Spatial relations exist within Space itself.
That was a profound maxim of Hume, when inquiring into the value or the real existence of an idea to seek for the impression to which the idea corresponded. In more general language it is the maxim to seek the empirical basis of our ideas. It is true that Hume himself overlooked in experience facts which were in the language of Plato's Republic rolling about before his feet; and hence failing to find in experience any impression of the self or of causality, he was compelled to refer the ideas of self or causality to the imagination, though in the case of self, for instance, we can see that while he noticed the substantive conditions he overlooked the transitive ones, and missed the essential continuity of mind against which the perceptions are merely standing out in relief. A thorough-going empiricism accepts his formula, but having no prejudice in favour of the separate and distinct existences which attract our attention, insists that in surveying experience no items shall be omitted from the inventory.
Following this maxim, if we ask what are relations in Space and Time the answer is not doubtful. They are themselves spaces and times. “Years ago,” says James in one of the chapters of his book, The Meaning of Truth (chap. vi. ‘A Word more about Truth,’ pp. 138 ff.), “when T. H. Green's ideas were most influential, I was much troubled by his criticisms of English sensationalism. One of his disciples in particular would always say to me, ‘Yes! terms may indeed be possibly sensational in origin; but relations, what are they but pure acts of the intellect coming upon the sensations from above, and of a higher nature?’ I well remember the sudden relief it gave me to perceive one day that space-relations at any rate were homogeneous with the terms between which they mediated. The terms were spaces and the relations were other intervening spaces.” The same kind of feeling of relief may have been felt by many besides myself who were nursed in the teaching of Green and remember their training with gratitude, when, they read the chapter in James's Psychology (vol. ii. pp. 148–53) where this truth was first stated by him; for example in the words, “The relation of direction of two points toward each other is the sensation of the line that joins the two points together.” Other topics are raised by the form of the statement, whether the alternative is merely between relations conceived as the work of the mind or as given in experience, and whether the relation which is a space is really a sensation. These matters do not concern us, at any rate at present. Nor have we yet to ask whether what is said of spatial is not true of all relations, namely that they are of the same stuff as their terms. What does concern us is that relations between bits of Space are also spaces. The same answer applies plainly to Time. If the bits of Space are points they are connected by the points which intervene. A relation of space or time is a transaction into which the two terms, the points or lines or planes or whatever they may be, enter; and that transaction is itself spatial. Relations in space are possible because Space is itself a connected whole, and there are no parts of it which are disconnected from the rest. The relation of continuity itself between the points of space is the original datum that the points are empirically continuous, and the conceptual relation translates into conceptual terms this original continuity, first regarding the points as provisionally distinct and then correcting that provisional distinctness. The “impression”—the empirical fact—to which the idea of continuity corresponds is this given character of Space which we describe by the sophisticated and reflective name of continuity. Relations in space or spatial relations are thus not mere concepts, still less mere words by which somehow we connect bits of space together. They are the concrete connections of these bits of space, and simple as Space is, it is (at least when taken along with its Time) as concrete as a rock or tree. Moreover, when we introduce into Space the element of Time which is intrinsic to it, relations of space become literally transactions between the spatial terms. All Space is process, and hence the spatial relation has what belongs to all relations, sense, so that the relation of a to b differs from the relation of b to a. Thus if a and b are points, the relation is the line between them, but that line is full of Time, and though it is the same space whether it relates a to b or b to a, it is not the same space-time or motion. The transaction has a different direction.
All relations which are spatial or temporal are thus contained within the Space and Time to which the terms belong. Space and Time, though absolute in the sense we have described, namely that spaces and times are in Newton's words their own places, are relational through and through, because it is one extension and it is one duration in which parts are distinguishable and are distinguished, not merely by us but intrinsically and of themselves: as we have seen through the action of Space and Time upon each other. Whether we call Space and Time a system of points and instants or of relations is therefore indifferent. Moreover, in any given case the relation may be of more interest than its terms. James has pointed out that while in general the relations between terms form fringes to the terms in our experience, so that the terms are substantive and the relations transitive, yet on occasion it may be the transition which is in the foreground—it may become substantival and the terms become its fringes. For instance the plot of a play may be distinct and impressive, and the persons shadowy, points of attachment to the plot. In a constitutional monarchy it is the relations of king and subjects which are substantive, the person of the king or of his subjects are merely the dim suggestions of things which the constitution unites.
Thus Space as extension and Time as duration are internally orderly, and they are orders, the one of coexistence and the other of succession, because order is a relation, and a comprehensive one, within extension and duration; or rather it is a relation within Space-Time, for it implies sense, and neither Space alone nor Time alone possesses sense. In other words, given empirical Space-Time, order of the parts of Space-Time is a relation, in the meaning of transition from part to part. Just as conceptual continuity corresponds to empirical or apprehended continuity, so conceptual order determined by some law or principle corresponds, as a relation between points or other bits of space and time themselves, to the empirical transitions between those bits. These empirical transitions in virtue of which one part of space and time is between others are the “impressions” which are the originals of the conceptual order.
How far a science of order could be founded on this bare conception of ordered parts of Space-Time I do not know. But at any rate the more comprehensive theorems of speculative mathematics at the present time do not thus proceed. They appear to use the conception of Space and Time not as being stuffs, as we have taken them to be, within which there are relations of the parts of Space and Time themselves, but as relational in the sense that they are relations between things or entities. This is the antithesis between absolute and relational Space and Time.
Absolute and relational Space and Time.
In the one philosophical view, the one which I have adopted, Space and Time are themselves entities, or rather there is one entity, Space-Time, and there are relations spatio-temporal within it. In the other, Space and Time are nothing but systems of relations between entities which are not themselves intrinsically spatio-temporal. In the simplest form of the doctrine they are relations between material points. They may be, as in some sense with Leibniz, relations between monads. But in every case the presupposition is of entities, which when the relations are introduced may then be said to be in Space and Time. We are, it seems, at once transported into a logical world of entities and their relations which subsist, but do not belong in themselves to either physical or mental empirical existence. For it must be admitted, I think, that it would be impossible to take Space and Time as relations between, say, material bodies, and at the same time to postulate an absolute Space and Time in which the bodies exist. The physical bodies, besides standing in spatial and temporal relations to one another, must then stand in a new relation to the places they occupy. But this offers an insuperable difficulty. Space and Time cannot at once be entities in their own right and at the same time merely be relations between entities; and the relation supposed between the place which is an entity and the physical body at that place is either a mere verbal convenience or it stands for nothing. All we can do is to define the place by means of relations between physical entities; and this it is which has been attempted by Messrs. Whitehead and Russell in a construction of extraordinary ingenuity, expounded in Mr. Russell's recent book on Our Knowledge of the External World. There the elements of the construction of a point are various perspectives of a thing, which is usually said to be at that point, arranged in a certain order, these perspectives being themselves physical objects.
Not to enter minutely into details for which I am not competent, I may illustrate the character of this mathematical method by reference to the number system, which shows how completely the method takes its start from assumed entities. Cardinal numbers are defined by the independent investigation of Messrs. Frege and Russell as the class of classes similar to a given class. The number 2 is the class of all groups of two things, which may be ordered in a one-to-one correspondence with each other. From this definition of number in neutral terms, for entity is any object of thought whatever, we can proceed to define the whole system of real numbers; first the fractions and then the surds, finally arriving at a purely logical definition of the system of real numbers, involving entities, certain relations of order, and certain operations.1 But once arrived at this point we may go farther. “It is possible, starting with the assumptions characterising the algebra of real numbers, to define a system of things which is abstractly equivalent to metric Euclidean geometry.”2 So that real algebra and ordinary geometry become abstractly identical. This is one stage in the arithmetisation of geometry which is the outstanding feature of recent mathematics. In the end, as I understand, there is but one science, arithmetic, and geometry is a special case of it.
It is no part of my purpose to question the legitimacy of this method. On the contrary, I take for granted that it is legitimate. Our question is whether it really does leave empirical Space behind it, and what light it throws on the difference, if any, between metaphysics and mathematics. For, as we have seen, in the simpler theory of mathematics which takes absolute Space and Time for granted, even if as fictions, geometry was concerned with the properties of figures and their relation to the principles adopted for convenience in the science, and the metaphysics of Space was an analysis of empirical Space; and the demarcation of the two sciences was fairly clear. But if it is claimed that mathematics at its best is not concerned with empirical Space at all, but with relations between entities, then we are threatened with one of two results. Either our metaphysics in dealing with empirical Space is concerned with a totally different subject from geometry, not merely treating the same topic in a different way or with a different interest, or else we must revise our conception of metaphysics and identify it in effect with mathematics or logic.
Assumptions of relational theory.
We may most clearly realise the contrast of this method with the empirical method of metaphysics if we recur to the importunate question, What then is a relation it Space and Time are relations? Empirical metaphysics explains what relations are.3 But the mathematical method can clearly not avail itself of the same answer. Relation is indeed the vaguest word in the philosophical vocabulary, and it is often a mere word or symbol indicating some connection or other which is left perfectly undefined; that is, relation is used as a mere thought, for which its equivalent in experience is not indicated. For Leibniz there is still an attachment left between the relations which are spatial and the Space we see. For empirical Space is but the confused perception by the senses of these intelligible relations. He never explains what the intelligible relations are. But our mathematical metaphysicians leave us in no doubt. “A relation,” says Mr. Russell (Principles of Mathematics, p. 95), “is a concept which occurs in a proposition in which there are two terms not occurring as concepts, and in which the interchange of the two terms gives a different proposition.” This is however a description of relation by its function in a proposition, and is a purely logical generalisation; it does not profess to say what relations are in themselves. To do this, we must have recourse to the method used in defining numbers, which gives us constructions of thought, in terms of empirical things, that are a substitute for the so-called things or relations of our empirical world. An admirable statement of the spirit of this method has been supplied by Mr. Russell himself in an article in Scientia.4 Thus, for instance, if we define a point, e.g. the point at which a penny is, by an order among perspectives of the penny, we are in fact substituting for the empirical point an intelligible construction which, as it is maintained, can take its place in science. When a thing is defined as the class of its perspectives, a construction is supplied which serves all the purposes of the loose idea of an empirical thing which we carry about with us. A relation is defined upon the same method.5 We are moving here in a highly generalised region of thoughts, used to indicate the empirical, but removed by thought from the empirical. The Humian question, What is the impression to which the idea of a relation (or that of a thing) corresponds, has lost its meaning. A thing or a relation such as we commonly suppose ourselves to apprehend empirically is replaced by a device of thought which enables us to handle them more effectively. Such constructions describe their object indirectly, and are quite unlike a hypothesis such as that of the ether, which however much an invention of thought professes to describe its object directly. As in the case of the theory of number, we seem to be in a logical or neutral world.
But we have cut our moorings to the empirical stuff of Space and Time only in appearance, and by an assumption the legitimacy of which is not in question, but which remains an assumption. The starting-point is entities or things which have being, and in the end this notion is a generalisation from material things or events. Now such things are supposed, on the relational doctrine, to be distinct from the Space and Time in which they are ordered. But there is an alternative hypothesis, the one which we have more than once suggested as involved with the empirical method here expounded. The hypothesis is that the simplest being is Space-Time itself, and that material things are but modes of this one simple being, finite complexes of Space-Time or motion, dowered with the qualities which are familiar to us in sensible experience. That hypothesis must justify itself in the sequel by its metaphysical success. But at least it is an alternative that cannot be overlooked. The neglect of it is traceable to the belief that we must choose between an absolute Spare and Time, which are alike the places of themselves and the places of material things, and on the other hand, a spatial and temporal world which is a system of relations between things. As we have seen, we cannot combine these notions. But if things are bits of Space-Time, they are not entities with mere thought relations which correspond to empirical Space and Time; rather, we only proceed to speak of relations between them because they are from the beginning spatio-temporal and in spatio-temporal relations to one another.
Contrast with emporocal theory.
I am not contending that this hypothesis, which is no new one but as old as the Timaeus of Plato with its construction of things out of elementary triangles, and theory. has been revived in physics in our own day in a different form,6 is established; but only that it is inevitable to an empirical metaphysics of Space and Time. Order is, as we have seen, a relation amongst these finite complexes within Space-Time. When we begin with developed material things, later in metaphysical (and actual) sequence than Space-Time itself, we are by an act of thought separating things from the matrix in which they are generated. When we do so we forget their origin, generalise them into entities, construct relations in thought between them, transport ourselves into a kind of neutral world by our thought, and elaborate complexes of neutral elements by which we can descend again to the spatio-temporal entities of sense. We can legitimately cut ourselves adrift from Space and Time because our data are themselves in their origin and ultimate being spatio-temporal, and the relations between them in their origin equally spatio-temporal. Thus we construct substitutes for Space and Time because our materials are thoughts of things and events in space and time. We appear to leave Space and Time behind us and we do so; but our attachments are still to Space and Time, just as they were in extending the idea of dimensionality. Only here our contact is less direct. For dimensionality or order is implied in Space and Time, but in this later method we are basing ourselves on entities which are not implied in Space and Time but which do presuppose it. Indirect as the attachment is, yet it persists. Consequently, though we construct a thought of order or of an operation and interpret Space and Time in terms of order, we are but connecting thought entities by a relation which those entities in their real attachments already contain or imply. If our hypothesis is sound, order is as much a datum of Space-Time apprehension as continuity is, and in the same sense.
Thus the answer to the question, are Space and Time relations between things, must be that they may be so treated for certain purposes; but that they are so, really and metaphysically, only in a secondary sense, for that notion refers us back to the nature of the things between which they are said to be relations, and that nature already involves Space and Time. Until we discover what reality it is for which the word relation stands and in that sense define it, the notion of relation is a mere word or symbol. It is an invention of our thought, not something which we discover. The only account we can give of it is that relation is what obtains between a king and his subjects or a town and a village a mile away or a father and his son. But such an account suffers from a double weakness. By using the word ‘between’ it introduces a relation into the account of relation; and it substitutes for definition illustration. We may legitimately use the unanalysed conception of relation and of entity as the starting-point of a special science. But there still remains for another science the question what relation and entity are, and that science is metaphysics. So examined, we find that relations of space and time are intrinsically for metaphysics relations within Space and Time, that is within extension and duration. Accordingly the relational view as opposed to the absolute view of Space and Time, whatever value it possesses for scientific purposes, is not intrinsically metaphysical.
Mathematics and metaphysics of Space.
We are now, however, in a position to contrast the metaphysical method with the mathematical. The method of metaphysics is analytical. It takes experience, that is, what is experienced (whether by way of contemplation or enjoyment), and dissects it into its constituents and discovers the relations of parts of experience to one another in the manner I have attempted to describe in the Introduction. But mathematics is essentially a method of generalisation. Partly that generalising spirit is evidenced by the extension of its concepts beyond their first illustrations. This has been noted already. But more than this, it is busy in discussing what may be learned about the simplest features of things. Mathematics as a science, says Mr. Whitehead, “commenced when first some one, probably a Greek, proved propositions about any things or about some things without specification of particular things. These propositions were first enunciated by the Greeks for geometry; and accordingly geometry was the great Greek mathematical science.”7 This is an admirable statement of the spirit of the science and of why it outgrew the limits of geometry. It also indicates why when mathematics is pushed to its farthest limits it becomes indistinguishable from logic. On this conception our starting-point is things, and we discuss their simplest and most general characters. They have being, are entities; they have number, order, and relation, and form classes. These are wide generalities about things. Accordingly geometry turns out in the end to be a specification of properties of number. In treating its subject mathematics proceeds analytically in the sense of any other science: it finds the simplest principles from which to proceed to the propositions it is concerned with. But it is not analytical to the death as metaphysics is. Existence, number and the like are for it simply general characters of things, categories of things, if the technical word be preferred. Now an analysis of things in the metaphysical sense would seek to show if it can what the nature of relation or quantity or number is, and in what sense it enters into the constitution of things. But here in mathematics things are taken as the ultimates under their generalised name of beings or entities. They are then designated by descriptions. What can be said about things in their character of being the elements of number? Hence we have a definition of number by things and their correspondences. But metaphysics does not generalise about things but merely analyses them to discover their constituents. The categories become constituents of things for it, not names of systems into which things enter. Its method is a method not so much of description as of acquaintance.
Mathematics deals with extension; metaphysics with intension.
The same point may be expressed usefully in a different way by reference to the familiar distinction in logic between the extension and the intension of names. Mathematics is concerned with the extension of its terms, while metaphysics is concerned with their intension, and of course with the connection between the two. The most general description of thing is entity, the most general description of their behaviour to each other is relation. Things are grouped extensionally into classes; intensionally they are connected by their common nature. Number is therefore for the mathematician described in its extensional aspect; so is relation.8 Now for metaphysics intension is prior to extension. When the science of extensional characters is completed, there still remains a science of intensional characters. It is not necessarily a greater or more important science. It is only ultimate.
The spirit by which mathematics has passed the limits of being merely the science of space and number, till it assumes the highly generalised form we have described, carries it still further, till in the end it becomes identical with formal logic. For logic also is concerned not with the analysis of things but with the forms of propositions in which the connections of things are expressed. Hence at the end pure mathematics is defined by one of its most eminent exponents as the class of all propositions of the form ‘p implies q,’ where p and q are themselves propositions.9
Mathematics is a term which clearly has different meanings, and the speculative conception of it endeavours to include the other meanings. But it is remarkable that as the science becomes more and more advanced, its affinity to empirical metaphysics becomes not closer but less intimate. The simple geometry and arithmetic which purported to deal with Space and quantity were very near to empirical metaphysics, for Space and Time of which they described the properties are for metaphysics the simplest characters of things. But in the more generalised conception, the two sciences drift apart. It is true that still mathematics deals with some of the most general properties of things, their categories. And so far it is in the same position towards metaphysics as before. But Space and Time have now been victoriously reduced to relations, while experiential metaphysics regards them as constituents and the simplest constituents of things. Hence it was that we were obliged to show that in cutting itself loose from Space and Time mathematics was like a captive balloon. It gained the advantage of its altitude and comprehensive view and discovered much that was hidden from the dweller upon the earth. But it needed to be reminded of the rope which held it to the earth from which it rose. Without that reminder either mathematics parts company from experiential metaphysics or metaphysics must give up the claim to be purely analytical of the given world.
Is metaohysics of the possible or the actual?
Now it is this last calamity with which metaphysics is threatened, and I add some remarks upon the point in order to illustrate further the conception of experiential metaphysics. For the mathematical philosopher, mathematics and logic and metaphysics become in the end, except for minor qualifications, identical. Hence philosophy has been described by Mr. Russell as the science of the possible.10 This is the inevitable outcome of beginning with things or entities and generalising on that basis. Our empirical world is one of many possible worlds, as Leibniz thought in his time. But all possible worlds conform to metaphysics. For us, on the contrary, metaphysics is the science of the actual world, though only of the a priori features of it. The conception of possible worlds is an extension from the actual world in which something vital has been left out by an abstraction. That vital element is Space-Time. For Space-Time is one, and when you cut things from their anchors in the one sea, and regard the sea as relations between the vessels which ride in it, without which they would not serve the office of ships, you may learn much and of the last value about the relations of things, but it will not be metaphysics. Thus the possible world, in the sense in which there can be many such, is not something to which we must add something in order to get the actual world. I am not sure whether Kant was not guilty of a mere pun when he said that any addition to the possible would be outside the possible and thus impossible. But at any rate the added element must be a foreign one, not already subsumed within the possible. And once more we encounter the difficulty, which if my interest here were critical or polemical it might be profitable to expound, of descending from the possible to the actual, when you have cut the rope of the balloon.
The need for metaphysics.
Nothing that I have written is intended to suggest any suspicion of the legitimacy or usefulness of the speculative method in mathematics. On the contrary I have been careful to say the opposite. Once more, as in the case of many-dimensional ‘Space,’ it would seem to me not only presumptuous on my part but idle on the part of any philosopher to question these achievements. Where I have been able to follow these speculations I have found them, as for instance in the famous definition of cardinal number and its consequences, illuminating. My business has consisted merely in indicating where the mathematical method in the treatment of such topics differs from that of empirical metaphysics; and in particular that the neutral world of number and logic is only provisionally neutral and is in truth still tied to the empirical stuff of Space-Time. Suppose it to be true that number is in its essence, as I believe, dependent on Space-Time, is the conception, we may ask, of Messrs. Frege and Russell to be regarded as a fiction? We may revert once more to the previous question, when a fiction is fictitious. If this doctrine is substituted for the analysis of number as performed by metaphysics as a complete and final analysis of that conception it would doubtless contain a fictitious element. Or, as this topic has not yet been explained, if the conception of Space as relations between things is intended not merely as supplying a working scientific substitute for the ordinary notion of extension but to displace empirical Space with its internal relations, the conception is fictitious. But if not, and if it serves within its own domain and for its own purpose to acquire knowledge not otherwise attainable, how can it be fictitious? I venture to add as regards the construction of points in space and time and physical things out of relations between sensibles proposed recently by Messrs. Whitehead and Russell, that if it bears out the hopes of its inventors and provides a fruitful instrument of discovery it will have irrespectively of its metaphysical soundness or sufficiency established its claim to acceptance. “Any method,” we may be reminded, “which leads to true knowledge must be called a scientific method.”11 Only, till its metaphysical sufficiency is proved it would needs have to be content with the name of science. For Space and Time may be considered as relations between things without distortion of fact. Now the sciences exist by selecting certain departments or features of reality for investigation, and this applies to metaphysics among the rest. They are only subject to correction so far as their subject matter is distorted by the selection. But to omit is not necessarily to distort.
On the other hand, if a method proper to a particular science is converted into a metaphysical method it may be defective or false. This is why I ventured to say of Minkowski's Space-Time,12 as a four-dimensional whole which admitted of infinite Spaces, that it was a mathematical representation of facts, but that it did not justly imply that the Universe was a four-dimensional one, because it overlooked the mutual implication of Space and Time with each other. If it were so understood it would contain a fictitious element. As it is, it contains an element which is not fictitious but only scientifically artificial.
We may then sum up this long inquiry in the brief statement that whether in physics, in psychology, or in mathematics, we are dealing in different degrees of directness with one and the same Space and Time; and that these two, Space and Time, are in reality one: that they are the same reality considered under different attributes. What is contemplated as physical Space-Time is enjoyed as mental space-time. And however much the more generalised mathematics may seem to take us away from this empirical Space-Time, its neutral world is filled with the characters of Space-Time, which for its own purposes it does not discuss. To parody a famous saying, a little mathematics leaves us still in direct contact with Space-Time which it conceptualises. A great deal more takes us away from it. But reviewed by metaphysics it brings us back to Space-Time again, even apart from its success in application. Thus if we are asked the question what do you mean by Space and Time? Do you mean by it physical Space and Time, extension and duration, or mental space and time which you experience in your mind (if Space be allowed so to be experienced), or do you mean by it the orders of relations which mathematics investigates? The answer is, that we mean all these things indifferently, for in the end they are one.
Young, loc. cit. p. 98.
Loc. cit. p. 182.
See later, Bk. II. ch. iv.
“Wherever possible, logical constructions are to be substituted for inferred entities [e.g., the cardinal number of two equally numerous collections]... The method by which the construction proceeds is closely analogous in these and all similar cases. Given a set of propositions nominally dealing with the supposed inferred entities, we observe the properties which are required of the supposed entities in order to make these propositions true. By dint of a little logical ingenuity, we then construct some logical function of less hypothetical entities which has the requisite properties. This constructed function we substitute for the supposed inferred entities, and thereby obtain a new and less doubtful interpretation of the body of propositions in question” (‘The Relation of Sense-data to Physics,’ Sec. vi. Scientia, 1914. The article is now reprinted in Mysticism and Logic (London, 1918); the reference is to pp. 155–6. What I imply in the text is that number, thing, relation, are directly experienced, and that metaphysics has to describe what is thus directly experienced. This is attempted in Bk. II.
Principia Mathematica, i. p. 211.
The reference is to the physical theory of the late Osborne Reynolds, according to which the universe is Space, and matter is comparable to a strain or a geological fault in this homogeneous medium. See his Rede Lecture, On an Inversion of Ideas as to the Structure of the Universe, Cambridge, 1903. Reynolds's theory that Space is granular in structure does not concern us here, but concerns the physicist.
Introduction to Mathematics, p. 14.
Whitehead and Russell, Princ. Math., Introduction, vol. i. p. 27. “Relations, like classes, are to be taken in extension, i.e. if R and S are relations which hold between the same pairs of terms, R and S are to be identical.” Compare ibid. p. 211.
Russell, Principles of Mathematics, p. 3.
“On Scientific Method in Philosophy” (Herbert Spencer Lecture), Oxford, 1914, p. 17. Reprinted in Mysticism and Logic, p. III.
A. Schuster, Presidential Address to British Association, 1916.
Above, Bk. I. ch. i. p. 59.