No empty or immoveable Space.
The physical universe is thus through and through historical, the scene of motion. Since there is no Space without Time, there is no such thing as empty Space or empty Time and there is no resting or immoveable Space. Space and Time may be empty of qualitative events or things, and if we are serious with Time there is no difficulty in the thought of a Space-Time which contained no matter or other qualities but was, in the language of Genesis, without form and void before there was light or sound. But though empty of qualities Space and Time are always full. Space is full of Time and Time is full of Space, and because of this each of them is a complete or perfect continuum. If this might seem a quibble of words, which it is not, let us say that Space-Time is a plenum. Its density is absolute or complete. There is no vacuum in Space-Time, for that vacuum would be itself a part of Space-Time. A vacuum is only an interval between bodies, material or other, which is empty of body; but it is full with space-time. Hence the old difficulty that if there were no vacuum motion would be impossible is without foundation, and was disposed of by Leibniz in answering Locke.1 If it were completely full of material bodies with their material qualities there would be no room for locomotion of those bodies with their qualities. But it is only full with itself. Material bodies can move in this absolute plenum of Space-lime, because their motion means merely that the time-coefficients of their spatial outlines change.
In the next place, there is no immoveable Space. In one sense, indeed, Space is neither immoveable (or at rest) nor in motion. Space as a whole is neither immoveable nor in motion. For that would suppose there was some Space in which it could rest or move and would destroy its infinitude. Even when we speak of Space as a whole we must observe that it is not a completed whole at any moment, for this would omit its temporality. Under a certain condition, to be explained presently, we may indeed contemplate Space as an infinite whole when we consider only the points it contains. Directly we allow for its Time, we realise that while there may be a complete whole of conceived timeless points there cannot be one of real point-instants or events. For incompleteness at any moment is of the essence of Time. Neither strictly can the universe be said to be in motion as a whole. It is motion, that is in so far as it is expressed in its simplest terms.
But it is not Space as a whole which is understood to be immoveable. The immoveable or absolute Space of Newton is the system of places which are immoveable. Now since every point is also, or rather as such, an instant, a resting place is only a place with its time left out. Rest, as we shall see more clearly presently, is only a relative term.
Perspectives and sections.
With this conception of the whole Space-Time as an infinite continuum of pure events or point-instants let us ask what the universe is at any moment of its history. The meaning of this obscure phrase will become clearer as we proceed. The emphasis rests upon the word history. Space-Time or the universe in its simplest terms is a growing universe and is through and through historical. If we resolve it into its phases, those phases must express its real life, and must be such as the universe can be reconstructed from in actual reality, they must be phases which of themselves grow each into the next, or pass over into each other. We are to take an instant which occupies a point and take a section of Space-Time through that point-instant in respect of its space or time. The point-instant in question we may call the point or centre of reference. What will this section of Space-Time be, or what would it look like to an observer supposed to be looking at it from the outside, if we make such an impossible assumption of an observer outside the whole universe? The natural and immediate answer would be, the time-section consists of the whole of Space as occupied in every point by events occurring at that moment. For we are accustomed to think of Space as so occupied. It is true that at this moment some event or other is occurring at every point of Space. I may not be aware of them directly, but I can know of them by report, and can anyhow think of all those events that occur at this moment. Accordingly it would seem that any moment a section of the universe would be nothing other than the whole of Space; and Space may then be described as the assemblage of events which occur at the same moment of time. Now I shall try to show that this Space so described is under certain conditions something real and legitimately conceived. It is a legitimate selection from the whole of Space-Time. But it does not represent what Space-Time is at any moment of its history. The fuller reasons will appear later. At present it is enough to observe that if Space is the assemblage of all events occurring now, it is open to the same objections as were urged against the notion of a single point or a single instant. It does not matter whether the instant occupies a point or the whole of Space; the universe cannot be composed in reality of such sections. An integration of such sections does not represent the history of the world. The world would need to be re-created at every moment. To insist on this is but repetition. For the moment which is now would be a now which perished utterly and was replaced by another now. Time would cease to be duration and would be nothing but a now, for the different nows would have no continuity. We should vanish utterly at each moment and be replaced by something like ourselves but new; to the greater glory perhaps of a Creator who would be completely unintelligible, but to the confounding of science in his creatures.
We have to distinguish from this legitimate but artificial selection a selection of point-instants which shall be the state of Space-Time at any historical moment of its continuous history. I shall describe—such a section as Space-Time considered with reference to the point-instant which is taken as the centre of reference, and I shall call it a ‘perspective’ of Space-Time taken from that point of reference; and for convenience I shall speak of the previous selection distinctively as a ‘section.’ Both are in fact sections of Space-Time, but in different senses; and it is useful to have different terms. The justification of the term perspective will appear presently. The perspectives of Space-Time are analogous to the ordinary perspectives of a solid body. They differ from them in that these are taken from some point outside the body, whereas the point or instant from which a perspective of Space-Time is taken is included in the perspective itself. The choice of the word is suggested, of course, by Mr. Russell's use of it in recent inquiries in his work on ‘External Reality,’ as that in its turn is affiliated with Leibniz' conception of the monads as mirroring the universe from their several points of view. Meantime we may contrast the two conceptions by illustrations. At any moment of a man's history his body is a perspective at that instant of his whole life. But it consists of cells at all degrees of maturity. We have the space of his body occupied by parts, some mature at this moment, and others which are immature or senescent. In other words, his space is of different dates of maturity. We might, on the other hand, think of his space as occupied with cells of the same maturity, and we should have the same space, and it would all be of the same date, but it would not be the man's body as it is at any moment whatever but a selection from various stages of his history. It would, however, give his shape. Once more the illustration limps because the man's space changes in volume with his growth. But we may suppose him not to alter; and in Space-Time since Space is infinite the difficulty does not arise. Or we may illustrate by a section of a tree. As mere dead wood the space of the section is given to us at one moment. But in the history of the plant, the concentric rings of the wood are of different dates. To the eye of the botanist the section is variable in its time; to the eye of the carpenter, or better still of the person who sits at it when it is a table, it presents no such variation.
Perspectives of Space-Time from an instant.
When therefore we consider Space-Time with reference to an instant of time, that is to a point-instant in respect of its time, we shall have the whole of Space, not occurring at one instant but filled with times of various dates. There is a continuum of events filling Space but divided by the point of reference into earlier and later, with the exception of those points in which the instant of the centre is intrinsically repeated, and which have the same instant. The other points will be earlier and later at various dates, and since any date is repeated in space there will be at each date points contemporaneous with each other,2 but earlier or later than the centre and its contemporaries. There are, if we choose to use a technical term, equitemporals or isochrones in space (just as there are in a perspective from a point equispatials or isochors in time). Call O the instant of reference. One of its points is o; there are points intrinsically contemporary with o. A point a is earlier than o, and if we call the time of o the present, a is past. The point a is of the same date as b and is earlier than c. For example, a and b may be contemporary points of the same structure, e.g. my hand; ac may represent a transaction of causality, for example a bullet killing a man, that is, with reference to o, a and c are occupied by the events in question. Now the meaning of such reference in date to o is that the events a, b, and c lie on lines of advance which connect them with o. Directly or indirectly o is connected by spatio-temporal events with every point in Space, as for instance the cells in a body are connected directly or indirectly with one another. The lines of advance need not necessarily be straight (as when, for instance, we see events in space by light, which proceeds, or is thought to proceed,3 in straight lines) but may be of the most complicated character. The comparison with light is the reason why the term perspective is appropriate to such a picture as we have drawn. For not only is it true that to an outside observer the various points of space would be at different dates, but he would get that perspective by being situated at the point of reference.
Accordingly I may illustrate the difference of dates in Space in the perspective from any instant by reference to a human percipient, supposed to be at the point of reference. Only whereas in his case the lines of advance by which he apprehends events outside him are the very developed and differentiated movements by which his senses are affected, with a pure event or point-instant the lines of advance are but the movements in Space-Time by which the centre is related however circuitously to the other points of Space. Moreover, in using the illustration, we assume according to our hypothesis that what the man perceives and the act of perceiving it are separate events whose reality is not dependent on or does not owe its existence to the reality of the other. As an example of a line of advance connecting the past or earlier point with the centre, I might take any sensation, for it is certain that the act of sensing a flash of light follows by a small but measurable interval the flash itself as a physical event. A better instance is the familiar case of apprehending Sirius and his place in the sky by means of the light from him which reaches my eyes some nine years after the event. What I see is an event which happened nine years ago at the place where I see it (though I see the distance very roughly); and Heaven knows what may have happened to Sirius between the date of what I see and now when I see him.
In the same way I may apprehend in my imagination a later event which, in reference to now, is future. Nine years hence I may apprehend what is taking place at Sirius at this moment, if Sirius now exists. (We have yet to see how we can with propriety speak of Sirius as existing at this present moment at all, since I only see him nine years late.) I mean by thinking of Sirius and his position in the future that there is a system of transactions now begun which will end by enabling me to see Sirius then. This system of transactions is begun on my side by the expectation in my mind of seeing Sirius; it was begun on the side of Sirius by the causes which lead to his continuance. Generally, the point c is future to o in that transactions in Space-Time are set up which will enable me at some future time to date c as contemporary with my present moment. There is a line of advance from o to c as well as a line of advance from some other event before c to c. Again, when a and b are contemporary events in the past they are connected by different lines of advance with o; and when a is before c the two points are successive in reference to o as when a percipient follows the causal succession in a bullet's hitting a man.
Continuing the human metaphor, which we shall find at long last4 to have its justification, we may personify Space, and having regard to the differing dates of its points with reference to the centre, which is the present of that perspective, we may say that Space at any moment is full of memory and expectation. The objection may be made, how can reality contain at this moment the past, for the past is past and exists no longer? But the difficulty is only apparent. It arises from identifying reality with the present or actual reality; it assumes in fact that Time is not real. The past event, it is true, does not exist now, and if existence is taken to be present existence, the past clearly does not exist. But if we avoid this error and take Time seriously, the past possesses such reality as belongs to the past, that is, to what is earlier than the point of reference; it does not exist now but it did exist then, and its reality is to have existed then. As to the later or future, there is at bottom no greater difficulty in speaking of the future as being real and existing really than there is in respect of the real existence of the past. A future or later point does not occur now, and therefore it is now not-yet, just as the past is now no longer. It has what reality belongs to it in the real Time. Thus in describing Space-Time in reference to a centre of reference which is now—its perspective from that point of view—we are not supposing that the universe is stopped at that moment artificially, in which case there would, as some think, be a now spread out over infinite Space. They are mistaken; for there would then be no Time and no Space. We are determining which among the instants of the whole of Time belong to the points of Space in their relation to the centre. We find that so far is it from being true that at any moment in its history Space is completely occurring now, that the only points which occur now or are filled with the present are the points in which the instant of reference is intrinsically repeated.
This proposition that Space considered at any moment is of various dates is very elementary and in that sense abstract. But before proceeding I may note that as an empirical fact Space, when we apprehend it through the senses and, therefore, as filled with ‘qualitied’5 events, and not merely with pure events, is not presented to us as simultaneous. I assume, for the reasons just mentioned, that what we sense is anterior to our act of sensing it, because of the time it takes the physical event to stimulate our organs. Bearing this in mind we can conclude what the time-relations are, not so much to ourselves, for that is not relevant to our purpose, but among the different objects perceived. If I am using eyes to apprehend Space through (I am not saying that we apprehend Space by sight but we do apprehend it through sight), it is clear that since different points of space are at very different distances from the eye, and the light reaches it from them in different times, however slight the difference of distance from my eyes may be, more distant points must in general have occurred earlier than nearer ones, in order that my acts of seeing the various sets may occur at the same moment. This applies not only to vast differences of distance, as between my lamp on the table and Sirius, but to points only slightly remote from one another. There will also be certain points which are equidistant from the eyes and are simultaneous with one another. So much for sight. Even with the hand it would be difficult to prove that all points touched by the hand can send their messages through to our mind in equal times, as they must if the sensing of them is to occur at the same time. Empirically then, though we may take in an immense space in an act which, however complex, occurs all together, the Space which we apprehend is presented with different dates, though to discover this may need reflection.
Two kinds of retort may be imagined to this statement. It is based on the deliverance of the senses; and the senses deceive. To which the answer is, that with all allowance for the feebleness and treachery of the senses, they have established themselves, if there is any truth in the doctrine of natural selection, by adaptation to the very objects which it is their office to observe. We are “miserably bantered by our senses,” and, moreover, we shall learn that, since we only apprehend Space and Time by the help of the senses, we pay for the privilege of seeing colours, and for the delicate touches and movements of the wood-worker or the etcher, by making mistakes about position in space or time. But we cannot believe that though the senses may confuse our apprehension in this respect, they are there to pervert it.
Let me add the application of this remark, or its extension, to the bare point-instant which is the point of reference. There, too, it might be asked how we can be sure that two contemporary points a and b, in which the same instant repeats itself intrinsically, are contemporary for O, or if a is really before c, that it will precede c for O. The date in reference to o is determined by the line of advance from the point to o. How can we know that the dates are, as it were, “apprehended” accurately by oO? The doubt is really suggested by humanising oO and treating it as if it were a sensitive subject, with all the drawbacks possessed by such. But consider o and a, b, and c in their purely spatio-temporal character. If a and b are intrinsically isochronous and aA and bA are in the perspective from O, that means that Space-Time is such, and its point-instants so connected with each other by lines of advance, that two intrinsically isochronous points belong to the same perspective of Space as o. If they were not isochronous relatively to o, they would not appear in the perspective from O. For o is itself part of the perspective. It is only because we suppose it to be looking on at Space from the outside, and endow it with something like our sensibility, that we think of it as open to misapprehension. Being so simple, it is infallible. On the other hand, if any two points x and y are not intrinsically isochronous, but only happen to be so for this perspective, they may not be so in a different perspective.
The second retort is that perhaps it is true that, perceptually, empirical space-positions occur at different times; but, conceptually, they are all simultaneous. Something will be said hereafter of the relation of concepts to percepts. But at least it is not the business of concepts to distort perceptual objects but to indicate the pattern on which they are built. If perceived Space is full of Time there is no conceived or conceptual Space which is unfilled with Time. On the contrary, the concept of Space must all the more urgently provide for the change of Time within Space. It is true that our familiar notion of Space as a framework in which events occur all over it at the same moment is, as we have said, a legitimate and real notion, and we are yet to explain how it arises. But though it implies thought, it does not rest on the difference of conceptual and perceptual Space but on another distinction, namely on the distinction between partial and total Space-Time, between spatial perspectives of Space-Time and Space-Time as a spatial whole.
Perspectives from a point.
Hitherto I have been dealing with perspectives of Space-Time from the point of view of a single instant as located at a point. But in the same way there are perspectives from the point of view of a point of space as located in its instant of time. Once more the section of Space-Time across a point might seem to be the whole of Time, and Time might be described as the assemblage of events which occupy a single point. Metaphysically this would be open to the same objections as the notion of Space as an assemblage of events occurring at the same time. The point would be discontinuous with other points, would be a mere “here,” and would require as before re-creation of the world in each “here.” But when we take not the section of the world through a point but its perspective, we shall have the whole of Time occupying not the same point but points of Space at all manner of distances from the central point of reference. That is, just as a perspective from an instant is spread out over the whole of Time and presents all variety of dates, a perspective from a point is spread out over the whole of Space and presents all varieties of locality. It would be tedious to enter into the details which correspond to the details of the preceding picture. I will only note that in our empirical experience this state of things is as much a fact as in the other case. Still, assuming the hypothesis that what I remember and what I expect are distinct existences from me, we realise that in thinking of the history of the past or divining the future, the events are located not in one place and still less in no place at all, but in the places where they occurred or will occur, however inaccurately we may apprehend their positions. The full development of these matters belongs more properly or more conveniently to the next chapter, and I must ask something from the sympathetic imagination or patience of the reader. There is a machinery of imagination and memory for sorting out events into the places to which they belong.
A perspective from an instant of time and one from a point of space are different perspectives, and cannot be combined into a single perspective. This may at first present a difficulty. The instant from which a perspective is taken occupies a point or points. O occupies o andits contemporaries. But o is itself intrinsically repeated in time. Why are these repetitions of o in time left out of the perspective? The answer is, that if o is repeated at O′ and oO′ is taken into the perspective, the perspective would be taken not from the instant O but from O′ as well. We include in the point of view O all the contemporary points occupied by O, but we cannot include the other times which occupy o. In fact, a perspective from an instant gives us a picture of Space; a perspective from a point gives us a picture of Time. If we attempted to combine the two pictures, and to get a “perspective” of Space-Time from the point of view both of the place and time of the point-instant oO, we should have, as a little consideration will show, not a perspective at all but the whole of Space-Time. Space-Time considered in reference to a point-instant from the point of view both of the point and the instant is nothing but Space-Time.6
Relation of the perspectives to one another.
Total Space-Time is the synthesis of all partial space-times or perspectives of Space-Time. I use the awkward word ‘total’ in order to avoid two others, either of which might be misleading. The one is the adjective ‘universal,’ which is ambiguous and might suggest just what it is desired to avoid, namely, that the whole continuum of point-instants is a concept derived from special bits of Space-Time or even from perspectives or partial space-times. Whereas it is, to use language borrowed from Kant, which may pass muster at present, a single infinite “individual.”7 The other adjective “absolute” I avoid because of its historical associations with Newton's doctrine of Absolute Space and Time.
What we have to see is first, what information we can draw from our experience, in pursuit of our empirical method, as to the differences between different perspectives; and secondly, that these perspectives are of themselves connected with one another, so that the synthesis of them is not an operation which we, human subjects who think, perform upon them, but one which they, as it were, perform on themselves. For a perspective of Space-Time is merely the whole of Space-Time as it is related to a point-instant by virtue of the lines of connection between it and other point-instants.
(1) Differences of perspectives.
The information we get from experience is first, that points of space which are simultaneous in one perspective may be successive in another, and points which are successive in one may be simultaneous in another. A simple instance of the first is that on one occasion two points in my hand may be isochronous for me or my brain but on another occasion, when the time has changed and the point-instant of reference therefore with it the one may precede the other. For example, in a new perspective an electric current may have been sent from one point to the other, and the points are successive. Observe that it is not the physical or qualified events at the two points of my hand which have changed their relation in Time; but only that in the two perspectives the points of Space have become differently dated. Again, suppose that a is earlier than the centre o. In a different perspective a may have the same date as o. Thus let us go back to Sirius, and merely for simplicity's sake (and because without some simplification, however impossible in fact, the mind is apt to reel before the complexity of things) let us assume that Sirius and I remain fixed with relation to each other. The event which I now see in him by the light from him is nine years old. But, on a different line of advance in the universe from the path of transmission of light, an event may be, and probably is, occurring in Sirius which is nine years later than the event there which I now see. If it comes into my present perspective at all it is as a future event. For simplicity let us suppose I am not expecting it and that it does not enter into the perspective. Still, it occurs in fact at the same instant as o; that is, from some point of reference different from oO the points o and a will be of the same date. Again I observe, that it is not the real physical events, my sight of Sirius now and the past physical event in Sirius, which will have, as it were, become contemporary, but only that the points at which they occur have become differently dated, that is, in the new perspective are occupied by different physical events. In the new perspective the future event in Sirius enters as contemporary with my present sight of him. It is not the same event as aA, but it occurs we are supposing at the same point. In other words, the points of Space are filled with different instants owing to that redistribution of instants among points which makes the history of Space-Time. Thus there are isochrones of o in the whole of Space, which are not related to it as its isochrones are in its own perspective, and which do not appear as isochrones in that perspective.
In general a perspective of Space-Time from one point-instant differs from the perspective from another point-instant, whether the perspectives be taken in respect of the instants or points. Points which were simultaneous in the one may be successive in the other; the interval of time or space may be altered, and even two points may reverse their dates in the different perspectives. For though a perspective takes in the whole of Space or the whole of Time, it does not take in the whole of Space-Time, the totality of point-instants; it would otherwise not be a perspective. If we endow point-instants with percipience, all these changes in the distribution of points among instants will be perceived accurately; for the percipient sensibility of so simple percipients must be also supposed perfectly simple.8 We are here within the region of Space-Time pure and simple, before qualitied events, like the fall of a stone or the birth of a flower, or the existence of complex percipients like plants or ourselves. Hence I have been obliged to repeat so often that qualitied events do not occur in different places or at different times because their dates and places may be changed in two perspectives. The place at which they occur or the time which they occupy, if the event in question remains within the perspective, alters its date or its place; their place or date in the first perspective is now occupied by some other event. We must add that such changes may not be noticeable to more complex percipients, because the extent of them may not fall within the limits of the discrimination of the percipient or class of percipients even with the help of instruments of precision, or because the difference is for them of no practical importance. Simultaneous events may seem to them still simultaneous, or the intervals, spatial or temporal, not to have altered, because the change is not perceived or is not interesting. One illustration we have already had, in the belief that what we see of Space is all contemporaneous.
(2) Their connection.
Next let us note that the various perspectives of the universe viewed from points or instants, or the contents of the universe as referred to a centre of reference, are of themselves connected, and together constitute the whole universe. Each perspective leads on to some other. The redistributions of dates among points are linked in one continuous process. For a point of reference is a point-instant. Its place is temporal, and merely an ideally separated position in a movement; is, in fact, a movement at its limit. Whether we consider the time-element or the space-element in it, both alike are transitional. Point merges into point and instant into instant, and each does so because of the other. Our centre oO is, the next instant, at the time O′ and becomes oO′, or to keep our notation uniform óO′, and the world referred to this new point-instant is a different selection from the world of point-instants.9 The mere fact that each perspective is from the beginning a selection from a whole, and not a construction by the centre to which it is referred, is enough to show that the perspectives are in their own nature united, and need no combining hand. It is in this sense that the whole of Space-Time is the synthesis of partial space-times or perspectives. At a later stage10 we shall see how important this consideration is for understanding the relation of perspectives, in the ordinary sense, of finite things like houses.
Corresponding remarks may be made about perspectives from points. Moreover, the two sets of perspectives are not-only internally connected but connected with one another. The instant from which a perspective is taken being located at a point, the perspective from it is connected with the perspective from the point.
Total Space and Time.
Total Space-Time is thus the synthesis of all perspectives, which is, in fact, only another way of saying that the perspectives are real perspectives of it or are its historical phases. Owing to the infinite interconnection of point-instants on different and independent lines of advance, independent, that is, of those which pass through any given point of reference, there is an infinity of such perspectives. Not limiting ourselves therefore to any one centre of reference but admitting infinite such centres, we can see first of all that when we are considering all the perspectives from every instant, any point of Space is occupied, not as in the single time-perspective by some one moment of Time but by the whole of Time. The whole of Time in the totality of such perspectives streams through each point of Space. Thus while the state of Sirius nine years hence may not enter into my present perspective (except in expectation) it occurs on some independent line of advance (not included in my perspective) at the present time in the total; and extending our view to all perspectives, we see that the position of Sirius is occupied by some time or other through infinite Time. The position in Space is occupied by only one time in a given time-perspective, but by all Time in the totality of perspectives.
In the same way consider the totality of point-perspectives, that is, perspectives from the point of view of a point. In a single such perspective an instant is localised in only one position of Space. But in the totality of them each instant is localised in all positions in Space. We saw that it was a condition of the very nature of Space-Time that each instant was repeated in space and each point in time. But we now see that while for any perspective (which is of course three-dimensional and possesses the corresponding characters of time) there is this intrinsic repetition, every time having its appropriate isochors and every point its appropriate isochrones; in total Space-Time each point is in fact repeated through the whole of Time and each instant over the whole of Space. Now when these particular selections are made of point-instants, the one from the total of one set of perspectives and the other from the other set, we have a total Space which occurs at one instant and a total Time which occupies one point.
The total Space and Time so arrived at are what we called, in distinction from perspectives, sections of Space-Time. They do not represent what the world of Space-Time is historically at any moment or at any point. For at any moment of its real history Space is not all of one date, and Time is not all at one point. But Space and Time so described can be got by an arbitrary selection from the infinite rearrangements of instants amongst points. And the result of the selection is to give us Space apart from its times and Time apart from its places. That Space and that Time are what is meant by the definitions of them as assemblages, the one of all events of the same date, the other of all events at the same place. Moreover real Space with its varying dates coincides with this total Space when the variation of dates is omitted; and correspondingly for Time. Hence from considering the true perspectives of Space-Time we can arrive at the notion of Space occurring at one time or Time occupying one place. But from these sections we cannot arrive at the notion of true perspectives or at true Space-Time. I need not now repeat the reasons why.
Absolute Space and Time.
It is because Time is intrinsically repeated in Space and Space in Time that it is possible at all to speak of Time or Space by themselves, when in fact neither exists apart from the other. They get shaken apart from each other in thought, just as the shape of billiard balls of varying colour gets shaken apart from the varying colour. But when we go on to consider the whole of Space-Time and discover that the whole of Time when you choose from all the perspectives, or when you make an arbitrary selection of space-points, streams through every point, and the whole of Space can be filled out with places of the same date, we then formulate the two conceptions, one of a Time which flows uniformly on and the other that of a Space immoveable: what are commonly known as Absolute Time and Absolute Space, and, so far as I can judge, the ordinary or “common-sense” notions of Time and Space. Arbitrary as the selections are, they are possible, and it is easy to see under what conditions the conceptions are valid, or the Space and Time in question can be regarded as real. They are valid so long as Absolute Space is understood to be total Space and not supposed to exclude Time, or Absolute Time to exclude Space, with their respective variations of date and place. They are the fully formulated Space and Time when these are shaken apart from each other. What is false in them is to suppose them real if Space is understood to occur at one instant or Time at one point. But if no such assumption is made, (and I believe no such assumption is made in mathematics, but the two are considered merely apart from one another without any ulterior view as to their relations,) then the whole of Space is the same framework as belongs alike to the real and the arbitrary selection from Space-Time at any instant; and the whole Time is the framework of the real and the arbitrary selection from Space-Time at any place. So understood, not only are they useful and valid conceptions, but they are real, in the same sense as the material body of an organism can be said to be real and the life of it also real, though the life does not exist without a body of a certain sort, and the body, to be the kind of body that it is, depends on life. In other words, the reality of Space-Time may be resolved into the elements total Space and total Time, provided only it be remembered that in their combination Space is always variously occupied by Time and Time spread variously over Space.
Hence we may note the impropriety of distinguishing total Space as conceptual from empirical Space (the only Space we know) as perceptual. Space and Time are shaken apart from each other. But total Space is no more the concept of Space than the shape of the billiard ball is its concept while the whole ball is a percept. Total Space is the same as real Space with the Time left out, by an abstraction which is legitimate or not according to the use made of it. If the concept of Space were got by omitting Time which is vital to it, the result would be not a concept but a false product of thought. The separation of the Space from its Time involves abstraction and, so far, thought, but concepts are not arrived at by abstraction.
I have so far spoken of total Space and total Time. In dealing with the conceptions of them as absolute I am partly beset with the historical difficulty of interpreting Newton's ideas, a task to which I am not equal, but mainly I am concerned with the question how far we can validly speak of an absolute Time and Space. I leave it to others to say whether it is not the idea of what I call total Space and total Time, Time and Space taken as wholes, which is in the background of Newton's mind. His familiar illustration of absolute motion at any rate, which I give in the note,11 suggests this interpretation. What is defective in Absolute Space is the notion of resting places. Space as a whole we have seen is neither immoveable nor in motion. But neither can a place be at rest if Space is only one element of Space-Time. Rest, in fact, appears to be purely relative and to have no real existence. Every place has its time-coefficient and is the seat of motion. In general, we speak of rest only wherever the motion is irrelevant for our purposes. This may arise from various reasons. Two motions may be the same, and the moving bodies, though each in motion, are at rest relatively to each other. Or I may rest in my chair while the mosquitoes move around me, but I am moving with the earth. I neglect that motion because I am interested in the mosquitoes, and because the mosquitoes also in following me move with the earth. But while I do not change my position relative to the earth they do. It seems, in fact, clear that if anything could be absolutely at rest everything must be at rest. For if any point in space retained its time, this would dislocate the whole system of lines of advance within Space-Time, a point being only a point on such a line.
Thus if absolute rest means the negation of motion, there is no such thing in reality. Rest is one kind of motion, or, better, it is a motion with some of its motional features omitted. But if absolute rest means merely position in space with its time left out, it is a legitimate abstraction if so understood. It may be gravely doubted whether anything else is ever intended by those who speak of absolute rest, though once more I do not enter into the interpretation of Newton, as being beyond my competence.
It is important to distinguish the different antitheses into which the idea of Absolute Space or Time enters. Absolute may be opposed to relational. Space may be treated as a stuff or as a system of relations. Or absolute may be opposed to relative. The two questions are not easily separable. It may be doubted if Newton, for whom Space and Time are non-relational, distinguished them; but they are distinct. Now with the relativity of position in time or space or of motion as commonly understood nothing in our conception of Space-Time conflicts. When motion or position is declared to be relative, we are thinking of the material bodies or the qualitative events which occupy times and places and are moving. Relative for Newton refers to the sensible measures of space or time. In regard to them all the commonplaces of the subject are evident. A thing is to the right of A and to the left of B; or more important, A which is to the right of B is also from the point of view of another observer to the left of B. An event is before another event and after a third. A train may be at rest with respect to another moving train but in motion with respect to the telegraph poles. Or the train may seem at rest and the telegraph poles to move. In fact a motion of A with respect to B which is at rest is equally a motion of B with respect to A which is at rest. This relativity has sometimes been urged by philosophers to demonstrate that Time or Space is self-contradictory and therefore unreal. A present event is next moment past and some other event is present, as if to call both events by the same name in different connections made any difference to the real position of an event.
Absolute position or motion.
But the case is different when instead of qualified bodies or events we think of the pure events or point-instants which in their continuity make up Space-Time. It is true that these events are related to each other. But to call position or motion absolute is merely to say that these positions and motions are what they are in their own right. It is simply untrue to say that two point-instants or pure events may be indifferently either before or after each other. The same points may be occupied by times which are before or after each other, but the two point-instants in total Space-Time are not the same in the two cases. We may help ourselves in this situation according to our custom by reference to human affairs. All good actions are relative to their circumstances and good under those circumstances; and it is sometimes thought that they cannot therefore be absolutely good or good in their own right. Absolute goodness is then regarded as some ideal which serves as a standard to which we can only approximate. On the contrary, it is because good actions are relative to or determined by their circumstances that they are absolutely good. Other good there is none. We fancy a perfect good because there are certain rules of action which apply to sets of circumstances comparatively so simple and perpetually recurrent, like telling truth and respecting life, that we confuse the universality of these rules with some special sort of absoluteness and construct an ideal of a perfect good. In analogous fashion point-instants and, what is the same thing, motions are related; to be an unrelated point-instant (absolute rest) is a contradiction and does not exist; it is in fact a contradiction because it is incompatible with the nature of Space-Time. A point-instant is essentially an element of a movement and is between other point-instants. Motion is related to other motion. But each point-instant and each motion is what it is and is in this sense absolute. The bare framework of such absolute order is Absolute Space or Time or Absolute Motion. Again, I leave it to others to judge if this is or is not the meaning of Newton.
From this point of view I may approach the old controversy whether it makes any difference to say the earth goes round the sun or the sun round the earth. To an influential school of thought, headed by the late E. Mach in our day, the difference is one of convenience and economy in description. Neither is truer than the other. Now it is quite true that a motion round the sun may be represented equally well by a motion round the earth. But in doing this we are representing either motion as merely a series of points in space, and omitting the intrinsic time. We are giving, in fact, a purely geometrical account instead of a physical one. Physically the two descriptions are not indifferent. It is, rather, because there is only one physical description that we can find two indifferent spatial descriptions. Let us say then that total Space-Time involves as two elements total Space and total Time, these two being the framework of places and instants within which point-instants (and with them the material or psychical events which occupy them) exist. Each of them is an abstraction from the real world of Space-Time; not an abstraction in the sense of a mere creation of the human mind, but each of them real under the limitations before described. Absolute Time and absolute Space mean for us only these two elements or factors in the whole, factors which are not juxtaposed but interrelated in the complex history of point-instants.
The Principle of Relativity.
Having ventured to suggest that absolute Space and Time, interpreted as of total Space and Time, have a very good meaning as understood within the one Space-Time or world from which they never do exist in abstraction, I am impelled in spite of natural hesitation to go further and make some brief remarks upon the philosophical bearing of the current principle of relativity which claims to displace the Newtonian conceptions of Space and Time. Our purely metaphysical analysis of Space-Time on the basis of ordinary experience is in essence and spirit identical with Minkowski's conception of an absolute world of four dimensions, of which the three-dimensional world of geometry omits the element of time. The principle of relativity as enunciated by Mr. A. Einstein is taken up, as I understand the matter, into the body of Minkowski's doctrine. And it would be strange, therefore, if our metaphysical doctrine should be in conflict with it, considered as a mathematical doctrine. The principle of relativity means that the laws of physics are the same for all observers in uniform motion with respect to each other; so that in Mr. Paul Langevin's phrase purely mechanical observations interior to two such systems would not reveal the motion of the systems relatively to each other.12 The principle was suggested by certain experimental evidence which need not be mentioned here, and it carries with it certain consequences of which, for the layman like the writer, the simplest and, at the same time, the most paradoxical are these. Time, it should be mentioned, is determined by means of clocks whose synchronism is tested in a certain method by means of a flash of light flashed from one clock to the second and then flashed back: the clocks are synchronous when the reading of the second clock is half the two readings on the first. Now it follows from the relativity principle that two clocks which are synchronous with one another in one system supposed to be at rest will for an observer who moves along with them not be synchronous, and hence events simultaneous to one set of observers are not so for the other set in uniform movement of translation with respect to the first. Secondly, a stick of a certain length lying in the direction of the translation will not be of the same length to the two sets of observers but will shrink for the resting observer in a certain ratio. The conclusion is that Space and Time are entirely relative and vary for each observer.
This is precisely what we should expect on the metaphysical statement (apart from the exact numerical determinations) if different sets of observers have different views or perspectives of the one Space-Time. Each such perspective is perfectly real and in no sense illusory, just as the perspectives we have of solid objects are the object as seen under certain aspects and are perfectly real.13 The motion of the one system S′ with regard to the other system S changes the perspective for the two sets of observers. Consequently though the material events of sending and receiving flashes of light at two stations are not altered in their relations in Space-Time, they will have different dates in the two cases; for the places at which the events occur will change their dates relative to the observers. In the same way, to take the case of the stick, times appropriate to the ends of the moving stick will occupy different places for the two sets of observers, and the stick will alter in length. There is no such thing as a purely spatial or temporal interval. A distance in space is a system of events, whether it is distance pure and simple or is occupied by a stick. The length of the stick in total Space-Time does not alter, but the dates of its points do, according to the perspective. Only when we forget this does it seem paradoxical to us that the length should vary to different observers. So, too, the retardation of the clock may seem paradoxical, for though we are familiar with the spatial character of Time, since we estimate it for purposes of accuracy by spatial marks, as in clocks, yet we forget that these spatial measures are themselves temporal.
The same thing may be expressed otherwise thus. We are dealing in the theory not with point-instants or pure events as such, but with the measurements of Space and Time by means of sensible or material events; in fact, by light-signals. What the theory does is to establish the relations between Space and Time as thus sensibly measured in such a way as to express the persistence in an identical form of physical laws for observers in uniform motion of translation with respect to each other. But this is not in any way inconsistent with there being pure events or point-instants which have their ‘absolute’ position in Space-Time. To illustrate the point, let me take the conclusion drawn by Mr. Langevin that under certain conditions events may have their order of succession reversed for the two sets of observers; if, that is to say, the distance between the events is greater than can be travelled by light in the time between them. It is inferred that there can be no causality between events at such a distance. But if the time of events were measured by sound-events at comparatively inconsiderable distances, events which are known to be in causal relation would have their succession reversed for observers in appropriate positions.14
Its metaphysical bearing.
Now the principle of relativity is a physical or mathematical principle, and is not primarily concerned with metaphysics (or even theory of knowledge, which for me is only a part of metaphysics). But we are concerned here with the metaphysics of it. It would seem at first sight to mean that while the laws of physics are the same for every observer, each one has his own Space and Time and lives in that world. But if that conclusion is drawn, and I do not feel sure that it is, the relativist seems compelled philosophically to go beyond his own Space-Time and arrive at a total Space-Time in our sense. It is sometimes said that the very reasonings which establish the relativist results (those paradoxes which are so beautifully natural) presuppose and postulate an absolute world; but I cannot find that this can be maintained. But they certainly seem to me to lead on to it. For the different sets of observers compare notes, or if they do not, the mathematician who supplies the formulas of transformation whereby equations expressed in the co-ordinates of one world can be expressed in the co-ordinates of another world, thereby contemplates a world in which the worlds of the two sets of observers are unified. Moreover, even within one world the various persons who read the clocks are supposed to communicate with each other, and they are not the same persons and may have a slightly different perspective. The only way in which the conclusion from this comparison of the observers at least in the two systems (to say nothing of observers within each system) can be turned is by the reply that the formulae are numerical and independent of Space and Time. For reasons which I cannot at present explain, I should regard this answer as unavailing, because number is itself dependent on Space and Time.
Thus the position metaphysically of the relativist is apparently one of solipsism, or rather the same question is raised as in solipsistic theories of knowledge. Solipsists, as has often been pointed out, could not talk to each other. Moreover, as Mr. Bradley has shown, a solipsist at one moment could not talk to himself as he was at a previous moment; he would have no continuous self. Now all such metaphysical difficulties are avoided if we start with the empirical fact that we do communicate with one another about a common world which each sees from his own view, and moreover that each remembers himself. If relativism means philosophically (and I repeat that I do not know that it does mean this) that Space-Time for each observer is his own, it inevitably leads on to a total Space-Time which combines these worlds.
I venture, therefore, to suggest that the importance of the doctrine does not lie in any supposed annihilation of absolute Space-Time as understood in the sense explained here, but in two other respects. Of the first a philosopher can judge. It is the truth that the world is not a geometrical but a physical one, and that Space and Time are indissoluble. This seems to me a result of the last importance and fundamental to metaphysics. The second is the exact determination on the basis of experimental evidence of how formulae are to be transformed in the case where one system moves in uniform translation with respect to another system. Such transformations are required in the Newtonian mechanics, but the contention of relativists is that they are only a first approximation. Later knowledge shows the transformations to be less simple.15 If this contention is well established, and this is a matter for physicists and certainly not for me, the principle means a vast advance for physics itself over and above the fundamental reconstruction of the relation of Space and Time. But whatever modifications it introduces into the Newtonian mechanics it leaves Time and Space and Motion in their ancient reality, or rather it leaves us still with Space-Time in itself as a total from which perspectives are selections; and therefore in that sense absolute and independent of the observers. And I do not feel sure that any relativist would object to this in a metaphysical sense. Time and Space in their ancient pure reality remain as the framework of history, and the new doctrine is a new doctrine of their sensible measures.16
Nouveaux Essais, Preface (Erdmann, p. 199b, Latta, p. 385).
E.g. all the points on the same spherical surface if the lines of advance are those of light.
I add the reservation because I understand that according to Mr. Einstein's most recent work light may not travel in perfectly straight lines.
See later, Bk. III. ch. ii. A, where it will be suggested that the instant of a point is its ‘mind.’
The word has Dr. Johnson's authority. “Lord Southwell was the highest-bred man without insolence that I ever was in company with, the most qualitied I ever saw.” (Boswell, March 23, 1783, G. B. Hill's ed., vol. iv. p. 174).
See later, Book IV. ch. i., for the connection of this with the so-called ontological argument.
Later it will be seen that Space-Time is not an individual.
For the questions raised by this idea (cp. above, p. 74) of the perfect accuracy of perception of point-instants, see later, Bk. III. ch. vii.
Not of course from the world of points. Every perspective includes the whole of Space and the whole of Time, but not the whole of Space-Time. That is, every point in Space is there with some date, and every instant in Time with some place; but not the whole mass of point-instants.
Bk. III. ch. vii.
Principia, Bk. I., Scholium to Definitions.
“If the earth is really at rest, the body which relatively rests in the ship will really and absolutely move with the same velocity which the ship has on the Earth. But if the earth also moves, the true and absolute motion of the body will arise partly from the true motion of the Earth in immoveable space; partly from the relative motion of the ship on the Earth: and if the body moves also relatively in the ship; its true motion will arise, partly from the true motion of the Earth in immoveable space, and partly from the relative motions as well of the ship on the Earth as of the body in the ship; and from these relative motions will arise the relative motion of the body on the Earth.”
In other words, the true or absolute motion is the motion when you take it in the whole of Space and not in relation to any one body in Space.
Le Temps, l'espace, et la causalité, p. 6. Referred to in note on p. 91.
See later, Bk. III. ch. vii.
I do not of course mean slightingly that the principle of relativity is a mere affair of measurements, but rather that measurement when you press it to its ultimate foundations always implies the introduction of time considerations into spatial quantities and space considerations into temporal quantities. This is a matter of the highest importance, and it is of a piece with the principle that a space without its time or a time without its space is a fiction. This second point is what I miss in Mr. Broad's treatment of the subject of relativity in his Appendix (Perception Physics and Reality, Cambridge, 1914, pp. 354 ff.), from which I have learnt much on the relation of the principle of relativity to measurement.
Possibly this new system may not be final See this suggestion as made by Mr. Langevin, p. 27, “Peut—être des expériences nouvelles nous obligeront-elles à retoucher le groupe de Lorentz, comme nous venons de retoucher le groupe de Galilée,” and Mr. Silberstein (Theory of Relativity, London, 1914, p. 108).
There is a very serviceable statement with very little mathematics of the relativity doctrine by Mr. Langevin in the Bulletin de la Société Francaise de Philosophie, 12me Année, No. I, Jan. 1912, whose language I have several times used. (See also his paper in ‘Report of 4th international philosophical congress at Bologna,’ 1911, vol. i. p. 193, and in Revue de Métaphysique.) I quote some words of his used, not in the paper but at the close of the discussion which followed it, p. 43. “We must conclude to the existence of a new reality, the Universe, of which the Space and Time particular to a group of observers are but perspectives, more immediately given, but relative and variable with the movement of the system of observation.” One of the speakers in the discussion of Mr. Langevin's paper suggests, only in order to disavow it, the idea I have used of different perspectives of a solid object. I imagine the disavowal to be based on the belief that such perspectives are therefore illusions, instead of being as they are realities and physical realities. See later, Bk. III. ch. vii. and Mr. Russell's treatment of perspectives in recent works.
The memoirs of Messrs. Lorentz, Einstein, and Minkowski are now conveniently collected in a single volume, Das Relativitätsprincip, Leipzig and Berlin, 191 3. On p. 58 above I have omitted to mention the name of the late P. Fitzgerald along with these writers.