b OS 'Si 4 CORNELL' UNIVERSITY LIBRARY BOUGHT WITH THE INCOME OF THE SAGE ENDOWACENT FUND GIVEN IN 1891 BY HENRY WILLIAMS SAGE Date Due y^i\ m ^^iiig mt-Br -PTTT m S-%r ik^ -mrt ^^ p^g^g f - \ The instrument of discovery throughout is modern logic, a very different science from the logic of the text-books . and also from the logic of idealism. Our second lecture has given a short account of modern logic and of its points of divergence from the various traditional kinds of logic. In our last lecture, after a discussion of causality and free will, we shall try to reach a general account of the logical-analytic method of scientific philosophy, and a tentative estimate of the hopes of philosophical progress which it allows us to entertain. In this lecture, I wish to apply the logical-analytic method to one of the oldest problems of philosophy, namely, the problem of our knowledge of the external world. What I have to say on this problem does not amount to an answer of a definite and dogmatic kind ; it amounts only to an analysis and statement of the questions involved, with an indication of the directions in which evidence may be sought. But although not yet a definite solution, what can be said at present seems to me to throw a completely new light on the problem, and to be indispensable, not only in seeking the answer, but also in the preliminary question as to what parts of our problem may possibly have an ascertainable answer. In every philosophical problem, our investigation starts from what may be called "data," by which I mean matters of common knowledge, vague, complex, inexact, as common knowledge always is, but yet some- how commanding our assent as on the whole and in some interpretation pretty certainly true. In the case of our /■ 5 66 SCIENTIFIC METHOD IN PHILOSOPHY present problem, the common knowledge involved is of various kinds. There is first our acquaintance with particular objects of daily life — furniture, houses, towns, other people, and so on. Then there is the extension of such particular knowledge to particular things outside our personal experience, through history and geography, newspapers, etc. And lastly, there is the systematisation of all this knowledge of particulars by means of physical science, which derives immense persuasive force from its astonishing power of foretelling the future. We are quite willing to admit that there may be errors of detail in this knowledge, but we believe them to be discoverable and corrigible by the methods which have given rise to our beliefs, and we do not, as practical men, entertain for a moment the hypothesis that the whole edifice may be built on insecure foundations. In the main, therefore, and without absolute dogmatism as to this or that special portion, we may accept this mass of common knowledge as affording data for our philosophical analysis. It may be said — and this is an objection which must be met at the outset — that it is the duty of the philosopher to call in question the admittedly fallible beliefs of daily life, and to replace them by something more solid and irrefragable. In a sense this is true, and in a sense it is effected in the course of analysis. But in another sense, and a very important one, it is quite impossible. While admitting that doubt is possible with regard to all our common knowledge, we must nevertheless accept that knowledge in the main if philosophy is to be possible at all. There is not any superfine brand of knowledge, obtainable by the philosopher, which can give us a stand- point from which to criticise the whole of the knowledge of daily life. The most that can be done is to examine and purify our common knowledge by an internal scrutiny, THE EXTERNAL WORLD 67 assuming the canons by which it has been obtained, and applying them with more care and with more precision. Philosophy cannot boast of having achieved such a degree of certainty that it can have authority to condemn the facts of experience and the laws of science. The philo- sophic scrutiny, therefore, though sceptical in regard to every detail, is not sceptical as regards the whole. That is to say, its criticism of details will only be based upon their relation to other details, not upon some external criterion which can be applied to all the details equally. The reason for this abstention from a universal criticism is not any dogmatic confidence, but its exact opposite ; it is not that common knowledge must be true, but that we possess no radically different kind of knowledge derived from some other source. Universal scepticism, though logically irrefutable, is practically barren ; it can only, therefore, give a certain flavour of hesitancy to our beliefs, and cannot be used to substitute other beliefs for them. Although data can only be criticised by other data, not by an outside standard, yet we may distinguish different grades of certainty in the different kinds of common knowledge which we enumerated just now. ^What does not go beyond our own personal sensible acquaintance must be for us the most certain : the " evidence (^f the senses " is proverbially the least open to question.^What depends on testimony, like the facts of history and geography which are learnt from books, has varying degrees of certainty according to the nature and extent of the testimony. Doubts as to the existence of Napoleon can only be maintained for a joke, whereas the historicity of Agamemnon is a legitimate subject of debate. In science, again, we find all grades of certainty short of the highest. The law of gravitation, at least 68 SCIENTIFIC METHOD IN PHILOSOPHY as an approximate truth, has acquired by this time the same kind of certainty as the existence of Napoleon, whereas the latest speculations concerning the constitu- tion of matter would be universally acknowledged to have as yet only a rather slight probability in their favour. These varying degrees of certainty attaching to different data may be regarded as themselves forming part of our data ; they, along with the other data, lie within the vague, complex, inexact body of knowledge which it is the business of the philosopher to analyse. The first thing that appears when we begin to analyse our common knowledge is that some of it is derivative, while some is primitive ; that is to say, there is some that we only believe because of something else from which it has been inferred in some sense, though not necessarily in a strict logical sense, while other parts are believed on their own account, without the support of any outside evidence. It is obvious that the senses give know- ledge of the latter kind : the immediate facts perceived by sight or touch or hearing do not need to be proved by argument, but are completely self-evident. Psychologists, however, have made us aware that what is actually given in sense is much less than most people would naturally suppose, and that much of what at first sight seems to be given is really inferred. This applies especially in regard to our space-perceptions. For instance, we instinctively infer the "real" size and shape of a visible object from its apparent size and shape, according to its distance and our point of view. When we hear a person speaking, our actual sensations usually miss a great deal of what he says, and we supply its place by unconscious inference ; in a foreign language, where this process is more difficult, we find ourselves apparently grown deaf, requiring, for example, to be much nearer the stage at a theatre than THE EXTERNAL WORLD 69 would be necessary in our own country. /Thus the first step in the analysis of data, namely, the discovery of what is really given in sense, is full of difficulty^' "We will, however, not linger on this point ; so long as its existence is realised, the exact outcome does not make any very great difference in our main problem. The next step in our analysis must be the consideration * of how the derivative parts of our common knowledge arise. Here we become involved in a somewhat puzzling ^ entanglement of logic and psychology. Psychologically, a belief may be called derivative whenever it is caused by one or more other beliefs, or by some fact of sense which is not simply what the belief asserts. Derivative beliefs in this sense constantly arise without any process of logical inference, merely by association of ideas or some equally extra-logical process. From the expression of a man's face we judge as to what he is feeling : we say we see that he is angry, when in fact we only see a frown. We do not judge as to his state of mind by any logical process : the judgment grows up, often without our being able to say what physical mark of emotion we actually saw. In such a case, the knowledge is derivative psychologically ; but logically it is in a sense primitive, since it is not the result of any logical deduction. There may or may not be a possible deduction leading to the same result, but whether there is or not, we certainly do not employ it. If we call a belief " logically primitive " ' when it is not actually arrived at by a logical inference, then innumerable beliefs are logically primitive which psychologically are derivative. The separation of these two kinds of primitiveness is vitally important to our present discussion. When we reflect upon the beliefs which are logically but not psychologically primitive, we find that, unless yo SCIENTIFIC METHOD IN PHILOSOPHY they can on reflection be deduced by a logical process from beliefs which are also psychologically primitive, our confidence in their truth tends to diminish the more we think about them. We naturally believe, for example, that tables and chairs, trees and mountains, are still there when we turn our backs upon them. I do not wish for a moment to maintain that this is certainly not the case, but I do maintain that the question whether it is the case is not to be settled ofF-hand on any supposed ground of [obviousness. The belief that they persist is, in all men except a few philosophers, logically primitive, but it is not psychologically primitive ; psychologically, it arises only through our having seen those tables and chairs, trees and mountains. As soon as the question is seriously raised whether, because we have seen them, we have a right to suppose that they are there still, we feel that some kind of argument must be produced, and that if none is forthcoming, our belief can be no more than a pious opinion. We do not feel this as regards the immediate objects of sense : there they are, and as far as their momentary existence is concerned, no further argument is required. There is accordingly more need of justifying our psychologically derivative beliefs than of justifying those that are primitive. " We are thus led to a somewhat vague distinction between what we may call " hard " data and " soft " data. This distinction is a matter of degree, and must not be pressed ; but if not taken too seriously it may help to make the situation clear. I mean by " hard " data those which resist the solvent influence of critical reflection, and by " soft " data those which, under the operation of this process, become to our minds more or less doubtful. The hardest of hard data are of two sorts : the particular facts of sense, and the general truths of THE EXTERNAL WORLD 71 logic. The more we refloat upon these, the more we realise exactly what they are, and exactly what a doubt concerning them really means, the more luminously J certain do they become. Verbal doubt concerning even these is possible, but verbal doubt may occur when what is nominally being doubted is not really in our thoughts, and only words are actually present to our minds. Real doubt, in these two cases, would, I think, be pathological. At any rate, to me they seem quite certain, and I shall assume that you agree with me in this. Without this assumption, we are in danger of falling into that universal scepticism which, as we saw, is as barren as it is irrefutable. 1 If we are to continue philosophising, we must make our bow to the sceptical hypothesis, and, while admitting the elegant terseness of its philosophy, proceed to the con- sideration of other hypotheses which, though perhaps not certain, have at least as good a right to our respect as the hypothesis of the sceptic. Applying our distinction of "hard" and "soft" data to psychologically derivative but logically primitive beliefs, we shall find that most, if not all, are to be classed as soft data. They may be found, on reflection, to be capable of logical proof, and they then again become believed, > but no longer as data. As data, though entitled to a certain limited respect, they cannot be placed on a level with the facts of sense or the laws of logic. The kind of respect which they deserve seems to me such as to warrant us in hoping, though not too confidently, that the hard data may prove them to be at least probable. Also, if the hard data are found to throw no light whatever upon their truth or falsehood, we are justified, I think, in giving rather more weight to the hypothesis of their truth than to the hypothesis of their falsehood. For the present, however, let us confine ourselves to the hard 72 SCIENTIFIC METHOD IN PHILOSOPHY data, with a view to discovering what sort of world can be constructed by their means alone. Our data now are primarily the facts of sense {i.e. of our own sense-data) and the laws of logic. But even the severest scrutiny will allow some additions to this slender stock. Some facts of memory — especially of recent memory — seem to have the highest degree of certainty. Some introspective facts are as certain as any facts of sense. And facts of sense themselves must, for our present purposes, be interpreted with a certain latitude. Spatial and temporal relations must sometimes be included, for example in the case of a swift motion falling wholly within the specious present. And some facts of com- parison, such as the likeness or unlikeness of two shades of colour, are certainly to be included among hard data. Also we must remember that the distinction of hard and soft data is psychological and subjective, so that, if there are other minds than our own— which at our present stage must be held doubtful — the catalogue of hard data may be different for them from what it is for us. Certain common beliefs are undoubtedly excluded from hard data. Such is the belief which led us to introduce the distinction, namely, that sensible objects in general persist when we are not perceiving them. Such also is the belief in other people's minds : this belief is obviously derivative from our perception of their bodies, and is felt to demand logical justification as soon 'as we become aware of its derivativeness. Belief in what is reported by the testimony of others, including all that we learn from books, is of course involved in the doubt as to whether other people have minds at all. Thus the world from which our reconstruction is to begin is very fragmentary. The best we can say for It is that it is slightly more extensive than the world at THE EXTERNAL WORLD 73 which Descartes arrived by a similar process, since that world contained nothing except himself and his thoughts. We are now in a position to understand and state the problem of our knowledge of the external world, and to remove various misunderstandings which have obscured the meaning of the problem. The problem really is : ^' Can the existence of anything other than our own hard / data be inferred from the existence of those data ? But before considering this problem, let us briefly consider what the problem is not. When we speak of the " external " world in this dis- cussion, we must not mean "spatially external," unless " space " is interpreted in a peculiar and recondite manner. The immediate objects of sight, the coloured' surfaces which make up the visible world, are spatially . external in the natural meaning of this phrase. We feer them to be " there " as opposed to " here " ; without making any assumption of an existence other than hard data, we can more or less estimate the distance of a coloured surface. It seems probable that distances, provided they are not too great, are actually given more or less roughly in sight ; but whether this is the case or not, ordinary distances can certainly be estimated ap- proximately by means of the data of sense alone. The immediately given world is spatial, and is further not wholly contained within our own bodies. Thus our knowledge of what is external in this sense is not open to doubt. , Another form in which the question is often put is : " Can we know of the existence of any reality which is independent of ourselves ? " This form of the question sufFers from the ambiguity of the two words "inde- pendent " and " self." To take the Self first : the 74 SCIENTIFIC METHOD IN PHILOSOPHY question as to what is to be reckoned part of the Self and what is not, is a very difficult one. Among many other things which we may mean by the Self, two may be selected as specially important, namely, (i) the bare subject which thinks and is aware of objects, (2) the whole assemblage of things that would necessarily cease to exist if our lives came to an end. The bare subject, if it exists at all, is an inference, and is not part of the data ; therefore this meaning of Self may be ignored in our present inquiry. The second meaning is difficult to make precise, since we hardly know what things depend upon our lives for their existence. And in this form, the definition of Self introduces the word " depend," which raises the same questions as are raised by the word "Independent." Let us therefore take up the word " Independent," and return to the Self later. When we say that one thing is "independent" of another, we may mean either that it is logically possible for the one to exist without the other, or that there Is no causal relation between the two such that the one only occurs as the effect of the other. The only way, so far as I know. In which one thing can be logically dependent upon another is when the other Is part of the one. The existence of a book, for example, is logically dependent upon that of Its pages : without the pages there would be no book. Thus In this sense the question, " Can we know of the existence of any reality which is independent of ourselves .'' " reduces to the question, " Can we know of the existence of any reality of which our Self is not part .'' " In this form, the question brings us back to the problem of defining the Self ; but I think, however the Self may be defined, even when It is taken as the bare subject, it cannot be supposed to be part of the Immediate object of sense ; thus in this form of the question we THE EXTERNAL WORLD 75 must admit that we can know of the existence of realities independent of ourselves. The question of causal dependence is much more difficult. To know that one kind of thing is causally inde- pendent of another, we must know that it actually occurs without the other. Now it is fairly obvious that, whatever legitimate meaning we give to the Self, our thoughts and feelings are causally dependent upon ourselves, i.e. do not occur when there is no Self for them to belong to. But in the case of objects of sense this is not obvious ;' indeed, as we saw, the common-sense view is that such| objects persist in the absence of any percipient. If this is the case, then they are causally independent of our- selves ; if not, not. Thus in this form the question' reduces to the question whether we can know that objects of sense, or any other objects not our own thoughts and feelings, exist at times when we are not perceiving thern^j This form, in which the difficult word " independent " no longer occurs, is the form in which we stated the problem a minute ago. Our question in the above form raises two distinct problems, which it is important to keep separate. First, l can we know that objects of sense, or very 'similar objects, exist at times when we are not perceiving them } Secondly, if this cannot be known, can we know that " other objects, inferable from objects of sense but not necessarily resembling them, exist either when we are perceiving the objects of sense or at any other time ? This latter problem arises in philosophy as the problem of the " thing in itself," and in science as the problem of matter as assumed in physics. We will consider this latter problem first. Owing to the fact that we feel passive in sensation, we naturally suppose that our sensations have outside catises. 76 SCIENTIFIC METHOD IN PHILOSOPHY Now it is necessary here first of all to distinguish between (i) our sensation, which is a mental event consisting in our being aware of a sensible object, and (2) the sensible obiect of which we are aware in sensation. When I speak of the sensible object, it must be understood that I do not mean such a thing as a table, which is both visible and tangible, can be seen by many people at once, and is more or less permanent. What I mean is just that patch of colour which is momentarily seen when we look at the table, or just that particular hardness which is felt when we press it, or just that particular sound which is heard when we rap it. Each of these I call a sensible object, and our awareness of it I call a sensation. Now our sense of passivity, if it really afforded any argument, would only tend to show that the sensation has an outside cause ; this cause we should naturally seek in the sensible object. Thus there is no good reason, so far, for supposing that sensible objects must have outside causes. But both the thing-in-itself of philosophy and the matter of physics present themselves as outside causes of the sensible object as much as of the sensation. What are the grounds for this common opinion .'' In each case, I think, the opinion has resxilted from the combination of a belief that something which can persist independently of our consciousness makes itself known in sensation, with the fact that our sensations often change in ways which seem to depend upon us rather than upon anything which would be supposed to persist independ- ently of us. At first, we believe unreflectingly that everything is as it seems to be, and that, if we shut our eyes, the objects we had been seeing remain as they were though we no longer see them. But there are arguments against this view, which have generally been thought THE EXTERNAL WORLD 77 conclusive. It is extraordinarily difficult to see just what the arguments prove ; but if we are to make any progress with the problem of the external world, we must try to make up our minds as to these arguments. A table viewed from one place presents a different appearance from that which it presents from another place. This is the language of common sense, but this language already assumes that there is a real table of which we see the appearances. Let us try to state what is known in terms of sensible objects alone, without any element of hypothesis. We find that as we walk round the table, we perceive a series of gradually changing visible objects. But in speaking of " walking round the table," we have still retained the hypothesis that there is a single table connected with all the appearances. What we ought to say is that, while we have those muscular and other sensations which make us say we are walking, our visual sensations change in a continuous way, so that, for example, a striking patch of colour is not suddenly replaced by something wholly different, but is replaced by an insensible gradation of slightly different colours with slightly different shapes. This is what we really; know by experience, when we have freed our minds from ' the assumption of permanent " things " with changing appearances. What is really known is a correlation of muscular and other bodily sensations with changes in visual sensations. But walking round the table is not the only way of altering its appearance. We can shut one eye, or put on blue spectacles, or look through a microscope. All these operations, in various ways, alter the visual appearance which we call that of the table. More distant objects will also alter their appearance if (as we say) the state of the atmosphere changes — if there is fog or rain or sun- 78 SCIENTIFIC METHOD IN PHILOSOPHY shine. Physiological changes also alter the appearances of things. If we assume the world of common sense, all these changes, including those attributed to physio- logical causes, are changes in thf intervening medium. It is not quite so easy as in the former case to reduce this set of facts to a form in which nothing is assumed beyond sensible objects. Anything inter-wening between ourselves and what we see must be invisibje : our view in every direction is bounded by the nearest visible object. It might be objected that a dirty pane of glass, for example, is visible although we can see things through it. But in this case we really see a spotted patchwork : the dirtier specks in the glass are visible, while the cleaner parts are invisible and allow us to see what is beyond. Thus the discovery that the intervening medium affects the appear- ances of things cannot be made by means of the sense of sight alone. Let us take the case of the blue spectacles, which is the simplest, but may serve as a type for the others. The frame of the spectacles is of course visible, but the blue glass, if it is clean, is not visible. The blueness, which we say is in the glass, appears as being in the objects seen through the glass. The glass itself is known by means of the sense of touch. In order to know that it is between us and the objects seen through it, we must know how to correlate the space of touch with the space of sight. This correlation itself, when stated in terms of the data of sense alone, is by no means a simple matter. But it presents no difficulties of principle, and may therefore be supposed accomplished. When it has been accomplished, it becomes possible to attach a mean- ing to the statement that the blue glass, which we can touch, is between us and the object seen, as we say, " through " it. THE EXTERNAL WORLD 79 But we have still not reduced our statement completely to what is actually given in sense. We have fallen into the assumption that the object of which we are conscious when we touch the blue spectacles still exists after we have ceased to touch them. So long as we are touching them, nothing except our finger can be seen through the part touched, which is the only part where we immediately know that there is something. If we are to account for the blue appearance of objects other than the spectacles, when seen through them, it might seem as if we must assume that the spectacles still exist when we are not touching them ; and if this assumption really is neces- sary, our main problem is answered : we have means of knowing of the present existence of objects not given in sense, though of the same kind as objects formerly given in sense. It may be questioned, however, whether this assump- tion is actually unavoidable, though it is unquestionably the most natural one to make. We may say that the object of which we become aware when we touch the spectacles continues to have effects afterwards, though perhaps it no longer exists. In this view, the supposed continued existence of sensible objects after they have ceased to be sensible will be a fallacious inference from the fact that they still have effects. It is often supposed that nothing which has ceased to exist can continue to have effects, but this is a mere prejudice, due to a wrong conception of causality. We cannot, therefore, dismiss our present hypothesis on the ground of a priori impossi- bility, but must examine further whether it can really account for the facts. It may be said that our hypothesis is useless in the case when the blue glass is never touched at all. How, in that case, are we to account for the blue appearance of 8o SCIENTIFIC METHOD IN PHILOSOPHY objects ? And more generally, what are we to make of the hypothetical sensations of touch which we associate with untouched visible objects, which we know would be verified if we chose, though in fact we do not verify them ? Must not these be attributed to permanent possession, by the objects, of the properties which touch would reveal ? ,/ Let us consider the more general question first. Experience has taught us that where we see certain kinds of coloured surfaces we can, by touch, obtain certain expected sensations of hardness or softness, tactile shape, t and so on. This leads us to believe that what is seen is usually tangible, and that it has, whether we touch it or not, the hardness or softness which we should expect to feel if we touched it. But the mere fact that we are able to infer what our tactile sensations would be shows that it is not logically necessary to assume tactile qualities before they are felt. All that is really known is that the ■ visual appearance in question, together with touch, will lead to certain sensations, which can necessarily be determined in terms of the visual appearance, since ■ otherwise they could not be inferred from it. We can now give a statement of the experienced facts concerning the blue spectacles, which will supply an interpretation of common-sense beliefs without assuming anything beyond the existence of sensible objects at the I times when they are sensible. By experience of the ! correlation of touch and sight sensations, we become able "■ ■ to associate a certain place in touch-space with a certain corresponding place in sight-space. Sometimes, namely in the case of transparent things, we find that there is a tangible object in a touch-place without there being any visible object in the corresponding sight-place. But in such a case as that of the blue spectacles, we find that THE EXTERNAL WORLD 8i whatever object is visible' beyond the empty sight-place in the same line of sight has a different colour from what it has when there is no tangible object in the intervening touch-place ; and as we move the tangible object in touch-space, the blue patch moves in sight-space. If now we find a blue patch moving in this way in sight- space, when we have no sensible experience of an intervening tangible object, we nevertheless infer that, if we put our hand at a certain place in touch-space, we should experience a certain touch-sensation. If we are to avoid non-sensible objects, this must be taken as the whole of our meaning when we say that the blue spectacles are in a certain place, though we have not touched them, and have only seen other things rendered ' blue by their interposition. v,- I think it may be laid down quite generally that, in so far as physics or common sense is verifiable, it must be capable of interpretation in terms of actual sense-data alone. The reason for this is simple. Verification con- sists always in the occurrence of an expected sense-datum. Astronomers teU us there will be an eclipse of the moon : we look at the moon, and find the earth's shadow biting into it, that is to say, we see an appearance quite different from that of the usual full moon. Now if an expected sense-datum constitutes a verification, what was asserted must have been about sense-data ; or, at any rate, if part of what was asserted was not about sense-data, then only the other part has been verified. There is in fact a certain regularity 9r conformity to law about the occurrence of sense-data, but the sense-data that occur at one time are often causally connected with those that occur at quite other times, and not, or at least not very closely, with those that occur at neighbouring times. If I look at the moon and immediately afterwards hear a 6 82 SCIENTIFIC METHOD IN PHILOSOPHY train coming, there is no very close causal connection between my two sense-data ; but if I look at the moon on two nights a week apart, there is a very close causal connection between the two sense-data. The simplest, or at least the easiest, statement of the connection is obtained by imagining a "real" moon which goes on whether I look at it or not, providing a series of possible sense-data of which only those are actual which belong to moments when I choose to look at the moon. But the degree of verification obtainable in this way is very small. It must be remembered that, at our present level of doubt, we are not at liberty to accept testimony. When we hear certain noises, which are those we should utter if we wished to express a certain thought, we assume that that thought, or one very like it, has been in another mind, and has given rise to the expression which we hear. If at the same time we see a body resembling our own, moving its lips as we move ours when we speak, we cannot resist the belief that it is alive, and that the feelings inside it continue when we are not looking at it. When we see our friend drop a weight upon his toe, and hear him say — what we should say in similar circumstances, the phenomena can no doubt be explained without assuming that he is anything but a series of shapes and noises seen and heard by us, but practically no man is so infected with philosophy as not to be quite certain that his friend has felt the same kind of pain as he himself would feel. We will consider the legitimacy of this belief presently ; for the moment, I only wish to point out that it needs the same kind of justification as our belief that the moon exists when we do not see it, and that, without it, testimony heard or read is reduced to noises and shapes, and cannot be regarded as evidence of the facts which it reports. The verification of physics THE EXTERNAL WORLD 83 which is possible at our present level is, therefore, only that degree of verification which is possible by one man's unaided observations, which will not carry us very far towards the establishment of a whole science. Before proceeding further, let us summarise the argu- ment so far as it has gone. The problem is : " Can the existence of anything other than our own hard data be inferred from these data ? " It is a mistake to state the problem in the form : " Can we know of the existence of anything other than ourselves and our states ? " or : " Can we know of the existence of anything independent of ourselves ? " because of the extreme difficulty of defining " self " and " independent " precisely. The felt passivity of sensation is irrelevant, since, even if it proved anything, it could only prove that sensations are caused by sensible objects. The natural naUve belief is that things seen persist, when unseen, exactly or approximately as they appeared when seen ; but this belief tends to be dispelled by the fact that what common sense regards as the appear- ] ance of one object changes with what common sense regards as changes in the point of view and in the inter- vening medium, including in the latter our own sense- ^ organs and nerves and brain. This fact, as just stated, assumes, however, the common-sense world of stable objects which it professes to call in question ; hence, before we can discover its precise bearing on our problem, we must find a way of stating it which does not involve any of the assumptions which it is designed to render doubtful. What we then find, as the bare outcome of experience, is that gradual changes in certain sense-data are correlated with gradual changes in certain others, or (in the case of bodily motions) with the other sense-data themselves. ■ . The assumption that sensible objects persist after they 84 SCIENTIFIC METHOD IN PHILOSOPHY have ceased to be sensible — for example, that the hard- ness * of a visible body, which has been discovered by- touch, continues when the body is no longer touched — may be replaced by the statement that the effects of sensible objects persist, i.e. that what happens now can only be accounted for, in many cases, by taking account of what happened at an earlier time. Everything that one man, by his own personal experience, can verify in the account of the world given by common sense and physics, will be explicable by some such means, since verification consists merely in the occurrence of an ex- pected sense-datum. But what depends upon testimony, whether heard or read, cannot be explained in this way, since testimony depends upon the existence of minds other than our own, and thus requires a knowledge of something not given in sense. But^ before examining the question of our knowledge of other minds, let us return to the question of the thing-in-itself, namely, to the theory that what "exists at times when we are not perceiving a given sensible object is something quite unlike that object, something which, together with us and our sense-organs, causes our sensations, but is never itself given in sensation. The thing-in-itself, when we start from common-sense assumptions, is a fairly natural outcome of the difficulties due to the changing appearances of what is supposed to be one object. It is supposed that the table (for example) causes our sense-data of sight and touch, but must, since these are altered by the point of view and the intervening medium, be quite different from the sense-data to which it gives rise. There is, in this theory, a tendency to a confusion from which it derives some of its plausibility, namely, the confusion between a sensation as a psychical occurrence and its object. A patch of colour, even if it THE EXTERNAL WORLD 85 only exists when it is seen, is still something quite different from the seeing of it : the seeing of it is mental, but the patch of colour is not. This confusion, however, can be avoided without our necessarily abandoning the theory we are examining. The objection to it, I think, lies in its failure to realise the radical nature of the recon- struction demanded by the difficulties to which it points. We cannot speak legitimately of changes in the point of view and the intervening medium until we have already constructed some world more stable than that of moment- ary sensation. Our discussion of the blue spectacles and the walk round the table has, I hope, made this clear. But what remains far from clear is the nature of the) reconstruction required. Although we cannot rest content with the above theory, in the terms in which it is stated, we must nevertheless treat it with a certain respect, for it is in outline the theory upon which physical science and physiology are built, and it must, therefore, be susceptible of a true interpretation. Let us see how this is to be done. The first thing to realise is that there are no such ' things as " illusions of sense." Objects of sense, even when they occur in dreams, are the most indubitably real , objects known to us. What, then, makes us call them unreal in dreams .? Merely the unusual nature of their connection with other objects of sense. I dream that I am in America, but I wake up and find myself in England without those intervening days on the Atlantic which, alas ! are inseparably connected with a " real " visit to America. Objects of sense are called " real " when they have the kind of connection with other objects of sense \ which experience has led us to regard as normal ; when they fail in this, they are called " illusions." But what ' is illusory is only the inferences to which they give rise ; 86 SCIENTIFIC METHOD IN PHILOSOPHY in themselves, they are every bit as real as the objects of waking life. And conversely, the sensible objects of waking life must not be expected to have any more ■. intrinsic reality than those of dreams. Dreams and \ waking life, in our first efforts at construction, must be /treated with equal respect ; it is only by some reality not merely sensible that dreams can be condemned. Accepting the indubitable momentary reality of objects of sense, the next thing to notice is the confusion under- lying objections derived from their changeableness. As we walk round the table, its aspect changes ; but it is thought impossible to maintain either that the table changes, or that its various aspects can all " really " exist in the same place. If we press one eyeball, we shall see two tables ; but it is thought preposterous to maintain that there are "really" two tables. Such arguments, however, seem to involve the assumption that there can be something more real than objects of sense. If we see two tables, then there are two visual tables. It is perfectly true that, at the same moment, we may discover by touch that there is only one tactile table. This makes us declare the two visual tables an illusion, because usually one visual object corresponds to one tactile object. But all that we are warranted in saying is that, in this case, the manner of correlation of touch and sight is unusual. Again, when the aspect of the table changes as we walk round it, and we are told there cannot be so many different aspects in the same place, the answer is simple : what does the critic of the table mean by " the same place " } The use of such a phrase presupposes that all our difficulties have been solved ; as yet, we have no right to speak of a " place " except with reference to one given set of momentary sense-data. When all are changed by a bodily movement, no place remains the THE EXTERNAL WORLD 87 same as It was. Thus the difficulty, if it exists, has at least not been rightly stated. We will now make a new start, adopting a different method. Instead of inquiring what is the minimum of assumption by which we can explain the world of sense, we will, in order to have a model hypothesis as a help for the imagination, construct one possible (not necessary) explanation of the facts. It may perhaps then be possible to pare away what is superfluous in our hypothesis, leaving a residue which may be regarded as the abstract answer to our problem. Let us imagine that each mind looks out upon the world, as in Leibniz's monadology, from a point of view peculiar to itself ; and for the sake of simplicity let us confine ourselves to the sense of sight, ignoring minds which are devoid of this sense. Each mind sees at each moment an immensely complex three-dimensional world ; but there is absolutely nothing which is seen by two minds simultaneously. When we say that two people^ see the same thing, we always find that, owing to difference of point of view, there are differences, how- ^ ever slight, between their immediate sensible objects. (I am here assuming the validity of testimony, but as we are only constructing a possible theory, that is a legitimate assumption.) The three-dimensional world seen by one mind therefore contains no place in common with that seen by another, for places can only be constituted by the things in or around them. Hence we may suppose, in spite of the differences between the different worlds, that each exists entire exactly as it is perceived, and might be exactly as it is even if it were not perceived. We may further suppose that there are an infinite number of such worlds which are in fact unperceived. If two men are sitting in a room, two somewhat similar worlds are 88 SCIENTIFIC METHOD IN PHILOSOPHY perceived by them ; If a third man enters and sits between them, a third world, intermediate between the two previous worlds, begins to be perceived. It is true that we cannot reasonably suppose just this world to have existed before, because it is conditioned by the sense- organs, nerves, and brain of the newly arrived man ; but we can reasonably suppose that some aspect of the universe existed from that point of view, though no one was perceiving it. The system consisting of all views of the universe perceived and unperceived, I shall call the system of " perspectives " ; I shall confine the expression "private worlds" to such views of the universe as are actually perceived. Thus a " private world " is a perceived " perspective " ; but there may be any number of unperceived perspectives. Two men are sometimes found to perceive very similar perspectives, so similar that they can use the same words to describe them. They say they see the same table, because the differences between the two tables they see are slight and not practically • nportant. Thus it is possible, sometimes, to establish a correlation by similarity between a great many of the things of one perspective, and a great many of the things of another. In case the similarity is very great, we say the points of view of the two perspectives are near together in space ; but this space in which they are near together is totally different from the spaces inside the two perspectives. It is a relation between the perspectives, and is not in either of them ; no one can perceive it, and if it is to be known it can be only by inference. Between two perceived per- spectives which are similar, we can imagine a whole series of other perspectives, some at least unperceived, and such that between any two, however similar, there are others still more similar. In this way the space which consists THE EXTERNAL WORLD 89 of relations between perspectives can be rendered con- tinuous, and (if we choose) three-dimensional. We can now define the momentary common-sense " thing," as opposed to its momentary appearances. By the similarity of neighbouring perspectives, many objects in the one can be correlated with objects in the other, namely, with the similar objects. Given an object in one perspective, form the system of all the objects correlated with it in all the perspestives ; that system may be identified with the momentary common-sense "thing." Thus an aspect of a " thing " is a member of the system of aspects which is the " thing " at that moment. (The correlation of the times of diiFerent perspectives raises certain complications, of the kind considered in the theory of relativity ; but we may ignore these at present.) All the aspects of a thing are real, whereas the thing is a mere logical construction. It has, however, the merit of being neutral as between different points of view, and of being visible to more than one person, in the only sense in which it can ever be visible, namely, in the sense that each sees one of its aspects. It will be observed that, while each perspective contains its own space, there is only one space in which the'i perspectives themselves are the elements. There are as ! many private spaces as there are perspectives ; there are therefore at least as many as there are percipients, and there may be any number of others which have a merely material existence and are not seen by anyone. But there is only one perspective-space, whose elements are single perspectives, each with its own private space. We have now to explain how the private space of a single perspective is correlated with part of the one all-embracing perspective space. Perspective space is the system of " points of view " 90 SCIENTIFIC METHOD IN PHILOSOPHY of private spaces (perspectives), or, since "points of view" have not been defined, we may say it is the system of the private spaces themselves. These private spaces will each count as one point, or at any rate as one element, in perspective space. They are ordered by means of their similarities. Suppose, for example, that we start from one which contains the appearance of a circular disc, such as would be called a penny, and suppose this appearance, in the perspective in question, is circular, not elliptic. We can then form a whole series of perspectives containing a graduated series of circular aspects of varying sizes : for this purpose we only have to move (as we say) towards the penny or away from it. The perspectives in which the penny looks circular will be said to lie on a straight line in perspective space, and their order on this line will be that of the sizes of the circular aspects. Moreover — though this statement must be noticed and subsequently examined — the perspectives in which the penny looks big will be said to be nearer to the penny than those in which it looks small. It is to be re- marked also that any other " thing " than our penny might have been chosen to define the relations of our perspectives in perspective space, and that experience shows that the same spatial order of perspectives would have resulted. In order to explain the correlation of private spaces with perspective space, we have first to explain what is meant by " the place (in perspective space) where a thing is." For this purpose, let us again consider the penny which appears in many perspectives. We formed a straight line of perspectives in which the penny looked circular, and we agreed that those in which it looked larger were to be considered as nearer to the penny. We can form another straight line of perspectives in which the penny is seen end-on and looks like a straight line of THE EXTERNAL WORLD 91 a certain thickness. These two lines will meet in a certain place in perspective space, i.e. in a certain perspective, which may be defined as " the place (in perspective space) where the penny is." It is true that, in order to prolong our lines until they reach this place, we shall have to make use of other things besides the penny, because, so far as experience goes, the penny ceases to present any appearance after we have come so near to it that it touches the eye. But this raises no real difficulty, because the spatial order of perspectives is found empirically to be independent of the particular " things " chosen for defining the order. We can, for example, remove our penny and prolong each of our two straight lines up to their inter- section by placing other pennies further ofF in such a way that the aspects of the one are circular where those of our original penny were circular, and the aspects of the other are straight where those of our original penny were straight. There will then be just one perspective in which one of the new pennies looks circular and the other straight. This will be, by definition, the place where the original penny was in perspective space. The above is, of course, only a first rough sketch of the way in which our definition is to be reached. It neglects the size of the penny, and it assumes that we can remove the penny without being disturbed by any simultaneous changes in the positions of other things. But it is plain that such niceties cannot affect the principle, and can only introduce complications in its application. Having now defined the perspective which is the place where a given thing is, we can understand what is meant by saying that the perspectives in which a thing looks large are nearer to the thing than those in which it looks small : they are, in fact, nearer to the perspective which is the place where the thing is. 92 SCIENTIFIC METHOD IN PHILOSOPHY We can now also explain the correlation between a private space and parts of perspective space. If there is an aspect of a given thing in a certain private space, then we correlate the place where this aspect is in the private space with the place where the thing is in perspective space. \' We may define "here" as the place, in perspective space, which is occupied by our private world. Thus we can now understand what is meant by speaking of a thing as near to or far from " here." A thing is near to " here " if the place where it is is near to my private world. We can also understand what is meant by saying that our private world is inside our head ; for our private world is a place in perspective space, and may be part of the place where our head is. It will be observed that two places in perspective space are associated with every aspect of a thing : namely, the place where the thing is, and the place which is the perspective of which the aspect in question forms part. Every aspect of a thing is a member of two difFerent classes of aspects, namely : (i) the various aspects of the thing, of which at most one appears in any given per- spective ; (2) the perspective of which the given aspect is a member, i.e. that in which the thing has the given aspect. The physicist naturally classifies aspects in the first way, the psychologist in the second. The two places associated with a single aspect correspond to the two ways of classifying it. We may distinguish the two places as that at which, and that from which, the aspect appears. The " place at which " is the place of the thing to which the aspect belongs ; the " place from which " is the place of the perspective to which the aspect belongs. Let us now endeavour to state the fact that the aspect which a thing presents at a given place is affected by the THE EXTERNAL WORLD 93 intervening medium. The aspects of a thing in different perspectives are to be conceived as spreading outwards from the place where the thing is, and undergoing various changes as they get further away from this place. The laws according to which they change cannot be stated if we only take account of the aspects that are near the thing, but require that we should also take account of the things that are at the places from which these aspects appear. This empirical fact can, therefore, be interpreted in terms of our construction. . .-3^t We have now constructed a largely hypothetical picture of the world, which contains and places the experienced facts, including those derived from testimony. The world we have constructed can, with a certain amount of trouble, be used to interpret the crude facts of sense, the facts of physics, and the facts of physiology. It is there- fore a world which may be actual. It fits the facts, and there is no empirical evidence against it ; it also is free : from logical impossibilities. But have we any good reason to suppose that it is real .'' This brings us back to our original problem, as to the grounds for believing in the existence of anything outside my private world. What we have derived from our hypothetical construction is that there are no grounds against the truth of this belief, but we have not derived any positive grounds in its favour. We will resume this inquiry by taking up again the question of testimony and the evidence for the existence of other minds. -"^ It must be conceded to begin with that the argument in favour of the existence of other people's minds cannot be conclusive. A phantasm of our dreams will appear to have a mind — a mind to be annoying, as a rule. It will give unexpected answers, refuse to conform to our desires, and show all those other signs of intelligence to 94 SCIENTIFIC METHOD IN PHILOSOPHY which we are accustomed in the acquaintances of our waking hours. And yet, when we are awake, we do not believe that the phantasm was, like the appearances of people in waking life, representative of a private world to which we have no direct access. If we are to believe this of the people we meet when we are awake, it must be on some ground short of demonstration, since it is obviously possible that what we call waking life may be only an unusually persistent and recurrent nightmare. It may be that our imagination brings forth all that other people seem to say to us, all that we read in books, all the daily, weekly, monthly, and quarterly journals that distract our thoughts, all the advertisements of soap and all the speeches of politicians. This may be true, since it cannot be shown to be false, yet no one can really believe it. Is there any logical ground for regarding this possibifity as improbable .'' Or is there nothing beyond habit and prejudice .'' The minds of other people are among our data, in the very wide sense in which we used the word at first. That is to say, when we first begin to reflect, we find ourselves already believing in them, not because of any argument, but because the belief is natural to us. It is, however, a psychologically derivative belief, since it results from observation of people's bodies ; and along with other such beliefs, it does not belong to the hardest of hard data, but becomes, under the influence of philosophic reflection, just sufficiently questionable to make us desire some argument connecting it with the facts of sense. The obvious argument is, of course, derived from analogy. Other people's bodies behave as ours do when we have certain thoughts and feelings ; hence, by analogy, it is natural to suppose that such behaviour is connected with thoughts and feelings like our own. Someone says, THE EXTERNAL WORLD 95 " Look out ! " and we find we are on the point of being killed by a motor-car ; we therefore attribute the words we heard to the person in question having seen the motor- car first, in which case there are existing things of which we are not directly conscious. But this whole scene, with our inference, may occur in a dream, in which case the inference is generally considered to be mistaken. Is there anything to make the argument from analogy more cogent when we are (as we think) awake ? The analogy in waking life is only to be preferred to that in dreams on the ground of its greater extent and consistency. If a man were to dream every night about a set of people whom he never met by day, who had consistent characters and grew older with the lapse of years, he might, like the man in Calderon's play, find it difficult to decide which was the dream-world and which was the so-called "real" world. It is only the failure of our dreams to form a consistent whole either with each other or with waking life that makes us condemn them. Certain uniformities are observed in waking life, while dreams seem quite erratic. The natural hypothesis would be that demons and the spirits of the dead visit us while we sleep ; but the modern mind, as a rule, refuses to entertain this view, though it is hard to see what could be said against it. On the other hand, the mystic, in moments of illumination, seems to awaken from a sleep which has filled all his mundane life : the whole world of sense becomes phantasmal, and he sees, with the clarity and convincingness that belongs to our morning realisa- tion after dreams, a world utterly different from that of our daily cares and troubles. Who shall condemn him ? Who shall justify him ? Or who shall justify the seeming solidity of the common objects among which we suppose ourselves to live ? 96 SCIENTIFIC METHOD IN PHILOSOPHY The hypothesis that other people have minds must, I think, be allowed to be not susceptible of any very strong support from the analogical argument. At the same time, it is a hypothesis which systematises a vast body of facts and never leads to any consequences which there is reason to think false. There is therefore nothing to be said against its truth, and good reason to use it as a working hypothesis. When once it is admitted, it en- ables us to extend our knowledge of the sensible world by testimony, and thus leads to the system of private worlds which we assumed in our hypothetical construc- tion. In actual fact, whatever we may try to think as philosophers, we cannot help believing in the minds of other people, so that the question whether our belief is justified has a merely speculative interest. And if it is justified, then there is no further difficulty of principle in that vast extension of our knowledge, beyond our own private data, which we find in science and common sense. This somewhat meagre conclusion must not be regarded as the whole outcome of our long discussion. The problem of the connection of sense with objective reality has commonly been dealt with from a standpoint which did not carry initial doubt so far as we have carried it ; most writers, consciously or unconsciously, have assumed that the testimony of others is to be admitted, and there- fore (at least by implication) that others have minds. Their difficulties have arisen after this admission, from the differences in the appearance which one physical object presents to two people at the same time, or to one person at two times between which it cannot be supposed to have changed. Such difficulties have made people doubtful how far objective reality could be known by sense at all, and have made them suppose that there were positive arguments against the view that it can be so THE EXTERNAL WORLD 97 known. Our hypothetical construction meets these arguments, and shows that the account of the world given by common sense and physical science can be interpreted in a way which is logically unobjectionable, and finds a place for all the data, both hard and soft. It is this hypothetical construction, with its reconciliation of psychology and physics, which is the chief outcome of our discussion. Probably the construction is only in part necessary as an initial assumption, and can be ob- tained from more slender materials by the logical methods of which we shall have an example in the definitions of points, instants, and particles ; but I do not yet know to what lengths this diminution in our initial assumptions can be carried. LECTURE IV THE WORLD OF PHYSICS AND THE WORLD OF SENSE LECTURE IV THE WORLD OF PHYSICS AND THE WORLD OF SENSE Among the objections to the reality of objects of sense, there is one which is derived from the apparent difference between matter as it appears in physics and things as they appear in sensation. Men of science, for the most part, are willing to condemn immediate data as " merely sub- jective," while yet maintaining the truth of the physics inferred from those data. But such an attitude, though it may be capable of justification, obviously stands in need of it ; and the only justification possible must be one which exhibits matter as a logical construction from sense- data — unless, indeed, there were some wholly a priori principle by which unknown entities could be inferred from such as are known. It is therefore necessary to find some way of bridging the gulf between the world of physics and the world of sense, and it is this problem which will occupy us in the present lecture. >' Physicists appear to be unconscious of the gulf, while psychologists, who are conscious of it, have not the mathematical know- ledge required for spanning it. The problem is difficult, and I do not know its solution in detail. All that I can hope to do is to make the problem felt, and to indicate the kind of methods by which a solution is to be sought. Let us begin by a brief description of the two con- trasted worlds. We will take first the world of physics, lOI I02 SCIENTIFIC METHOD IN PHILOSOPHY for, though the other world is given while the physical world is inferred, to us now the world of physics is the more familiar, the world of pure sense having become strange and difficult to rediscover. Physics started from the common-sense belief in fairly permanent and fairly rigid bodies — tables and chairs, stones, mountains, the earth and moon and sun. This common-sense belief, it should be noticed, is a piece of audacious metaphysical theorising ; objects are not continually present to sensa- tion, and it may be doubted whether they are there wheq they are not seen or felt. This problem, which has been acute since the time of Berkeley, is ignored by common sense, and has therefore hitherto been ignored by physicists. We have thus here a first departure from the immediate data of sensation, though it is a departure merely by way of extension, and was probably made by our savage ancestors In some very remote prehistoric epoch. But tables and chairs, stones and mountains, are not quite permanent or guite rigid. Tables and chairs lose their legs, stones are split by frost, and mountains are cleft by earthquakes and eruptions. Then there are other things, which seem material, and yet present almost no permanence or rigidity. Breath, smoke, clouds, are examples of such things — so, in a lesser degree, are ice and snow ; and rivers and seas, though fairly permanent, are not in any degree rigid. Breath, smoke, clouds, and generally things that can be seen but not touched, were thought to be hardly real ; to this day the usual mark of a ghost is that it can be seen but not touched. Such objects were peculiar in the fact that they seemed to dis- appear completely, not merely to be transformed into something else. Ice and snow, when they disappear, are replaced by water ; and it required no great theoretical WORLDS OF PHYSICS AND OF SENSE 103 eflFort to invent the hypothesis that the water was the same thing as the ice and snow, but in a new form. Solid bodies, when they break, brealc into parts which are practically the same in shape and size as they were before. A stone can be hammered into a powder, but the powder consists of grains which retain the character they had before the pounding. Thus the ideal of absolutely rigid and absolutely permanent bodies, which early physi- cists pursued throughout the changing appearances, seemed attainable by supposing ordinary bodies to be composed of a vast number of tiny atoms. This billiard-ball view of matter dominated the imagination of physicists until quite modern times, until, in fact, it was replaced by the electromagnetic theory, which in its turn is developing into a new atomism. Apart from the special form of the atomic theory which was invented for the needs of chem- istry, some kind of atomism dominated the whole of traditional dynamics, and was implied in every statement of its laws and axioms. The pictorial accounts which physicists give of the material world as they conceive it undergo violent changes under the influence of modifications in theory which are much slighter than the layman might suppose from the alterations of the description. Certain features, however, have remained fairly stable. It is always assumed that there is something indestructible which is capable of motion in space ; what is indestructible is always very small, but does not always occupy a mere point in space. There is supposed to be one all-embracing space in which the motion takes place, and until lately we might have assumed one all-embracing time also. But the principle of rela- tivity has given prominence to the conception of " local time," and has somewhat diminished men's confidence in the one even-flowing stream of time. Without dogma- I04 SCIENTIFIC METHOD IN PHILOSOPHY tising as to the ultimate outcome of the principle of relativity, however, we may safely say, I think, that it does not destroy the possibility of correlating different local times, and does not therefore have such far-reaching philosophical consequences as is sometimes supposed. In fact, in spite of difficulties as to measurement, the one all-embracing time still, I think, underlies all that physics has to say about motion. We thus have still in physics, as we had in Newton's time, a set of indestructible entities which may be called particles, moving relatively to each other in a single space and a single time. - The world of immediate data is quite different from this. Nothing is permanent; even- the things that we think are fairly permanent, such as mountains, only be- come data when we see them, and are not immediately given as existing at other moments. So far from one all-embracing space being given, there are several spaces for each person, according to the different senses which give relations that may be called spatial. Experience teaches us to obtain one space from these by correlation, and experience, together with instinctive theorising, teaches us to correlate our spaces with those which we believe to exist in the sensible worlds of other people. The con- struction of a single time offers less difficulty so long as we confine ourselves to one person's private world, but the correlation of one private time with another is a matter of great difficulty. Thus, apart from any of the fluctuat- ing hypotheses of physics, three main problems arise in connecting the world of physics with the world of sense, namely (i) the construction of permanent "things," (2) the construction of a single space, and (3) the construc- tion of a single time. We will consider these three problems in succession. (i) The belief in indestructible "things" very early WORLDS OF PHYSICS AND OF SENSE 105 took the form of atomism. The underlying motive In atomism was not, I think, any empirical success In Inter- preting phenomena, but rather an instinctive belief that beneath all the changes of the sensible world there must be something permanent and unchanging. This belief was, no doubt, fostered and nourished by Its practical successes, culminating In the conservation of mass ; but It was not produced by these successes. On the con-j trary, they were produced by It. Philosophical writers on physics sometimes speak as though the conservation of something or other were essential to the possibility of science, but this, I believe, is an entirely erroneous opinion. If the a priori belief in permanence had not existed, the same laws which are now formulated In terms of this belief might just as well have been formulated without it. Why should we suppose that, when ice melts, the water which replaces it is the same thing in a new form } Merely because this supposition enables us to state the phenomena In a way which is consonant with our prejudices. What we really know is that, under' certain conditions of temperature, the appearance we call Ice Is replaced by the appearance we call water. We can give laws according to which the one appearance will be succeeded by the other, but there Is no reason except prejudice for regarding both as appearances of the same substance. One task, if what has just been said is correct, which confronts us in trying to connect the world of sense with the world of physics, Is the task of reconstructing the conception of matter without the a priori beliefs which historically gave rise to it. In spite of the revolutionary results of modern physics, the empirical successes of the conception of matter show that there must be some legiti- mate conception which fulfils roughly the same functions. io6 SCIENTIFIC METHOD IN PHILOSOPHY The time has hardly come when we can state precisely what this legitimate conception is, but we can see in a general way what it must be like. For this purpose, it ' is only necessary to take our ordinary common-sense statements and reword them without the assumption of , permanent substance. We say, for example, that things ' change gradually — sometimes very quickly, but not with- out passing through a continuous series of intermediate states. What this means is that, given any sensible appearance, there will usually be, if we watch, a con- tinuous series of appearances connected with the given one, leading on by imperceptible gradations to the new appearances which common-sense regards as those of the same thing. Thus a thing may be defined as a certain series of appearances, connected with each other by ; continuity and by certain causal laws. In the case of slowly changing things, this is easily seen. Consider, say, a wall-paper which fades in the course of years. It is an effort not to conceive of it as one " thing " whose colour is slightly different at one time from what it is at another. But what do we really know about it .'' We know that under suitable circumstances — i.e. when we are, as is said, " in the room " — we perceive certain colours in a certain pattern : not always precisely the same colours, but sufficiently similar to feel familiar7 If we can state the laws according to which the colour varies, we can state all that is empirically verifiable ; the assump- tion that there is a constant entity, the wall-paper, which " has " these various colours at various times, is a piece Lof gratuitous metaphysics. We may, if we like, define the wall-paper as the series of its aspects. These are collected together by the same motives which led us to regard the wall-paper as one thing, namely a combination of sensible continuity and causal connection. More WORLDS OF PHYSICS AND OF SENSE 107 generally, a " thing " will be defined as a certain series of aspects, namely those which would commonly be said to be of the thing. To say that a certain aspect is an aspect o/a certain thing will merely mean that it is one of those which, taken serially, are the thing. Everything will then proceed as before : whatever was verifiable is unchanged, but our language is so interpreted as to avoid an unnecessary metaphysical assumption of permanence, j The above extrusion of permanent things afFords an example of the maxim which inspires all scientific philo- sophising, namely " Occam's razor " : Entities are not to be multiplied without necessity. In other words, in dealing with any subject-matter, find out what entities are undeniably involved, and state everything in terms of these entities. Very often the resulting statement is more complicated and difficult than one which, like common sense and most philosophy, assumes hypothetical entities whose existence there is no good reason to believe in. We find it easier to imagine a wall-paper with changing colours than to think merely of the series of colours ; but it is a mistake to suppose that what is easy and natural in thought is what is most free from un- warrantable assumptions, as the case of " things " very aptly illustrates. The above summary account of the genesis of " things," though it may be correct in outline, has omitted some serious difficulties which it is necessary briefly to consider. Starting from a world of helter-skelter sense-data, we' wish to collect them into series, each of which can be regarded as consisting of the successive appearances of one " thing." There is, to begin with, some conflict between what common sense regards as one thing, and what physics regards an unchanging collection of particles. To common sense, a human body is one thing, but to io8 SCIENTIFIC METHOD IN PHILOSOPHY science the matter composing it is continually changing. This conflict, however, is not very serious, and may, for our rough preliminary purpose, be largely ignored. The problem is : by what principles shall we select certain ' data from the chaos, and call them all appearances of the ; same thing ? A rough and approximate answer to this question is not very difficult. There are certain fairly stable collec- tions of appearances, such as landscapes, the furniture of rooms, the faces of acquaintances. In these cases, we have little hesitation in regarding them on successive occasions as appearances of one thing or collection of things. But, as the Comedy of Errors illustrates, we may be led astray if we judge by mere resemblance. This shows that something more is involved, for two different things may have any degree of likeness up to exact similarity. Another insufficient criterion of one thing is continuity. As we have already seen, if we watch what we regard as one changing thing, we usually find its changes to be con- tinuous so far as our senses can perceive. We are thus led to assume that, if we see two finitely diffi;rent appear- ances at two different times, and if we have reason to regard them as belonging to the same thing, then there was a continuous series of intermediate states of that thing during the time when we were not observing it. And so it comes to be thought that continuity of change is necessary and sufficient to constitute one thing. But in fact it is neither. It is not necessary^ because the unobserved states, in the case where our attention has not been concentrated on the thing throughout, are purely hypothetical, and cannot possibly be our ground for supposing the earlier and later appearances to belong to the same thing ; on the contrary, it is because we sup- WORLDS OF PHYSICS AND OF SENSE 109 pose this that we assume intermediate unobserved states. Continuity is also not sufficient, since we can, for example, pass by sensibly continuous gradations from'any one drop of the sea to any other drop. The utmost we can say is that discontinuity during uninterrupted observation is as a rule a mark of difference between things, though even this cannot be said in such cases as sudden explosions. The assumption of continuity is, however, successfully made in physics. This proves something, though not anything of very obvious utility to our present problem : it proves that nothing in the known world is inconsistent with the hypothesis that all changes are really continuous, though from too great rapidity or from our lack of observation they may not always appear continuous. In this hypothetical sense, continuity may be allowed to be a necessary condition if two appearances are to be classed as appearances of the same thing. But it is not a sufficient condition, as appears from the instance of the drops in the sea. Thus something more must be sought before we can give even the roughest definition of a '^hing." What is wanted further seems to be something in the nature of fulfilment of causal laws. This statement, as it stands, is very vague, but we will endeavour to give it precision. When I speak of " causal laws," I mean any laws which connect events at different times, or even, as a limiting case, events at the same time provided the connection is not logically demonstrable. In this very general sense, the laws of dynamics are causal laws, and so are the laws correlating the simultaneous appearances of one " thing " to different senses. The question is : How do such laws help in the definition of a " thing " } To answer this question, we must consider what it is that is proved by the empirical success of physics. What is proved is that its hypotheses, though unverifiable no SCIENTIFIC METHOD IN PHILOSOPHY where they go beyond sense -data, are at no point in contradiction with sense-data, but, on the contrary, are ideally such as to render all sense-data calculable from a sufficient collection of data all belonging to a given period of time. Now physics has found it empirically possible to collect sense-data into series, each series being regarded as belonging to one " thing," and behaving, with regard to the laws of physics, in a way in which series not belonging to one thing would in general not behave. If it is to be unambiguous whether two appearances belong to the same thing or not, there must be only one way of grouping appearances so that the resulting things obey the laws of physics. It would be very difficult to prove that this is the case, but for our present purposes we may let this point pass, and assume that there is only one way. We must include in our definition of a " thing " those of its aspects, if any, which are not observed. Thus we may lay down the following definition : Things are those series of aspects which obey the laws of physics. That such series exist is an empirical fact, which constitutes the verifiability of physics. It may still be objected that the " matter " of physics is something other than series of sense-data. Sense-data, it may be said, belong to psychology and are, at any rate in some sense, subjective, whereas physics is quite independent of psychological considerations, and does not assume that its matter only exists when it is perceived. To this objection there are two answers, both of some importance. (a) We have been considering, in the above account, the question of the verifiability of physics. Now verifi- ability is by no means the same thing as truth ; it is, in fact, something far more subjective and psychological. For a proposition to be verifiable, it is not enough that WORLDS OF PHYSICS AND OF SENSE in it should be true, but it must also be such as we can discover to be true. Thus verifiability depends upon our capacity for acquiring knowledge, and not only upon the objective truth. In physics, as ordinarily set forth, there is much that is un verifiable : there are hypotheses as to (a) how things would appear to a spectator ^™^^^ portion Pj P2 of the path of the particle, this portion being one which contains "P. We say then that. If the motion of the particle is continuous at the ' See next lecture. THE THEORY OF CONTINUITY 137 time tj it must be possible to find two instants /j, 4, one earlier than / and one later, such that throughout the whole time from t-^ to 4 (both included), the particle lies between P^ and Pg. And we say that this must still hold however small we make the portion Pj Pg. When this is the case, we say that the motion is continuous at the time / ; and when the motion is continuous at all times, we say that the motion as a whole is continuous. It is obvious that if the particle were to jump suddenly from P to some other point Q, our definition would fail for all intervals Pj Pj which were too small to include Q. Thus our definition affords an analysis of the continuity of motion, while admitting points and instants and denying infinitesimal distances in space or periods in time. Philosophers, mostly in ignorance of the mathe- matician's analysis, have adopted other and more heroic methods of dealing with the primd facie difficulties of continuous motion. A typical and recent example of philosophic theories of motion is afforded by Bergson, whose views on this subject I have examined elsewhere.^ Apart from definite arguments, there are certain feelings, rather than reasons, which stand in the way of an acceptance of the mathematical account of motion. To begin with, if a body is moving at all fast, we see its motion just as we see its colour. A slow motion, like that of the hour-hand of a watch, is only known in the way which mathematics would lead us to expect, namely by observing a change of position after a lapse of time ; but, when we observe the motion of the second-hand, we do not merely see first one position and then another — we see something as directly sensible as colour. What is this something that we see, and that we call visible mo- ' Monist, July 19 12, pp. 337-341- 138 SCIENTIFIC METHOD IN PHILOSOPHY tlon ? Whatever it is, it is not the successive occupation of successive positions : something beyond the mathe- matical theory of motion is required to account for it. Opponents of the mathematical theory emphasise this fact. "Your theory," they say, "may be very logical, and might apply admirably to some other world ; but in this actual world, actual motions are quite diiFerent from what your theory would declare them to be, and require, there- fore, some different philosophy from yours for their adequate explanation.". The objection thus raised is one which I have no wish to underrate, but I believe it can be fully answered with- out departing from the methods and the outlook which have led to the mathematical theory of motion. Let us, however, first try to state the objection more fully. If the mathematical theory is adequate, nothing happens when a body moves except that it is in different places at different times. But in this sense the hour-hand and the second-hand are equally in motion, yet in the second-hand there is something perceptible to our senses which is absent in the hour-hand. We can see, at each moment, that the second-hand is moving, which is different from seeing it first in one place and then in another. This seems to involve our seeing it simultaneously in a number of places, although it must also involve our seeing that it is in some of these places earlier than in others. If, for example, I move my hand quickly from left to right, you seem to see the whole movement at once, in spite of the fact that you know it begins at the left and ends at the right. It is this kind of considera- tion, I think, which leads Bergson and many others to regard a movement as really one indivisible whole, not the series of separate states imagined by the mathe- matician. THE THEORY OF CONTINUITY 139 To this objection there are three supplementary- answers, physiological, psychological, and logical. We will consider them successively. (i) The physiological answer merely shows that, if the physical world is what the mathematician supposes, its sensible appearance may nevertheless be expected to be what it is. The aim of this answer is thus the modest one of showing that the mathematical account is not im- possible as applied to the physical world ; it does not even attempt to show that this account is necessary, or that an analogous account applies in psychology. When any nerve is stimulated, so as to cause a sensa- tion, the sensation does not cease instantaneously with the cessation of the stimulus, but dies away in a short finite time. A flash of lightning, brief as it is to our sight, is briefer still as a physical phenomenon : we continue to see it for a few moments after the light-waves have ceased to strike the eye. Thus in the case of a physical motion, if it is sufficiently swift, we shall actually at one instant see the moving body throughout a finite portion of its course, and not only at the exact spot where it is at that instant. Sensations, however, as they die away, grow gradually fainter ; thus the sensation due to a stimulus which is recently past is not exactly like the sensation due to a present stimulus. It follows from this that, when we see a rapid motion, we shall not only see a number of positions of the moving body simultan- eously, but we shall see them with different degrees of intensity — the present position most vividly, and the others with diminishing vividness, until sensation fades away into immediate memory. This state of things accounts fully for the perception of motion. A motion is perceived, not merely inferred, when it is sufficiently swift for many positions to be sensible at one time ; and HO SCIENTIFIC METHOD IN PHILOSOPHY the earlier and later parts of one perceived motion are distinguished by the less and greater vividness of the sensations. This answer shows that physiology can account for our perception of motion. But physiology, in speaking of stimulus and sense-organs and a physical motion distinct from the immediate object of sense, is assuming the truth of physics, and is thus only capable of showing the physical account to be possible, not of showing it to be necessary. This consideration brings us to the psycho- logical answer. (2) The psychological answer to our difficulty about motion is part of a vast theory, not yet worked out, and only capable, at present, of being vaguely outlined. We considered this theory in the third and fourth lectures ; for the present, a mere sketch of its application to our present problem must suffice. The world of physics, which was assumed in the physiological answer, is ob- viously inferred from what is given in sensation ; yet as soon as we seriously consider what is actually given in sensation, we find it apparently very different from the world of physics. The question is thus forced upon us : Is the inference from sense to physics a valid one ? I believe the answer to be affirmative, for reasons which I suggested in the third and fourth lectures ; but the answer cannot be either short or easy. It consists, broadly speak- ing, in showing that, although the particles, points, and instants with which physics operates are not themselves given in experience, and are very likely not actually exist- ing things, yet, out of the materials provided in sensation, it is possible to make logical constructions having the mathematical properties which physics assigns to particles, points, and instants. If this can be done, then all the propositions of physics can be translated, by a sort of THE THEORY OF CONTINUITY 141 dictionary, into propositions about the kinds of objects which are given in sensation. Applying these general considerations to the case of motion, we find that, even within the sphere of immediate sense-data, it is necessary, or at any rate more consonant with the facts than any other equally simple view, to distinguish instantaneous states of objects, and to regard such states as forming a compact series. Let us consider a body which is moving swiftly enough for Its motion to be perceptible, and long enough for its motion to be not wholly comprised in one sensation. Then, in spite of the fact that we see a finite extent of the motion at one instant, the extent which we see at one instant is diiFerent from that which we see at another. Thus we are brought back, after all, to a series of momentary views of the \ moving body, and this series will be compact, like the former physical series of points. In fact, though the terms of the series seem different, the mathematical char- acter of the series is unchanged, and the whole mathe- matical theory of motion will apply to it verbatim. When we are considering the actual data of sensation in this connection, it is important to realise that two sense-data may be, and must sometimes be, really different when we cannot perceive any difference between them. An old but conclusive reason for believing this was emphasised by Poincar^.^ In all cases of sense-data capable of gradual change, we may find one sense-datum indistinguishable from another, and that other indis- tinguishable from a third, while yet the first and third are quite easily distinguishable. Suppose, for example, a person with his eyes shut is holding a weight in his hand, and someone noiselessly adds a small extra weight. If 1 " Le continu mathdmatique," Revue de Mdtaphysique et de Morale, vol. i. p. 29. 142 SCIENTIFIC METHOD IN PHILOSOPHY the extra weight is small enough, no difference will , be perceived in the sensation. After a time, another small extra weight may be added, and still no change wiU be perceived ; but if both extra weights had been added at once, it may be that the change would be quite easily perceptible. Or, again, take shades of colour. It would be easy to find three stuffs of such closely similar shades that no difference could be perceived between the first and second, nor yet between the second and third, while yet the first and third would be distinguishable. In such a case, the second shade cannot be the same as the first, or it would be distinguishable from the third ; nor the same as the third, or it would be distinguishable from the first. It must, therefore, though indistinguishable from both, be really intermediate between them. Such considerations as the above show that, although we cannot distinguish sense-data unless they differ by more than a certain amount, it is perfectly reasonable to suppose that sense-data of a given kind, such as weights or colours, really form a compact series. The objections which may be brought from a psychological point of view against the mathematical theory of motion are not, there- fore, objections to this theory properly understood, but only to a quite unnecessary assumption of simplicity in the momentary object of sense. Of the immediate object of sense, in the case of a visible motion, we may say that at each instant it is in all the positions which remain sensible at that instant ; but this set of positions changes continuously from moment to moment, and is amenable to exactly the same mathematical treatment as if it were a mere point. When we assert that some mathematical account of phenomena is correct, all that we primarily assert is that something definable in terms of the crude phenomena satisfies our formulae ; and in this sense the THE THEORY OF CONTINUITY 143 mathematical theory of motion is applicable to the data of sensation as well as to the supposed particles of abstract physics. There are a number of distinct questions which are apt to be confused when the mathematical continuum is said to be inadequate to the facts of sense. We may state these, in order of diminishing generality, as follows : — {a) Are series possessing mathematical continuity logically possible .'' (b) Assuming that they are possible logically, are they not impossible as applied to actual sense-data, because, among actual sense-data, there are no such fixed mutually external terms as are to be found, e.g., in the series of fractions } {c) Does not the assumption of points and in- stants make the whole mathematical account fictitious .'' {d) Finally, assuming that all these objections have been answered, is there, in actual empirical fact, any sufficient reason to believe the world of sense continuous .? Let us consider these questions in succession. {a) The question of the logical possibility of the mathematical continuum turns partly on the elementary misunderstandings we considered at the beginning of the present lecture, partly on the possibility of the mathe- matical infinite, which will occupy our next two lectures, and partly on the logical form of the answer to the Bergsonian objection which we stated a few minutes ago. I shall say no more on this topic at present, since it is desirable first to complete the psychological answer. {¥) The question whether sense-data are composed of mutually external units is not one which can be decided by empirical evidence. It is often urged that, as a 144 SCIENTIFIC METHOD IN PHILOSOPHY matter of immediate experience, the sensible flux is devoid of divisions, and is falsified by the dissections of the intellect. Now I have no wish to argue that this view is contrary to immediate experience : I wish only to maintain that it is essentially incapable of being proved by immediate expeiience. As we saw, there must be among sense-data differences so slight as to be imperceptible : the fact that sense-data are immediately given does not mean that their differences also must be immediately given (though they may be). Suppose, for example, a coloured surface on which the colour changes gradually — so gradually that the difference of colour in two very neighbouring portions is imperceptible, while the difference between more widely separated portions is quite noticeable. The effect produced, in such a case, will be precisely that of " interpenetration," of transition which is not a matter of discrete units. And since it tends to be supposed that the colours, being immediate data, must appear different if they are different, it seems easily to follow that " interpenetration " must be the ultimately right account. But this does not follow. It is unconsciously assumed, as a premiss for a reductio ad absurdum of the analytic view, that, if A and B are immediate data, and A differs from B, then the fact that they differ must also be an immediate datum. It is difficult to say how this assumption arose, but I think it is to be connected with the confusion between " acquaintance " and " knowledge about." j^^quaintanee, which is what we derive from sense, does not, theoretically at least, imply even the smallest " knowledge about," i.e. it does not imply knowledge of any proposition concern- .ing the object with which we are acquainted. It is a jmistake to speak as if acquaintance had degrees : there is merely acquaintance and non-acquaintance. When we THE THEORY OF CONTINUITY 145 speak of Jbe.coining" better acquainted," as for instance with a person, what we must mean is, becoming acquainted with more parts of a certain^ whole ; but the acquaintance with each part is either complete or non- existent. Thus it is a mistake to say that if we were perfectly acquainted with an object we should know all about it. "Knowledge about" is knowledge of pro- positions, which is not involved necessarily in acquaintance with the constituents of the propositions. To know that^ two shades of colour are different is knowledge about; them ; hence acquaintance with the two shades does not\ in any way necessitate the knowledge that they are/ different. From what has just been said it follows that the nature of sense-data cannot be validly used to prove that they are not composed of mutually external units. It may be admitted, on the other hand, that nothing in their empirical character specially necessitates the view that they are composed of mutually external units. This view, if it is held, must be held on logical, not on empirical, grounds. 1 believe that the logical grounds are adequate to the conclusion. They rest, at bottom, upon the impossibility of explaining complexity without assuming constituents. It is undeniable that the visual field, for example, is complex ; and so far as I can see, there is always self-contradiction in the theories which, while admitting this complexity, attempt to deny that it results from a combination of mutually external units. But to pursue this topic would lead us too far from our theme, and I shall therefore say no more about it at present. (c) It is sometimes urged that the mathematical account of motion is rendered fictitious by its assumption of points and instants. Now there are here two different 10 146 SCIENTIFIC METHOD IN PHILOSOPHY questions to be distinguished. There is the question of absolute or relative space and time, and there is the question whether what occupies space and time must be composed of elements which have no extension or duration. And each of these questions in turn may take two forms, namely : (a) is the hypothesis consistent with the facts and with logic .? (/3) is it necessitated by the facts or by logic ? I wish to answer, in each case, yes to the first form of the question, and no to the second. But in any case the mathematical account of motion will not be fictitious, provided a right interpretation is given to the words " point " and " instant." A few words on each alternative will serve to make this clear. Formally, mathematics adopts an absolute theory of space and time, i.e. it assumes that, besides the things which are in space and time, there are also entities, called " points " and " instants," which are occupied by things. This view, however, though advocated by Newton, has long been regarded by mathematicians as merely a convenient fiction. There is, so far as I can see, no conceivable evidence either for or against it. It is logically possible, and it is consistent with the facts. But the facts are also consistent with the denial of spatial and temporal entities over and above things with spatial and temporal relations. Hence, in accordance with Occam's razor, we shall do well to abstain from either assuming or denying points and instants. This means, so far as practical working out is concerned, that we adopt the relational theory ; for in practice the refusal to assume points and instants has the same efFect as the denial of them. But in strict theory the two are quite different, since the denial introduces an element of unverifiable dogma which is wholly absent when we merely refrain from the assertion. Thus, although we shall derive THE THEORY OF CONTINUITY 147 points and instants from things, we shall leave the bare possibility open that they may also have an independent existence as simple entities. We come now to the question whether the things in space and time are to be conceived as composed of elements without extension or duration, i.e. of elements which only occupy a point and an instant. Physics, formally, assumes in its diiFerential equations that things consist of elements which occupy only a point at each instant, but persist throughout time. For reasons ex- plained in Lecture IV., the persistence of things through time is to be regarded as the formal result of a logical construction, not as necessarily implying any actual per sistence. The same motives, in fact, which lead to the division of things into point-particles, ought presumably to lead to their division into instant-particles, so that the ultimate formal constituent of the matter in physics will be a point-instant-particle. But such objects, as well as the particles of physics, are not data. The same economy of hypothesis, which dictates the practical adoption of a relative rather than an absolute space and time, also dictates the practical adoption of material elements which have a finite extension and duration. Since, as we saw in Lecture IV., points and instants can be constructed as logical functions of such elements, the mathematical account of motion, in which a particle passes continuously through a continuous series of points, can be interpreted in a form which assumes only elements which agree with our actual data in having a finite extension and duration. Thus, so far as the use of points and instants is concerned, the mathematical account of motion can be freed from the charge of employing fictions. id) But we must now face the question : Is there, in actual empirical fact, any sufficient reason to believe the 148 SCIENTIFIC METHOD IN PHILOSOPHY world of sense continuous ? The answer here must, I think, be in the negative. We may say that the hypothesis of continuity is perfectly consistent with the facts and with logic, and that it is technically simpler than any other tenable hypothesis. But since our powers of discrimination among very similar sensible objects are not infinitely precise, it is quite impossible to decide between different theories which only diiFer in regard to what is below the margin of discrimination. If, for example, a coloured surface which we see consists of a finite number of very small surfaces, and if a motion which we see consists, like a cinematograph, of a large finite number of successive positions, there will be nothing empirically discoverable to show that objects of sense are not continuous. In what is called experienced continuity, such as is said to be given in sense, there is a large negative element : absence of perception of difference occurs in cases which are thought to give perception of absence of difference. When, for example, we cannot distinguish a colour A from a colour B, nor a colour B from a colour C, but can distinguish A from C, the indistinguishability is a purely negative fact, namely, that we do not perceive a difference. Even in regard to immediate data, this is no reason for denying that there is a difference. Thus, if we see a coloured surface whose colour changes gradually, its sensible appearance if the change is continuous will be indistinguishable from what it would be if the change were by small finite jumps. If this is true, as it seems to be, it follows that there can never be any empirical evidence to demonstrate that the sensible world is continuous, and not a collection of a very large finite number of elements of which each differs'^ from its neighbour in a finite though very small degree. The continuity of space and tiroe,, the. infintite number of THE THEORY OF CONTINUITY 149 differenL..§hadjes- in -the^. spectrum, and so on, are allin the^nature of unyerifiable hypotheses — perfectly possible logically, perfectly consistent with the known facts, and simpler' technically than any other tenable hypotheses, but not the sole ^hypotheses which are logically and em- pirically adequate. If a relational theory of instants is constructed, in which an " instant " is defined as a group of events simultaneous with each other and not all simultaneous with any event outside the group, then if our resulting series of instants is to be compact, it must be possible, if X wholly precedes y, to find an event z, simultaneous with part of x, which wholly precedes some event which wholly precedes y. Now this requires that the number of events concerned should be infinite in any finite period of time. If this is to be the case in the world of one man's sense-data, and if each sense-datum is to have not less than a certain finite temporal extension, it will be necessary to assume that we always have an infinite number of sense-data simultaneous with any given sense-datum. Applying similar considerations to space, and assuming that sense-data are to have not less than a certain spatial extension, it will be necessary to suppose that an infinite number of sense-data overlap spatially with any given sense-datum. This hypothesis is possible, if we suppose a single sense-datum, e.g. in sight, to be a finite surface, enclosing other surfaces which are also single sense-data. But there are difl!iculties in such a hypothesis, and I do not know whether these difficulties could be successfully met. If they cannot, we must do one of two things : either declare that the world of one man's sense-data is not continuous, or else refuse to admit that there is any lower limit to the duration and extension of a single sense-datum. I do not know what ISO SCIENTIFIC METHOD IN PHILOSOPHY is the right course to adopt as regards these alternatives. The logical analysis we have been considering provides the apparatus for dealing with the various hypotheses, and the empirical decision between them is a problem for the psychologist. (3) We have now to consider the logical answer to the alleged difficulties of the mathematical theory of motion, or rather to the positive theory which is urged on the other side. The view urged explicitly by Bergson, and implied in the doctrines of many philosophers, is, that a motion is something indivisible, not validly analysable into a series of states. This is part of a much more general doctrine, which holds that analysis always falsifies, because the parts of a complex whole are different, as combined in that whole, from what they would otherwise be. It is very difficult to state this doctrine in any form which has a precise meaning. Often arguments are used which have no bearing whatever upon the question. It is urged, for example, that when a man becomes a father, his nature is altered by the new relation in which he finds himself, so that he is not strictly identical with the man who was previously not a father. This may be true, but it is a causal psychological fact, not a logical fact. The doctrine would require that a man who is a father cannot be strictly identical with a man who is a son, because he is modified in one way by the relation of fatherhood and in another by that of sonship. In fact, we may give a precise statement of the doctrine we are combating in the form : There can never be two facts concerning the same thing. A fact concerning a thing always is or involves a relation to one or more entities ; thus two facts concerning the same thing would involve two relations of the same thing. But the doctrine in question holds that a thing is so modified by its relations that it cannot be the same THE THEORY OF CONTINUITY 151 in one relation as in another. Hence, if this doctrine is true, there can never be more than one fact concerning any one thing. I do not think the philosophers in question have realised that this is the precise statement of the view they advocate, because in this form the view is so contrary to plain truth that its falsehood is evident as soon as it is stated. The discussion of this question, however, involves so many logical subtleties, and is so beset with difficulties, that I shall not pursue it further at present. When once the above general doctrine is rejected, it is obvious that, where there is change, there must be a succession of states. There cann ot be change — and motionjs_on]yLa~pardQilar,..case_.Q|_ diang.e=™-ttftless -there _is_ something different at one time from what there is at some other tinie." Change, therefore, must involve relations and complexity, and must demand analysis. So long as our analysis has only gone as far as other smaller changes, it is not complete ; if it is_to Jie_carapleteTit must end with terms that are not changes, but are related by a relation of earlier and later: In the case of changes which appear continuous, such as motions, it seems to be impos- sible to find anything other than change so long as we deal with finite periods of time, however short. We are thus driven back, by the logical necessities of the case, to the conception of instants without duration, or at any rate without.^ny duration which even the most delicate instruments can reveal. This conception, though it can be made to seem difficult, is really easier than_ any other that the facts allow. It is a kind of logical framework into which any tenable theory must fit — not necessarily itself the statement of the crude facts, but a form in which statements which are true of the crude facts can be made by a suitable interpretation, The direct con- 152 SCIENTIFIC METHOD IN PHILOSOPHY sideration of the crude facts of the physical world has been undertaken in earlier lectures ; in the present lecture, we have only been concerned to show that nothing in the crude facts is inconsistent with the mathe- matical doctrine of continuity, or demands a continuity of a radically different kind from that of mathematical motion. LECTURE VI THE PROBLEM OF INFINITY CONSIDERED HISTORICALLY LECTURE VI THE PROBLEM OF INFINITY CONSIDERED HISTORICALLY It will be remembered that, when we enumerated the grounds upon which the reality of the sensible world has been questioned, one of those mentioned was the supposed impossibility of infinity and continuity. In view of our earlier discussion of physics, it would seem that no conclusive empirical evidence exists in favour of infinity or continuity in objects of sense or in matter. Nevertheless, the explanation which assumes infinity and continuity remains incomparably easier and more natural, from a scientific point of view, than any other, and since Georg Cantor has shown that the supposed contradictions are illusory, there is no longer any reason to struggle after a finitist explanation of the world. I The supposed difficulties of continuity all have their {source in the fact that a continuous series must have an infinite number of terms, and are in fact difficulties con- ; cerning infinity. Hence, in freeing the infinite from contradiction, we are at the same time showing the logical •i possibility of continuity as assumed in science. The kind of way in which infinity has been used to discredit the world of sense may be illustrated by Kant's first two antinomies. In the first, the thesis states : " The world has a beginning in time, and as regards space is enclosed within limits " ; the antithesis states : 156 SCIENTIFIC METHOD IN PHILOSOPHY " The world has no beginning and no limits in space, but is infinite in respect of both time and space." Kant professes to prove both these propositions, whereas, if what we have said on modern logic has any truth, it must be impossible to prove either. In order, however, to rescue the world of sense, it is enough to destroy the proof of one of the two. For our present purpose, it is the proof that the world is finite that interests us. Kant's argument as regards space here rests upon his argument as regards time. We need therefore only examine the argument as regards time. What he says is as follows : " For let us assume that the world has no beginning as regards time, so that up to every given instant an eternity has elapsed, and therefore an infinite series of successive states of the things in the world has passed by. But the infinity of a series consists just in this, that it can never be completed by successive synthesis. Therefore an infinite past world-series is impossible, and accordingly a beginning of the world is a necessary condition of its existence ; which was the first thing to be proved." Many different criticisms might be passed on this argument, but we will content ourselves with a bare minimum. To begin with, it is a mistake to define the infinity of a series as "impossibility of completion by successive synthesis." The notion of infinity, as we shall see in the next lecture, is primarily a property of classes, and only derivatively applicable to series ; classes which are infinite are given all at once by the defining property of their members, so that there is no question of " completion " or of " successive synthesis." And the word " synthesis," by suggesting the mental activity of synthesising, introduces, more or less surreptitiously, that reference to mind by which all Kant's philosophy was infected. In the second place, when Kant says that THE PROBLEM OF INFINITY 157 an infinite series can " never " be completed by successive synthesis, all that he has even conceivably a right to say is that Jt cannot be completed in a finite time. Thus what he really proves is, at most, that if the world had no beginning, it must have already existed for an infinite time. This, however, is a very poor conclusion, by no means suitable for his purposes. And with this result we might, if we chose, take leave of the first antinomy. It is worth while, however, to consider how Kant came to make such an elementary blunder. What happened in his imagination was obviously something like this : Starting from the present and going backwards in time, we have, if the world had no beginning, an infinite series of events. As we see from the word " synthesis," he imagined a mind trying to grasp these successively, in the reverse order to that in which they had occurred, i.e. going from the present backwards. This series is obviously one which has no end. But the series of events up to the present has an end, since it ends with the present. Owing to the inveterate subjectivism of his mental habits, he failed to notice that he had reversed the sense of the series by substituting backward synthesis for forward happening, and thus hesupposed that it was necessary to identify the mental series, which had no end, with the physical series, which had an end but no beginning. It was this mistake, I think, which, operating unconsciously, led him to attribute validity to a singularly flimsy piece of fallacious reasoning. The second antinomy illustrates the dependence of the problem of continuity upon that of infinity. The thesis states : " Every complex substance in the world consists of simple parts, and there exists everywhere nothing but the simple or what is composed of it." The antithesis states : " No complex thing in the world consists of 158 SCIENTIFIC METHOD IN PHILOSOPHY simple parts, and everywhere in it there exists nothing simple." Here, as before, the proofs of both thesis and antithesis are open to criticism, but for the purpose of vindicating physics and the world of sense it is enough to find a fallacy in one of the proofs. We will choose for this purpose the proof of the antithesis, which begins as follows : " Assume that a complex thing (as substance) consists of simple parts. Since all external relation, and therefore all composition out of substances, is only possible in space, the space occupied by a complex thing must consist of as many parts as the thing consists of. Now space does not consist of simple parts, but of spaces." The rest of his argument need not concern us, for the nerve of the proof lies in the one statement : " Space does not consist of simple parts, but of spaces." This is like Bergson's objection to " the absurd proposition that motion is made up of immobilities." Kant does not tell us why he holds that a space must consist of spaces rather than of simple parts. Geometry regards space as made up of points, which are simple ; and although, as we have seen, this view is not scientifically or logically necessary^ it remains primd facie possible, and its mere possibility is enough to vitiate Kant's argu- ment. For, if his proof of the thesis of the antinomy were valid, and if the antithesis could only be avoided by assuming points, then the antinomy itself would afford a conclusive reason in favour of points. Why, then, did Kant think it impossible that space should be composed of points ^ I think two considerations probably influenced him. In the^firs^ place,.the-£SsejitiaLthing about space is spatial order, and mere points, by themselves, will not account for spatial order. It is" obvious that- his argument THE PROBLEM OF INFINITY 159 assumes absolute space ; but it is spatial relations that are alone important, and they cannot be reduced to points. This ground for his view depends, therefore, upon his ignorance of the logical theory of order and his oscilla- tions between absolute and relative space. But there is also another ground for his opinion, which is more relevant to our present topic. This is the ground derived from infinite divisibility. A space may be halved, and then halved again, and so on ad infinitum, and at every stage of the process the parts are still spaces, not points. In order to reach points by such a method, it would be necessary_.to come to the _end of an unending process, which is impossible. But just as an infinite class I can be given all at once by its defining concept, though it cannot be reached by successive enumeration, so an infinite set of_points can be given all at once as making up a line or area or volume, though they can never be reached by the process of ^ successive division. Thus the' infinite divisibility of space gives no ground for denying that space is composed of points. Kant does not give his grounds^ for this denial, and we can therefore only con- jecture what they were. But the above two grounds, which we have seen to be fallacious, seem sufficient to account for his opinion, and we may therefore conclude , that the antithesis of the second antinomy is unproved. The above illustration of Kant's antinomies has only been introduced in order to show the relevance of the problem of infinity to the problem of the reality of objects of sense. In the remainder of the present lecture, I wish to state and explain the problem of infinity, to show how it arose, and to show the irrelevance of all the solutions proposed by philosophers. In the following lecture, I shall try to explain the true solution, which has been discovered by the mathematicians, but nevertheless belongs essentially i6o SCIENTIFIC METHOD IN PHILOSOPHY to philosophy. The solution is definitive, in the sense that it entirely satisfies and convinces all who study it carefully. For over two thousand years the human intellect was baffled by the problem ; its many failures and its ultimate success make this problem peculiarly apt for the illustration of method. I The problem appears to have first arisen in some such way as the following.^ Pythagoras and his followers, who were interested, like Descartes, in the application of number to geometry, adopted in that science more arithmetical methods than those with which Euclid has made us familiar. They, or their contemporaries the atomists, believed, apparently, that space is composed of indivisible points, while time is composed of indivisible instants.^ This belief would not, by itself, have raised the difficulties which they encountered, but it was pre- sumably accompanied by another belief, that the number of points In any finite area or of instants in any finite period must be finite. I do not suppose that this latter belief was a conscious one, because probably no other possibility had occurred to them. But the belief never- theless operated, and very soon brought them into conflict with facts which they themselves discovered. Before explaining how this occurred, however, it is necessary to say one word in explanation of the phrase " finite number." The exact explanation Is a matter for our next lecture ; for the present, it must suffice to say that I mean o and i and 2 and 3 and so on, for ever — in other words, any number that can be obtained by 1 In what concerns the early Greek philosophers, my knowledge is largely derived from Burnet's valuable work, Early Greek Philosophy (2nd ed., London, 1908). I have also been greatly assisted by Mr D. S. Robertson of Trinity College, who has supplied the deficiencies of my knowledge of Greek, and brought important references to my notice. ^ Cf. Aristotle, Metaphysics, M. 6, 1080^, 18 sqq., and 1083^, 8 sqq. THE PROBLEM OF INFINITY i6i Successively adding ones. This includes all the numbers that can be expressed by means of our ordinary numerals, and since such numbers can be made greater and' greater, without ever reaching an unsurpassable maximum, it is easy to suppose that there are no other numbers. But this supposition, natural as it is, is mistaken. Whether the Pythagoreans themselves believed space and time to be composed of indivisible points and instants is a debatable question.^ It would seem that the distinction between space and matter had not yet been clearly made, and that therefore, when an atomistic view is expressed, it is difficult to decide whether particles of matter or points of space are intended. There is an interesting passage^ in Aristotle's Physics,^ where he says : " The Pythagoreans all maintained the existence of the void, and said that it enters into the heaven itself from the boundless breath, inasmuch as the heaven breathes in the void also ; and the void differentiates natures, as ' There is some reason to think that the Pythagoreans distinguished between discrete and continuous quantity. G. J. AUman, in his Greek Geometry from Thales to Euclid, says (p. 23) ; " The Pythagoreans made a fourfold division of mathematical science, attributing one of its parts to the how many, t4 irrf' a-(t>aipiKiiv) contemplates continued quantity so far as it is of a self-motive nature. (Proclus, ed. Friedlein, p. 35. As to the distinction between rh -rniAiKov, continuous, and rh viirov, discrete quantity, see Iambi., t'n Nicomachi Gera- seni Arithmeticam introductionem, ed. Tennulius, p. 148.)" Cf- p. 48. ^ Referred to by Burnet, op. cit., p. 120, ' iv., 6. ^l},b, 22 ; H. Ritter and L. Preller, Historia Philosophic Graces, 8th ed., Gotha, 1898, p. 75 (this work will be referred to in future as " R. P."). II 1 62 SCIENTIFIC METHOD IN PHILOSOPHY if it were a sort of separation of consecutives, and as if it were their differentiation ; and that this also is what is first in numbers, for it is the void which differentiates them." This seems to imply that they regarded matter as con- sisting of atoms with empty space in between. But if so, they must have thought space could be studied by only paying attention to the atoms, for otherwise it would be hard to account for their arithmetical methods in geometry, or for their statement that " things are numbers." The difficulty which beset the Pythagoreans in their attempts to apply numbers arose through their discovery of incommensurables, and this, in turn, arose as follows. Pythagoras, as we all learnt in youth, discovered the proposition that the sum of the squares on the sides of a right-angled triangle is equal to the square on the hypotenuse. It is said that he sacrificed an ox when he discovered this theorem ; if so, the ox was the first martyr to science. But the theorem, though it has remained his chief claim to immortality, was soon found to have a consequence fatal to his whole philosophy. Consider the case of a right-angled triangle whose two sides are equal, such a triangle as is formed by two sides of a square and a diagonal. Here, in virtue of the theorem, the square on the diagonal is double of the square on either of the sides. But Pythagoras or his early followers easily proved that the square of one whole number cannot be double of the square of another. ^ 1 The Pythagorean proof is roughly as follows. If possible, let the ratio of the diagonal to the side of a square be mjn, where m and n are whole numbers having no common factor. Then we must have m'^=2n^. Now the square of an odd number is odd, but tn^, being equal to 2«2, is even. Hence m must be even. But the square of an even number divides by 4, therefore n\ which is half of m^, must be even. Therefore n must THE PROBLEM OF INFINITY 163 Thus the length of the side and the length of the diagonal are incommensurable ; that is to say, however small a unit of length you take, if it is contained an exact number of times in the side, it is not contained any exact number of times in the diagonal, and vice versa. Now this fact might have been assimilated by some philosophies without any great difficulty, but to the philosophy of Pythagoras it was absolutely fatal. Pytha- goras held that number is the constitutive essence of all things, yet no two numbers could express the ratio of the side of a square to the diagonal. It would seem probable that we may expand his difficulty, without departing from his thought, by assuming that he regarded the length of a line as determined by the number of atoms contained in it — a line two inches long would contain twice as many atoms as a line one inch long, and so on. But if this were the truth, then there must be a definite numerical ratio between any two finite lengths, because it was supposed that the number of atoms in each, how- ever large, must be finite. Here there was an insoluble contradiction. The Pythagoreans, it is said, resolved to keep the existence of incommensurables a profound secret, revealed only to a few of the supreme heads of the sect ; and one of their number, Hippasos of Metapontion, is even said to have been shipwrecked at sea for impiously disclosing the terrible discovery to their enemies. It must be remembered that Pythagoras was the founder of a new religion as well as the teacher of a new science : if the science came to be doubted, the disciples might fall into sin, and perhaps even eat beans, which according to Pythagoras is as bad as eating parents' bones. be even. But, since m is even, and m and n have no common factor, « must be odd. Thus n must be both odd and even, which is impossible ; and therefore the diagonal and the side cannot have a rational ratio. 1 64 SCIENTIFIC METHOD IN PHILOSOPHY The problem first raised by the discovery of incom- mensurables proved, as time went on, to be one of the most severe and at the same time most far-reaching problems that have confronted the human intellect in its endeavour to understand the world. It showed at once that numerical measurement of lengths, if it was to be made accurate, must require an arithmetic more advanced and more difficult than any that the ancients possessed. They therefore set to work to reconstruct geometry on a basis which did not assume the universal possibility of numerical measurement — a reconstruction which, as may be seen in Euclid, they effected with extraordinary skill and with great logical acumen. The moderns, under the influence of Cartesian geometry, have reasserted the universal possibility of numerical measurement, extending arithmetic, partly for that purpose, so as to include what are called " irrational " numbers, which give the ratios of incommensurable lengths. But although irrational numbers have long been used without a qualm, it is only in quite recent years that logically satisfactory definitions of them have been given. With these definitions, the first and most obvious form of the difficulty which con- fronted the Pythagoreans has been solved ; but other forms of the difficulty remain to be considered, and it is these that introduce us to the problem of infinity in its pure form. We saw that, accepting the view that a length is com- posed of points, the existence of incommensurables proves that every finite length must contain an infinite number of points. In other words, if we were to take away points one by one, we should never have taken away all the points, however long we continued the process. The number of points, therefore, cannot be counted^ for counting is a process which enumerates things one by one. '' The THE PROBLEM OF INFINITY 165 property of being unable to be counted is characteristic of infinite collections, and is a source of many of their paradoxical qualities. ' So paradoxical are these qualities that until our own day they were thought to constitute logical contradictions. A long line of philosophers, from Zeno ^toM. Bergson, have based much of their metaphysics upon the supposed impossibility of infinite collections. Broadly speaking, the difficulties were stated by Zeno, and nothing material was added until we reach Bolzano's Paradoxien des C7«f«d?7/c^f«, a little work written in 1847-8, and published posthumously in 1851. Intervening at- tempts to deal with the problem are futile and negligible. The definitive solution of the diflSculties is due, not to Bolzano, but to Georg Cantor, whose work on this subject first appeared in 1882. In order to understand Zeno, and to realise how little modern orthodox metaphysics has added to the achieve- ments of the Greeks, we must consider for a moment his master Parmenides, in whose interest the paradoxes were invented.^ Parmenides expounded his views in a poem divided into two parts, called " the way of truth " and " the way of opinion " — like Mr Bradley's " Appearance " and " Reality," except that Parmenides tells us first about reality and then about appearance. " The way of opinion," in his philosophy, is, broadly speaking, Pythagorean ism ; it begins with a warning : " Here I shall close my trust- worthy speech and thought about the truth. Hence- forward learn the opinions of mortals, giving ear to the deceptive ordering of my words." What has gone before has been revealed by a goddess, who tells him what ' In regard to Zeno and the Pythagoreans, I have derived much valuable information and criticism from Mr P. E. B. Jourdain. ^ So Plato makes Zeno say in the Parmenides, apropos of his philosophy as a whole ; and all internal and external evidence supports this view. 1 66 SCIENTIFIC METHOD IN PHILOSOPHY really is. Reality, she says, is uncreated, indestructible, unchanging, indivisible ; it is " immovable in the bonds of mighty chains, without beginning and without end ; since coming into being and passing away have been driven afar, and true belief has cast them away." The funda- mental principle of his inquiry is stated in a sentence which would not be out of place in Hegel : ^ " Thou canst not know what is not — that is impossible — nor , utter it ; for it is the same thing that can be thought and that can be." And again : " It needs must be that what can be thought and spoken of is ; for it is possible for it to be, and it is not possible for what is nothing to be." The impossibility of change follows from this principle ; for what is past can be spoken of, and therefore, by the principle, still is. The great conception of a reality behind the passing illusions of sense, a reality one, indivisible, and unchang- ing, was thus introduced into Western philosophy by Parmenides, not, it would seem, for mystical or religious reasons, but on the basis of a logical argument as to the impossibility of not-being. All the great metaphysical systems — notably those of Plato, Spinoza, and Hegel — are the outcome of this fundamental idea. It is difficult to disentangle the truth and the error in this view. The contention that time is unreal and that the world of sense is illusory must, I think, be regarded as based upon fallacious reasoning. Nevertheless, there is some sense — easier to feel than to state — in which time is an unim- portant and superficial characteristic of reality. Past and future must be acknowledged to be as real as the present, and a certain emancipation from slavery to time is essential to philosophic thought. The importance of ' "With Parmenides," Hegel says, "philosophising proper began.'' Werke (edition of 1840), vol. xiii. p. 274. THE PROBLEM OF INFINITY 167 time is rather practical than theoretical, rather in relation to our desires than in relation to truth. A truer image of the world, I think, is obtained by picturing things as entering into the stream of time from an eternal world outside, than from a view which regards time as the devouring tyrant of all that is. Both in thought and in feeling, to realise the unimportance of time is the gate of wisdom. But unimportance is not unreality ; and therefore what we shall have to say about Zeno's arguments in support of Parmenides must be mainly critical. The relation of Zeno to Parmenides is explained by Plato ^ in the dialogue in which Socrates, as a young man, learns logical acumen and philosophic disinterested- ness from their dialectic. I quote from Jowett's translation : " I see, Parmenides, said Socrates, that Zeno is your second self in his writings too ; he puts what you say in another way, and would fain deceive us into believing that he is telling us what is new. For you, in your poems, say All is one, and of this you adduce excellent proofs ; and he on the other hand says There is no Many ; and on* behalf of this he offers overwhelming evidence. To deceive the world, as you have done, by saying the same thing in different ways, one of you affirming the one, and the other denying the many, is a strain of art beyond the reach of most of us. " Yes, Socrates, said Zeno. But although you are as keen as a Spartan hound in pursuing the track, you do not quite apprehend the true motive of the composition, which is not really such an ambitious work as you imagine ; for what you speak of was an accident ; I had no serious intention of deceiving the world. The truth ' Parmenides, 128 a-d. 1 68 SCIENTIFIC METHOD IN PHILOSOPHY is, that these writings of mine were meant to protect the arguments of Parmenides against those who scoff at him and show the many ridiculous and contradictory results which they suppose to follow from the affirmation of the one. My answer is an address to the partisans of the many, whose attack I return with interest by retorting upon them that their hypothesis of the being of the many if carried out appears in a still more ridiculous light than the hypothesis of the being of the one." Zeno's four arguments against motion were intended to exhibit the contradictions that result from supposing that there is such a thing as change, and thus to support the Parmenidean doctrine that reality is unchanging.^ Unfortunately, we only know his arguments through Aristotle,^ who stated them in order to refute them. Those philosophers in the present day who have had their doctrines stated by opponents will realise that a just or adequate presentation of Zeno's position is hardly to be expected from Aristotle ; but by some care in inter- pretation it seems possible to reconstruct the so-called "sophisms" which have been "refuted" by every tyro from that day to this. Zeno's arguments would seem to be " ad hominem " ; that is to say, they seem to assume premisses granted by his opponents, and to show that, granting these premisses, it is possible to deduce consequences which his opponents must deny. In order to decide whether they are valid arguments or " sophisms," it is necessary to guess at the tacit premisses, and to decide who was the " homo " at whom they were aimed. Some maintain that they were ' This interpretation is combated by Milhaud, Les philosophes-g^omitres de la Grice, p. 140 n., but his reasons do not seem to nie convincing. All the interpretations in what follows are open to question, but all have the support of reputable authorities. ^ Physics, vi. 9. 2396 (R.P. 136-139). THE PROBLEM OF INFINITY 169 aimed at the Pythagoreans,* while others have held that they were intended to refute the atomists.^ M. Evellin, on the contrary, holds that they constitute a refutation of infinite divisibility,* while M. G. Nod, in the interests of Hegel, maintains that the first two arguments refute infinite divisibility, while the next two refute indivisibles.* Amid such a bewildering variety of interpretations, we can at least not complain of any restrictions on our liberty of choice. The historical questions raised by the above-mentioned discussions are no doubt largely insoluble, owing to the very scanty material from which our evidence is derived. The points which seem fairly clear are the following : (i) That, in spite of MM. Milhaud and Paul Tannery, Zeno is anxious to prove that motion is really impossible, and that he desires to prove this because he follows Parmenides in denying plurality ; ^ (2) that the third and fourth arguments proceed on the hypothesis of indi- visibles, a hypothesis which, whether adopted by the Pythagoreans or not, was certainly much advocated, as may be seen from the treatise On Indivisible Lines attri- buted to Aristotle. As regards the first two arguments, they would seem to be valid on the hypothesis of indi- visibles, and also, without this hypothesis, to be such as ■ Cf. Gaston Milhaud, Les philosophes-g^omitres de la Grke, p. 140 n. ; Paul Tannery, Pour Phistoire de la science helline, p. 249 ; Burnet, op. cit., p. 362. 2 CJC R. K. Gaye, " On Aristotle, Physics, Z ix." Journal of Philology, vol. xxxi., esp. p. III. Also Moritz Cantor, Vorlesungen iiber Geschichte der Mathemaiik, ist ed., vol. i., 1880, p. 168, who, however, subsequently adopted Paul Tannery's opinion, Vorlesungen, 3rd ed. (vol. i. p. 200). ^ " Le mouvement et les partisans des indivisibles," Revue de MUa- physique et de Morale, vol. i. pp. 382-395. * " Le mouvement et les arguments de Zdnon d'El^e," Revue de Mdta- pkysigue et de Morale, vol. i. pp. 107-125. ' Q^ M. Brochard, " Les prdtendus sophismes de Z^nbn d'El^e," Revue de Mitaphysique et de Morale, vol. i. pp. 209-215 I70 SCIENTIFIC METHOD IN PHILOSOPHY would be valid if the traditional contradictions in infinite numbers were insoluble, which they are not. We may conclude, therefore, that Zeno's polemic is directed against the view that space and time consist of -points and instants; and that as against the view that a finite stretch of space or time consists of a finite number of points and instants, his arguments are not sophisms, \but perfectly valid. The conclusion which Zeno wishes us to draw is that plurality is a delusion, and spaces and times are really indivisible. The other conclusion which is possible, namely, that the number of points and instants is infinite, was not tenable so long as the infinite was infected with ..^.contradictions. In a fragment which is not one of the four famous arguments against motion, Zeno says : "If things are a many, they must be just as many as they are, and neither more nor less. Now, if they are as many as they are, they will be finite in number. " If things are a many, they will be infinite in number ; for there will always be other things between them, and others again between these. And so things are infinite in number." ^ This argument attempts to prove that, if there are many things, the number of them must be both finite and infinite, which is impossible ; hence we are to con- clude that there is only one thing. But the weak point in the argument is the phrase: "If they are just as many as they are, they will be finite in number." This phrase is not very clear, but it is plain that it assumes the impossibility of definite infinite numbers. Without this assumption, which is now known to be false, the arguments of Zeno, though they suflSce (on certain very reasonable assumptions) to dispel the hypothesis of finite ' Simplicius, Phys., 140, 28 D (R.P. 133) ; Burnet, op. cit., pp. 364-365. THE PROBLEM OF INFINITY 171 indivisibles, do not suffice to prove that motion and change and plurality are impossible. They are not, how- ever, on any view, mere foolish quibbles : they are serious arguments, raising difficulties which it has taken two thousand years to answer, and which even now are fatal to the teachings of most philosophers. < ■ The first of Zeno's arguments is the argument of the race-course, which is paraphrased by Burnet as follows : ^ " You cannot get to the end of a race-course. You cannot traverse an infinite number of points in a finite time. You must traverse the half of any given distance before you traverse the whole, and the half of that again before you can traverse it. This goes on ad infinitum, so that there are an infinite number of points in any given space, and you cannot touch an infinite number one by one in a finite time." ^ Zeno appeals here, in the first place, to the fact that ' Op. cit., p. 367. ^ Aristotle's words are : " The first is the one on the non-existence of motion on the ground that what is moved must always attain the middle point sooner than the end-point, on which we gave our opinion in the earlier part of our discourse.'' Phys., vi. 9. 939B (R.P. 136). Aristotle seems to refer to Phys., vi. 2. 223AB [R.P. 136A] : "All space is continu- ous, for time and space are divided into the same and equal divisions. . . . Wherefore also Zeno's argument is fallacious, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time. For there are two senses in which the term 'infinite' is applied both to length and to time, and in fact to all continuous things, either in regard to divisibility, or in regard to the ends. Now it is not possible to touch things infinite in regard to number in a finite time, but it is possible to touch things infinite in regard to divisibility : for time itself also is infinite in this sense. So that in fact we go through an infinite, [space] in an infinite [time] and not in a finite [time], and we touch infinite things with infinite things, not with finite things." Philoponus, a sixth- century commentator (R.P. I36a, Exc. Paris Philop. in Arist. Phys., 803, 2. Vit.), gives the following illustration : " For if a thing were moved the space of a cubit in one hour, since in every space there are an infinite number of points, the thing moved must needs touch all the points of the space ; it will then go through an infinite collection in a finite time, which is impossible." 172 SCIENTIFIC METHOD IN PHILOSOPHY any distance, however small, can be halved. From this it follows, of course, that there must be an infinite number of points in a line. But, Aristotle represents him as arguing, you cannot touch an infinite number of points one by one in a finite time. The words " one by one" are important, (i) If all the points touched are concerned, then, though you pass through them continu- ously, you do not touch them " one by one." That is to say, after touching one, there is not another which you touch next : no two points are next each other, but between any two there are always an infinite number of others, which cannot be enumerated one by one. (2) If, on the other hand, only the successive middle points are concerned, obtained by always halving what remains of the course, then the points are reached one by one, and, though they are infinite in number, they are in fact all reached in a finite time. His argument to the contrary may be supposed to appeal to the view that a finite time must consist of a finite number of instants, in which case what he says would be perfectly true on the assumption that the possibility of continued dichotomy is undeniable. If, on the other hand, we suppose the argument directed against the partisans of infinite divisibility, we must suppose it to proceed as follows : ^ " The points given by successive halving of the distances still to be traversed are infinite in number, and are reached in succession, each being reached a finite time later than its predecessor ; but the sum of an infinite number of finite times must be infinite, and therefore the process will never be completed." It is very possible that this is historically the right inter- pretation, but in this form the argument is invalid. If half the course takes half a minute, and the next quarter » Cf. Mr C. D. Broad, " Note on Achilles and the Tortoise," Mind, N.S., vol. xxii. pp. 31S-9. THE PROBLEM OF INFINITY 173 takes a quarter of a minute, and so on, the whole course will take a minute. The apparent force of the argument, on this interpretation, lies solely in the mistaken supposi- tion that there cannot be anything beyond the whole of ' an infinite series, which can be seen to be false by observ- ing that I is beyond the whole of the infinite series |-, f, ¥> T¥> • • • The second of Zeno's arguments is the one concerning Achilles and the tortoise, which has achieved more notoriety than the others. It is paraphrased by Burnet as follows : ^ " Achilles will never overtake the tortoise. He must first reach the place from which the tortoise started. By that time the tortoise will have got some way ahead. Achilles must then make up that, and again the tortoise will be ahead. He is always coming nearer, but he never makes up to it." ^ This argument is essentially the same as the previous one. It shows that, if Achilles ever overtakes the tortoise, \ it must be after an infinite number of instants have elapsed since he started. This is in fact true ; but the view that ; an infinite number of instants make up an infinitely long time is not true, and therefore the conclusion that Achilles will never overtake the tortoise does not follow. The third argument,^ that of the arrow, is very interest- ing. The text has been questioned. Burnet accepts the alterations of Zeller, and paraphrases thus : " The arrow in flight is at rest. For, if everything is 1 Op. cit. 2 Aristotle's words are: "The second is the so-called Achilles. It consists in this, that the slower will never be overtaken in its course by the quickest, for the pursuer must always come first to the point from which the pursued has just departed, so that the slower must necessarily be always still more or less in advance." Phys., vi. g. 239B (R.P. 137). 3 Phys., vi. 9. 239B (R.P. 138). 174 SCIENTIFIC METHOD IN PHILOSOPHY at rest when it occupies a space equal to itself, and what is in flight at any given moment always occupies a space equal to itself, it cannot move." But according to Prantl, the literal translation of the unemended text of Aristotle's statement of the argu- ment is as follows : " If everything, when it is behaving in a uniform manner, is continually either moving or at rest, but what is moving is always in the kow, then the moving arrow is motionless." This form of the argument brings out its force more clearly than Burnet's paraphrase. Here, if not in the first two arguments, the view that a finite part of time consists of a finite series of successive '■ instants seems to be assumed ; at any rate the plausibility of the argument seems to depend upon supposing that there are consecutive instants. Throughout an instant, ( it is said, a moving body is where it is : it cannot move j during the instant, for that would require that the instant \ should have parts. Thus, suppose we consider a period consisting of a thousand instants, and suppose the arrow is in flight throughout this period. At each of the thousand instants, the arrow is where it is, though at the next instant it is somewhere else. It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever. This is what M. Bergson calls the cinemato- graphic representation of reality. The more the difficulty is meditated, the more real it becomes. The solution lies in the theory of continuous series : we find it hard to avoid supposing that, when the arrow is in flight, there is a next position occupied at the next moment ; but in ; fact there is no next position and no next moment, and when once this is imaginatively realised, the difficulty is seen to disappear. THE PROBLEM OF INFINITY 175 The fourth and last of Zeno's arguments is ^ the argu- ment of the stadium. The argument as stated by Burnet is as follows : Ji ' i ' gl Position. S a soi t d Position. " Half the time may be equal to double the time. Let us suppose three rows of bodies, one of which (A) is at rest while the other two (B, C) are moving with equal velocity in opposite directions. By the time they are all in the same part of the course, B will have passed twice as many of the bodies in C as in A. Therefore the time which it takes to pass C is twice as long as the time it takes to pass A. But the time which B and C take to reach the position of A is the same. Therefore double the time is equal to the half." Gaye ^ devoted an interesting article to the interpreta- tion of this argument. His translation of Aristotle's statement is as follows : " The fourth argument is that concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course, and the other that between the middle point and the starting- post. This, he thinks, involves the conclusion that half a given time is equal to double the time. The fallacy of the reasoning lies in the assumption that a body occupies an equal time in passing with equal velocity a body that is in motion and a body of equal size that is at rest, an j j 1 P/tys., vi. 9. 239B (R.P. 139). 2 f^i- g^f 176 SCIENTIFIC METHOD IN PHILOSOPHY assumption which is false. For instance (so runs the argument), let A A ... be the stationary bodies of equal size, B B . . . the bodies, equal in number and in size to A A . . ., originally occupying the half of the course from the starting-post to the middle of the A's, and C C . . . those originally occupying the other half from the goal to the middle of the A's, equal in number, size, and velocity, to B B . . . Then three consequences follow. First, as the B's and C's pass one another, the first B reaches the last C at the same moment at which the first C reaches the last B. Secondly, at this moment the first C has passed all the A's, whereas the first B has passed only half the A's and has consequently occupied only half the time occupied by the first C, since each of the two occupies an equal time in passing each A. Thirdly, at the same moment all the B's have passed all the C's : for the first C and the first B will simultaneously reach the opposite ends of the course, since (so says Zeno) the time occupied by the first C in passing each of the B's is equal to that occupied by it in passing each of the A's, because an equal time is occupied by both the first B and the first C in passing all the A's. This is the argument : but it presupposes the aforesaid fallacious assumption." This argument is not quite easy to follow, and it is only valid as against the assumption that a finite time consists of a finite First Position. Second Position. , j. . B B' B^ B B' B" '^""^°^'" °* mstants. We may re-state it A A' A" A A' A" ^^ different language. Let us suppose three C C' C"^ C C' C", drill-sergeants, A, A', and A", standing in a row, while the two files of soldiers march past them in opposite directions. At the first moment which we con- THE PROBLEM OF INFINITY 177 sider, the three men B, B', B" in one row, and the three men C, C, C" in the other row, are respectively opposite to A, A', and A". At the very next moment, each row has moved on, and now B and.C" are opposite A'. Thus B and C" are opposite each other. When, then, did B 1 pass C ? It must have been somewhere between the • two moments which we supposed consecutive, and there- | fore the two moments cannot really have been consecutive. y It follows that there must be other moments between any j two given moments, and therefore that there must be an I infinite number of moments in any given interval of time, j The above difficulty, that B must have passed C at some time between two consecutive moments, is a genuine one, but is not precisely the difficulty raised by Zeno. What Zeno professes to prove is that " half of a given time is equal to double that time." The most intelligible explanation of the argument known to me is that of Gaye.^ Since, however, his explanation is not easy to set forth shortly, I will re-state what seems to me to be the logical essence of Zeno's contention. If we suppose that time consists of a series of consecutive instants, and that motion consists in passing through a series of con- secutive points, then the fastest possible motion is one which, at each instant, is at a point consecutive to that at which it was at the previous instant. Any slower motion must be one which has intervals of rest inter- spersed, and any faster motion must wholly omit some points. All this is evident from the fact that we cannot have more than one event for each instant. But now, in the case of our A's and B's and C's, B is opposite a fresh A every instant, and therefore the number of A's passed gives the number of instants since the beginning of the motion. But during the motion B has passed twice as ' Loc. cit., p. 105. 12 1 78 SCIENTIFIC METHOD IN PHILOSOPHY many C's, and yet cannot have passed more than one each instant. Hence the number of instants since the motion began is twice the number of A's passed, though we pre- viously found it was equal to this number. From this result, Zeno's conclusion follows. Zeno's arguments, in some form, have aiForded grounds for almost all the theories of space and time and infinity which have been constructed from his day to our own. We have seen that all his arguments are valid (with certain reasonable hypotheses) on the assumption that finite spaces and times consist of a finite number of points and instants, and that the third and fourth almost certainly in fact proceeded on this assumption, while the first and second, which were perhaps intended to refute the opposite assumption, were in that case fallacious^/ We may therefore escape from his paradoxes either by maintaining that, though space and time do consist of points and instants, the number of them in any finite interval is infinite ; or by denying that space and time consist of points and instants at all ; or lastly, by denying the reality of space and time altogether. It would seem that Zeno himself, as a supporter of Parmenides, drew the last of these three possible deductions, at any rate in regard to time. In this a very large number of philoso- phers have followed him. Many others, like M. Bergson, have preferred to deny that space and time consist of points and instants. Either of these solutions will meet the difficulties in the form in which Zeno raised them. But, as we saw, the difficulties can also be met if infinite numbers are admissible. And on grounds which are independent of space and time, infinite numbers, and series in which no two terms are consecutive, must in any case be admitted. Consider, for example, all the fractions less than i, arranged in order of magnitude. THE PROBLEM OF INFINITY 179 Between any two of them, there are others, for example, the arithmetical mean of the two. Thus no two fractions are consecutive, and the total number of them is infinite. It will be found that much of what Zeno says as regards the series of points on a line can be equally well applied to the series of fractions. And we cannot deny that there are fractions, so that two of the above ways of escape are closed to us. It follows that, if we are to solve the whole class of difficulties derivable from Zeno's by analogy, we must discover some tenable theory of infinite numbers. What, then, are the difficulties which, until the last thirty years, led philosophers to the belief that infinite numbers are impossible .'' 4 The diflSculties of infinity are of two kinds, of which the first may be called sham, while the others involve, for their solution, a certain amount of new and not altogether easy thinking. The sham difficulties are those suggested by the etymology, and those suggested by confusion of the mathematical infinite with what philosophers im- pertinently call the " true " infinite. Etymologically, "infinite" should mean "having no end." But in fact some infinite series have ends, some have not ; while some collections are infinite without being serial, and can therefore not properly be regarded as either endless or having ends. The series of instants from any earlier one to any later one (both included) is infinite, but has two ends ; the series of instants from the beginning of time to the present moment has one end, but is infinite. Kant, in his fii"st antinomy, seems to hold that it is harder for the past to be infinite than for the future to be so, on the ground that the past is now completed, and that nothing infinite can be completed. It is very difficult to see how he can have imagined that there was any sense in this remark ; but it seems most probable that he was L 1 80 SCIENTIFIC METHOD IN PHILOSOPHY thinking of the infinite as the " unended." It is odd that he did not see that the future too has one end at the present, and is precisely on a level with the past. His regarding the two as different in this respect illustrates just that kind of slavery to time which, as we agreed in speaking of Parmenides, the true philosopher must learn to leave behind him. The confusions introduced into the notions of philoso- phers by the so-called " true " infinite are curious. They see that this notion is not the same as the mathematical infinite, but they choose to believe that it is the notion which the mathematicians are vainly trying to reach. They therefore inform the mathematicians, kindly but firmly, that they are mistaken in adhering to the " false " infinite, since plainly the " true " infinite is something quite different. The reply to this is that what they call the " true " infinite is a notion totally irrelevant to the problem of the mathematical infinite, to which it has only a fanci- ful and verbal analogy. So remote is it that I do not propose to confuse the issue by even mentioning what the " true " infinite is. It is the " false " infinite that concerns us, and we have to show that the epithet " false " is undeserved. There are, however, certain genuine difficulties in understanding the infinite, certain habits of mind derived from the consideration of finite numbers, and easily extended to infinite numbers under the mistaken notion that they represent logical necessities. For example, every number that we are accustomed to, except o, has another number immediately before it, from which it results by adding i ; but the first infinite number does not have this property. The numbers before it form an infinite series, containing all the ordinary finite numbers, having no maximum, no last finite number, after which THE PROBLEM OF INFINITY i8i one little step would plunge us into the infinite. If it is assumed that the first infinite number is reached by a succession of small steps, it is easy to show that it is self- contradictory. The first infinite number is, in fact, beyond the whole unending series of finite numbers. " But," it will be said, " there cannot be anything beyond the whole of an unending series." This, we may point out, is the very principle upon which Zeno relies in the arguments of the race-course and the Achilles. Take the race-course : there is the moment when the runner still has half his distance to run, then the moment when he still has a quarter, then when he still has an eighth, and so on in a strictly unending series. Beyond the whole of this series is the moment when he reaches the goal. Thus there certainly can be something beyond the whole of an unending series. But it remains to show that this fact is only what might have been expected. The diflliculty, like most of the vaguer difficulties besetting the mathematical infinite, is derived, I think, from the more or less unconscious operation of the idea of counting. If you set to work to count the terms in an infinite collection, you will never have completed your task. Thus, in the case of the runner, if half, three- quarters, seven-eighths, and so on of the course were marked, and the runner was not allowed to pass any of the marks until the umpire said " Now," then Zeno's conclusion would be true in practice, and he would never reach the goal. But it is not essential to the existence of a collection, or even to knowledge and reasoning concerning it, that we should be able to j)ass its terms in review one by one. This may be seen in the case of fijiite collections ; we can speak of " mankind " or " the human race," though many of the individuals in this collection are not personally 1 82 SCIENTIFIC METHOD IN PHILOSOPHY known to us. We can do this because we know of various characteristics which every individual has if he belongs to the collection, and not if he does not. And exactly the same happens in the case of infinite collections : they may be known by their characteristics although their terms cannot be enumerated. In this sense, an unending series may nevertheless form a whole, and there may be new terms beyond the whole of it. Some purely arithmetical peculiarities of infinite numbers have also caused perplexity. For instance, an infinite number is not increased by adding one to it, or by doubling it. Such peculiarities have seemed to many to contradict logic, but in fact they only contradict con- firmed mental habits. The whole difficulty of the subject lies in the necessity of thinking in an unfamiliar way, and in realising that many properties which we have thought inherent in number are in fact peculiar to finite numbers. If this is remembered, the positive theory of infinity, which will occupy the next lecture, will not be found so difficult as it is to those who cling obstinately to the prejudices instilled by the arithmetic which is learnt in childhood. LECTURE VII THE POSITIVE THEORY OF INFINITY LECTURE VII THE POSITIVE THEORY OF INFINITY The positive theory of infinity, and the general theory of number to which it has given rise, are among the triumphs of scientific method in philosophy, and are therefore specially suitable for illustrating the logical- analytic character of that method. The work in this subject has been done by mathematicians, and its results can be expressed in mathematical symbolism. Why, then, it may be said, should the subject be regarded as philosophy rather than as mathematics ? This raises a difficult question, partly concerned with the use of words, but partly also of real importance in understand- ing the function of philosophy. Every subject-matter, it would seem, can give rise to philosophical investigations as well as to the appropriate science, the difference between the two treatments being in the direction of movement and in the kind of truths which it is sought to establish. In the special sciences, when they have become fully developed, the movement is forward and synthetic, from the simpler to the more complex. But in philosophy we follow the inverse direction : from the complex and relatively concrete we proceed towards the simple and abstract by means of analysis, seeking, in the process, to eliminate the particularity of the original subject-matter, and to confine our attention entirely to the logical yyrw of the facts concerned, i8S 1 86 SCIENTIFIC METHOD IN PHILOSOPHY Between philosophy and pure mathematics there is a certain affinity, in the fact that both are general and a prion. Neither of them asserts propositions which, like those of history and geography, depend upon the actual concrete facts being just what they are. We may illustrate this characteristic by means of Leibniz's conception of many possible worlds, of which one only is actual. In all the many possible worlds, philosophy and mathematics will be the same ; the difFerences will only be in respect of those particular facts which are chronicled by the descrip- tive sciences. Any quality, therefore, by which our actual world is distinguished from other abstractly possible worlds, must be ignored by mathematics and philosophy alike. Mathematics and philosophy difFer, however, in their manner of treating the general properties in which all possible worlds agree ; for while mathematics, starting from comparatively simple propositions, seeks to build up more and more complex results by deductive synthesis, philosophy, starting from data which are common know- ledge, seeks to purify and generalise them into the simplest statements of abstract form that can be obtained from them by logical analysis. The difference between philosophy and mathematics may be illustrated by our present problem, namely, the nature of number. Both start from certain facts about numbers which are evident to inspection. But mathe- matics uses these facts to deduce more and more com- plicated theorems, while philosophy seeks, by analysis, to go behind these facts to others, simpler, more fundamental, and inherently more fitted to form the premisses of the science of arithmetic. The question, " What is a number ? " is the pre-eminent philosophic question in this subject, but it is one which the mathematician as such need not ask, provided he knows enough of the properties THE POSITIVE THEORY OF INFINITY 187 of numbers to enable him to deduce his theorems. We, since our object is philosophical, must grapple with the philosopher's question. The answer to the question, " What is a number .'' " which we shall reach in this lecture, will be found to give also, by implication, the answer to the difficulties of infinity which we considered in the previous lecture. The question " What is a number .'' " is one which, until quite recent times, was never considered in the kind of way that is capable of yielding a precise answer. Philosophers were content with some vague dictum such as, " Number is unity in plurality." A typical definition of the kind that contented philosophers is the following from Sigwart's Logic (§ 66, section 3) : " Every number is not merely a plurality, but a plurality thought as held togethei and closed, and to that extent as a unity." Now there is in such definitions a very elementary blunder, of the same kind that would be committed if we said " yellow is a flower " because some flowers are yellow. Take, for example, the number 3. A single collection of three things might conceivably be described as " a plurality thought as held together and closed, and to that extent as a unity " ; but a collection of three things is not the number 3. The number 3 is something which all collec- tions of three things have in common, but is not itself a collection of three things. The definition, therefore, apart from any other defects, has failed to reach the necessary degree of abstraction : the number 3 is something more abstract than any collection of three things. Such vague philosophic definitions, however, remained inoperative because of their very vagueness. What most men who thought about numbers really had in mind was that numbers are the result of counting. " On the consciousness of the law of counting," says Sigwart is 8 SCIENTIFIC METHOD IN PHILOSOPHY at the beginning of his discussion of number, " rests the possibility of spontaneously prolonging the series of numbers ad infinitum." It is this view of number as generated by counting which has been the chief psychological obstacle to the understanding of infinite numbers. Counting, because It is familiar, is erroneously supposed to be simple, whereas it is in fact a highly complex process, which has no meaning unless the numbers reached in counting have some significance independent of the process by which they are reached. And infinite numbers cannot be reached at all In this way. The mistake is of the same kind as if cows were defined as what can be bought from a cattle-merchant. To a person who knew several cattle-merchants, but had never seen a cow, this might seem an admirable defini- tion. But if In his travels he came across a herd of wild cows, he would have to declare that they were not cows at all, because no cattle-merchant could sell them. So infinite numbers were declared not to be numbers at all, because they could not be reached by counting. It will be worth while to consider for a moment what counting actually is. We count a set of objects when we let our attention pass from one to another, until we have attended once to each, saying the names of the numbers in order with each successive act of attention. The last number named in this process is the number of the objects, and therefore counting is a method of finding out what the number of the objects is. But this operation is really a very complicated one, and those who Imagine that It Is the logical source of number show themselves remarkably incapable of analysis. In the first place, when we say " one, two, three . . . " as we count, we cannot be said to be discovering the number of the objects counted unless we attach some meaning THE POSITIVE THEORY OF INFINITY 189 to the words one, two, three, ... A child may learn to know these words in order, and to repeat them correctly like the letters of the alphabet, without attach- ing any meaning to them. Such a child may count correctly from the point of view of a grown-up listener, without having any idea of numbers at all. The operation of counting, in fact, can only be intelligently performed by a person who already has some idea what the numbers are ; and from this it follows that counting does not give the logical basis of number. Again, how do we know that the last number reached in the process of counting is the number of the objects counted .'' This is just one of those facts that are too familiar for their significance to be realised ; but those who wish to be logicians must acquire the habit of dwelling upon such facts. There are two propositions involved in this fact : first, that the number of numbers from I up to any given number is that given number — for instance, the number of numbers from i to 100 is a hundred ; secondly, that if a set of numbers can be used as names of a set of objects, each number occurring only once, then the number of numbers used as names is the same as the number of objects. The first of these propositions is capable of an easy arithmetical proof so long as finite numbers are concerned ; but with infinite numbers, after the first, it ceases to be true. The second proposition remains true, and is in fact, as we shall see, an immediate consequence of the definition of number. But owing to the falsehood of the first proposition where infinite numbers are concerned, count- ing, even if it were practically possible, would not be a valid method of discovering the number of terms in an infinite collection, and would in fact give different results according to the manner in which it was carried out. I90 SCIENTIFIC METHOD IN PHILOSOPHY There are two respects in which the infinite numbers that are known differ from finite numbers : first, infinite numbers have, while finite numbers have not, a property which I shall call reflexiveness ; secondly, finite numbers have, while infinite numbers have not, a property which I shall call inductiveness. Let us consider these two properties successively. (i) Reflexiveness. — A number is said to be reflexive when it is not increased by adding r to it. It follows at once that any finite number can be added to a reflexive number without increasing it. This property of infinite numbers was always thought, until recently, to be self- contradictory ; but through the work of Georg Cantor it has come to be recognised that, though at first astonishing, it is no more self-contradictory than the fact that people at the antipodes do not tumble ofF. In virtue of this property, given any infinite collection of objects, any finite number of objects can be added or taken away without increasing or diminishing the number of the collection. Even an infinite number of objects may, under certain conditions, be added or taken away without altering the number. This may be made clearer by the help of some examples. Imagine all the natural numbers o, i, 2, 3, . . . to be written down in a row, and immediately beneath them o, I, 2, 3, ...... . t, 2, 3, 4, . . . «+i . . . write down the numbers i, 2, 3, 4, . . ., so that i is under o, 2 is under i, and so on. Then every number in the top row has a number directly under it in the bottom row, and no number occurs twice in either row. It follows that the number of numbers in the two rows must be the same. But all the numbers that occur in the bottom row also occur in the top THE POSITIVE THEORY OF INFINITY 191 row, and one more, namely o ; thus the number of terms in the top row is obtained by adding one to the number of the bottom row. So long, therefore, as it was supposed that a number must be increased by adding i to it, this state of things constituted a con- tradiction, and led to the denial that there are infinite numbers. The following example is even more surprising. Write the natural numbers i, 2, 3, 4, ... in the top row, and the even numbers 2, 4, 6, 8, . . . in the bottom row, so that under each number in the top row stands its double in the bottom row. Then, as before, the number of numbers in the two rows is the same, yet the second row results from taking away all the odd numbers — an infinite collection — from the top row. This example is given by Leibniz to prove that there can be no infinite numbers. He believed in infinite collections, but, since he thought that a number must always be increased when it is added to and diminished when it is subtracted from, he maintained that infinite collections do not have numbers. " The number of all numbers," he says, " implies a contradiction, which I show thus : To any number there is a corresponding number equal to its double. Therefore the number of all numbers is not greater than the number of even numbers, i.e. the whole is not greater than its part." ^ In dealing with this argument, we ought to substitute " the number of all finite numbers " for " the number of all numbers " ; we then obtain exactly the illustration given by our two rows, one containing all the finite numbers, the other only the even finite numbers. It will be seen that Leibniz regards it as self-contradictory to maintain that the whole is not ' Phil. Werke, Gerhardt's edition, vol. i. p. 338. 192 SCIENTIFIC METHOD IN PHILOSOPHY greater than its part. But the word "greater" is one which is capable of many meanings ; for our purpose, we must substitute the less ambiguous phrase " containing a greater number of terms." In this sense, it is not self- contradictory for whole and part to be equal ; it is the realisation of this fact which has made the modern theory of infinity possible. There is an interesting discussion of the reflexiveness of infinite wholes in the first of Galileo's Dialogues on Motion. I quote from a translation published in 1730.^ The personages in the dialogue are Salviati, Sagredo, and Simplicius, and they reason as follows : " Simp. Here already arises a Doubt which I think is not to be resolv'd ; and that is this : Since 'tis plain that one Line is given greater than another, and since both contain infinite Points, we must surely necessarily infer, that we have found in the same Species something greater than Infinite, since the Infinity of Points of the greater Line exceeds the Infinity of Points of the lesser. But now, to assign an Infinite greater than an Infinite, is what I can't possibly conceive. " Sah. These are some of those Difficulties which arise from Discourses which our finite Understanding makes about Infinites, by ascribing to them Attributes which we give to Things finite and terminate, which I think most improper, because those Attributes of Majority, Minority, and Equality, agree not with Infinities, of which we can't say that one is greater than, less than, or equal to another. For Proof whereof I have something come ' Mathematical Discourses concerning two new sciences relating to tnechanics and local motion^ in four dialogues. By Galileo Galilei, Chief Philosopher and Mathematician to the Grand Duke of Tuscany. Done into English from the Italian, by Tho. Weston, late Master, and now published by John Weston, present Master, of the Academy at Greenwich, See pp. 46 ff. I THE POSITIVE THEORY OF INFINITY 193 into my Head, which (that I may be the better under- stood) I will propose by way of Interrogatories to Simplicius, who started this Difficulty. To begin then : I suppose you know which are square Numbers, and which not? ^^ Simp. I know very well that a square Number is that which arises from the JMultiplication of any Number into itself ; thus 4 and 9 are square Numbers, that arising from 2, and this from 3, multiplied by themselves. " Salv. Very well ; And you also know, that as the Products are call'd Squares, the Factors are call'd Roots : And that the other Numbers, which proceed not from Numbers multiplied into themselves, are not Squares. Whence taking in all Numbers, both Squares and Not Squares, if I should say, that the Not Squares are more than the Squares, should I not be in the right .'' " Simp. Most certainly. " Salv. If I go on with you then, and ask you. How many squar'd Numbers there are .'* you may truly answer. That there are as many as are their proper Roots, since every Square has its own Root, and every Root its own Square, and since no Square has more than one Root, nor any Root more than one Square. " Simp. Very true. "Salv. But now, if I should ask how many Roots there are, you can't deny but there are as many as there are Numbers, since there's no Number but what's the Root to some Square. And this being granted, we may likewise affirm, that there are as many square Numbers, as there are Numbers ; for there are as many Squares as there are Roots, and as many Roots as Numbers. And yet in the Beginning of this, we said, there were many more Numbers than Squares, the greater Part of Numbers being not Squares : And tho' the Number of 13 % 194 SCIENTIFIC METHOD IN PHILOSOPHY Squares decreases in a greater proportion, as we go on to bigger Numbers, for count to an Hundred you'll find lo Squares, viz. i, 4, 9, 16, 25, 36, 49, 64, 81, 100, which is the same as to say the loth Part are Squares ; in Ten thousand only the looth Part are Squares ; in a Million only the loooth : And yet in an infinite Number, if we can but comprehend it, we may say the Squares are as many as all the Numbers taken together. " Sagr. What must be determin'd then in this Case .? " Salv. I see no other way, but by saying that all Numbers are infinite ; Squares are Infinite, their Roots Infinite, and that the Number of Squares is not less than the Number of Numbers, nor this less than that : and then by concluding that the Attributes or Terms of Equality, Majority, and Minority, have no Place in Infinites, but are confin'd to terminate Quantities." The way in which the problem is expounded in the above discussion is worthy of Galileo, but the solution suggested is not the right one. It is actually the case that the number of square (finite) numbers is the same as the number of (finite) numbers. The fact that, so long as we confine ourselves to numbers less than some given finite number, the proportion of squares tends towards zero as the given finite number increases, does not contradict the fact that the number of all finite squares is the same as the number of all finite numbers. This is only an instance of the fact, now familiar to mathe- maticians, that the limit of a function as the variable approaches a given point may not be the same as its value when the variable actually reaches the given point. But although the infinite numbers which Galileo discusses are equal. Cantor has shown that what Simplicius could not conceive is true, namely, that there are an infinite number of different infinite numbers, and that the conception of THE POSITIVE THEORY OF INFINITY 195 greater and less can be perfectly well applied to them. The whole of Simplicius's difficulty comes, as is evident, from his belief that, if greater and less can be applied, a part of an infinite collection must have fewer terms than the whole ; and when this is denied, all contradictions disappear. As regards greater and less lengths of lines, which is the problem from which the above discussion starts, that involves a meaning of greater and less which is not arithmetical. The number of points is the same in a long line and in a short one, being in fact the same as the number 'of points in all space. The greater and less of metrical geometry involves the new metrical concep- tion of congruence, which cannot be developed out of arithmetical considerations alone. But this question has not the fundamental importance which belongs to the arithmetical theory of infinity. (2) Non-inductiveness. — The second property by which infinite numbers are distinguished from finite numbers is the property of non-inductiveness. This will be best explained by defining the positive property of inductive- nes.s which characterises the finite numbers, and which is named after the method of proof known as " mathe- matical induction." Let us first consider what is meant by calling a property " hereditary " in a given series. Take such a property as being named Jones. If a man is named Jones, so is his son ; we will therefore call the property of being called Jones hereditary with respect to the relation of father and son. If a man is called Jones, all his de- scendants in the direct male line are called Jones ; this follows from the fact that the property is hereditary. Now, instead of the relation of father and son, consider the relation of a finite number to its immediate successor, that is, the relation which holds between o and i, between 196 SCIENTIFIC METHOD IN PHILOSOPHY I and 2, between 2 and 3, and so on. If a property of numbers is hereditary with respect to this relation, then if it belongs to (say) 100, it must belong also to all finite numbers greater than 100 ; for, being hereditary, it belongs to lor because it belongs to 100, and it belongs to 102 because it belongs to 10 1, and so on — where the " and so on " will take us, sooner or later, to any finite number greater than 100. Thus, for example, the property of being greater than 99 is hereditary in the series of finite numbers ; and generally, a property is hereditary in this series when, given any number that possesses the property, the next number must always also possess it. It will be seen that a hereditary property, though it must belong to all the finite numbers greater than a given number possessing the property, need not belong to all the numbers less than this number. For example, the hereditary property of being greater than 99 belongs to 100 and all greater numbers, but not to any smaller number. Similarly, the hereditary property of being called Jones belongs to all the descendants (in the direct male line) of those who have this property, but not to all their ancestors, because we reach at last a first Jones, before whom the ancestors have no surname. It is obvious, however, that any hereditary property possessed by Adam must belong to all men ; and similarly any hereditary property possessed by o must belong to all finite numbers. This is the principle of what is called " mathematical induction." It frequently happens, when we wish to prove that all finite numbers have some property, that we have first to prove that o has the property, and then that the property is hereditary, i.e. that, if it belongs to a given number, then it belongs to the next number. Owing to the fact that such proofs THE POSITIVE THEORY OF INFINITY 197 are called " inductive," I shall call the properties to which they are applicable "inductive" properties. Thus an inductive property of numbers is one which is hereditary and belongs to o. Taking any one of the natural numbers, say 29, it is easy to see that it must have all inductive properties. For since such properties belong to o and are hereditary, they belong to i ; therefore, since they are hereditary, they belong to 2, and so on ; by twenty-nine repetitions of such arguments we show that they belong to 29. We may define the " inductive " numbers as all those that possess all inductive properties ; they will be the same as what are called the " natural " numbers, i.e. the ordinary finite whole numbers. To all such numbers, proofs by mathematical induction can be validly applied. They "are those numbers, we may loosely say, which can be reached from o by successive additions of i ; in other words, they are all the numbers that can be reached by counting. But beyond all these numbers, there are the infinite numbers, and infinite numbers do not have all inductive properties. Such numbers, therefore, may be called non- inductive. All those properties of numbers which are proved by an imaginary step-by-step process from one number to the next are liable to fail when we come to infinite numbers. The first of the infinite numbers has no immediate predecessor, because there is no greatest finite number ; thus no succession of steps from one number to the next will ever reach from a finite number to an infinite one, and the step-by-step metnod of proof fails. This is another reason for the supposed self- contradictions of infinite number. Many of the most familiar properties of numbers, which custom had led people to regard as logically necessary, are in fact only 198 SCIENTIFIC METHOD IN PHILOSOPHY demonstrable by the step-by-step method, and fail to be true of infinite numbers. But so soon as we realise the necessity of proving such properties by mathematical induction, and the strictly limited scope of this method of proof, the supposed contradictions are seen to contra- dict, not logic, but only our prejudices and mental habits. The property of being increased by the addition of I — i.e. the property of non-reflexiveness — may serve to illustrate the limitations of mathematical induction. It is easy to prove that o is increased by the addition of i, and that, if a given number is increased by the addition of I, so is the next number, i.e. the number obtained by the addition of i . It follows that each of the natural numbers is increased by the addition of i. This follows generally from the general argument, and follows for each particular case by a sufficient number of applications of the argu- ment. We first prove that o is not equal to i ; then, since the property of being increased by i is hereditary, it follows that i is not equal to 2 ; hence it follows that 2 is not equal to 3 ; if we wish to prove that 30,000 is not equal to 30,001, we can do so by repeating this reasoning 30,000 times. But we cannot prove in this way that all numbers are increased by the addition of i ; we can only prove that this holds of the numbers attain- able by successive additions of i starting from o. The reflexive numbers, which lie beyond all those attainable in this way, are as a matter of fact not increased by the addition of i. The two properties of reflexiveness and non-inductive- ness, which we have considered as characteristics of infinite numbers, have not so far been proved to be always found together. It is known that all reflexive numbers are non-inductive, but it is not known that all non-inductive numbers are reflexive. Fallacious proofs THE POSITIVE THEORY OF INFINITY 199 of this proposition have been published by many writers, including myself, but up to the present no valid proof has been discovered. The infinite numbers actually known, however, are all reflexive as well as non-induc- tive ; thus, in mathematical practice, if not in theory, the two properties are always associated. For our pur- poses, therefore, it will be convenient to ignore the bare possibility that there may be non-inductive non-reflexive numbers, since all known numbers are either inductive or reflexive. When infinite numbers are first introduced to people, they are apt to refuse the name of numbers to them, because their behaviour is so difi^erent from that of finite numbers that it seems a wilful misuse of terms to call them numbers at all. In order to meet this feeling, we must now turn to the logical basis of arithmetic, and consider the logical definition of numbers. The logical definition of numbers, though it seems an essential support to the theory of infinite numbers, was in fact discovered independently and by a different man. The theory of infinite numbers — that is to say, the arith- metical as opposed to the logical part of the theory — was discovered by Georg Cantor, and published by him in 1882—3.^ The definition of number was discovered about the same time by a man whose great genius has not received the recognition it deserves — I mean Gottlob Frege of Jena. His first work, Begriffsschrifi, published in 1 879, contained the very important theory of hereditary properties in a series to which I alluded in connection with inductiveness. His definition of number is con- tained in his second work, published in 1884, and entitled Die Grundlagen der Arithmetik, eine logisch-mathematische ' In his Grundlagen dner allgemeinen Mannichfaltigkeitshhrt and in articles in Acta Mathematica, vol. ii. 200 SCIENTIFIC METHOD IN PHILOSOPHY Untersuchung Uber den Begriff der Zahl} It is with this book that the logical theory of arithmetic begins, and it will repay us to consider Frege's analysis in some detail. Frege begins by noting the increased desire for logical strictness in mathematical demonstrations which distin- guishes modern mathematicians from their predecessors, and points out that this must lead to a critical investiga- tion of the definition of number. He proceeds to show the inadequacy of previous philosophical theories, especially of the " synthetic a priori " theory of Kant and the empirical theory of Mill. This brings him to the ques- tion : What kind of object Is it that number can properly be ascribed to .'' He points out that physical things may be regarded as one or many : for example, if a tree has a thousand leaves, they may be taken altogether as con- stituting its foliage, which would count as one, not as a thousand ; and one pair of boots is the same object as two boots. It follows that physical things are not the subjects of which number is properly predicated ; for when we have discovered the proper subjects, the number to be ascribed must be unambiguous. This leads to a discussion of the very prevalent view that number is really something psychological and subjective, a view which Frege emphatically rejects. " Number," he says, " is as little an object of psychology or an outcome of pscyhical processes as the North Sea. . . . The botanist wishes to state something which is just as much a fact when he gives the number of petals in a flower as when he gives its colour. The one depends as little as the other upon our caprice. There is therefore a certain > The definition of number contained in this book, and elaborated in the Grundgesetze der Arithmetik (vol. i., 1893 ; vol. ii., 1903) was re- discovered by me in ignorance of Frege's work. I wish to state as emphatically as possible— what seems still often ignored— that his dis- covery antedated mine by eighteen years. THE POSITIVE THEORY OF INFINITY 201 similarity between number and colour ; but this does not consist in the fact that both are sensibly perceptible in external things, but in the fact that both are objective " (P- 34)- " I distinguish the objective," he continues, " from the palpable, the spatial, the actual. The earth's axis, the centre of mass of the solar system, are objective, but I should not call them actual, like the earth itself" (p. 35). He concludes that number is neither spatial and physical, nor subjective, but non-sensible and objective. This conclusion is important, since it applies to all the subject- matter of mathematics and logic. Most philosophers have thought that the physical and the mental between them exhausted the world of being. Some have argued that the objects of mathematics were obviously not sub- jective, and therefore must be physical and empirical ; others have argued that they were obviously not physical, and therefore must be subjective and mental. Both sides were right in what they denied, and wrong in what they asserted ; Frege has the merit of accepting both denials, and finding a third assertion by recognising the world of logic, which Is neither mental nor physical. The fact is, as Frege points out, that no number, not even i, is applicable to physical things, but only to general terms or descriptions, such as " man," " satellite of the earth," "satellite of Venus." The general term "man" is applicable to a certain number of objects : there are in the world so and so many men. The unity which philo- sophers rightly feel to be necessary for the assertion of a number is the unity of the general term, and it is the general term which is the proper subject of number. And this applies equally when there Is one object or none which falls under the general term. " Satellite of the earth " is a term only applicable to one object, namely, 202 SCIENTIFIC METHOD IN PHILOSOPHY the moon. But " one " is not a property of the moon itself, which may equally well be regarded as many molecules : it is a property of the general term " earth's satellite." Similarly, o is a property of the general term "satellite of Venus," because Venus has no satellite. Here at last we have an intelligible theory of the number o. This was impossible if numbers applied to physical objects, because obviously no physical object could have the number o. / 120. Enumeration, 2012. Euclid, 160, i64.\ 244 SCIENTIFIC METHOD IN PHILOSOPHY Evellin, 169. Evolutionism, 4, 1 1 fif. Extension, 146, 149. External world, knowledge of, 63 ff. Fact, 51. atomic, 52. Finalism, 13. Form, logical, 42 ff., 185, 208. Fractions, 132, 179, Free will, 213, 227 ff. Frege, 5, 40, 199 ff. Galileo, 4, 59, 192, 194, 239, 240. Gaye, 169 n., 175, 177. Geometry, 5. Giles, 206 n. Greater and less, 195. Harvard, 4. Hegel, 3, 37 ff., 46, 166. " Here," 73, 92. Hereditary properties, 195. Hippasos, 163, 237. Hui Tzu, 206. Hume, 217, 221. Hypotheses in philosophy, 239. Illusions, 85. Incommensurables, 162 ff., 237. Independence, 73, 74. causal and logical, 74, 75. IndiscernibiUty, 141, 148. Indivisibles, 160. Induction, 34, 222. mathematical, 195 ff. Inductiveness, 190, 195 ff. Inference, 44, 54. Infinite, vi, 64, 133, 149. historically considered, 155 ff. "true," 179, 180. positive theory of, 185 ff. Infinitesimals, 135. Instants, 116 ff., 129, 151, 216. defined, 118. Instinct v. Reason, 20 ff. Intellect, 22 ff. Intelligence, how displayed by friends, 93. inadequacy of display, 96. Interpretation, 144. James, 4, 10, 13. Jourdain, 165 n. Jowett, 167. Judgment, 58. Kant, 3, 112, ii6, 155 ff., 200. Kaowiedgaihoat, ua. Language, bad, 82, 135. Laplace, 12. Laws of nature, 218 ff. Leibniz, 13, 40, 87, 186, 191. Logic, 201. analytic not constructive, 8. Aristotelian, 5. and fact, 53. inductive, 34, 222. mathematical, vi, 40 ff. mystical, 46. and philosophy, 8, 33 ff., 239. Logical constants, 208, 213. Mach, 123, 224. Macran, 39 n. Mathematics, 40, 57. Matter, 75, loi ff. permanence of, 102 ff. Measurement, 164. Memory, 230, 234, 236. Method, deductive, 5. logical-analytic, v, 65, 211, 236 ff. Milhaud, 168 n., 169 n. Mill, 34, 200. Montaigne, 28. Motion, 130, 216. continuous, 133, 136. mathematical theory of, 133. perception of, 137 ff. Zeno's arguments on, 168 ff. Mysticism, 19, 46, 63, 95. Newton, 30, 146. Nietzsche, 10, 11. Noel, 169. Number, cardinal, 131, 186 ff. defined, 199 ff. finite, 160, 190 ff. inductive, 197. infinite, 178, 180, 188 ff., 197. reflexive, 190 ff. Occam, 107, 146. One and many, 167, 170. Order, 131. Parmenides, 63, 165 ff., 178. Past and future, 224, 234 ff. INDEX 245 P eano, 40. Perspectives, 88ff,, iii. Philoponus, 171 n. Philosophy and ethics, 26 ff. and mathematics, 185 ff. province of, 17, 26, 185, 236. scientific, 11, 16, i8, 29, 236 ff. Physics, loi ff., 147, 239, 242. descriptive, 224. verifiabiUty of, 81, no. Place, 86, 90. at axiAfrotn, 92. Plato, 4, 19, 27, 46, 63, 165 n., 166, 167. Poincar6, 123, 141. Points, 113 ff., 129, 158. . definition of, vi, 115. Pragmatism, 11. Prantl, 174. Predictability, 229 ff. Premisses, 211. Probability, 36. Propositions, 52. atomic, 52. general, 55. molecular, 54. Pythagoras, 19, 160 ff., 237. Race-course, Zeno's argument of, 171 ff. Realism, new, 6. Reflexiveness, 190 ff. Relations, 45. asymmetrical, 47. Bradley's reasons against, 6. external, 150. intransitive, 48. multiple, 50. one-one, 203. reality of, 49. symmetrical, 47, 124. transitive, 48, 124. Relativity, 103, 242. Repetitions, 230 ff. Rest, 136. Ritter and Preller, 161 n. Robertson, D. S., 160 n. Rousseau, 20. Royce, so- Santayana, 46. Scepticism, 66, 67. Seeing double, 86. Self, 73- Sensation, 25, 75, 123. and stimulus, 139. Sense-data, 56, 63, 67, 75, no, 141, 143, 213. and physics, v, 64, 81, 97, loi ff., 140. infinitely numerous ? 149, 159. Sense-perception, 53. Series, 49. compact, 132, 142, 178. continuous, 131, 132. Sigwart, 187. Simplicius, 170 n. Simultaneity, 116. Space, 73, 88, 103, 112 ff., 130. absolute and relative, 146, 159. antinomies of, 155 ff. perception of, 68. of perspectives, 88 ff. private, 89, 90. of touch and sight, 78, 113. Spencer, 4, 12, 236. Spinoza, 46, 166. Stadium, Zeno's argument of, 134 n., 175 ff Subject-predicate, 45. Synthesis, 157, 185. Tannery, Paul, 169 n. Teleology, 223. Testimony, 67, 72, 82, 87, 96, 212. Thales, 3. Thing-in-itself, 75, 84. Things, 89 ff., 104 ff., 213. Time, 103, 116 ff., 130, 155 ff., 166, 215. absolute or relative, 146. local, 103. private, 121. Uniformities, 217. Unity, organic, 9. Universal and particular, 39 n. Volition, 223 ff. Whitehead, vi, 207. Wittgenstein, vii, 208 n. Worlds, actual and ideal, in. possible, 186. private, 88. Zeller, 173. Zeno, 129, 134, 136, 165 ff. PRINTED BY NEILL AND CO., LTD., EDINBURGH. ^'^-tpr'i^rjT