, 52.\n    atomic, 52.\n    general, 55.\n    molecular, 54.\n\n  Pythagoras, 19, 160 ff., 237.\n\n\n  Race-course, Zeno's argument of, 171 ff.\n\n  Realism, new, 6.\n\n  Reflexiveness, 190 ff.\n\n  Relations, 45.\n    asymmetrical, 47.\n    Bradley's reasons against, 6.\n    external, 150.\n    intransitive, 48.\n    multiple, 50.\n    one-one, 203.\n    reality of, 49.\n    symmetrical, 47, 124.\n    transitive, 48, 124.\n\n  Relativity, 103, 242.\n\n  Repetitions, 230 ff.\n\n  Rest, 136.\n\n  Ritter and Preller, 161 n.\n\n  Robertson, D. S., 160 n.\n\n  Rousseau, 20.\n\n  Royce, 50.\n\n\n  Santayana, 46.\n\n  Scepticism, 66, 67.\n\n  Seeing double, 86.\n\n  Self, 73.\n\n  Sensation, 25, 75, 123.\n    and stimulus, 139.\n\n  Sense-data, 56, 63, 67, 75, 110, 141, 143, 213.\n    and physics, v, 64, 81, 97, 101 ff., 140.\n    infinitely numerous? 149, 159.\n\n  Sense-perception, 53.\n\n  Series, 49.\n    compact, 132, 142, 178.\n    continuous, 131, 132.\n\n  Sigwart, 187.\n\n  Simplicius, 170 n.\n\n  Simultaneity, 116.\n\n  Space, 73, 88, 103, 112 ff., 130.\n    absolute and relative, 146, 159.\n    antinomies of, 155 ff.\n    perception of, 68.\n    of perspectives, 88 ff.\n    private, 89, 90.\n    of touch and sight, 78, 113.\n\n  Spencer, 4, 12, 236.\n\n  Spinoza, 46, 166.\n\n  Stadium, Zeno's argument of, 134 n., 175 ff.\n\n  Subject-predicate, 45.\n\n  Synthesis, 157, 185.\n\n\n  Tannery, Paul, 169 n.\n\n  Teleology, 223.\n\n  Testimony, 67, 72, 82, 87, 96, 212.\n\n  Thales, 3.\n\n  Thing-in-itself, 75, 84.\n\n  Things, 89 ff., 104 ff., 213.\n\n  Time, 103, 116 ff., 130, 155 ff., 166, 215.\n    absolute or relative, 146.\n    local, 103.\n    private, 121.\n\n\n  Uniformities, 217.\n\n  Unity, organic, 9.\n\n  Universal and particular, 39 n.\n\n\n  Volition, 223 ff.\n\n\n  Whitehead, vi, 207.\n\n  Wittgenstein, vii, 208 n.\n\n  Worlds, actual and ideal, 111.\n    possible, 186.\n    private, 88.\n\n\n  Zeller, 173.\n\n  Zeno, 129, 134, 136, 165 ff.\n\n\n    PRINTED BY NEILL AND CO., LTD., EDINBURGH.\n\n\n\n\n  [ Transcriber's Note:\n\n    The following is a list of corrections made to the original.\n    The first line is the original line, the second the corrected one.\n\n    Second Impression\n    Second Impression.\n\n  more often than the impossibility of alternatives which seemed _prima\n  more often than the impossibility of alternatives which seemed _prima\n\n  with it. We will call these the \"initial contemporaries of the given\n  with it. We will call these the \"initial contemporaries\" of the given\n\n  the next are liable to fail when we come to infinite number. The first\n  the next are liable to fail when we come to infinite numbers. The first\n\n  of psychology or an outcome of pscyhical processes as the North Sea....\n  of psychology or an outcome of psychical processes as the North Sea....\n\n  according to the length and direction of the interval. \"A quarter of a\n  according to the length and direction of the interval. \"A quarter of an\n\n    Intelligence, how displayed by friends, 93\n    Intelligence, how displayed by friends, 93.\n\n    Number, cardinal, 131 186 ff.\n    Number, cardinal, 131, 186 ff.\n\n  ]\n\n\n\n\n\n\n<pre id=\"pg-footer\">","title":"Our Knowledge of the External World as a Field for Scientific Method in Philosophy","tradition":"philosophy-ethics","translated_text":null,"word_count":71494}